tag:blogger.com,1999:blog-36285423351682311612024-03-17T17:40:04.771-07:00GM Jackson Physics and MathematicsGM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.comBlogger148125tag:blogger.com,1999:blog-3628542335168231161.post-44422035178565806702024-01-06T12:58:00.000-08:002024-01-08T13:00:01.986-08:00Was the Speed of Gravity Successfully Measured?ABSTRACT:
This paper shows mathematically and experimentally why it is highly unlikely that the speed of gravity was successfully measured.
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Consider the rubber-sheet analogy. If you place an iron ball on a rubber sheet, you will see the ball depress and curve the rubber sheet. If you roll the ball accross the sheet, you will see the sheet flatten out at the ball's previous position and the sheet will begin to curve at the ball's current position. Over a given distance, it takes time for the curve to flatten and reform. We can calculate the speed of this process by dividing the distance by the time. One might assume we can calculate the speed of gravity in an analogous manner. In fact, I had an email exchange with Sergei Kopeikin who claimed that during the Jovian Deflection Experiment he observed Jupiter's gravity fading and reforming as Jupiter moved through space. The speed of this process turned out to be the speed of light or close to it. He also informed me that the gravity was stronger at Jupiter's previous (retarded) position and weaker at Jupiter's current position due to the light-time delay.
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That got me thinking. If Jupiter moved at light speed, there would be zero gravity at its instant position, since if takes time for the spacetime to curve and when it does, Jupiter has moved to its next position. Its maximum gravity would be at a previous position. This is true even if Jupiter moves much slower than light speed. And so ... Houstin, we have a problem: Newton and Einstein created equations that assume Jupiter's maximum gravity is located at Jupiter's current position, not its previous position. If we plug in zero (or minimum gravity) at Jupiter's current position, or, plug in maximum gravity where there is no Jupiter (mass Mj), the equations break down and become inequalities:
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</p><p>Additionally, the Jovian observations are inconsistent with observations of the solar system orbiting the center of the Milky Way galaxy. According to Ethan Siegel (see reference below), the sun and the planets orbit the galaxy's center on the same plane. This implies that the solar system's curved spacetime moves in sync with the sun and planets. If the sun were to get ahead of its gravity (like Jupiter), the planets would lag behind the sun and form what looks like a vortex.
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There is also a physical and thought experiment that can verify whether or not curved spacetime lags behind a planet's motion: Imagine two sky divers (Alice and Bob) jumping from a jet. Alice holds a target and Bob holds and aims a paintball gun. Bob has perfect aim and takes aim and fires along the horizontal axis (x). The paintball accelerates and hits the bullseye. This would not be possible if Alice, Bob, the target and the paintball were not falling at the same rate. Here is a crude illustration of what has happened so far:
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If the paintball's acceleration vector lagged behind, it would have followed the path of the dotted line. It didn't; it followed the path of Earth's spacetime curvature along with Alice and Bob. Not only does gravity pull down matter, it also pulls down acceleration vectors. Additionally, the fact the paintball hit the bullseye implies that Earth's curved spacetime vector, along with everything else, follows the sun's curved spacetime as the Earth orbits the sun. Imagine the sun's gravity pulling everything forward along the z axis:
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From this experiment we can infer that the solar system orbits the galaxy center on the same plane because the solar system's curved spacetime vectors follow the galaxy's curved spacetime. Or, another way to put it, the solar system's gravity falls at the same rate as the solar system. This means where there is mass, there is gravity and vice versa. Gravity does not lag behind a mass's movement. This is consistent with gravity equations, but not consistent with the rubber-sheet analogy or the Jovian experiment.
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OK, so the Jovian experiment is called into question. So what? Surely LIGO's discovery of gravitational waves clinches the notion that scientists have successfully measured the speed of gravity. There's even a nice quadrupole-moment equation that gives the strain or amplitude (h) of such waves:
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As the black holes' orbits decay, gravitational waves carry away energy and momentum. Unfortunately, these quantities are conserved. Why is this unfortunate? It is the hope of many physicists that gravitons make up gravitational waves. It is believed that when gravitons interact with matter, this interaction will be indistinguishable from gravity, but gravity does not appear to conserve force and momentum. For example, if you consider falling objects at rest and the earth accelerating to them, the earth accelerates more if it gains mass and accelerates less if it loses mass. Or, consider the earth at rest and drop any two objects with different masses in a vacuum chamber and they will appear to have virtually the same velocity at any point in spacetime:
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Notice at equations 6 and 6a there's squared momentums in the numerators and they are not conserved because the velocity c is constant. By contrast, if gravitational waves interact with masses M and M', we have the following:
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Since the momentum is conserved, we can expect the strain h to change when the waves interact with different masses. If the strain is gravitational, it should not change at all. Gravitational waves behave somewhat like a Newtonian force. If there is just enough lost energy to move a feather, that energy will not move a mack truck. Gravity has no problem moving both the feather and the truck.
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Additionally, any quadrupole-moment force can cause gravitational waves! Let me demonstrate. Take equation 4 and make some substitutions:
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Equation 16 shows that any force with a quadrupole moment can cause gravitational waves. An example would be a rotating dumbbell powered by an electric motor. Perhaps such waves should be relabeled "vacuum waves." It is highly doubtful they are made up of gravitons. If they were, there would be a strong correlation between gravitational waves and the strength of gravity. Earth is the strongest source of gravity we experience; yet, its gravitational waves are nil. By contrast, the gravity we experience from black holes lightyears away is nil, but their gravitational waves are significant. It
is also highly doubtful the speed of gravity was successfully measured. However, we can sate with confidence that vacuum waves propagate at or close to the speed of light.
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References:
</p><p>
1. Ibison, Michael, Puthoff, Harold E., Little, Scott. The Speed of Gravity Revisited.
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2. Kopeikin, Sergei, Fomalont, Edward B. 27 Mar 2006. Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment.
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3. Flanagan, Eanna. Hughes, Scott A. 2005. The Basics of Gravitational Wave Theory. New Journal of Physics.
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4. Carlip, S. Aberration and the Speed of Gravity. December 1999.
</p><p>
5. Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research
University of Maryland Physics Army Research Lab.
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6. Siegel, Ethan. August 30, 2018. Our Motion Through Space Isn't A Vortex, But Something Far More Interesting. Forbes
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7. Galileo's Leaning Tower of Pisa experiment. Wikipedia.
</p><p>
8. David Scott does the feather hammer experiment on the moon | Science News. Youtube.com
</p><p>
9. Tzortzakakis, Filippos, LIGO Analysis: Direct Detection of Gravitational Waves. Journal of Research Progress Vol. 1.
</p><p> GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-54628307059842636472023-05-23T10:07:00.000-07:002023-05-23T10:38:43.383-07:00Why Entanglement and Faster Than Light Speed Are Consistent with Relativity<p>
ABSTRACT:
</p><p>
This paper shows why entanglement is not limited to the quantum realm, and shows how entanglement and superluminal speed is not only possible, but consistent with special and general relativity.
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Imagine two photons. Photon A and photon B are propagating in opposite directions. According to the velocity addition formula, their combined velocity v is as follows:
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Now, imagine two observers, Alice and Bob. Alice looks at each photon individually and notices that they each propagate at c. Bob looks at both photons at once and notes that their combined velocity is c. At this point, you probably have some questions: How does photon A seem to know that photon B is propagating in the opposite direction? It's not like photon B can send a signal to photon A (a signal that would have to be faster than light) to let photon A know that it needs to cut its velocity in half along with photon B so their combined velocity will be no faster than light. Further, how do A and B seem to know that Bob is watching them both? They also seem to know that Alice is watching only one of them. The one she's watching seems to adjust its velocity to c just for her benefit. It's as if the photons are entangled with each other and also entangled with all observers.
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Einstein described quantum entanglement as "spooky action at a distance"--yet, where would the velocity addition formula be without "spooky action at a distance"? Below is a mathematical derivation of the entanglement of two particles with velocities v1 and v2:
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</p><p>
Equation 7 above shows that, at any distance r (the distance between the two particles), any change of velocities (v1, v2) must lead to an instantaneous change in velocity v; otherwise, light speed c would not be constant in a vacuum. Note that the terms on the right side are in units of frequency and wavelength. To maintain a constant light speed requires any change of frequency to be instantly offset by a change of wavelength. Additionally, this entangled relationship between frequency and wavelength is shown by equation 9 below:
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At 11 above is a scalar version of Einstein's field equations. Equation 10 shows that velocity v can be infinite if distance r drops to zero. How is this possible given that infinite energy is required to accelerate mass m to light speed? Equation 9 provides the answer: the infinite velocity is achieved with just the rest-mass energy (E). No force acts on mass m. If a force acts on mass m, then momentum p will be greater than zero. It is this momentum that requires infinite energy to reach light speed. Since infinite energy is not available, mass m cannot reach light speed in this way--and--here is the ironic part: to reach a speed faster than light requires no outside force or energy--just the rest-mass! Albeit, equation 9 shows that superluminal speed is offset by extreme curvature of spacetime. This offset happens instantaneously (yes, more "spooky action at a distance") to ensure that the rest-mass energy is conserved.
</p><p>
Below is a proof that shows the absurdity of assuming it takes a time of r/c seconds for a change of frequency (a change of velocity or mass density) to update a change of wavelength (velocity or curved spacetime), where r is distance and c is the speed of a graviton:
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Ironically, the very speed of light itself depends on instantaneous "spooky action at a distance." We can conserve the energy of our two-particle system in the following manner:
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The speed of light also depends on the speed of our expanding universe--even if that speed is faster than light:
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</p><p>
Equation 19 above shows that velocity Hr could be faster than light; yet the right side of the equation never exceeds c or light speed.
</p><p>
So far, it appears that gravity and dark energy have infinite velocity potential and that spacetime and matter are entangled--which enables "spooky action at a distance" beyond the quantum realm. So ... are there any experiments or observations that lend support to such weirdness? At the time of this writing, I know of no direct observation of superluminal speed. However, black holes lead to the inference that light speed is not enough to escape a black hole's gravity that has a potential meeting or exceeding light speed (see equation 10 above). Additionally, no light can reach us from galaxies that are beyond the cosmological horizon. The "spooky action ..." on a cosmological scale is consistent with astronomical observations cited by Laplace and Van Flandern.
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At equation 21 below we define the frequency (f) of an electric field. Albeit, there is a problem. It is assumed that the electric field extends to infinity! At any distance r, an observer allegedly experiences an electric field. If the electric charge q is beyond the cosmological horizon, i.e., r > c/H, an observer at that distance never observes q's electric field nor its frequency f. So at 22 we create a new variable s that equals zero if r > c/H. Equation 23 shows that the observer observes zero evidence of frequency f. Equations 24 through 26 show that variable s should also be applied to gravitational waves (GWs) (and their frequencies), since they are limited to light speed and can't reach an observer if they originate beyond the cosmological horizon.
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It is clear that an electric field and GWs have a limited observable range. There is one field, however, that truly has an unlimited range: the vacuum field or "dark energy" if you prefer. An observer at any distance r would never claim there is no evidence of such a field. Thus our new variable s is inapplicable. Velocity v at equation 27 below never equals zero unless r equals zero. At 28 we create a new variable Sv that is always equal to one. As a result equation 29 can be substituted for equation 27.
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</p><p>
We bring back the Friedmann equation at 30 below. According to the WMAP spacecraft, space is nearly flat, so we set k to zero.
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Let's assume a gravitational field has a limited range of r = c/H. The diagram below shows a sphere with volume V divided into an alpha section and a beta section. The alpha section is within the c/H limit for observer O; the beta section is not. This creates an inequality at 31. If gravity depends on gravitons limited to light speed, the Friedmann equation is invalid if distance r is greater than c/H.
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Next, lets assume the vacuum and gravitational fields both exist everywhere. The equality is restored and the Friedmann equation is always valid:
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This seems inconsistent with GWs that cannot penetrate the c/H barrier. Let's examine GW equations and see if we can reconcile this apparent inconsistency.
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At 33 above we begin with a GW power equation for two rotating black holes. With a little algebra we derive equation 37. At 37 we assume gravity has an unlimited range, so we multiply that part by Sv which equals 1. We further assume GWs that are more than c/H meters away from an observer cannot be detected. So we multiply P and the frequency by s, where s equals 0. Equation 37 confirms that gravity overcomes the c/H barrier and GWs may not. GWs do not carry gravitational information. If there was zero frequency, the black holes would still have gravity and there would be no GWs. The source of GWs is the kinetic energy needed to maintain the orbits of the black holes. Over time this energy is converted to massless waves that propagate no faster than light. Equation 38 below shows how gravity can exist in the absence of GWs (notice that the s's cancel):
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</p><p>
Conclusion:
</p><p>
It appears that vacuum and gravitational fields extend to infinity unlike the electric field. Both gravity and dark energy have the potential for speed greater than light; yet, ironically, this does not violate the light-speed limit. In fact, the light speed limit itself depends on "spooky action at a distance"--i.e.--entanglement of frequency and wavelength. This entanglement is also essential to the velocity addition formula that ensures that two velocities never exceed light speed.
</p><p>
</p><p>
References:
</p><p>
1. Flanagan, Eanna. Hughes, Scott A. 2005. The Basics of Gravitational Wave Theory. New Journal of Physics.
</p><p>
2. Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research
University of Maryland Physics Army Research Lab.
</p><p>
3. Siegel, Ethan. August 30, 2018. Our Motion Through Space Isn't A Vortex, But Something Far More Interesting. Forbes
</p><p>
4. Tzortzakakis, Filippos, LIGO Analysis: Direct Detection of Gravitational Waves. Journal of Research Progress Vol. 1.
</p><p>
5. Roberts, Tom, Schleif, Siegmar. 2007. What is the experimental basis of Special Relativity?
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6. Friedmann equations. Wikipedia
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-54633913337275669642023-04-29T16:05:00.002-07:002023-04-29T16:44:42.624-07:00Why the Speed of Gravity and the Speed of Gravitational Waves Are Not the Same<p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/Oo8TaPVsn9Y" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe>
</p><p>
ABSTRACT:
</p><p>
According to Relativity Theory, everything propagates through spacetime at light speed. However, a mass at rest propagates solely through time and experiences zero velocity. A massless photon propagates solely through space and experiences no time. Other objects propagate through time and space, and, experience both time and subluminal velocities. This paper demonstrates that both gravity (the fundamental interaction) and gravitational waves propagate at light speed through spacetime, but with varying degrees through time and space, i.e., they each have different velocities through space.
</p><p>
The year was 1971. Via the Apollo 15 mission, David Scott performed the following experiment on the moon: With a hammer in one hand and a feather in the other, he held them the same distance from the moon's surface. He dropped them. They hit the ground simultaneously. This experiment might not seem like a big deal, but it confirms that Galileo was correct. More importantly, it shows that gravitational interactions (with the exception of gravitational waves) should never be modeled after the electromagnetic (EM) force.
</p><p>
The EM force, like Newtonian forces, conserves energy, momentum and itself in the following manner:
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</p><p>
Equations 1 and 2 above show that different masses have different velocities and different rates of acceleration when acted on by the same magnitude of force. Since momentum is conserved, we can use equations 2 above and 3 below to predict the speed of the photons that mediate the force:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmykFeehE1HBFg8d8n8SMUQHvIuVF4SS143b4qtkprljvhjaxrHBYsz52hhqSXLfePtTLmX7TOZBBSyHZbWGl2ETrEFaKq3A065sbwTKHthFNrvFWv4nh8dtTGy3OIzK3Ip-cFGwdDnMHSTXxWoOLGS2DAo1QU7QeBELZOsQEcOi6_deYvFlsQ04DO/s350/GS2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="117" data-original-width="350" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmykFeehE1HBFg8d8n8SMUQHvIuVF4SS143b4qtkprljvhjaxrHBYsz52hhqSXLfePtTLmX7TOZBBSyHZbWGl2ETrEFaKq3A065sbwTKHthFNrvFWv4nh8dtTGy3OIzK3Ip-cFGwdDnMHSTXxWoOLGS2DAo1QU7QeBELZOsQEcOi6_deYvFlsQ04DO/s320/GS2.png"/></a></div>
</p><p>
No surprises here. Photons propagate at c as expected. Gravity, unlike EM, is full of surprises. Let's model gravity after the EM force and see what happens. Let's assume there is a gravitational field of gravitons that mediate the "force" of gravity. Here is the math:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit_pmO5kXYW4A78Nzk4HVftU012nInvsimqHxCwAtRYbDiZh1bK9N3yuoJzH-sT0CeZJnrFqmnEen-9kSxQSa1e6ofnv3dIPjo-1KAsjBYvBS7WspHO6EIFdA2QxFiQ-qdvQ7gaABN94Z0DQAGFzbYkdLhp3PCwr3tpo5ZnEBkv-ztw2VpCbhde_dD/s773/GS3a.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="773" data-original-width="559" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit_pmO5kXYW4A78Nzk4HVftU012nInvsimqHxCwAtRYbDiZh1bK9N3yuoJzH-sT0CeZJnrFqmnEen-9kSxQSa1e6ofnv3dIPjo-1KAsjBYvBS7WspHO6EIFdA2QxFiQ-qdvQ7gaABN94Z0DQAGFzbYkdLhp3PCwr3tpo5ZnEBkv-ztw2VpCbhde_dD/s400/GS3a.png"/></a></div>
</p><p>
At 6 and 7 above, force and momentum are not conserved. If we assume momentum is conserved, at 8 the speed of gravitons depends on the mass of the falling object. We cannot count on their speed being c. What about gravitational waves? Why do they consistently propagate at or near the speed of light? Consider the following thought experiment:</p><p>
</p><p>
You throw a baseball with enough force to place it into orbit around the earth. The force you use is independent of and counters gravity. It is also conserved and so is the ball's angular momentum. The ball's tangential velocity depends on its mass and vice versa. As the ball orbits earth its velocity changes, i.e., the ball accelerates. An accelerating mass emits gravitational waves (GWs). The energy converted to GWs is the same conserved energy you put into the ball when you threw it into orbit. We can predict the speed of these GWs the same way we predicted the speeds of photons and gravitons:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7n3polb50PIscDzSWUOBbhlHjmA2oiTPOOlRSs49cKGlK2zc68_q53OwptEkzPc1tPjeKhfCmnwSajDjPxVYhhJyeZBuT4wA6FLBAB_X61zfV-3RmMQfxw2irysdWlEx0OI31fJX-VsGNp20UF5STg7JARc8DPuTCfDfW1fSG71Ha1xdMR6RnqOV6/s798/GS4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="691" data-original-width="798" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7n3polb50PIscDzSWUOBbhlHjmA2oiTPOOlRSs49cKGlK2zc68_q53OwptEkzPc1tPjeKhfCmnwSajDjPxVYhhJyeZBuT4wA6FLBAB_X61zfV-3RmMQfxw2irysdWlEx0OI31fJX-VsGNp20UF5STg7JARc8DPuTCfDfW1fSG71Ha1xdMR6RnqOV6/s400/GS4.png"/></a></div>
Equation 9 shows that the ball's velocity happens to equal the gravitational potential velocity at distance r. If the ball had more mass (m), its velocity would be less and its orbit would decay. If the ball had less mass, it would have more velocity and would rise out of orbit. Equation 10 shows how much power P is emitted. Over time velocity v will be reduced and the ball will spiral into the earth. At any time, momentum p equals mv. Equation 11 predicts the speed of GWs to be ~c, the speed of light. This is possible because the baseball was originally accelerated to v by you, not gravity. To predict the speed of gravity, absent the influence of another force (you throwing a baseball), requires a model that is different from the EM or force model. Over a century ago, Einstein realized this and had a big idea:
</p><p>
From an airplane flying 10,000 feet above the earth's surface, drop several items with different masses. If we assume they are falling to earth, they all fall at the same rate, so momentum at any instant is not conserved. But what if those items are at rest and it is the earth with mass M falling or accelerating to the items? Clearly, an independent force accelerating the earth would be conserved. With more mass, the earth would accelerate less. The problem is, with more mass the earth really accelerates more. This fact implies that there is no independent force causing earth to accelerate. So there is apparently no independent force acting on the earth or the items that appear to be falling. </p><p>
If the earth simply accelerates to the items, what need is there for a graviton? As shown at 6 through 8 above, gravitons, if they exist, fail to either conserve force, energy and momentum, or, they don't have a consistent speed if force, energy and momentum are conserved. David Scott's experiment showed us this is true. The feather and the hammer fell at the same rate, not different rates.
</p><p>
Laplace and Van Flandern, based on observations, concluded that the speed of gravity must be several orders of magnitude faster than light. Perhaps infinite! (Masses simply accelerating towards one another combined with independant forces causing angular momentum could certainly provide that impression.) Other physicists hate the idea of infinite speed and insist the speed of gravity is c. To accommodate the Laplace and Van Flandern observations, they point to a model of moving charges, where one charge's vector is lined up with the another charge's instant position rather than its retarded position, creating the illusion that there is no light-time delay, i.e., infinite photon speed when in reality photon speed is c. This model is then projected onto a cosmological scale, and thus moving planets and stars work in a similar fashion and create the illusion of infinite graviton speed when in reality graviton speed is allegedly c. The problem with this model is it completely ignores the Heisenberg Uncertainty Principle. For the model to work, one has to know the position and velocity of the charges with precision. And, as demonstrated above, charges (or EM) conserve force and momentum in a way that gravity does not.
</p><p>
Laplace, Van Flandern and the physicists who criticize them have one thing in common: they all think of gravity as a force consisting of bosons that either propagate at c or much faster than c. Because of General Relativity, it makes perfect sense that most physicists want to limit the speed of gravity to c; however, when two black holes collide and form a new more massive singularity, it is not clear how a boson propagating at c can escape that singularity and inform the rest of the universe of the event. Since nothing propagating at c can escape a black hole, how does this new black hole singularity reset the curvature of its surrounding spacetime?
</p><p>
Let's start with what we know. At 12 below we have the Compton wavelength equation. Note that when mass m changes, the wavelength must change instantaneously, since a mass and its wavelength are essentially the same entity. If we think of the wavelength as spacetime, then spacetime is updated the instant mass changes. At 14 we convert the equation to Planck units with an alpha scale factor. At 15 we derive a Scharzschild radius. Equation 16 shows that a change in alpha instantaneously causes a change in beta, since their sum times a Planck length make up distance r. (Distance r, of course, along with the mass, determines the rate of Newtonian acceleration.) Equation 17 is a scalar version of Einstein's field equations. On the right side we have curvature units.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWQVYvtLYs3s2_h0k-xTqHj05Qibr5QIZ3aejyx4F02uWHqkuYfA5GpNLMyoFMZOYKxe5eJpiIhAaISO4yRNBcY8OH96WvPbIyy1WXE3DvN0H9wrTlvuSOWbYkJbOVHW8iVgb_-DNH5vdCh3XyQy7YDqa7zNsn4BeWbgNM9ZhYVb7oF1wYvwtcTAZK/s843/GS5.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="843" data-original-width="603" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWQVYvtLYs3s2_h0k-xTqHj05Qibr5QIZ3aejyx4F02uWHqkuYfA5GpNLMyoFMZOYKxe5eJpiIhAaISO4yRNBcY8OH96WvPbIyy1WXE3DvN0H9wrTlvuSOWbYkJbOVHW8iVgb_-DNH5vdCh3XyQy7YDqa7zNsn4BeWbgNM9ZhYVb7oF1wYvwtcTAZK/s400/GS5.png"/></a></div>
</p><p>
When two black holes merge, alpha increases everywhere it appears at equation 17. This causes an instantaneous decrease of beta at any distance r from the new black hole's singularity. Thus the new black hole doesn't have to send information at light speed or any speed to update its surrounding spacetime. Mass and spacetime have an entangled relationship. Energy and momentum equations show this to be true. Where would energy and momentum be without mass entangled with velocity? Of course velocity is in units of space and time. Thus one can conclude that the only valid speed for gravity is how fast matter moves at an arbitrary distance from a falling observer:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1np_i1Q9CEzSdr3-gkLE6JtnQDH_id4GCTBEUPpJdxskv98BA-0uuttHj76Gd0ZuwaxVciCssbg93vvWJDTYe0X6tGsAHA8Hzh4sF4ecUJKkyyA-kEUuS2MOn6C90bBN4PO1kFAIwkA-FbMJ7M3CCnfkKrJkyqcIQXbSb9SAY4yI9SK30LXT0Ospd/s614/GS6.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="614" data-original-width="529" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1np_i1Q9CEzSdr3-gkLE6JtnQDH_id4GCTBEUPpJdxskv98BA-0uuttHj76Gd0ZuwaxVciCssbg93vvWJDTYe0X6tGsAHA8Hzh4sF4ecUJKkyyA-kEUuS2MOn6C90bBN4PO1kFAIwkA-FbMJ7M3CCnfkKrJkyqcIQXbSb9SAY4yI9SK30LXT0Ospd/s400/GS6.png"/></a></div>
</p><p>
Equation 18 above shows how fast gravity moves through space. Equation 19 shows how fast gravity moves through time. Finally, equation 20 shows how fast gravity moves through spacetime--the speed of light.
</p><p>
</p><p>
</p><p>
</p><p>
<!--Premise: momentum and wavelength or entangled. Show black hole formula and GWs emitted from a black hole acting like one solid mass. action at a distance divorce papers. Then derive rs/r and space and time equation for final equation where c is speed through space and time. Since no force or bosons, the only speed for gravity is GM/r through space.-->
</p><p>
References:
</p><p>
1. Ibison, Michael, Puthoff, Harold E., Little, Scott. The Speed of Gravity Revisited.
</p><p>
2. Kopeikin, Sergei, Fomalont, Edward B. 27 Mar 2006. Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment.
</p><p>
3. Flanagan, Eanna. Hughes, Scott A. 2005. The Basics of Gravitational Wave Theory. New Journal of Physics.
</p><p>
4. Carlip, S. Aberration and the Speed of Gravity. December 1999.
</p><p>
5. Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research
University of Maryland Physics Army Research Lab.
</p><p>
6. Siegel, Ethan. August 30, 2018. Our Motion Through Space Isn't A Vortex, But Something Far More Interesting. Forbes
</p><p>
7. Galileo's Leaning Tower of Pisa experiment. Wikipedia.
</p><p>
8. David Scott does the feather hammer experiment on the moon | Science News. Youtube.com
</p><p>
9. Tzortzakakis, Filippos, LIGO Analysis: Direct Detection of Gravitational Waves. Journal of Research Progress Vol. 1.
</p><p> GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-15333350850583487882022-12-31T17:12:00.003-08:002022-12-31T17:33:23.890-08:00What Really Happens If the Sun Disappears?<p>
To demonstrate that the speed of gravity is no faster than light, physicists love to point out that if the sun suddenly vanished, it would take approximately eight minutes for the information to reach earth, and over five hours for the information to reach Pluto. It is assumed that earth and Pluto would remain under the influence of the sun's gravity for eight minutes or over five hours respectively. After all, it is currently understood that information, including gravitational information cannot exceed light speed.
</p><p>
But here is the irony: to make the point that nothing is faster than light, the sun (poof!) disappears faster than light! Further, such a thought experiment only takes into account an observer on earth (or Pluto). Allegedly, both gravitational and electromagnetic information reach the observer simultaneously, so said observer (Alice) observes nothing out of the ordinary. She sees the sun vanish and notices that the sun's gravity has vanished along with it. However, another observer (Bob) has parked his spaceship telescope halfway between the sun and the orb where Alice is located (see drawing below).
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjikMsBbJTJAKNgNJgEGDzjApqXGWsngg81_Lc4jj50UnnWV8mg5RMxrl8mdYzNI00eCBJ-BH_rx2LGUKgRkx8tSav7jxob54-xAlc2LgM2sbwVTU93akn6DwAjxzBu8VonJPrP1SjNARiN4x-gc3N0y0dyJO2k-RhdNjl7IJP8bEW8G_clVTPs87yx/s432/0.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="367" data-original-width="432" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjikMsBbJTJAKNgNJgEGDzjApqXGWsngg81_Lc4jj50UnnWV8mg5RMxrl8mdYzNI00eCBJ-BH_rx2LGUKgRkx8tSav7jxob54-xAlc2LgM2sbwVTU93akn6DwAjxzBu8VonJPrP1SjNARiN4x-gc3N0y0dyJO2k-RhdNjl7IJP8bEW8G_clVTPs87yx/s320/0.png"/></a></div>
</p><p>
The orb is distance r from the sun and Bob's telescope is distance r from the sun and the orb. Information from the sun and the orb reach Bob simultaneously within a time of r/c seconds, where c is light speed. Now, let's suspend disbelief and pretend the sun vanishes instantaneously (see drawing below).
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0_wLbdn7p8M8T-RiiNZaHV8KSs61JNMgy-8RXW3vcl4nakwKUozzUAECusBg9Amz_fUw2V9-GuW3PG_RVCZFWUln2hkdL65PeHoX7s13qG-W5VZ3KNhofcKVF3yf9O3l9v31YsPMbjijrPyITKTowZOFwGGfwQE-qiuoZpaEUt-BW8q8pz4tWRAj-/s432/0a.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="367" data-original-width="432" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0_wLbdn7p8M8T-RiiNZaHV8KSs61JNMgy-8RXW3vcl4nakwKUozzUAECusBg9Amz_fUw2V9-GuW3PG_RVCZFWUln2hkdL65PeHoX7s13qG-W5VZ3KNhofcKVF3yf9O3l9v31YsPMbjijrPyITKTowZOFwGGfwQE-qiuoZpaEUt-BW8q8pz4tWRAj-/s320/0a.png"/></a></div>
</p><p>
Because information from the sun and the orb reach Bob simultaneously, Bob must wait another r/c seconds to observe what Alice observes. Suppose he doesn't wait. Suppose he plugs in the numbers he observes when he first receives information from the sun and the orb. Here is what Bob's math reveals:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK1v4NFbJyic7pXkeB8ieiCtwpV57dfEB9gI69ZqqDkAttSS07mAnYaUPXkptGFa19cx2wLi_dOaPZnVxUMdA7AYBFbbtEq586t6_DJ-q0Y_sLi784MrxhKx0Eq-vjNr3jbnqVcD_euKdX0fIz1KLSeVcUYI0bJiC1s-GgGu4Yp8hsOpeySlifaLiA/s813/1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="813" data-original-width="722" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK1v4NFbJyic7pXkeB8ieiCtwpV57dfEB9gI69ZqqDkAttSS07mAnYaUPXkptGFa19cx2wLi_dOaPZnVxUMdA7AYBFbbtEq586t6_DJ-q0Y_sLi784MrxhKx0Eq-vjNr3jbnqVcD_euKdX0fIz1KLSeVcUYI0bJiC1s-GgGu4Yp8hsOpeySlifaLiA/s600/1.png"/></a></div>
</p><p>
At 1) we have Einstein's field equation. If the sun vanishes, the stress-energy tensor (T) value would be severely reduced, so at 2) T has a limit of zero. A little algebra gives us a value that is substantially greater than the gravitational constant. Both 2) and 3) reveal that Newton's constant is not constant, i.e., has a much greater value than expected for a time period of r/c, the time it takes for the information to reach Alice.
</p><p>
If we assume that gravitational information is necessary and that it propagates at light speed, then we must abandon the idea that Newton's constant is constant for all observers. Clearly it is not if gravitational information is necessary to cause gravity.
</p><p>
On the other hand, if we assume gravity is more analogous with the equivalence-principal thought experiment, where the dropped pen seems to fall to the spaceship's floor but the floor really accelerates to the pen, then gravitational information is not necessary to cause gravity. Further, the math shows that Newton's constant remains constant, because there is no r/c time delay--the spaceship does not send a signal to the pen--it simply accelerates to the pen, creating the "persistent illusion" of gravity, where reality, according to Einstein, is a "persistent illusion." </p><p>
Of course nothing I have written here addresses gravitational waves or how gravity is supposed to work outside an equivalence-principle spaceship. I address those concerns in "The Beautiful Destruction of the Graviton" and "Quantizing Gravity without the Graviton."--which you can find at my profile page at Accademia.edu.
</p><p>
OK, then ... if the sun should magically disappear what really happens? If Newton's constant is really a constant, the impact on the solar system will be immediate but this will not violate the light-speed limit because no gravitational information needs to propagate from the sun.
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-28816643885940690682022-08-06T21:56:00.001-07:002022-11-06T21:20:30.337-08:00P = NP? A Solution to the Clay Mathematics Institute's Sample NP Problem<p>
ABSTRACT:
</p><p>
This paper offers a solution to the sample NP problem that can be found at the Clay Mathematics Institute's website. A python program is included along with commentary on whether or not P = NP.
</p><p>
At their P vs. NP page, the Clay Mathematics Institute (CMI) provides the following NP problem example:
</p><p>
"Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice. This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a coworker is satisfactory (i.e., no pair taken from your coworker's list also appears on the list from the Dean's office), however the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe! Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students."
</p><p>
To make this problem even more difficult and more explicit, let's assume there is only one set of 100 students who qualify for places in the dormitory, and, let's assume we can randomly select any two students from this set--and the pair will not appear on the dean's list of pairs of incompatible students. Thus, no pair from this set will match any pair on the dean's list. The following math shows that this ideal set is buried in a haystack of approximately 10^96 (not 10^200) sets of 100 students! </p><p>
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</p><p>
Below we derive equation 4 which shows that not only k increases as n increases, but the exponent increases as well:
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</p><p>
A brute-force algorithm will undoubtedly exceed polynomial time (P). However, we can convert this NP problem to a P problem if we replace student names with numbers ranging from 0 to 399, then pair with each student number an incompatibility score. We do this by examining how many times each student number appears on the dean's list of student pairs. For example, if the list shows [[1,2], [1,3] ...], student 1 appears twice and receives a score of 2. Students 2 and 3 only appear once, so they receive a score of 1. Having the lowest incompatibily scores, students 2 and 3 will appear on the final 100 list along with 98 other low-scoring students.
</p><p>
The final list of 100 students can be checked against the dean's list in polynomial time. The process of checking the dean's list to give each student a score can also be performed in polynomial time. This can be mathematically shown. Let's start by calculating the size of the dean's list of pairs:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLhfVcxVpEAs3n8YLFZyuk9bZ391K5nMNJHIIVMJW_4FuBl9ee_3u5BUG5wpY0MSGg9B1-WsHiY-7brl_UZaR9QzoKJ2Ma9ce4g1wKjaBQvp2WMfRtW7-HJy2XFo1LKPvOGSExIcNo-64NsauLKxgrG7I_BBWO_kH6wpIhhHO_pHK_9fxoCjbjtE7H/s563/3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="269" data-original-width="563" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLhfVcxVpEAs3n8YLFZyuk9bZ391K5nMNJHIIVMJW_4FuBl9ee_3u5BUG5wpY0MSGg9B1-WsHiY-7brl_UZaR9QzoKJ2Ma9ce4g1wKjaBQvp2WMfRtW7-HJy2XFo1LKPvOGSExIcNo-64NsauLKxgrG7I_BBWO_kH6wpIhhHO_pHK_9fxoCjbjtE7H/s400/3.png"/></a></div>
</p><p>
The first term on the right side of equation 5 is the total number of student pairs possible with no repeats. We subtract from that the total possible pairs (with no repeats) from the final 100 list. Thus, the size and complexity of the problem has been reduced from approximately 10^96 arrangements of 100 students to less than 100,000 arrangements of student pairs. Below is the relevant math showing the problem is solved in polynomial time:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQMhQGxtzKjyehszZIClOefgcX60xN62Mpb3OHTyHfb8s3CpDypfJRYrLG6t3YWp_-p6Lc55xzq3foAasu7lsXDNDFvceJffU5M7tJ1cWx4i4041lRFc_QBzYPu88F82k0i9AgBJFoLvT0-499XF0eGTDlLPDJ0LLl2G1cLxpt-dFgfMdpn1Z2mePe/s808/4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="808" data-original-width="512" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQMhQGxtzKjyehszZIClOefgcX60xN62Mpb3OHTyHfb8s3CpDypfJRYrLG6t3YWp_-p6Lc55xzq3foAasu7lsXDNDFvceJffU5M7tJ1cWx4i4041lRFc_QBzYPu88F82k0i9AgBJFoLvT0-499XF0eGTDlLPDJ0LLl2G1cLxpt-dFgfMdpn1Z2mePe/s400/4.png"/></a></div>
</p><p>
The term on the right side of equation 11 does not have the number of students (n) as an exponent. The variable k' increases as n increases but the exponent stays fixed at 2--unlike equation 4 where the exponent increases as n increases.
</p><p>
Now, on paper it looks as though the NP problem can be reduced to a P problem, but what about in practice? Below is a python program that solves the CMI's sample NP problem:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyItsJ3_u4xJfg5oydxPrQozNuxQsGyDAF4lApy9VFSsSbAwYZjHuAuMAMdks3Ohh9Y2cq3P2mxkc0mzeN6J3FN_I3oaIt8QfBgDRpRTKMZC8vg6lvywlpBB52vEpg5iyyMmD8AWC-YL0iBg70ElZEnnXMMBdGaefc3WHED67q2z3vR4Hgv_eGfvCy/s990/final_NP1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="583" data-original-width="990" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyItsJ3_u4xJfg5oydxPrQozNuxQsGyDAF4lApy9VFSsSbAwYZjHuAuMAMdks3Ohh9Y2cq3P2mxkc0mzeN6J3FN_I3oaIt8QfBgDRpRTKMZC8vg6lvywlpBB52vEpg5iyyMmD8AWC-YL0iBg70ElZEnnXMMBdGaefc3WHED67q2z3vR4Hgv_eGfvCy/s600/final_NP1.png"/></a></div>
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhCLcjVhiJ_J-Rvsezy4itqicPXzjZuI6-otfYIhyZtbq05M-MYQZ2Ehh-uAHeAPz36LySN1tPFzEgssF24ic-D6_xJfAStz5TDARjZBHDLoCkfqss6FBynR3HJHp7RC8kua-2IB5OmRzOSKoNygwCVMTk6hVNHFCX0YjhAcbzX2rBgsmNKRQSk0iu/s1189/final_NP2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="575" data-original-width="1189" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhCLcjVhiJ_J-Rvsezy4itqicPXzjZuI6-otfYIhyZtbq05M-MYQZ2Ehh-uAHeAPz36LySN1tPFzEgssF24ic-D6_xJfAStz5TDARjZBHDLoCkfqss6FBynR3HJHp7RC8kua-2IB5OmRzOSKoNygwCVMTk6hVNHFCX0YjhAcbzX2rBgsmNKRQSk0iu/s600/final_NP2.png"/></a></div>
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirxN_HAMiXy9SebG0b3UkriL64Gc975Zta7yBljZY3DnM0fKLrAadegpUT3rNYYg_okiqLX74uJUlziYNRJhI7l7P5JITn2rDAjIimHAknY3TSLWyqpvTeo3pnKqiOzBdENcZa7blQuOYpLN2vRoDSyUF8gIvSA-6Oo6V46eJgXq12SAhwypxjcFMA/s1485/final_NP3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="608" data-original-width="1485" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirxN_HAMiXy9SebG0b3UkriL64Gc975Zta7yBljZY3DnM0fKLrAadegpUT3rNYYg_okiqLX74uJUlziYNRJhI7l7P5JITn2rDAjIimHAknY3TSLWyqpvTeo3pnKqiOzBdENcZa7blQuOYpLN2vRoDSyUF8gIvSA-6Oo6V46eJgXq12SAhwypxjcFMA/s600/final_NP3.png"/></a></div>
</p><p>
Here is the final output:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOok_7W8k57agwTLRSQ042PK-ZKiAP9hhcWqLxPLDeQ83_OCC1erwIPXKPWLz-BgEac2evB68j5K8UFnm06sJr0PkwRHg4tFl_z1XQOuZMaiLkQoI6fQHAkYWwQQzj9vqkDV7xxG8gj6MDCgs7gL_iutpUG1-mXvHd628FwfttaySfnywWAZCg9JSZ/s1553/final_NP_output.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="328" data-original-width="1553" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOok_7W8k57agwTLRSQ042PK-ZKiAP9hhcWqLxPLDeQ83_OCC1erwIPXKPWLz-BgEac2evB68j5K8UFnm06sJr0PkwRHg4tFl_z1XQOuZMaiLkQoI6fQHAkYWwQQzj9vqkDV7xxG8gj6MDCgs7gL_iutpUG1-mXvHd628FwfttaySfnywWAZCg9JSZ/s600/final_NP_output.png"/></a></div>
</p><p>
Lines 1 to 21 are devoted to creating a random dean's list. The remaining program solves the puzzle by giving each student number an incompatibility score. The 100 students with the lowest scores make the final cut. As you can see, there is only 45 lines of code, most of which are used merely to set up the problem. Only the final 22 lines of code are used to solve the problem. The total run time on my personal computer is about one minute. Now, does this mean that P = NP?
</p><p>
At the very least it can be shown that an NP problem can be reduced to a P problem. But is this always the case? One thing I have noticed is that some NP problems have more clues than others. The CMI's example has 74,850 clues, i.e., the student pairs on the dean's list. These clues tell which pairs don't belong to the set of pairs of students on the final 100 list. So there's no need to guess which of the approximately 10^200 sets of 100 students is the right answer. We can use the power of deduction.
</p><p>
Now, imagine this same NP problem without any dean's list, without any clues. The dean simply tells you whether or not you have the right answer. Even the great Sherlock Holmes would be forced to test up to approximately 10^96 sets of 100 students. His powers of deduction would be useless. So even if we can make NP equal to P, we can always make the NP problem harder--the hardest problem has no clues. Thus it seems reasonable to conclude that P doesn't equal NP where there are no clues. Where there are clues, there is opportunity!
</p><p>
References:
</p><p>
1. Cook, Stephen. The P Versus NP Problem. Clay Mathematics Institute
</p><p>
2. Stewart, Ian. Ian Stewart on MInesweeper. Clay Mathematics Institute
</p><p>
3. P vs NP Problem. Clay Mathematics Institute
</p><p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-16585787090502123752022-07-16T13:29:00.026-07:002022-07-17T13:52:46.552-07:00A Solution to the Continuum Hypothesis<p>
ABSTRACT:
</p><p>
Re: the Continuum Hypothesis: Is there any set which has more members than the set of natural numbers (N), but fewer members than the set of real numbers (R)? The short answer is no. The long answer, that includes mathematical proof, shall be set forth in this paper.
</p><p>
Imagine a finite set of natural numbers (N') with n' members. Imagine another finite set of natural numbers (N) with 2^n' members. Also imagine a set of real numbers (R) with c members:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZLzMyjW8W8k4JrNjCzJKaF0yKnLzLMVev-sVum5cHpBlCnbSxswaJuBs2d87MrKH4p1nFBwluC07l8QoDmp6lKEDVP8-HjO8VxfbR9qaiA0vTa2FHwwYVeVj8HGBoy9k50g_9mpy4W5yLdC-_rxp4GYChTd_6mmEFiSvBIAoLGl5AnmaIAygA8UJa/s486/0a.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="288" data-original-width="486" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZLzMyjW8W8k4JrNjCzJKaF0yKnLzLMVev-sVum5cHpBlCnbSxswaJuBs2d87MrKH4p1nFBwluC07l8QoDmp6lKEDVP8-HjO8VxfbR9qaiA0vTa2FHwwYVeVj8HGBoy9k50g_9mpy4W5yLdC-_rxp4GYChTd_6mmEFiSvBIAoLGl5AnmaIAygA8UJa/s400/0a.png"/></a></div>
</p><p>
Even though N and N' are both sets of natural numbers, they do not bijectively map to each other. However, there is a bijective mapping between N and the power set of N' (P(N')), since P(N') also contains 2^n' members:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3k-GGVzakhsBslgiVb1ehypCUZqXwbHyq0Rp8UXsRqui8DvQR1SafNF2xgDyZycNBmoQdwwVD48XlVb5Wfj2zxBY_8Q-DoAjuuYtfb3w2A9Ye8JLr3hJBPq7CHscp4zb2xeDNZC-FtFAW3E1gYSADzxybm531lsaG4AUoRKVXDuR4mbWmPFCFuive/s564/0b.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="215" data-original-width="564" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3k-GGVzakhsBslgiVb1ehypCUZqXwbHyq0Rp8UXsRqui8DvQR1SafNF2xgDyZycNBmoQdwwVD48XlVb5Wfj2zxBY_8Q-DoAjuuYtfb3w2A9Ye8JLr3hJBPq7CHscp4zb2xeDNZC-FtFAW3E1gYSADzxybm531lsaG4AUoRKVXDuR4mbWmPFCFuive/s400/0b.png"/></a></div>
</p><p>
Now let's take n' to its infinite limit. R has c members, where c = 2^n'. N and R have the same cardinality:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjthPjeZgE2_mW3QZpz3SGfn75CdtCDx8xo5V32k9T_uxZl8gP5_BjC7lT80H34hna45aeLaaxdlZYMiadVHIyeP8fHoFoHTj3JB7AB_mAkVOUkArNwKV000lAHdv_8eJLR7F1JeTW0wqg6xCyf75pqYNLuCydjYSBrLCCnUAZlN5BAn1MFf_1B9pBR/s449/0c.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="268" data-original-width="449" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjthPjeZgE2_mW3QZpz3SGfn75CdtCDx8xo5V32k9T_uxZl8gP5_BjC7lT80H34hna45aeLaaxdlZYMiadVHIyeP8fHoFoHTj3JB7AB_mAkVOUkArNwKV000lAHdv_8eJLR7F1JeTW0wqg6xCyf75pqYNLuCydjYSBrLCCnUAZlN5BAn1MFf_1B9pBR/s400/0c.png"/></a></div>
</p><p>
N and R have the same cardinality? This is not consistent with Cantor's theorem which states: ""Let f be a map from set N' to its power set P(N'). Then f: N'-->P(N') is not surjective. As a consequence, the cardinality of N' is less than the cardinality of P(N') holds for any set N'."
</p><p>
Clearly P(N') has more members than N'. We can prove this by counting the members of P(N'). We start with 1 and count all the way to 2^n'... but this implies the existence of a bigger set of natural numbers (N) with 2^n' members. Albeit, the power set of N (P(N)) has more members than N, but these "more members" imply the existence of an even larger set of natural numbers and so on and so on to infinity. Speaking of infinity, the cardinality of R (c) equals aleph-1 which is believed to be greater than aleph-0:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGPEaQ7-d0WhePM3f-KlMwPta2tOAiHGnUoVib4NJbTCEu8qHMwEECIvDDoejXGNwohBdC8PFhRC7k4apcm0BdyrDLBf_lkLF_7HYvPgtQCfrv6dTohjAr2jC2XhljZVHvUxIifBHHcHkACEmdZAEFEp5dGiM-HwIlUQeG5cseTfPknj5VktR1vwX_/s561/0d.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="438" data-original-width="561" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGPEaQ7-d0WhePM3f-KlMwPta2tOAiHGnUoVib4NJbTCEu8qHMwEECIvDDoejXGNwohBdC8PFhRC7k4apcm0BdyrDLBf_lkLF_7HYvPgtQCfrv6dTohjAr2jC2XhljZVHvUxIifBHHcHkACEmdZAEFEp5dGiM-HwIlUQeG5cseTfPknj5VktR1vwX_/s400/0d.png"/></a></div>
</p><p>
If aleph-1 is greater than aleph-0, then the reciprocal of aleph-1 is less than the reciprocal of aleph-0 (see 5 below). However, if the absolute value of the reciprocal of aleph-0 equals zero, then the absolute value of aleph-1's reciprocal is less than zero (see 6). But no number has an absolute value less than zero, so aleph-1 can't be greater than aleph-0 (see 7, 8, 9).
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCXgA8TWGvQaaEPBZMLUshImO8Eeth1facZJIlH7sK7o13OGM1R6Iovoarq9gOklh4AO386dcHSmYbuZUXZP5MZ7I7kbwpNAkcVHmNEROAmBfaPGelXJQVTyfJP_6elDSo6tWIaryMFqm0wga4YkC4KzBRY5FtAQ0cou2LzQEjN7XYZCzBns_hUijs/s601/000a.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="586" data-original-width="601" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCXgA8TWGvQaaEPBZMLUshImO8Eeth1facZJIlH7sK7o13OGM1R6Iovoarq9gOklh4AO386dcHSmYbuZUXZP5MZ7I7kbwpNAkcVHmNEROAmBfaPGelXJQVTyfJP_6elDSo6tWIaryMFqm0wga4YkC4KzBRY5FtAQ0cou2LzQEjN7XYZCzBns_hUijs/s600/000a.png"/></a></div>
</p><p>
When comparing infinite sets, it appears that N can have the same cardinality as P(N) and R. Cantor, of course, set forth a brilliant counter-argument in support of his theorem. His argument (or a contemporary version of it) begins with a complete list of members of the power set:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyPkkLPjZqm0bCLnv2wy-Mw_hM-DHE3wesiByTNH1lUqIGC1Y595sue3-boufEH20aFGYFmMGSH_TmXzcJm7XuWO-voaT5feW6ChtAO-rJgMJ8DSGRS8_GyedVfR8YZNa6_X9D-3nRHeIwzcH41juZ23Mnmu1WvxvfOr2LAJErlcnfh0g0JBL_sTrc/s470/C4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="397" data-original-width="470" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyPkkLPjZqm0bCLnv2wy-Mw_hM-DHE3wesiByTNH1lUqIGC1Y595sue3-boufEH20aFGYFmMGSH_TmXzcJm7XuWO-voaT5feW6ChtAO-rJgMJ8DSGRS8_GyedVfR8YZNa6_X9D-3nRHeIwzcH41juZ23Mnmu1WvxvfOr2LAJErlcnfh0g0JBL_sTrc/s600/C4.png"/></a></div>
</p><p>
Each number is classified as either "selfish" or "non-selfish." A number is selfish if it is a member of a subset and that same number is the natural number paired with the subset; otherwise, it is non-selfish. In the list above, rows 1 and 3 are examples where 1 and 3 are selfish. They are each members of their respective sets as well as natural numbers paired with those sets. Rows 2 and 4 are examples of 2 and 4 being non-selfish numbers. Now, by definition of the power set, there allegedly should be a subset S that contains all non-selfish numbers. Here's where Cantor creates the template for Russel's paradox: the natural number s can't be a member of S or S would not be the set of all non-selfish numbers. Also, the natural number s can't be a non-selfish number either, since it would be a member of S. A third option is a selfish number that is not a member of S. Unfortunately, such a number is a natural number paired with a different subset. Thus, Cantor argues that there is no possible natural number s that can pair with S. Therefore, there is no bijective mapping between N and P(N).
</p><p>
Yes, a brilliant argument! But Cantor had to move the goalpost to make it. If we move the goalpost back to its original position we see that a bijective mapping between two sets only requires that the sets have the same cardinality, i.e., the same number of members. It does not require the numbers and subsets to have certain attributes like "selfishness" or "non-selfishness." As an illustration, suppose we have a subset S={1, 2, 3, ..., n} that is the only subset that hasn't been paired with a natural number. Subset S is defined as the set of all non-selfish numbers. If we stay true to that definition, we will never find a number to pair it with, notwithstanding the fact that number 3 hasn't been paired with any subset. We break down and decide to map 3 to S. Now there is a complete bijective mapping between N and P(N) (where N and P(N) each have infinite members)--so why should anyone care that S is no longer "the set of all unselfish numbers"?
</p><p>
Let's now address the other proofs (including the famous diagonalization proof) that seem to support the claim that R and N don't have the same cardinality. These proofs start with a list of, say, all the real numbers. Each real number is paired with a natural number. (It is assumed that all natural numbers are listed.) A new real number is then produced that is not on the list. This supposedly contradicts the assumption that the real numbers are countable. If the real numbers are countable, the new real number would be on the list--so the argument goes. However, if the real numbers are countable, then it should be possible to produce a new counting number (natural number) that is not on the list for every new real number that is not on the list. Here's how it is possible:
</p><p>
Translate the natural numbers and/or Cantor ordinals to ASCII or unicode. This will produce natural numbers that will bijectively map to the real numbers. Now, use any proof that will produce a real number that is not on the list. This new real number will map to a unique natural number that is also not on the list. Below is an example of a possible translation scheme:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPW4Qa28WHzsKSSy_d7Yj3P565Hsqq1otjuvC76uKgo0Dcip7gack9fvJ7KMeoW6JeDA0uh9uvnZsR_IqBK3rTFiZZpYYIzG4mdoUdkM5eds95mOllFPBqP14oslOHKMfOTkoZLOoQN8DlAay_UVw_U9dfmJdkoq3r6iPu-A3bLh5PQb9XM-nvv5eR/s378/00a.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="334" data-original-width="378" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPW4Qa28WHzsKSSy_d7Yj3P565Hsqq1otjuvC76uKgo0Dcip7gack9fvJ7KMeoW6JeDA0uh9uvnZsR_IqBK3rTFiZZpYYIzG4mdoUdkM5eds95mOllFPBqP14oslOHKMfOTkoZLOoQN8DlAay_UVw_U9dfmJdkoq3r6iPu-A3bLh5PQb9XM-nvv5eR/s400/00a.png"/></a></div>
</p><p>
The following is the bijective mapping scheme:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1P2uV8JJhYOnRYWu82D5EhmNeZ80sCYQBrqOOLVO720v6WdXAIdasGzSpN5HQyHOqjZ0NMan_Xpxea9CozNNu-13DWfFSTmCxINHGvzxbdxQoj9b7AgyLwafOC0L_3cjLJ_pXJWe1pZtCzYDyiQnqnd1d2sP-1jOG3HDRjzTG5LbypSYQvFU_u7fo/s774/002.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="599" data-original-width="774" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1P2uV8JJhYOnRYWu82D5EhmNeZ80sCYQBrqOOLVO720v6WdXAIdasGzSpN5HQyHOqjZ0NMan_Xpxea9CozNNu-13DWfFSTmCxINHGvzxbdxQoj9b7AgyLwafOC0L_3cjLJ_pXJWe1pZtCzYDyiQnqnd1d2sP-1jOG3HDRjzTG5LbypSYQvFU_u7fo/s400/002.png"/></a></div>
</p><p>
Note that omega+1 circled in red is translated to 9694349. This is a unique natural number that is not on the list (that includes omega). All the natural numbers listed (translated to unicode) have a left-leading digit of 4 or 5. The unlisted natural number has a left-leading digit of 9, and, unlike omega (969) has a 43. It can be mapped to the unlisted real number. Also note that every order of infinity can be translated into a unique, finite natural number. One concern is the unicode numbers can become quite large compared to untranslated natural numbers. Will the larger unicode numbers reach infinity sooner than their smaller counterparts as the count increases, causing a more incomplete list? The following shows a comparison between n and much larger n^n. If n is the largest number short of infinity, is n^n infinity or beyond?
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn5kjiQBqpadCha2RtrJz8ragDfbWefduB-wOo6c1JIB65olXnxkvLQ4bjAs3QUQDI4Z5yW9qYxY3NLQQfDFqbdD-uS0EvZzVaa11Mib8Bf01QVXYI0FezroBPUPhwWDzXXBBN2_D5frBu_Gce1yls8k8LIvD4SfptT6EeC-JB3evI3f65a3iBR4he/s555/000b.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="422" data-original-width="555" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgn5kjiQBqpadCha2RtrJz8ragDfbWefduB-wOo6c1JIB65olXnxkvLQ4bjAs3QUQDI4Z5yW9qYxY3NLQQfDFqbdD-uS0EvZzVaa11Mib8Bf01QVXYI0FezroBPUPhwWDzXXBBN2_D5frBu_Gce1yls8k8LIvD4SfptT6EeC-JB3evI3f65a3iBR4he/s400/000b.png"/></a></div>
</p><p>
It appears that n^n will always be finite as long as n is finite and n^n won't be infinite until n is infinite. Thus the unicode natural numbers can map to the same infinite (or finite) list as the smaller untranslated natural numbers. This is good news! There is simply no finite or infinite number that can't be translated into a unique natural number. For every unlisted real number there is a corresponding unlisted natural number. To falsify this conclusion, all one needs to do is show that a Cantor ordinal or combination of Cantor ordinals and natural numbers can't be translated into a unique unicode natural number.
</p><p>
With all the forgoing in mind, let's review the question again: "Is there any set which has more members than the set of natural numbers (N), but fewer members than the set of real numbers (R)?" No, because R and N have the same cardinality.
</p><p>
ADDENDUM:
</p><p>
Is Infinity Real?
</p><p>
By definition, no number is greater than infinity, and, infinity seems unobtainable. The largest number imaginable can always be made larger. If infinity is obtainable, what is the largest finite number just short of infinity? We can agree that an arbitrary line segment has an infinite number of points. We might infer that a longer line segment has a larger infinity of points--but it too has an infinite number of points. How is that possible? If we think of the points as infinite zeros added together, these infinite zeros can add up to any finite number n. The value of n can be determined by how zero is expressed:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv46sup7sSIs5jXvbvlL35YcWp7eowVoMUpZDUUbB8RbbW0rEWh7OXe0T9lhNE9KmkKuBSnOdxdY14UZxUL1MSMzRUHqNah5crTnrAluQ22hP6E2fp38knen71TLL_ZJUwJrcCBjqjgpAeneZXmgEG_fpimDqQnuE4QRwRNDAdlM98ymkKnJObYv6A/s638/000c.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="404" data-original-width="638" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv46sup7sSIs5jXvbvlL35YcWp7eowVoMUpZDUUbB8RbbW0rEWh7OXe0T9lhNE9KmkKuBSnOdxdY14UZxUL1MSMzRUHqNah5crTnrAluQ22hP6E2fp38knen71TLL_ZJUwJrcCBjqjgpAeneZXmgEG_fpimDqQnuE4QRwRNDAdlM98ymkKnJObYv6A/s400/000c.png"/></a></div>
</p><p>
Clearly infinity is obtainable when the point length is zero and we are not willing to traverse one point at a time. Instead, we choose to slide past those infinite points to reach a finite number. Also, there doesn't appear to be more than one infinity, since the same infinite sum of zeros leads to any number n. On a grander scale, an infinite number of members leads to any infinite set. Like finite numbers, different infinite sets appear to be different, but involve the same infinity.
</p><p>
Limits and Infinite Precision
</p><p>
If infinity does not exist, then no precise or approximate number would exist? Wait ... approximate numbers should exist, since they don't require infinite precision, right? Wrong. Each approximation is a precise value of itself. To illustrate, suppose we want to expand the exponent e^x using a Taylor series. Let's assume there is no such thing as infinity, so the expansion only has a limited number of terms:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjI4xD8E_qvmIsj9U5j-8gJBi8409UsspdEHNpJaEFvsU0LhqlRo7fm74GhbBC3ukCxr6vF1mwH1oenW5PISqb5TBPSwc9yQqoEg0g_6L8VDs88N6KWgydLzxi4rmvkImK081CuTHhqmbEw5k8WKR-TLCWns3rX1pjGJoWW-yQi-wDeSBgoC_TbUiok/s399/000d.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="104" data-original-width="399" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjI4xD8E_qvmIsj9U5j-8gJBi8409UsspdEHNpJaEFvsU0LhqlRo7fm74GhbBC3ukCxr6vF1mwH1oenW5PISqb5TBPSwc9yQqoEg0g_6L8VDs88N6KWgydLzxi4rmvkImK081CuTHhqmbEw5k8WKR-TLCWns3rX1pjGJoWW-yQi-wDeSBgoC_TbUiok/s320/000d.png"/></a></div>
</p><p>
We have an approximation of e^x, but how can we if we don't believe in infinity? Let's set the approximation equal to k, then we shall expand k:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqKpZGMyA_Pr6FLT_kVN0O91JZDFcEtwQOPNUhjo5MaCvACaEMYTYPfkN1-KV570iIqIyq5aMkxHCOiBFusr_xzF9TfZ1IxV6zqXDgESK5CyW4gLd2bRP_vm7F15YDF6RXTa3VrBBx8oIiY8uvtKzo-Hkac4UEZqI1KSCLHTGMxY2BxvuTc1Me5JSD/s506/000e.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="186" data-original-width="506" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqKpZGMyA_Pr6FLT_kVN0O91JZDFcEtwQOPNUhjo5MaCvACaEMYTYPfkN1-KV570iIqIyq5aMkxHCOiBFusr_xzF9TfZ1IxV6zqXDgESK5CyW4gLd2bRP_vm7F15YDF6RXTa3VrBBx8oIiY8uvtKzo-Hkac4UEZqI1KSCLHTGMxY2BxvuTc1Me5JSD/s320/000e.png"/></a></div>
</p><p>
Without an infinite number of terms, we can't have an approximation of e^x--only an approximation of an approximation of e^x. Plus, we can carry out this insanity further with a limited expansion of the approximation of the approximation ... and so on. To claim that we have an approximation is to claim we don't have the number we were shooting for, but we do have the precise value of a number that is close. If we come up short or overshoot a precise point on a number line, we land on a different point that has its own infinite precision. When we have an error margin, that margin is bound by two numbers. Whether they are deemed precise or approximate, they are precisely what they are. That precision implies a Taylor expansion with infinite terms. Even if we insist that a number can't be precisely had, we can at least slide past it along the number line. If there is no infinity, there are no numbers.
</p><p>
Circling back to the question, what is the largest finite number short of infinity? Using the line-segment analogy, there are infinite points between the start and finish. If we can't pin down the exact point that qualifies as the largest finite number, we can at least slide past it.
</p><p>
References:
</p><p>
1. Cantor, Georg. Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a General Theory of Sets). jamesrmeyer.com.
</p><p>
2. Cantor, Georg. Uber eine elemtare Frage de Mannigfaltigketslehre (On an Elementary Question of Set Theory). jamesmeyer.com.
3. Cantor's Theorem. Wikipedia
</p><p>
4. 2020. SP20:Lecture 9 Diagonalization. courses.cs.cornell.edu
</p><p>
5. Cantor's Diagonal Argument. Wikipedia
</p><p>
6. Cardinality of the Continuum. Wikipedia
</p><p>
7. Continuum Hypothesis. Wikipedia
</p><p>
8. Huge List of Unicode Symbols. vertex42.com
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-91964582048016785032022-06-30T14:15:00.001-07:002022-07-02T11:17:37.250-07:00Why Cantor's Diagonalization Proof Comes Up Short<p>
ABSTRACT:
</p><p>
This paper reveals why Cantor's diagonalization argument fails to prove what it purportedly proves and the logical absurdity of "uncountable sets" that are deemed larger than the set of natural numbers.
</p><p>
Cantor's diagonalization proof (CDP) is used to prove various things like Gödel's Incompleteness theorem (GIT) and the uncountability of real numbers (URN). In the case of GIT, it is assumed that you can have a complete list of provable statements. You use CDP to prove a statement that is not on the list, then conclude the statement is not a provable statement; otherwise, it would be part of the list of all provable statements. However, if your complete list is not really complete, then it makes sense that you can prove a new provable statement not on the list. In the case of proving the URN, you use proof by contradiction: assume the real numbers are countable and assume a complete list of them between, say, .000 ... and .111 ... The list is believed to be complete because an infinite number of counting numbers (1 to n) each allegedly have a one-to-one correspondence to each real number on the list. CDP is then used to show there is a real number that is not on the list (a contradiction). Therefore, according to the proof, there is no bijection between the real numbers and the counting numbers.
</p><p>
Below, we will work through an example proof using CDP and demonstrate why it comes up short. Let's begin with an nXn matrix of real numbers in base 2. We go along the diagonal and flip each bit from 0 to 1 or 1 to 0 to create a new number not on the list. If we think the number is on the list, say, at row r, we can check it and see that its bit at column r doesn't match. We can repeat this exercise for all rows and see that we truly have a new real number not on the list.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq9wRkPWwlGKMPM1fo80eLIjSRRIGuJUCBwDePW6neN3-yIOh8IGE_ZFKyhZ2ldyA8OoSLZlQD07TJeabWjJtyw7vZ6T6WOzTGg6Ts7_hbxKKB5YK5H2iVL0UY6DhtLY8uSRTxbmoCLFxWRrmeKTWr6riVE-E9kTqhJZ9FvAeJv5EOTo12bn3IAhoP/s440/1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="440" data-original-width="391" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq9wRkPWwlGKMPM1fo80eLIjSRRIGuJUCBwDePW6neN3-yIOh8IGE_ZFKyhZ2ldyA8OoSLZlQD07TJeabWjJtyw7vZ6T6WOzTGg6Ts7_hbxKKB5YK5H2iVL0UY6DhtLY8uSRTxbmoCLFxWRrmeKTWr6riVE-E9kTqhJZ9FvAeJv5EOTo12bn3IAhoP/s400/1.png"/></a></div>
</p><p>
Assuming we have used up all the counting numbers to make the complete list of reals from .000 ... to .111 ..., we are forced to conclude that the number of reals exceeds the number of counting numbers by at least one. Therefore, the reals are uncountable (contrary to an initial assumption, via proof by contradiction, that the reals are countable). However, a list of positive integers (counting numbers, including 0) can also be shown to be uncountable:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPyKNlK4hFcwDl_-YA7yZevg04TuN3xkBDvBCb1MPi2BMjzvVbloESM63I-4dI7_y8nToq__SIUBFaXVOvv7jGQKdx7Bnk86nip9hq5dmtTXfQz4UTkAIoNMrtaigMmZ69yn8xVMpoPyZ0LS54kCX1GB334_zds3CP4z9eiS4bd1pcFmvdekBAbDRe/s473/2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="473" data-original-width="442" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPyKNlK4hFcwDl_-YA7yZevg04TuN3xkBDvBCb1MPi2BMjzvVbloESM63I-4dI7_y8nToq__SIUBFaXVOvv7jGQKdx7Bnk86nip9hq5dmtTXfQz4UTkAIoNMrtaigMmZ69yn8xVMpoPyZ0LS54kCX1GB334_zds3CP4z9eiS4bd1pcFmvdekBAbDRe/s400/2.png"/></a></div>
</p><p>
Using CDP we can create a positive integer that is not on the list! The good news is we can use this new integer to count the new real number, so could it be that the real numbers are countable after all? If your alarm bells are going off, I feel your pain. You may have noticed that the new integer is 0...11 which is 3 in base 10. How can the number 3 be absent from a complete list of positive integers?! Answer: the complete list is not really complete. An nXn matrix only provides a subset of the set of positive integers. This is why CDP can produce numbers that are not part of the list. The assumption that we have a "complete list" of members of a given set is a false premise. Here's why:
</p><p>
Each matrix column corresponds to a digit. Thus we have a list of numbers with n digits. Where there's n digits, there's B^n (B is the number base) numbers with n digits. For example, if B = 2, and there are 3 digits, there are eight possible numbers, not three. If we only list three numbers, there are five more numbers that are not on the list. Once included, though, no amount of diagonal gimmickry will create a three-digit number not part of the list. This is not only true for integers but also real numbers, since reals are just integers with decimal points. Below is a mathematical proof showing that a complete list has 2^n members, not n members:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin4Nh-tcBK7KlQSWLsKPvGFKp_qmsWQQUZpK4NS9MswHODlqP4htnYcw2JqJqVFddl9espglS-1CdNCzo_LliKp1jedW7k7mQuL8XU8JSmWemBOOoFYeRpJzBowoEWL5CwWtKE9lsY9lUCmY3yYhhT8MeNKCvg0Y__bBsc9ORN7H3NuTqjDc3z73R4/s812/2.1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="812" data-original-width="701" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin4Nh-tcBK7KlQSWLsKPvGFKp_qmsWQQUZpK4NS9MswHODlqP4htnYcw2JqJqVFddl9espglS-1CdNCzo_LliKp1jedW7k7mQuL8XU8JSmWemBOOoFYeRpJzBowoEWL5CwWtKE9lsY9lUCmY3yYhhT8MeNKCvg0Y__bBsc9ORN7H3NuTqjDc3z73R4/s600/2.1.png"/></a></div>
</p><p>
If the goal is to have a complete list, then we need a (2^n)Xn matrix:
</p><p>
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</p><p>
Note that 0...11 is now included on the list. Now, just for the sake of argument, let's assume the list of reals we started with is complete, i.e., countable. By some method m, we derive a new real number not on the list. Let's assume method m is an indisputable method. We conclude that the set of real numbers is larger than the set of counting numbers, and, cannot be counted. But this implies that we run out of counting numbers before we run out of reals, i.e., it is possible to have a higher "count" of real numbers than the the numbers that count them. If we can't count the extra real numbers, then it is meaningless to say we have more, or, a higher count of real numbers and a lower count of the numbers that do the counting. The following diagram provides a visual representation of this absurdity:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTJZ7E5TtSWRf9FuMjKO4RlLjZOH0tSLeAPVQj8F27C3gbRh07gC6xGfU75pF70q1IXFcu6LTIve1mOCs8gm_TwNDu0cNsQC5rqWmidSkNSgA1-w37J9AOn_WQfioFeJBb1BnoBbvP0kUQoJNLTquu4SdnUW-rQ1btQZlw1EZW05c5eH81BrcxD6G8/s646/3.1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="523" data-original-width="646" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTJZ7E5TtSWRf9FuMjKO4RlLjZOH0tSLeAPVQj8F27C3gbRh07gC6xGfU75pF70q1IXFcu6LTIve1mOCs8gm_TwNDu0cNsQC5rqWmidSkNSgA1-w37J9AOn_WQfioFeJBb1BnoBbvP0kUQoJNLTquu4SdnUW-rQ1btQZlw1EZW05c5eH81BrcxD6G8/s400/3.1.png"/></a></div>
</p><p>
If we have a list of n real numbers and derive one more, what does "one more" mean if not n+1 real numbers? If "one more" means n+1, then for every new real number created, there is an additional natural number added as well. With this in mind, we can map the positive integers (including 0) to real numbers in a range of 0 to 1:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6sRNoJmpxJS2iYKzZkQvkvVogIyfhnhhNGcnjbLLEQvgkRt1GfuSx8bwAC3BtEORBCgQRGALcT2FrButJgR69nW4CbjcL-vCHpsKlaTGeCyfq-kBJipJgk6xmR2oE9XTrxx5DeYGbeJggP0fMTZSnmVWscUak9r4MqpG09Q58nC1ypxctAoFzSYcs/s329/4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="220" data-original-width="329" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6sRNoJmpxJS2iYKzZkQvkvVogIyfhnhhNGcnjbLLEQvgkRt1GfuSx8bwAC3BtEORBCgQRGALcT2FrButJgR69nW4CbjcL-vCHpsKlaTGeCyfq-kBJipJgk6xmR2oE9XTrxx5DeYGbeJggP0fMTZSnmVWscUak9r4MqpG09Q58nC1ypxctAoFzSYcs/s400/4.png"/></a></div>
</p><p>
Notice that if we take n to the limit of infinity, the number of real numbers grows and so does the number of positive integers. To count the reals between 0 and 1 requires 2^n + 1 counting numbers as n moves to infinity. Suppose we expand the range of reals: 0 to 2^n:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYDxNW_YzK7BbjeMw10GcFE7iSH_huOxE1EHxYEC42t8Rqs6PGlD-b1V_6DTAO_SCvycgVKU2HkPp6c3RQyCoTmQ8-kVaRL0u1WcJT8LN4am08pTWrGkQDHs1Uvnk_epothZ3pXDJ8nwZcWxr3btC3OnOEUqq5CmuNIcagSTjIWF3utJLBOfntflCL/s360/5.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="199" data-original-width="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYDxNW_YzK7BbjeMw10GcFE7iSH_huOxE1EHxYEC42t8Rqs6PGlD-b1V_6DTAO_SCvycgVKU2HkPp6c3RQyCoTmQ8-kVaRL0u1WcJT8LN4am08pTWrGkQDHs1Uvnk_epothZ3pXDJ8nwZcWxr3btC3OnOEUqq5CmuNIcagSTjIWF3utJLBOfntflCL/s400/5.png"/></a></div>
</p><p>
We now need 2^(2n) + 1 (n-->infinity) natural numbers to count this expanded range. If we continue to map 0 to 0, we could map the even positive integers to the positive real numbers, and, the odd positive integers to the negative real numbers:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN0AQ2pCgRNRAPos9lTxNoppa6cc2D_PISTYnAxURUslwMmBgC708YqKE6P2_G1BF6-9FuSkmRuhm6cXeF8RdFjNqS_Bj9Dm_ICGx6Q0eWIxdh05AGy9O_nGN2-QgqSe0I7LWr0-77pNVxMNeZpvDEqhtBve1YAJ5sRJdoW1uLS321BpX8Ruyb1aDp/s819/6.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="220" data-original-width="819" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN0AQ2pCgRNRAPos9lTxNoppa6cc2D_PISTYnAxURUslwMmBgC708YqKE6P2_G1BF6-9FuSkmRuhm6cXeF8RdFjNqS_Bj9Dm_ICGx6Q0eWIxdh05AGy9O_nGN2-QgqSe0I7LWr0-77pNVxMNeZpvDEqhtBve1YAJ5sRJdoW1uLS321BpX8Ruyb1aDp/s600/6.png"/></a></div>
</p><p>
We now need 2^(2n+1) + 1 (n-->infinity) natural numbers to count real numbers from -2^n to 2^n.
</p><p>
The great thing about the concept of infinity is infinity isn't a finite number. If we think the natural number set N is too small and can't count a bigger real number set R, we can make N bigger without using up its infinite potential. Both sets are forever growing and neither is fixed in stone. Therefore, for every increase in set R there is a corresponding increase in set N. If not, then it is meaningless to claim there are more members in R when its extra members can't be counted.
</p><p>
In conclusion, Cantor's diagonalization result makes perfect sense once we are aware of the fact that the "complete list" is only a partial list of members of a set. This should cast doubt on whatever Cantor's diagonal argument allegedly proves. Additionally, all methods of proving the uncountability of real numbers, etc. fail to address the absurdity of claiming that a set R has more members than the set of counting numbers needed to confirm that R does in fact have more members. It is obvious that there are more real numbers between, say, 1 and 2 than natural numbers. Albeit, the phrase "more real numbers" implies a higher count, i.e., countability.
</p><p>
References:
</p><p>
1. 2020. SP20:Lecture 9 Diagonalization. courses.cs.cornell.edu
</p><p>
2. Cantor's Diagonal Argument. Wikipedia
</p><p>
3. Dr. Pevam. 2018. R is Uncountable. youtube.com
</p><p>
4. Zap, Elvis. 2010. The Cantor Set is Uncountable. youtube.com
</p><p>
5. Hartl, Werner (and a comment from Pendaran Roberts.) Cantors Diagonal Argument Fails. youtube.com
</p>
<!--Let's test the hypothesis: k/2^n is always rational. If we start with the increment formula k = k +1, we increase k until it reaches the following integer: 314 ..... In case it isn't clear to you the ellipse represents more digits. In this case we've switched to base 10, so let's divide k by 10^n, where n--infinity. So we have k/10^n = 3.14 .... = pi, an irrational number.>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-91719503842803789472022-05-29T21:20:00.010-07:002022-05-30T21:35:16.275-07:00Why Gödel Failed to Prove His First Incompleteness Theorem<p>
ABSTRACT:
</p><p>
This paper shows how Gödel failed to prove his first incompleteness theorem and shows a standard of proof that completes a system containing genuine, not contrived, undecidable statements.
</p><p>
Gödel's First Incompleteness Theorem is often stated as follows:
</p><p>
"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."
</p><p>
On its face, this theorem appears to be asserting that any consistent formal system (F) cannot consistently prove its statements or their negations, i.e., it is incomplete. The statements are of the language of F, which may be constructed from Peano arithmetic axioms and Gödel's numbering system where letters, words, symbols, etc. are encoded into natural numbers. Whether or not you agree Gödel proved his theorem, consider the following proof of the transitive property:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhi8kgCdBnAVoDMjxhVrcXlxnGPEklDDOL_lXx93OZ0uLueDVj2bvvxuhFAU2Om6AJH2d_rGOBjd20MmnQxnx5ldRp1IX2DSOlWk1-QIJM5lT9lJz6PeEinAJ8BxVZMGS8stl9PkxYnMLC1tRDzpSpWFzMXwrBvvzPBe5tXAw-QpHDBOBaiLc_LV-Q9/s534/1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="139" data-original-width="534" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhi8kgCdBnAVoDMjxhVrcXlxnGPEklDDOL_lXx93OZ0uLueDVj2bvvxuhFAU2Om6AJH2d_rGOBjd20MmnQxnx5ldRp1IX2DSOlWk1-QIJM5lT9lJz6PeEinAJ8BxVZMGS8stl9PkxYnMLC1tRDzpSpWFzMXwrBvvzPBe5tXAw-QpHDBOBaiLc_LV-Q9/s400/1.png"/></a></div>
</p><p>
After stepping through it, are you convinced that A = C? Well, you shouldn't be, because the main characters in this proof are not what they seem--they are each code for a statement in a similar but different language. Let's label the original characters' language "X." and the different language "Y." Below is the Rosetta Stone or table that translates X into Y and vice versa:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy9MK3pOB_Zl_8FJ0uzfSQlCB3GZmuVIU9Iljplt66sJl0ybK2GIdXNyK1q_FQV3eu8MDP0MJfpHb_MUUgvlm3tUKwvvNXwQyC-RSWujFGqAUiZFNOBYO3E74Gx-BvDd2YOLjXQvWgfetscTQC1UNvZ2ajNJZoUXLo4wk52boxfAfmi4iKP6-Kkt2-/s442/2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="442" data-original-width="382" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy9MK3pOB_Zl_8FJ0uzfSQlCB3GZmuVIU9Iljplt66sJl0ybK2GIdXNyK1q_FQV3eu8MDP0MJfpHb_MUUgvlm3tUKwvvNXwQyC-RSWujFGqAUiZFNOBYO3E74Gx-BvDd2YOLjXQvWgfetscTQC1UNvZ2ajNJZoUXLo4wk52boxfAfmi4iKP6-Kkt2-/s600/2.png"/></a></div>
</p><p>
Now, examine the proof in the Y-language:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOYWFZ1IwH8BkCOLL9Cilfw9yTLoSSsfQ3oTF6UFBg-ozLdkHhwmkDRT6f-qP4T9_D9sqsqI89TPfCJusSblGmvW4rW8z2tqrMJV87nag52JCkXx86YFyePYUHC2DVvU1hCVbH65WKq8u-s_0dY153yPCRs3Uv_ymXByYWBpREgBbdKYQSGsDRuO3T/s555/3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="253" data-original-width="555" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOYWFZ1IwH8BkCOLL9Cilfw9yTLoSSsfQ3oTF6UFBg-ozLdkHhwmkDRT6f-qP4T9_D9sqsqI89TPfCJusSblGmvW4rW8z2tqrMJV87nag52JCkXx86YFyePYUHC2DVvU1hCVbH65WKq8u-s_0dY153yPCRs3Uv_ymXByYWBpREgBbdKYQSGsDRuO3T/s600/3.png"/></a></div>
</p><p>
As you can plainly see, A doesn't equal C in the Y-language. Let's assume languages X and Y are subsets of a greater language. Let's call it language F. In system F, which contains the language of F, we have the following:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzgGZfHTXX0Qo8HXj3tRDmmxVTAwASeoOT2x1L1NW2ISoJog8egTld9dmmA4ws4PrJkfddAoey6JHoCOYvr8_AP3Izn_dHx4CCF8U2vR4yXd01STgOShh5WjRH0Fd6J3kLh5r_R1MaIwNYSCdCHWFK6aJrAj4Vl725Jivh8XW_0pliU7TtA1EX7-f4/s524/4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="80" data-original-width="524" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzgGZfHTXX0Qo8HXj3tRDmmxVTAwASeoOT2x1L1NW2ISoJog8egTld9dmmA4ws4PrJkfddAoey6JHoCOYvr8_AP3Izn_dHx4CCF8U2vR4yXd01STgOShh5WjRH0Fd6J3kLh5r_R1MaIwNYSCdCHWFK6aJrAj4Vl725Jivh8XW_0pliU7TtA1EX7-f4/s400/4.png"/></a></div>
</p><p>
Let's assume system F is consistent. That being the case, we can't assert that we proved A equals C, and A doesn't equal C, so we declare that neither is proved, i.e., the statement A = C is unprovable. I know this seems absurd, but it should have a familiar ring if you are familiar with the process Gödel used to prove his theorem. It is the process of encoding one language into another and labeling the result the "language of F," making sure the conclusion in language Y negates the conclusion in language X. Such a process violates the law of non-contradiction, and, if it is applied throughout all mathematics and logic, then nothing can ever be proved or disproved!
</p><p>
The corollary, of course, is all statements that can be proved are provable again, including perhaps the infamous Gödel sentence (G) if we insist on conforming to the law of non-contradiction. At line 6 below is the diagnal lemma used to prove that G is unprovable. Translated into English it says, "System F proves G if and only if G has the property of unprovability."
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikSVBBsfWAP54am2KHx5zanE5XjQQZfKQw_HqBXYqH03RgLyA6QY2U28zjyeuTs2c6g19-bteDJyfJMM_IwkRigcBBNU6yzyOxJQ4TNRQoq4IL3fdMN2NTCiZJLRtv5SMbtAKAuJcuDqcOlAdLsKJtPnRnhoDlxOxJIpAJLpKsqP65-vvBJooXWWt4/s600/6.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="120" data-original-width="600" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikSVBBsfWAP54am2KHx5zanE5XjQQZfKQw_HqBXYqH03RgLyA6QY2U28zjyeuTs2c6g19-bteDJyfJMM_IwkRigcBBNU6yzyOxJQ4TNRQoq4IL3fdMN2NTCiZJLRtv5SMbtAKAuJcuDqcOlAdLsKJtPnRnhoDlxOxJIpAJLpKsqP65-vvBJooXWWt4/s400/6.png"/></a></div>
</p><p>
Not only does this lemma violate the law of non-contradiction, but its statement was arbitrarily chosen (not proved). If one examines the proof of the original lemma, the proved lemma reads, "System F proves G if and only if G has the the property of A." Any property can be assigned to the placeholder A. Gödel chose "unprovability." It is fortunate he didn't choose "vampire powers," otherwise his theorem might claim that some statements in F have vampire powers.
</p><p>
Anyway, claiming that F proves G only if F can't prove G is contradictory and should not be validated. The contradiction is done away with if we stick with the original language of F (language X) where natural numbers are not code for statements (in language Y). In language Y the statement might read, "Statement G can't be proved in system F." In language X it is just #G', the Gödel number of the statement. At 7 below we make that adjustment:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2vjZsgP1FI_BfblQ1Q0ne1X7UMFl0TwZqMtCWNbc8iUYAAfE9Td1xuGv26wghj9DMNXY0Ng7FUtc0EP1fDJKksuBRV1qEFv9WoPH__XD5S4Ms324Qz8rcfyHrlD4eRJs2qhKvvMI_WK8TuAsFG6ht6rziDC2Cmr3UrDZc6-S-0R88Ibh_xeX6QLgN/s591/7.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="107" data-original-width="591" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2vjZsgP1FI_BfblQ1Q0ne1X7UMFl0TwZqMtCWNbc8iUYAAfE9Td1xuGv26wghj9DMNXY0Ng7FUtc0EP1fDJKksuBRV1qEFv9WoPH__XD5S4Ms324Qz8rcfyHrlD4eRJs2qhKvvMI_WK8TuAsFG6ht6rziDC2Cmr3UrDZc6-S-0R88Ibh_xeX6QLgN/s400/7.png"/></a></div>
</p><p>
After making a substitution, the new statement reads, "F proves G if and only if there is a natural number #G'." No contradiction here.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtMEc798pZWjs2uVH1SGV7kCaMxjCICzofwgEb8K72SWfkHWW3jhIPTiUB5XjR8_AFWd3NVYVvXDVmIwXhfZjzO2O5aIJXfCZ2Jtx0tleGh2OyhxYhIch9YAlJKH0CerJKYEK66PpbKueZBGdt2pxWjK4W5itKglYAX7WA3sEAK9YgiaGe0XwevIvL/s590/8.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="120" data-original-width="590" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtMEc798pZWjs2uVH1SGV7kCaMxjCICzofwgEb8K72SWfkHWW3jhIPTiUB5XjR8_AFWd3NVYVvXDVmIwXhfZjzO2O5aIJXfCZ2Jtx0tleGh2OyhxYhIch9YAlJKH0CerJKYEK66PpbKueZBGdt2pxWjK4W5itKglYAX7WA3sEAK9YgiaGe0XwevIvL/s400/8.png"/></a></div>
</p><p>
To prove there is a natural number #G' in system F, use Peano's fifth axiom: "If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers." That brings us to line 9 which forces the conclusion at 10 which is system F proves G.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjMz2sXbUyS99hnW7B_PqmuhSQ9BQK74MV72_iB-8sdA_MrlTvFXhUwhOIQ_MX5Dm--UEA6SCNasCKS-skdFnbvWnTWjdYy5eaBLVrdevKPeto-GYmHORxFKba_hLelUANhYmCUEAM3tpwZPBs-LnwYOWnl_5MEEBaL4HMQrHMD3TL0E2ihii7Qr1E/s592/9.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="159" data-original-width="592" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjMz2sXbUyS99hnW7B_PqmuhSQ9BQK74MV72_iB-8sdA_MrlTvFXhUwhOIQ_MX5Dm--UEA6SCNasCKS-skdFnbvWnTWjdYy5eaBLVrdevKPeto-GYmHORxFKba_hLelUANhYmCUEAM3tpwZPBs-LnwYOWnl_5MEEBaL4HMQrHMD3TL0E2ihii7Qr1E/s400/9.png"/></a></div>
</p><p>
"System F proves G," of course, is not the result Gödel wanted, so he made certain that the natural number #G' is code for a different statement, a statement that violates the law of non-contradiction and negates "F proves G."
</p><p>
Given the preceding arguments and evidence, it appears Gödel failed to prove his first incompleteness theorem. Notwithstanding this, his theorem may have some empirical support. For example, if you toss a coin in the air, while it is spinning, the statement "It is heads," can't be proved, nor can the statement's negation "It is tails," can be proved. The coin is in a fuzzy state of superposition. The usual standard of proof, where a statement is either true (heads) or false (tails) is inadequate, i.e., the system is incomplete. There is, however, a way to remedy this.
</p><p>
The negation of "true" is "not true." We can decide that a statement is only "true" if it is proved true, and, a statement is "not true" if it is proved undecidable or false. Doing this completes the system if a complete system is where any statement or its negation can be proved. Circling back to the spinning coin, the statement "It's heads," is "not true." If and when the coin lands and shows tails, the statement is still "not true." It's only true if heads is face up.
</p><p>
In conclusion, Gödel failed to prove his first incompleteness theorem because no statement in logic or mathematics can be proved or disproved if that statement is code for a different statement that negates the proof. By contrast, all that is decidable remains decidable if the law of non-contradiction is not violated. In circumstances where there is genuine undecidability, we can impose a standard of proof where a statement is not either "true" or "false," but either "true" or "not true." Such a standard of proof completes the system.
</p><p>
References:
</p><p>
1. Wolchover, Natalie. 07/19/2020. How Godel's Proof Works. wired.com.
</p><p>
2. Godel's Incompleteness Theorems. 11/11/2013. Stanford Encyclopedia of Philosophy.
</p><p>
3. Craig, Robin. 1992. Holes in the Heart of Reason. TableAus.
</p><p>
4. Godel's Incompleteness Theorems. Wikipedia.
</p><p>
5. Raatikainen, Panu. 2020. The Diagonal Lemma. Stanford Encyclopedia.
</p><p>
6. Hosch, William L. Peano Axioms. Britannica Encyclopedia.
</p><p>
7. Raatikainen, Panu. 2020. Godel Numbering.
</p><p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-47526576678764303562022-05-18T16:57:00.038-07:002022-05-19T09:34:14.429-07:00The Incompleteness of Gödel's Theorem<p>
ABSTRACT:
</p><p>
This paper shows what happens when Gödel's theorem is turned on itself, and the logical inconsistency that occurs when self-referencing statements are converted to Gödel numbers.
</p><p>
Gödel's First Incompleteness Theorem (GFIT) is often stated as follows:
</p><p>
"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F."
</p><p>
Here's another way GFIT is stated: "In any reasonable mathematical system there's always true statements that can't be proved."
</p><p>
What happens if we turn the mirror of GFIT towards itself? Is the theorem still valid? Let's assume GFIT is a theorem in system F. We want to prove that a Gödel statement G is true yet unprovable. We can express G in the form of an equation: G = "This statement cannot be proved or disproved in system F."
</p><p>
Let's assume that every statement in F is proved or disproved except for G. If GFIT is true, then G is also true, since there must be at least one statement in F that can't be proved or disproved. But if G is true, then GFIT is false because G was proved by a theorem in F, namely GFIT. Now, if GFIT is false, then its negation ~GFIT is true: there are no statements in F that can't be proved or disproved. Since ~GFIT is true, G is false. If G is false ~GFIT is still true and GFIT is still false.
</p><p>
Of course one might argue that what I have presented so far is unfair. GFIT should exist outside of F as some sort of elite theorem exempt from its own rules and with zero self-awareness. If it could talk, it might say, "Rules for thee but not for me." Perhaps the best argument in favor of placing GFIT outside of F is it clearly fails to work inside F. So let's try removing it from F to see what happens.
</p><p>
So G is still inside F and there are no axioms or theorems that prove or disprove it (because we also removed ~GFIT), thus G is true and so is GFIT. Albeit, G is now its own axiom and it proves itself, so now GFIT is false. But wait ... G also turns false, since its message is consistent with GFIT, so nothing proved it--and it along with GFIT is true again, then false again ... ad nauseum.
</p><p>
Placing GFIT outside of F creates a paradox. Further, it makes sense to place GFIT (better yet, ~GFIT) inside F because GFIT is an F-system theorem. Its language makes that very clear. But let's ignore that and see what happens if we place both GFIT and G outside of system F.
</p><p>
Being outside of system F, G is not proved or disproved by any axioms or theorems inside F, so it's true and so is GFIT. So far, so good. Now, here's the beautiful part: Since G is true, it's its own axiom again, but it is not an F-axiom, so it remains true and so does GFIT! Look ma, no paradox! Unfortunately, none of this proves the validity of GFIT, since according to GFIT, "statements that cannot be proved or disproved" are inside system F.
</p><p>
So far we've tested GFIT with words which have shown GFIT to be paradoxical at best and false at worst. Let's try using Gödel numbers and express our statements mathematically and see what happens. Let's focus on the Gödel sentence G. G = "This statement can't be proved." Function N converts the statement to a Gödel number: N(G) = N("This statement can't be proved."). Now let's convert the statement into variables. "This statement" = G; " can't be proved" = P. Making some substitutions we get N(G) = N(G,P), but we erred. G has two different values: G = "This statement" and G = "This statement can't be proved." However, if we choose just one of these values to make our equation consistent, then it is no longer an equation: N(G) != N(G,P). The Gödel number on the left side no longer equals the Gödel number on the right side. This invalidates the G statement and shows that self-referential statements like G are logically inconsistent. Thus using G in a proof of GFIT would only make that proof logically inconsistent and the truth of GFIT logically inconsistent.
</p><p>
In conclusion, it seems reasonable to assume that our systems are not perfect. If a theorem is required to make the point, let's not assume the theorem was created by magical beings from Galaxy M64. Let's turn the critical eye of that theorem on itself and dare to see what happens.
</p><p>
References:
</p><p>
1. Wolchover, Natalie. 07/19/2020. How Godel's Proof Works. wired.com.
</p><p>
2. Godel's Incompleteness Theorems. 11/11/2013. Stanford Encyclopedia of Philosophy.
</p><p>
3. Craig, Robin. 1992. Holes in the Heart of Reason. TableAus.
</p><p>
4. Godel's Incompleteness Theorems. Wikipedia.
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-66725821211486758492022-04-26T16:20:00.146-07:002022-05-10T13:47:22.845-07:00A Simple Four Color Theorem Proof Etc. <p>
Abstract:
</p><p>
The four color map theorem states that no more than four colors are needed to color the regions of any map so that no two adjacent regions have the same color. More specifically, the theorem states that no more than four colors are required to color the vertices of any planar graph where any two adjacent vertices don't have the same color. This paper shows how the theorem arises from a broken correlation between the number of colors and the number of vertices (dots), and, how repairing this broken correlation leads to maps requiring more than four colors. However, if three dimensions are not invoked on a on planer graph, it can be shown that no more than four colors are needed to color it.
</p><p>
Premise: All maps, regardless of their shapes and sizes, can be represented by planer dot-and-line diagrams. Such drawings make analysis easier and more universal. Each dot (vertex) represents a country and each line represents a border. The rules are any two dots connected by a line cannot be the same color and lines cannot intersect.
</p><p>
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Suppose there is a single country. We can represent it with one dot and one color:
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Take two randomly-positioned dots. Draw a line (border) between them. These two dots and the line can be mapped to a straight horizontal figure. This will allow us to make apples-to-apples comparisons between different random maps. Also note that we now have two dots and two colors.
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizFaYVlp69JTQENsXi8uJr1NQhK0hvtG3_nBPcJUuTFRcKv6sfogvTX3REKzUlhzdHr6b0Q0-bXwCNdvFxws2HFsnjkgHGlYtM93ckpjOBtnI5NWN7zayu0mN039t_wPjJOAeasxHJPhFnPKDF1N4weezEU59aC4-p8ttRsh7S5mcpRjdYWWJZzVe8/s262/b2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="200" data-original-height="262" data-original-width="203" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizFaYVlp69JTQENsXi8uJr1NQhK0hvtG3_nBPcJUuTFRcKv6sfogvTX3REKzUlhzdHr6b0Q0-bXwCNdvFxws2HFsnjkgHGlYtM93ckpjOBtnI5NWN7zayu0mN039t_wPjJOAeasxHJPhFnPKDF1N4weezEU59aC4-p8ttRsh7S5mcpRjdYWWJZzVe8/s200/b2.png"/></a></div>
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Next we have three randomly-positioned dots mutually-connected by three lines. Here again we can map this to a horizontal figure. We make sure the lines do not intersect since they represent borders on a two-dimensional (2D) map. There is now three dots and three colors.
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ9lrg9FGm0-qU1HLYlbhvx-wAyd8NRNWFcipa97rBDvN7094kQP7XptEnnUlImiVFgPGCUEXEAHVp0QP5L4W-wYATWwArBSNepgoad4LqMAQq8yaLUmAHCxS9t0WNo9Ls0XXnswiKvlmqJOhxc2yEOIoUYoKOQC0Xozh7if7JY8GokVPH1t66Mosl/s346/b3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="200" data-original-height="272" data-original-width="346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ9lrg9FGm0-qU1HLYlbhvx-wAyd8NRNWFcipa97rBDvN7094kQP7XptEnnUlImiVFgPGCUEXEAHVp0QP5L4W-wYATWwArBSNepgoad4LqMAQq8yaLUmAHCxS9t0WNo9Ls0XXnswiKvlmqJOhxc2yEOIoUYoKOQC0Xozh7if7JY8GokVPH1t66Mosl/s200/b3.png"/></a></div>
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We repeat the previous process with four dots and four colors:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi57lRmdXITmr_y6OgJyBBccZyZQy0LjOP1WXX4KQgtZsyx_ApekqU1bqFIs1JD5lp16P_-QGtIjn-bU2xD6ByCQoCyrNC0to5rmoCma3OUFJ1vnzd1XEG92B1_davIc4VbqrYXLFTr_tIAwAAxagsB01fXqUTX1sVRSFCAF-uVVomhRy-Nq-XzJL5S/s354/b4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="200" data-original-height="354" data-original-width="331" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi57lRmdXITmr_y6OgJyBBccZyZQy0LjOP1WXX4KQgtZsyx_ApekqU1bqFIs1JD5lp16P_-QGtIjn-bU2xD6ByCQoCyrNC0to5rmoCma3OUFJ1vnzd1XEG92B1_davIc4VbqrYXLFTr_tIAwAAxagsB01fXqUTX1sVRSFCAF-uVVomhRy-Nq-XzJL5S/s200/b4.png"/></a></div>
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There appears to be a pattern: one dot requires one color; two dots require two colors: three mutually-connected dots require three colors--and four dots, mutually-connected, require at least four colors. If we connect five dots, will we need at least five colors? According to the diagrams below, we can't mutually connect all five dots without intersecting lines:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtU4mxpH35439Hlm2ZxlDS3rwPLj7BtLM9ADQYneipOSUUXbq_OwQ_7EUe18a8onZTCH2MvVWlLx1xPJlC9Yk6oKd-2ElMDvfXb2VcGxbC_MRAvepE0o0wiSt3sELHo7m9KA-wYrHbG_sKcLmsYa1Mxs5HHLQW82dzqiNzDJTvWqsbXhpUIUPZHWWM/s388/b5.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="200" data-original-height="373" data-original-width="388" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtU4mxpH35439Hlm2ZxlDS3rwPLj7BtLM9ADQYneipOSUUXbq_OwQ_7EUe18a8onZTCH2MvVWlLx1xPJlC9Yk6oKd-2ElMDvfXb2VcGxbC_MRAvepE0o0wiSt3sELHo7m9KA-wYrHbG_sKcLmsYa1Mxs5HHLQW82dzqiNzDJTvWqsbXhpUIUPZHWWM/s200/b5.png"/></a></div>
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Since the purple dot and the blue dot are not connected, they can be the same color:
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</p><p>
The pattern we observed earlier is broken. Five dots don't require a minimum of five colors as expected--they only require four. Perhaps six dots require at least five colors:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ifSjVjdqVhzrJcacLeOZH1YgloYCQ0k7LMcxTaWL5krLEKCJht78xf3_fmPV3VhlgtUfvpLrRi8cqNIkfqhdVRq9exCL0CC67kflafhS27Zxg8OsNoeNf9nHnayhs94Mk9WFKZXSVavqW2x8TVwhA5oCjxebF33T82mL_QX9l4rOHsBufyxO_17d/s408/b7.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="200" data-original-height="226" data-original-width="408" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_ifSjVjdqVhzrJcacLeOZH1YgloYCQ0k7LMcxTaWL5krLEKCJht78xf3_fmPV3VhlgtUfvpLrRi8cqNIkfqhdVRq9exCL0CC67kflafhS27Zxg8OsNoeNf9nHnayhs94Mk9WFKZXSVavqW2x8TVwhA5oCjxebF33T82mL_QX9l4rOHsBufyxO_17d/s200/b7.png"/></a></div>
</p><p>
Nope! Maybe seven or eight dots require at least five colors:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_qvfgUg_oEU5qgeHyvqNEun5SzhJ-0pBmSk-Vkjgz1yjXjOV677xXJE7NeKvyBNyiCoOZOWH4joPUZ0OEqzaj-tvJpaYCIMImXVj5HHyfd-ahJ-kdAx9NKqhiitvZcfPrHlpsKdgB5bkj6HpKz17-Cz09zcFcwF2zi81NM4wf0yR0gAYVBcmBj7dm/s605/b8.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="200" data-original-height="605" data-original-width="515" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_qvfgUg_oEU5qgeHyvqNEun5SzhJ-0pBmSk-Vkjgz1yjXjOV677xXJE7NeKvyBNyiCoOZOWH4joPUZ0OEqzaj-tvJpaYCIMImXVj5HHyfd-ahJ-kdAx9NKqhiitvZcfPrHlpsKdgB5bkj6HpKz17-Cz09zcFcwF2zi81NM4wf0yR0gAYVBcmBj7dm/s200/b8.png"/></a></div>
</p><p>
Wrong again. But take note of a new pattern: when we add a new dot beyond four, and do our best to mutually connect all dots so their colors cannot be reused, at least two dots are not connected. As a result, they can be the same color, and, the total number of required colors never exceeds four no matter how many dots we have, no matter how big and complex the map. However, this is only true if a 2D map never invokes a third dimension (3D), i.e, there is only north, south, east, west, and no elevation or up and down. At the diagram below, we invoke 3D by allowing the orange line to cross over the black line. Try to imagine the orange line lifting off the page to cross over.
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMD_GON-fguPNPwBWLT9ITUWYgapK7ag4i1zhwo8zE0MBzyi-XDl7fZGFUfO14zuVTREjrl0wxXqViXx1FLFXV85Bt6HaTAoiM8ths_836XRyR9xQy5qJU35ITvt37lxFLukyXk6hUo3NfbL5DPSImbyUi4m1sciRkvicKUxEwxZLKHX9HkKeKebct/s360/b9.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="200" data-original-height="246" data-original-width="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMD_GON-fguPNPwBWLT9ITUWYgapK7ag4i1zhwo8zE0MBzyi-XDl7fZGFUfO14zuVTREjrl0wxXqViXx1FLFXV85Bt6HaTAoiM8ths_836XRyR9xQy5qJU35ITvt37lxFLukyXk6hUo3NfbL5DPSImbyUi4m1sciRkvicKUxEwxZLKHX9HkKeKebct/s200/b9.png"/></a></div>
</p><p>
Now all five dots are mutally-connected. Five dots now require a minimum of five colors. In 3D, the ratio of colors (c) to dots (d) is always one. In 2D, this ratio breaks down when there's more than four dots:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwEB45t0pS0m7j4RpBMAYrBJp_fV7i4ktAKvAwpu1V_YPy1v9P9t9op0tP-elR0gdEECmm11u-f-JGbTEWG6H0_FkG7bGCpe5uADTChUZ0pHeno28PTT-UbhbJKhvi2JcgACNXeu6QYqzf3LVRVa95d94nuISoOIYT9GEG-OJRMO8N3-VNOxJ7vNnh/s711/b10.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="616" data-original-width="711" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwEB45t0pS0m7j4RpBMAYrBJp_fV7i4ktAKvAwpu1V_YPy1v9P9t9op0tP-elR0gdEECmm11u-f-JGbTEWG6H0_FkG7bGCpe5uADTChUZ0pHeno28PTT-UbhbJKhvi2JcgACNXeu6QYqzf3LVRVa95d94nuISoOIYT9GEG-OJRMO8N3-VNOxJ7vNnh/s600/b10.png"/></a></div>
</p><p>
Below is an example of eight dots, mutually-connected in 3D, requiring at least eight colors:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTubZvCq21eQQp8j27X6lAu_Sx5jWM0OXxe_bXK2BIc_Os8b4g6SiMNOQtHKu7mSkwXc1OWwQvJfq-xeiKmTcC6NYxwFngU9jLT0jTQMBa4biGAOLTI_f2gTzymHv1U-GKnPMGElmQrTzOG_CApmHMTc2tqVgUTFpv9YRYlwEJ-m0LS_IZKUfcaMyw/s466/b11.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="200" data-original-height="182" data-original-width="466" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTubZvCq21eQQp8j27X6lAu_Sx5jWM0OXxe_bXK2BIc_Os8b4g6SiMNOQtHKu7mSkwXc1OWwQvJfq-xeiKmTcC6NYxwFngU9jLT0jTQMBa4biGAOLTI_f2gTzymHv1U-GKnPMGElmQrTzOG_CApmHMTc2tqVgUTFpv9YRYlwEJ-m0LS_IZKUfcaMyw/s200/b11.png"/></a></div>
</p><p>
If 3D is invoked on a 2D plane, lines are allowed to cross. This potentially allows up to an infinite number of mutual connections between an infinite number of dots, and such a map requires up to an infinite number of colors.
</p><p>
If we examine other dimensions (D) of space, we make the following observations: if D = 0, there is always one dot and one color. If D = 1, there is only one type of map: a straight line. It is fairly obvious that only two colors are needed for this map, no matter how big it is.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlLzFzI-YOG60UF2cEZMvM8pb4jQh0Jo0zko0-ZyIetXE7Dlu32bSlEGkpuBuNns02DzI3YAP4sdcYspStlDIxCqVUmXwEzHpiOCFl0rbl7HxCTfSeADQBezZhjbccBl5nFomrkn-ZobzLEPghs3BujvUVZBoYfPgRg2LOdGx3GLlXdkFhKeP__59y/s393/c1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="299" data-original-width="393" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlLzFzI-YOG60UF2cEZMvM8pb4jQh0Jo0zko0-ZyIetXE7Dlu32bSlEGkpuBuNns02DzI3YAP4sdcYspStlDIxCqVUmXwEzHpiOCFl0rbl7HxCTfSeADQBezZhjbccBl5nFomrkn-ZobzLEPghs3BujvUVZBoYfPgRg2LOdGx3GLlXdkFhKeP__59y/s400/c1.png"/></a></div>
</p><p>
When D = 2, and lines are not allowed to cross, it appears that no more than four mutual connections can be made between dots, and no more than four colors are ever needed no matter how big or complex the map. As mentioned earlier, a 3D map could require up to an infinite number of colors. Further, we should note that 0D only accommodates 0D maps (a single dot), 1D accommodates 0D and 1D maps, 2D accommodates 0D, 1D, 2D maps, and finally, our 3D space can accommodate all maps from 0D to 3D. Below is a graph that illustrates our findings:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghps11CFvIFYaEEXPThOlUuDRO2h9VwBkUqM1TiDnn1MbxPzJMu08PvXagL7hvxZW3rm-1TZQ0lGevCFl3Mz0Tf1PzXka6EiqfK_3XKZD-3xrLHNOW8-yfbt8fjwyH84cBGOT-CwrNhoY9VK_XjLAQwnANZlNYLZXxC__6J6i-MpCRzwc-xm019gR7/s438/c2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="438" data-original-width="391" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghps11CFvIFYaEEXPThOlUuDRO2h9VwBkUqM1TiDnn1MbxPzJMu08PvXagL7hvxZW3rm-1TZQ0lGevCFl3Mz0Tf1PzXka6EiqfK_3XKZD-3xrLHNOW8-yfbt8fjwyH84cBGOT-CwrNhoY9VK_XjLAQwnANZlNYLZXxC__6J6i-MpCRzwc-xm019gR7/s600/c2.png"/></a></div>
</p><p>
Take a look at the 2D column. It indicates that 1-color, 2-color, 3-color, and 4-color maps are possible. But are four colors really the maximum needed for any possible 2D map that needs more than three colors? It would take a great deal of computing power to test an infinite number of different types of maps that can exist in 2D space. The data table below lists what we know and a question mark where there is uncertainty:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMHm7vvJ-4QeHEKFoNGH6qEb5F_wIVyK5L95KH9AZwLW1_sNErSDqeu2zIn9RNYyn8D0_l5wm-C9ZDLVJxub32GRUKcmE21vX4e7VEhYZ-4H3yidMwKSRBto8_uLiDw_6UmfnZYzWtczoQxDTY3O4XCh49837MEENQPD5C6TZtxkYq_NiNlP2iJyv5/s913/c3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="484" data-original-width="913" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMHm7vvJ-4QeHEKFoNGH6qEb5F_wIVyK5L95KH9AZwLW1_sNErSDqeu2zIn9RNYyn8D0_l5wm-C9ZDLVJxub32GRUKcmE21vX4e7VEhYZ-4H3yidMwKSRBto8_uLiDw_6UmfnZYzWtczoQxDTY3O4XCh49837MEENQPD5C6TZtxkYq_NiNlP2iJyv5/s600/c3.png"/></a></div>
</p><p>
The data table suggests a pattern where we can predict that the question mark should equal four, i.e., no more than four colors are required to color any 2D map. But is there really a pattern or are the numbers listed coincidental? Can we show that the numbers are logically and mathematically connected? The answer is yes:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZO-kxIKyAJZeC90U5g3xBa0DD1NONe16eYweMhWXQn_b3GQs3_tuA0cppEUp15viNHgvoDcHkhn6JX1_x38wa86-J54PHXanrz0VqQ3TanDrGyD2mj8aeXlvr_W8m_Z9JaQSXjjMsjRM-WVsu2-kOZKAzLX5Yfc44SCD26bKTqDoskbfSeFgm7xTh/s818/c4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="818" data-original-width="725" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZO-kxIKyAJZeC90U5g3xBa0DD1NONe16eYweMhWXQn_b3GQs3_tuA0cppEUp15viNHgvoDcHkhn6JX1_x38wa86-J54PHXanrz0VqQ3TanDrGyD2mj8aeXlvr_W8m_Z9JaQSXjjMsjRM-WVsu2-kOZKAzLX5Yfc44SCD26bKTqDoskbfSeFgm7xTh/s600/c4.png"/></a></div>
</p><p>
Dimensions zero through three each have a maximum number of colors necessary to color maps in their respective spaces. The pattern between them is a tan function (see equation 3). So we can logically conclude that no more than four colors are needed to color any 2D map? Yes, as long as 3D is not invoked on a 2D map. If 3D is invoked, here's what is possible:
</p><p>
Imagine an exoplanet approximately 50 light-years from earth. It orbits a yellow star within the Goldilocks zone. It has a vast ocean. In the middle of that ocean is an island continent with five countries: purple, red, yellow, green, and brown.
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In the middle of the island is a mountain. The red and purple countries share a border at the mountain top. The green country created a tunnel through the mountain to connect with and share a border with the yellow country. Each country shares a common border with the remaining four. Not counting the ocean, at least five colors are needed to color this 2D map.
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References:
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1. Four Color Map Theorem. Wikipedia
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2. Weisstein, EW. 2002. Four Color Theorem. Wolfram MathWorld
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3. Najera, Jesus. 10/22/2019, The Four-Color Theorem. Cantor's Paradise
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4. Planer Graph. javatpoint.com.
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<!--Premise: All maps, regardless of their shapes and sizes, can be represented by stick drawings. Such drawings make analysis easier and make this simple proof more universal.
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Premise: Any random four countries, with common boundaries to each other (e.g. the left diagram), are equivalent, can be mapped to, and represented by the right diagram. (Note that no lines cross.)
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Premise: More countries can be added (dotted lines indicate additional countries without limit) and all four drawings are equivalent. In all cases, the yellow is surrounded. To keep this proof simple we shall use the drawing on the far right.
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Whether a fifth country is added inside the triangle or outside, it makes no difference in the analysis. Both drawings are equivalent. This is also true when more countries are added. For ease of illustration, we choose to add countries to the outside.
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When a fifth country is added, we might be tempted to color it purple (a fifth color), but this isn't necessary. We can color it yellow, since our yellow country shares no border with our fifth country. If we add a sixth country, we can color it red, since our previous red country has no border in common. We can add countries seven, eight and color them blue and green respectively. No need to ever add a fifth color, since there is at least one of the four colors that is isolated from any new country we add and that color can therefore be reused. This exercise can be done an infinite number of times. So no matter how large and complex the map, no more than four colors are required.
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When can a four-color map NOT be reduced to a three-color map? When the fourth color (yellow) is surrounded by an odd-numbered perimeter of dots (countries) as shown in diagram A. (The dashed lines represent multiple countries, alternating blue and red.) Or, when the fourth color (yellow) is on the odd-numbered perimeter as shown in diagram B. Diagram C is an example where it looks like the fourth color (yellow) is surrounded by an even-numbered perimeter (see D). But because it is on the odd-numbered perimeter of an adjacent cluster (see E), the map can not be reduced to three colors. A more detailed explanation or proof will be provided in the near future.
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Here's the skinny on when a 4-color map can't be reduced to 3 colors. Let's start at the beginning and work forward. At 1), we have the simplest map of all--a 1-color map. Add a single country at 2)--now we have a 2-color map. Add a an even number of countries at 3). Because the number is even, we can alternate between 2 colors--now we have a 3-color map. If we take any country and surround it with an even number of countries (see 4), we always get a 3-color map.
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At 5) we stack countries together and get a 4-color map. At 6) we surround a country with an odd number of countries. This also makes a 4-color map. An even number allows just two colors to be alternated, but an odd number requires a third color which forces the surrounded country to have a fourth color. (Note how the maps at 5) and 6) have similar looking stick-and-dot representations even though they look different.)
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At 7), 8 and 9) we have stick representations for 3 and 4-color maps. (The dashed lines represent a repeating 2-color series of any size.) (The diagram at 10) shows that a 5th color requires a 3D surface.) If we combine just the 3-color maps to make a more complex map, it will still be a 3-color map. If, however, we add at least one 4-color cluster to an otherwise 3-color map, the whole map becomes a 4-color map. (See examples at next post.)
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Diagram A is a complex 3-color map made from simple 3-color clusters. The yellow dots on the outside represent the background color. Notice how each surrounded dot is a source of an even number of lines. This implies that each dot is enclosed by an even number of other dots. As we showed previously, an even number requires only two colors, so the dot enclosed has a third color. This is the essence of the 3-color map.
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Diagram B shows how simple it is to make diagram A into a 4-color map. Just add a dot surrounded by an odd number of other dots. Odd numbers can't have less than three colors, so the enclosed dot needs color four. This is the essence of the 4-color map.
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At diagram A, not just the interior countries of a 3-color map are surrounded by an even number of colors, but so are countries at the map's perimeter if the background color is included. At diagram B, we can convert the 3-color map into a 4-color by simply adding a fourth color at the perimeter. Note that where a fourth color is added, it is surrounded by an odd number of colors, even at the perimeter if the background color is counted.
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At A, the dot drawing models the most upper-right red country. The green dot furthest from the diagram represents the green background. This red country is surrounded by an even number of dots or line segments. At diagram B, when fourth color is added, the number of surrounding dots or line segments is odd. Again, the green dot furthest from the diagram represents the green background.
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In 2D space you can have the following correlation: the minimum number of countries needed to make a 1-color irreducible map is 1 or b(1) = 1, where b is the number of needed countries, and a = 1 (a 1-color map). This correlation holds for 2-color, 3-color and 4-color maps. b(a) = a. But the correlation breaks down for 5-color maps and higher. The 1 through 4 color maps are irreducible (can't use fewer colors), but 5 and higher are always reducible in 2D. However, in 3D, any n-color map can be made irreducible. The upper limit of n is infinity.
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Circled in red is a formula that calculates how colored maps behave in different dimensions. For example, in 1D space, only 2 colors are needed. The possible map types include 0-color, 1-color and 2-color. That's a total of 3, so according to the formula, N = 3. In 2D space, you can have 0 through 4-color maps or N = 5. A 0-color map is essentially no map. Zero countries are needed. b(0) = 0.
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Circled in blue is the simplified version of the formula in 3D space. Cut off the last two terms and you have an equation of quantum energies! The other thing that is fascinating about 3D is how the number of map types jumps to infinity! I adjusted the formula so the value of N will always be an integer as it tends to infinity.
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Further notes: Leaving out the background, a minimum of one dot is needed for a one-color map; a minimum of two dots are needed for a two-color map; a minimum of three dots are needed for a three color map; a minimum of four dots are needed for a four-color map. What is the minimum number of dots needed for a five-color map? The answer is unknown. So far, five or more dots can be reduced to a four-color map. Refine triangulation algorithm and research this topic to get more ideas.-->
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-62312716217511603112022-04-22T15:24:00.002-07:002022-04-22T15:50:09.315-07:00Proof that Various Infinities Have a Finite Value<p>
In the late 19th century, German mathematician Georg Cantor demonstrated that there are a variety of infinities of various sizes. Equations 1 and 2 below compare two different arbitrary infinities:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjctbiR3ZOSiWyIS9H9Hyu8a9LmAedAnAL7Q_sMdzW3ztYsMVElS-M_5OCxUW5EUbWmYmP8BGjNuh7_O8mMUlvczIee9pGAcus9Q1R_OfO8XpRy7UqjGxyodG8wUzuYhFuMgWdu00n6LxnsOWAaEtVIVz8eThWiouXP2lpmd9yasmDJu0fwbK9IA5P9/s428/1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="171" data-original-width="428" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjctbiR3ZOSiWyIS9H9Hyu8a9LmAedAnAL7Q_sMdzW3ztYsMVElS-M_5OCxUW5EUbWmYmP8BGjNuh7_O8mMUlvczIee9pGAcus9Q1R_OfO8XpRy7UqjGxyodG8wUzuYhFuMgWdu00n6LxnsOWAaEtVIVz8eThWiouXP2lpmd9yasmDJu0fwbK9IA5P9/s400/1.png"/></a></div>
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The infinity at equation 2 is n times bigger than the infinity at equation 1.
</p><p>
Srinivasa Ramanujan, an Indian mathematician (1887–1920), was cleverly able to extract finite values from various divergent infinite series. If having various infinities isn't weird enough, imagine various infinities having a finite value! Normally, the process of extracting a finite value from an infinity requires tackling that infinity with a unique set of steps. Thus it is not clear if all infinities have a finite value. Perhaps there are some that do and some that don't. Can mathematics provide any clues? Let's begin with an arbitrary function f that diverges to infinity:
</p><p>
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</p><p>
What makes this particular infinity unique is not that variable N tends to infinity, but that N is multiplied by a coefficient alpha. This infinity is alpha times the size of a benchmark infinity that is simply N with an infinite limit. Below is another infinity (function s) with a coefficient of beta, where beta may not equal alpha:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsVAURSorgxOmDTy9_vck2NPKztpUVLPMvlfGphlaAlNNNctKL0oRlVtS1Q2zgrPufOcvu94patywaBrjzyK603imVYDjpZ7k1quZT3IgKdun_oLgRAopqwpin5o_RQj3YuIfBMP-f0yj7sBbGk4RhRNn3OckEJag-6cqn2sk4ADWT5-xXYrULAZr5/s622/3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="146" data-original-width="622" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsVAURSorgxOmDTy9_vck2NPKztpUVLPMvlfGphlaAlNNNctKL0oRlVtS1Q2zgrPufOcvu94patywaBrjzyK603imVYDjpZ7k1quZT3IgKdun_oLgRAopqwpin5o_RQj3YuIfBMP-f0yj7sBbGk4RhRNn3OckEJag-6cqn2sk4ADWT5-xXYrULAZr5/s400/3.png"/></a></div>
</p><p>
The relationship between f and s is as follows:
</p><p>
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</p><p>
Using some Ramanujanian algebra, we can solve s:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQ0e0uayTswmWPunUlSbBd_EngjI4RdYF8tJDso24G4XE4Aa6xN2J6ajO6IHJ_0yna2WJ6U1aTtW5EWSKTZwovR696NTTXtdnKbRDrWI4aHpVAcInkePbtLPchbYmBwdvO_x-Dq-WOjVzuTTVJ2pte7IjC_w3dEQ58aNhKWPMBG9YzqHbV3D60mRoV/s366/5.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="366" data-original-width="362" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQ0e0uayTswmWPunUlSbBd_EngjI4RdYF8tJDso24G4XE4Aa6xN2J6ajO6IHJ_0yna2WJ6U1aTtW5EWSKTZwovR696NTTXtdnKbRDrWI4aHpVAcInkePbtLPchbYmBwdvO_x-Dq-WOjVzuTTVJ2pte7IjC_w3dEQ58aNhKWPMBG9YzqHbV3D60mRoV/s600/5.png"/></a></div>
</p><p>
We can now solve f by making a substitution:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAITO_3CXD2gfh_AiAcfGqJjO8Qy7HmW6EzatTyNPFAtRyznL6rezMS6ypl_3CThT5AqDryX8wzTFN4_iP2UVEgaXrfPGf7OvQyG7SsC1Y1G0y7BnDAfZ7PW4gpbyztSyKSnjLHUmVzDKIyF23bLxl79Qqs24NJE2zoUD45OaiNMFKLDkfy2e8R_ue/s364/6.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="269" data-original-width="364" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAITO_3CXD2gfh_AiAcfGqJjO8Qy7HmW6EzatTyNPFAtRyznL6rezMS6ypl_3CThT5AqDryX8wzTFN4_iP2UVEgaXrfPGf7OvQyG7SsC1Y1G0y7BnDAfZ7PW4gpbyztSyKSnjLHUmVzDKIyF23bLxl79Qqs24NJE2zoUD45OaiNMFKLDkfy2e8R_ue/s400/6.png"/></a></div>
</p><p>
If coefficient k is finite and x doesn't equal 1, the solution to any arbitrary infinite function is finite. Otherwise it is infinite.
</p><p>
References:
</p><p>
1. Matson, John. 07/19/2007. Strange but True: Infinity Comes in Different Sizes. Scientific American
</p><p>
2. Dodds, Mark. 09/02/2018. The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12? Cantor's Paradise
</p><p>
</p><p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-11897462573714300402022-04-21T13:49:00.004-07:002022-04-21T14:57:07.749-07:00How to Have Unlimited Orthogonal Space and Time Dimensions within 4D Spacetime <p>
ABSTRACT:
</p><p>
By means of a thought experiment and a mathematical proof, it can be shown that unlimited space and time dimensions are possible, and, that only three space dimensions are mutually perpendicular.
</p><p>
Imagine you are throwing a party. You have invited n number of guests. You want to know the following: the starting location and time of each guest and the time each guest arrives at your party. You know the location of the party and you know what time the guests are supposed to arrive. All of this information consists of 2n+1 bits of time and n+1 bits of location. Assuming each location has coordinates x, y and z, the total bits of space information is 3n + 3.
</p><p>
Each bit of information is statistically independent, i.e., orthogonal to all the rest. We can think of any two bits as having a 90 degree separation. In the case of coordinates x, y, and z, such separation can be easily drawn on graph paper. In all other cases, such separation may be purely abstract and unimaginable. In any case, we can argue that 3n+3 space dimensions and 2n+1 time dimensions are necessary. To have all the information you want, you need 5n+4 dimensions. If n = 100 guests, you only need 504 spacetime dimensions!
</p><p>
It's fairly obvious that the above thought experiment only involves three space dimensions that are mutually perpendicular. But is this universally true? Is there, say, a mathematical proof? The following equation suggests there's no upper limit to how many space dimensions you can have:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrOTgpvpUXZAzKsdtNxRGILQGOe-aGB5afHpf_3jVG5F5lHPDe8OehoSLzcJgH1Br9uNDHBvqxVJGhXfw6LTORWAWOTi55oINyE2TvkuRKVgc-ZC2VZvYYtZ9uwVqAS5KnVG5Bw6HzBeFTJvc88XzNNIKw_Kdw14rsyE2XLKyIk-hP1FD7_rzP1ARe/s446/1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="202" data-original-width="446" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrOTgpvpUXZAzKsdtNxRGILQGOe-aGB5afHpf_3jVG5F5lHPDe8OehoSLzcJgH1Br9uNDHBvqxVJGhXfw6LTORWAWOTi55oINyE2TvkuRKVgc-ZC2VZvYYtZ9uwVqAS5KnVG5Bw6HzBeFTJvc88XzNNIKw_Kdw14rsyE2XLKyIk-hP1FD7_rzP1ARe/s400/1.png"/></a></div>
</p><p>
It seems like the only constraint re: the number of space dimensions is an empirical one. Let's see if we can find a mathematical one. Let's begin with the following premises:
</p><p>
1. w, x, y and z are unit vectors.
</p><p>
2. w is an arbitrary extra space dimension. What is true for w is true for any extra space dimension.
</p><p>
3. A unit vector is consistent with 1D space and points in only one direction.
</p><p>
4. if two unit vectors (x, y) are perpendicular, they define a plane that is consistent with 2D space. Thus plane xy is not perpendicular to plane xy.
</p><p>
Following these premises we have:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6s_q9fBGSt1IqHXfDrTSzrIocq17QJKcQINjGbyl-t2xitR9YLg_6Id7DreWEcRBmHMP2wRYOOomYyN-Eod_UHgZ1m44PGFnyV_nuYDuPYc0vJCCHoOYZjgXa43rT0YROBkEQTbx4ZTTphZ6Fu9lCyHVZiCkEq8fk7hu-a-0vRhhiKL58WVdFqMq1/s584/2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="584" data-original-width="503" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6s_q9fBGSt1IqHXfDrTSzrIocq17QJKcQINjGbyl-t2xitR9YLg_6Id7DreWEcRBmHMP2wRYOOomYyN-Eod_UHgZ1m44PGFnyV_nuYDuPYc0vJCCHoOYZjgXa43rT0YROBkEQTbx4ZTTphZ6Fu9lCyHVZiCkEq8fk7hu-a-0vRhhiKL58WVdFqMq1/s600/2.png"/></a></div>
</p><p>
At steps 2 through 4 we assume that all four unit vectors are mutually perpendicular. At 5 we assume w is perpendicular to plane xy. At 6 we assume z is also perpendicular to plane xy. At 7 we conclude that either w is parallel to z or if w is perpendicular to z, then plane xy must be perpendicular to plane xy--which violates premise 4. So w is not an extra dimension. According to premise 2, what is true for w is true for any alleged extra space dimension. Thus, we can further conclude there are only three space dimensions that are mutually perpendicular. If such a conclusion is valid, we should be able to falsify the following 7D cross product table:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoVa6AyJIZQ6Ob_ap2psZQDyIn1oE2OIuuwTvzhEKKUj4xNeoLHXwcM9agj1gE5vV7icu6A7kOvegh1AqwV-d3GFyPdtXMNbbY_WgNYFiVAM_r90M-F0Muc5YwOlneknAc54i1tDZR1t-W_UeeJhMj78YPy5pwzHqADdQJM0ZKm955yPFjfNP4fMJE/s836/3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="623" data-original-width="836" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoVa6AyJIZQ6Ob_ap2psZQDyIn1oE2OIuuwTvzhEKKUj4xNeoLHXwcM9agj1gE5vV7icu6A7kOvegh1AqwV-d3GFyPdtXMNbbY_WgNYFiVAM_r90M-F0Muc5YwOlneknAc54i1tDZR1t-W_UeeJhMj78YPy5pwzHqADdQJM0ZKm955yPFjfNP4fMJE/s600/3.png"/></a></div>
</p><p>
From this table we can gather that the unit vector e1 is the solution to three cross-products involving 6 dimensions (see equation 8 below). Premise 3 stipulates that a unit vector only points in one direction. It is also obvious that e2e3, e4e5, e6e7 make three planes persuant to premise 4. Vector e1 can't point in just one direction if it is normal to all three planes, unless all three planes are subsets of the same plane. The inevitable conclusion is not all of these dimensions are mutually perpendicular.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUlnqRupfU8l6SEMujX2HnB8YauUHcOQFZvtVOmZ_SBGARi0G1HhjqzziamKfdL8Latr_zSDG3teo6Yuu9zr0sPkGoAZuW1WNxtXemFS_zt9kVy2dgqW9FBTRgf9k885cX_vCz1O6B3v2dvB9IpxMESHRen247YtW8UaIjLs4HV7SXsjsTfsf6g2BJ/s558/4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="430" data-original-width="558" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUlnqRupfU8l6SEMujX2HnB8YauUHcOQFZvtVOmZ_SBGARi0G1HhjqzziamKfdL8Latr_zSDG3teo6Yuu9zr0sPkGoAZuW1WNxtXemFS_zt9kVy2dgqW9FBTRgf9k885cX_vCz1O6B3v2dvB9IpxMESHRen247YtW8UaIjLs4HV7SXsjsTfsf6g2BJ/s400/4.png"/></a></div>
</p><p>
Now, let's take a look at the so-called extra dimensions 4 through 7. At each of the equations 9 through 12 below, the unit vectors circled in red contribute to planes with normal vectors pointing in different directions. Thus they can't all have the same normal vector or cross-product solution.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfszaf3QYlGubKW-q8hvU_H6KMtWs7AC3ki61vq8AFdK2A7PyORwpSYSBXI12ELxSy6jmOQwYRElAZwZpSctYq2icCwZI6sfAh-7eFX4gO3dEO8wSfLhMzzIWi6LSzy0UBGcBiP9jjlJiQLxm6elJdYaX5KTdX_sAvmaCkRgrsc3XKknhIfCmqw53-/s593/5.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="336" data-original-width="593" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfszaf3QYlGubKW-q8hvU_H6KMtWs7AC3ki61vq8AFdK2A7PyORwpSYSBXI12ELxSy6jmOQwYRElAZwZpSctYq2icCwZI6sfAh-7eFX4gO3dEO8wSfLhMzzIWi6LSzy0UBGcBiP9jjlJiQLxm6elJdYaX5KTdX_sAvmaCkRgrsc3XKknhIfCmqw53-/s600/5.png"/></a></div>
</p><p>
Therefore, it is safe to say that dimensions 4 through 7 do not behave like mutually perpendicular dimensions.
</p><p> Circling back to our hypercube at equation 1, we can conclude that there is no upper limit to how many dimensions the cube can have, but only three are mutually perpendicular. The rest may or may not be orthogonal in the sense that they are statistically independent.
</p><p>
References:
</p><p>
1. Seven-dimensional Cross Product. Wikipedia
</p><p>
2. Octonian. Wikipedia
</p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-28411485837282061332022-04-16T14:30:00.011-07:002022-04-17T09:16:21.747-07:00Resolving the Liar's Paradox: "This Statement is False"<p>
"I am a liar," is the original liar's paradoxical statement, but we're going to focus on one of its variations: "This statement is false." If true, then it is false. If false, it must be true. To resolve this paradox, we begin with the following premises:
</p><p>
1. If a statement is true or false, it is one or the other and not both, and, we can determine whether it is true or false.
</p><p>
2. If a statement is true and false, it is a challenge to give it a truth value.
</p><p>
Now let's define some statements: Statement A = "This statement has five words." Statements B and C = "This statement is false." Statement D = "This statement has four words."
</p><p>
Given the foregoing premises, we can determine the truth value of each statement using an AND-gate truth table and by following these steps:
</p><p>
1. Read the statement and determine if it is true or false. If this can't be done, assume it is true (or false if you prefer). Mark the left side of the truth table T for true or F for false.
</p><p>
2. Then ask, "If I assert the statement is true (false), does it become false (true)? If not, repeat your previous mark. If so, add the opposite mark.
</p><p>
If these steps are followed, the truth table for statements A, B, C, D should look like this:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8uas6zNxbhYQgoKLGDkdBz4GJQHfre3chJ-6ijTVwGiVxIIVbmMjec_hNcTBr3lvpqt9q-BQJ1_vLcPq_FLWus6_1TSq3iYv9V6lcIY2VcnWjOONICUzTrP6TfnDSCo8-BCNbuXdcDC4VuSipXZCc2OTHMTrJ0YShpsfhJD8RWZYAEs4Y4qQn43HW/s685/L1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="639" data-original-width="685" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8uas6zNxbhYQgoKLGDkdBz4GJQHfre3chJ-6ijTVwGiVxIIVbmMjec_hNcTBr3lvpqt9q-BQJ1_vLcPq_FLWus6_1TSq3iYv9V6lcIY2VcnWjOONICUzTrP6TfnDSCo8-BCNbuXdcDC4VuSipXZCc2OTHMTrJ0YShpsfhJD8RWZYAEs4Y4qQn43HW/s400/L1.png"/></a></div>
</p><p>
For the true statement A, there is no contradiction or paradox so the two marks are TT. Statement B equals statement C. At B we start out assuming the statement is true, and, at C we begin assuming it is false. That leads to scores TF and FT, respectively. Statement D is determined to be false and remains false, so we have FF. At the far right column are the final truth values. The true statement is true; the false statement is false, and, the paradoxical statement is false. This last result is consistent with the law of non-contradiction. Statements that are contradictory or lead to a contradiction are not credible and should be considered false, notwithstanding any claim to the contrary.
</p><p>
References:
</p><p>
1. Mano, M. Morris and Charles R. Kime. Logic and Computer Design Fundamentals, Third Edition. Prentice-Hall, 2004. p. 73.
</p><p>
2. Epimenides paradox has "All Cretans are liars." Titus 1:12
</p><p>
3. Jan E.M. Houben (1995). "Bhartrhari's solution to the Liar and some other paradoxes". Journal of Indian Philosophy. 23 (4): 381–401.
</p><p>
4. Hájek, P.; Paris, J.; Shepherdson, J. (Mar 2000). "The Liar Paradox and Fuzzy Logic". The Journal of Symbolic Logic. 61 (1): 339–346.
</p><p>
5. Mills, Eugene (1998). "A simple solution to the Liar". Philosophical Studies. 89 (2/3): 197–212.
</p><p>
</p><p>
</p><p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-39757162189949091472022-04-09T12:33:00.000-07:002022-04-09T12:33:30.310-07:00Debunking Extra Space Dimensions and Minimum Distance<p>
ABSTRACT:
</p><p>
By means of two thought experiments and some mathematics this paper shows that extra space dimensions are untenable. This paper also shows that the minimum distance is many orders of magnitude shorter than the Planck length.
</p><p>
Imagine a 2D universe on an x-y plane (see diagram below). Imagine a normal vector intersecting this plane at point p. 2D-guy inhabits this universe. He can't see the vector that intersects point p. He can only detect point p, so he has no reason to believe the normal vector exists. Now, to avoid point p, he goes around it (see red arrows).
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOMYY3QX364TUC_ZUy0hjnWnHiQYuqWdmBW08Js82d7Qdgnb__jZY3PMXZxJvVQTpLDqLSvp0kn73dmgGuU62PcMjPuMm9qogJWkx6CPDbHL3P2j4wt1vWuLZUrkWX5G_zosEmiAAPhMRLzs0H_iSTKfuyaF0xCkb5K0hj3iX7zJaOMBzvZNBdQ5yF/s404/01.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="357" data-original-width="404" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOMYY3QX364TUC_ZUy0hjnWnHiQYuqWdmBW08Js82d7Qdgnb__jZY3PMXZxJvVQTpLDqLSvp0kn73dmgGuU62PcMjPuMm9qogJWkx6CPDbHL3P2j4wt1vWuLZUrkWX5G_zosEmiAAPhMRLzs0H_iSTKfuyaF0xCkb5K0hj3iX7zJaOMBzvZNBdQ5yF/s400/01.png"/></a></div>
</p><p>
He knows it's possible to draw an imaginary line through point p that can serve as an axis. He also notices when he goes around point p he's not encircling the x-axis or the y-axis--the two dimensions of his space. Thus, he infers that the imaginary axis he's going around does not belong to his 2D universe. He realizes he has discovered a new dimension!
</p><p>
Now, what happens if we apply 2D-guy's process to 3D space? Will we discover a fourth dimension? Let's try it. First we must scale everything up one dimension: The universe becomes 3D; the normal vector becomes a normal plane; Point p becomes line L. Let's assume there's a fourth dimension w, and let's define the normal plane as wx. Plane wx intersects our universe at line L which runs along the x-axis. We should not be able to detect the w-axis nor the bulk of the wx plane. We illustrate this with broken lines at the diagram below:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKGlxYbwyPxWjzoCA3Z8mmm-MsB_FH9tg8-5BszK2nlqTPRQmcoldWKAgZGHu7a0Hh3-r7LYA3pyMAmmZpPDUgFLsejPoxV1AiXe3_yQkLAWxHpfzh6O_KYLy1r0CF0H7iPoFVHH4KwYPJeUJNGynaFSRC-Z_PKR7GKGf6-NbZQTHOgs3wK9PtEnSI/s401/02.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="401" data-original-width="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKGlxYbwyPxWjzoCA3Z8mmm-MsB_FH9tg8-5BszK2nlqTPRQmcoldWKAgZGHu7a0Hh3-r7LYA3pyMAmmZpPDUgFLsejPoxV1AiXe3_yQkLAWxHpfzh6O_KYLy1r0CF0H7iPoFVHH4KwYPJeUJNGynaFSRC-Z_PKR7GKGf6-NbZQTHOgs3wK9PtEnSI/s400/02.png"/></a></div>
</p><p>
To avoid line L, we circle around it (see red circular path). We know we can draw an imaginary plane through line L. We know that x is one dimension of the plane. We know the axis we are circling (to avoid line L) is the plane's other dimension. We note we are not going around the x-axis nor the z-axis. That leaves the w-axis, but notice that the w-axis is indistinguishable from the y-axis. Therefore, our assumption that w is a new dimension and is undectable beyond line L is false. Unlike 2D-guy, we have not discovered a new dimension. However, we learned from 2D-guy that if a new dimension exists, it should be possible to do a rotation around an axis that does not exist in our universe. Until someone demonstrates such a rotation, we can conclude, for now, that the highest dimension of space is 3D.
</p><p>
But what if there are extra dimensions that are very small and curled up? If that's the case we should be able to enter alternate universes and those from alternate universes should be able to enter ours. Let me demonstrate what I mean. Imagine a line and pretend it is 3D space. Extending from it is a small extra curled-up dimension:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHtYONzImm-3JHdAiACTLAZeM6q7aX-bCGMZ3lXVbihg2-acD43icTozF5_4J3VNn7YDn4X7EVTuCl5h3RiixFpFr2_aqDFJXZ1cj1cBkNB--Id9gsGr6_QNJqRQ74SLG4Y-vWj3r1tGyQass5vQthp8-jwAb_GzMm5VvKYa_z94S1pg98LhGHG6Pe/s519/03.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="167" data-original-width="519" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHtYONzImm-3JHdAiACTLAZeM6q7aX-bCGMZ3lXVbihg2-acD43icTozF5_4J3VNn7YDn4X7EVTuCl5h3RiixFpFr2_aqDFJXZ1cj1cBkNB--Id9gsGr6_QNJqRQ74SLG4Y-vWj3r1tGyQass5vQthp8-jwAb_GzMm5VvKYa_z94S1pg98LhGHG6Pe/s400/03.png"/></a></div>
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Let's introduce an arbitrary red object that is way too big to enter the tiny curled-up dimension:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWJUNM0Sn0wH1GXMOLiiBYs_1AMKEQh48wQJaYd85rT_T8bzo_-YZON6ysMbn_ZT52UKg64juiy3Rerh_y_N2Xk8C9aJefet7qkje4Ih1Hr__hO6RbLEIz30G6ouD7mar0g0BQ9CqgjW2z9iEhwRZUjCd3pWAXjaqja7nAVktOevMamUN5PMMSA_w1/s465/04.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="130" data-original-width="465" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWJUNM0Sn0wH1GXMOLiiBYs_1AMKEQh48wQJaYd85rT_T8bzo_-YZON6ysMbn_ZT52UKg64juiy3Rerh_y_N2Xk8C9aJefet7qkje4Ih1Hr__hO6RbLEIz30G6ouD7mar0g0BQ9CqgjW2z9iEhwRZUjCd3pWAXjaqja7nAVktOevMamUN5PMMSA_w1/s400/04.png"/></a></div>
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Because the red object is too big to fit, it is assumed there is no way for the big red object to enter or detect the existence of the curled-up dimension. But didn't Euclid say something about a line existing between any two points? (In this case the line would be 3D.)
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGhUqyzJRB7NUmNWlm8HBAovLuegTIDW8u9mhEpuZb2OVwidDLgcazwYPvfSLWAQmHyOdbH9L64T-hBJfDCSejtSgnecIkOVmEYpPOHJ8BuTyji2Oap3N_h-AwcZZyKDx1EJ0S6yeynwW5fTGTbhzyqbuP-ueOVlrBLERI83HnfMCXrDQvCrbRd64B/s464/05.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="120" data-original-width="464" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGhUqyzJRB7NUmNWlm8HBAovLuegTIDW8u9mhEpuZb2OVwidDLgcazwYPvfSLWAQmHyOdbH9L64T-hBJfDCSejtSgnecIkOVmEYpPOHJ8BuTyji2Oap3N_h-AwcZZyKDx1EJ0S6yeynwW5fTGTbhzyqbuP-ueOVlrBLERI83HnfMCXrDQvCrbRd64B/s400/05.png"/></a></div>
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There's no reason why the big red object can't follow the path of this new line (3D space)and wind up in an alternate universe adjacent to ours:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh83fdrORGsbKzcMKzqezi9qKK8IvMm0F1MY-1Y6IzHYANTpCRJmZ4FbdkjQp01mFcW_t7AHmakVF8wuEJXXqGgKqjHYGSWMWCmiqDqKdE6F-VvZh37P2YT9d8wYVgobwYVcY4l96tzjgT3T0tgz5Uc4AZVdKEwiNvce_9A58ZED_spFgLziNrXSYz6/s469/06.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="102" data-original-width="469" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh83fdrORGsbKzcMKzqezi9qKK8IvMm0F1MY-1Y6IzHYANTpCRJmZ4FbdkjQp01mFcW_t7AHmakVF8wuEJXXqGgKqjHYGSWMWCmiqDqKdE6F-VvZh37P2YT9d8wYVgobwYVcY4l96tzjgT3T0tgz5Uc4AZVdKEwiNvce_9A58ZED_spFgLziNrXSYz6/s400/06.png"/></a></div>
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As you can see, the big red object still can't enter the small, curled-up dimension, but the curled dimension facilitates access to alternate universes. The fact that big objects don't disappear from our universe and don't seemingly emerge from nowhere is strong evidence that microscopic curled-up dimensions don't exist. But wait! Quantum particles pop into existence and vanish all the time. It is hypothetically believed they enter a curled-up dimension (vanish), then leave that dimension and re-enter our universe. However, there's an alternate hypothesis: particles are really particle-waves. Waves experience constructive and destructive interference. When there's an excitation of a field, a particle pops into existence. That excitation could be or is equivalent to constructive interference. When there's destructive interference, energy vanishes--leaving the impression that the particle has disappeared.
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The foregoing arguments seem to kill any notion that there are more than three space dimensions, but what about 4D spacetime? Or, what about the 6D object that can be found in Las Vegas? Let's address the 6D object first.
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The 6D object I'm referring to is the die. The die has six orthogonal sides. Each side is statistically independent. We can change the value of a side without impacting the value of the other sides. If we change, say, the one to a seven, the other sides will still be two, three, four, five, and six. The most important point we can take away from the die is it is possible to have more than three orthogonal dimensions within 3D space! The die is a 6D object but it is also a 3D cube.
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Spacetime, on the other hand, involves three dimensions of space and one dimension of time. If time is multiplied by a velocity, it has units of distance and is treated as a fourth space dimension. But is it really? Let's see what the math has to say:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxrQE-J02vfGUcLx4BOWrwHWtt8NNw6oEN22NQISrK6zM9OET5-BK0fklT0D_B6qhiTjZRpvEUtpdfMDPEFO-ctPXbmPEMAkzDtap2R--ZZMZu85Vw8QHYUbWGoq7zrrXdbBmg6bvZOhCcL9v9f3SmzR-JNNB29dz27bXrDsitsRc73XbOykv1cbfQ/s521/07.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="521" data-original-width="494" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxrQE-J02vfGUcLx4BOWrwHWtt8NNw6oEN22NQISrK6zM9OET5-BK0fklT0D_B6qhiTjZRpvEUtpdfMDPEFO-ctPXbmPEMAkzDtap2R--ZZMZu85Vw8QHYUbWGoq7zrrXdbBmg6bvZOhCcL9v9f3SmzR-JNNB29dz27bXrDsitsRc73XbOykv1cbfQ/s400/07.png"/></a></div>
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Equation 1 represents a photon propagating through dimensions x, y, and z over a period of time t. It covers a distance of ct or r. For the sake of keeping the math simple, at equation 2 we rotate the path r so it is along the x-axis. Equation 3 reveals that space and time are not statistically independent, i.e., orthogonal to each other. The the value of time t depends on how far the photon propagates along x, and the value of x depends on how much time t lapses. This is the consequence of converting t into distance units by multiplying it by velocity c. So ct is not a true space dimension that is orthogonal to x. However, time t without c is a very useful statistically-independent parameter. For example, coordinates x, y, z tell you where to be for your dentist appointment and time t tells you when. A change in location does not have to change the time of the appointment, nor does a change in time have to change the location. So what can be done to make ct orthogonal to x? How about multiplying ct and x by factors of g? (See equation 6.) A change in x still causes a change in t, but g-sub-tt can be adjusted so the term stays constant. By the same token, the other term stays constant if g-sub-xx is adjusted when a change in t changes x.
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So can we now credibly argue that (g-sub-tt)ct is a genuine fourth space dimension? Well, no 3D space dimension (x,y,z) has to be a function of (depend on) the others. We can, for example, eliminate y and z and still have x. But we can't eliminate a photon's path (x, y and/or z) and still have ct--the distance along a non-existent path. And, if there's no ct, then there's no (g-sub-tt)ct. Therefore, (g-sub-tt)ct is a pseudo-dimension at best.
</p><p>
So far, it seems we've only debunked a fourth dimension of space. What about dimensions five through infinity? Well, how we label a dimension is arbitrary. Any extra dimension can be labeled the fourth dimension. Thus, all arguments we have made against dimension four apply to any extra space dimension.
</p><p>
Now let's turn our attention to the concept of the shortest distance. The popular choice is the Planck length. In fact some theorists quantize space with Planck-size cubes or Planck-size tetrahedrons or Planck-size strings:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhv--zlPUJk0P-s86kvBiqfrZ3561revxuBLZQAEYRSg_FOuM7bhr4VUXI0TCx57K_4g7_n9ffRYbbU_6ytaN7dmjsfBDwIR0vbeD1I44X9ftNveUgxsjv7NsNP0KLOCKEg6j5Moz2aJs8rNNC7R5pNTw09-3qeMDh6d5GAr0qNThSc1FLFWM4Y54yJ/s815/08.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="815" data-original-width="544" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhv--zlPUJk0P-s86kvBiqfrZ3561revxuBLZQAEYRSg_FOuM7bhr4VUXI0TCx57K_4g7_n9ffRYbbU_6ytaN7dmjsfBDwIR0vbeD1I44X9ftNveUgxsjv7NsNP0KLOCKEg6j5Moz2aJs8rNNC7R5pNTw09-3qeMDh6d5GAr0qNThSc1FLFWM4Y54yJ/s400/08.png"/></a></div>
</p><p>
In the above diagram, the cube and tetrahedron have sides that are each one Planck length. However, the red diagonal lines reveal shorter lengths all the way down to a single point. These shorter lengths are absolutely necessary to create the shapes desired. Without a zero-length point, for example, there can be no corners for cubes and tetrahedrons. Additionally, there can be no strings in any string theory, since a string is a 1D object. A 1D object implies a zero cross-section or single point. A minimum-distance-greater-than-zero requirement would be a nightmare for M-theorists, since all D-branes would have to be 10 dimensions (including strings!). To have less than 10 dimensions requires zero distance for one or more dimensions. So it can be argued that the minimum distance is really zero, at least on paper. What about the physical world?
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzSKngYAfRKKWVUilquzqZQM8_4FyHVfR5Kkx-TQpaibx3pX1i-33etUY2sLo7We2tfLlzhIYpWQuQpiWBUO4P-XQciFn98PZlOZKSOaPALgnLqHuq6oSVUg9y35vPzhUOnS96hqh_2tMSt79-FmSp2Yo2Yw1amlO-xX1XmIk02Spcz1rdkWPolJln/s266/09.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="146" data-original-width="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzSKngYAfRKKWVUilquzqZQM8_4FyHVfR5Kkx-TQpaibx3pX1i-33etUY2sLo7We2tfLlzhIYpWQuQpiWBUO4P-XQciFn98PZlOZKSOaPALgnLqHuq6oSVUg9y35vPzhUOnS96hqh_2tMSt79-FmSp2Yo2Yw1amlO-xX1XmIk02Spcz1rdkWPolJln/s400/09.png"/></a></div>
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Equation 7 tells us that the shortest wavelength is determined by the highest energy. When the universe was a singularity, how short was the singularity's wavelength? If we only account for the energy in the known universe, that wavelength would be approximately a Planck length of a Plank length of a Planck length! Not exactly zero, but far less than a Planck length. Add energy beyond our known universe, and the distance is even shorter.
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From a philosophical standpoint, the very concept of length implies a 1D object in the same manner the concept of area implies a 2D object. To measure length requires that we ignore all but one dimension, i.e., we set all but one dimension to zero. So zero distance is necessary, at least in the mind's eye. Since the mind's eye lives in this universe, we can infer that the minimum distance in this universe is zero.
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In conclusion, any extra space dimension would allow rotations around an imaginary axis that is not part of 3D space. It would also allow any object access to an alternate universe. The shortest distance is many orders of magnitude shorter than the Planck length, and the Planck length may only be a lower limit of what we can successfully measure.
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References:
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1. Greene, Brian. 2003. The Elegant Universe. W. W. Norton
</p><p>
2. Irwin, Klee. 04/23/2017. The Tetrahedron. Quantum Gravity Research.
</p><p>
3. Sutter, Paul. 02/23/2022. Loop Quantum Gravity: Does Space-time Come in Tiny Chunks? Space.com
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GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-27140453034541610882022-03-31T11:13:00.004-07:002022-03-31T14:17:25.822-07:00Curing Divergences without Supersymmetry and Renormalization<p>
Abstract:
</p><p>
Supersymmetry or ad hoc methods such as renormalization are often used to tame infinities that result from divergent functions in quantum physics. Although SUSY particles have yet to be discovered and may be too massive to fulfill their purpose, and, renormalization seems to lack mathematical rigor. Here we offer an alternative method that employs the least-action and Heisenberg uncertainty principles.
</p><p>
Imagine a Lagrangian with divergent terms. One strategy is to renormalize it. Simply discard the divergent terms, especially if they are infinite. However, Paul Dirac had this to say about such methods: "I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves disregarding a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!"
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Another strategy is to add superpartners that each have the same mass as their respective standard-model counterparts but make an opposite contribution to the Lagrangian. As a result, the divergence vanishes. Albeit, there is a slight problem: the symmetry of Super-symmetry is broken--the superpartners are believed to be more massive than their standard-model partners. This deflates the balloon of vanishing divergences. To make matters worse, there is a complete and total lack of empirical evidence supporting these superpartners.
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If renormalization seems like bad math and SUSY particles are nowhere to be found, what other options are there? How about the least-action and Heisenberg uncertainty principles? Let's first examine the least-action principle:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjL27uY3dauHaNRhgwF6-J75kWUQM6d53P72wnkbwi6xGa8Rk62G3skahO_zjp_B-CkpY8A_qztUGFzkK7DJiUIPZ6bEyPiByNx3lQL_MO-EfVXQ3d8_6O7-D4sPOepws25jstFJpW0H5HLits_RTp8jWJ4LNLl7oLTJo94gtRDaelzQZvN6kg7GBL_/s280/S1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="119" data-original-width="280" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjL27uY3dauHaNRhgwF6-J75kWUQM6d53P72wnkbwi6xGa8Rk62G3skahO_zjp_B-CkpY8A_qztUGFzkK7DJiUIPZ6bEyPiByNx3lQL_MO-EfVXQ3d8_6O7-D4sPOepws25jstFJpW0H5HLits_RTp8jWJ4LNLl7oLTJo94gtRDaelzQZvN6kg7GBL_/s400/S1.png"/></a></div>
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A particle typically takes the shortest path possible between two points. For that to happen, delta-s, at equation 1, cannot be a large, divergent quantity. It should be zero units of action or time multiplied by energy. However, the following is true:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxkr14tMSu7dR4oXuLfjoZPwNGXMSrMByoa3wyCcPtu3ewm_ngd_dZEjSvb-VZRuxSMBUHxboMMOrMtNQXDfJseBjE9t0_TIPg3x7axfUYX6BFi0JGx3dWJmKNSyArMNenQ7I9WdyyTVinzrFZOa8aBndHbEKzZJjowQzowR_fZVwl-3y4hZKlkTKh/s359/S2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="190" data-original-width="359" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxkr14tMSu7dR4oXuLfjoZPwNGXMSrMByoa3wyCcPtu3ewm_ngd_dZEjSvb-VZRuxSMBUHxboMMOrMtNQXDfJseBjE9t0_TIPg3x7axfUYX6BFi0JGx3dWJmKNSyArMNenQ7I9WdyyTVinzrFZOa8aBndHbEKzZJjowQzowR_fZVwl-3y4hZKlkTKh/s400/S2.png"/></a></div>
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Line 3 shows that time multiplied by energy is greater than or equal to h-bar. To get delta-s to equal zero requires steps 4 through 6:</p><p>
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM_9h0rjJKogSNJq2kFOQb7xXWQ-wu8Jtnlj0le4sHoEXITAj_3NWIPOboxEj7xjf-wAf0jEB_z3zsOfuMSeX6pTaO14nTZjoI5VQf7qF6SP0m26JCtuD1CjcN86SmQjryDs7ohVfLkGLuDdbtmFIg1ugDREHgAN599w7qT_jogbA6c0dwG7w2Q_sZ/s443/S3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="244" data-original-width="443" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM_9h0rjJKogSNJq2kFOQb7xXWQ-wu8Jtnlj0le4sHoEXITAj_3NWIPOboxEj7xjf-wAf0jEB_z3zsOfuMSeX6pTaO14nTZjoI5VQf7qF6SP0m26JCtuD1CjcN86SmQjryDs7ohVfLkGLuDdbtmFIg1ugDREHgAN599w7qT_jogbA6c0dwG7w2Q_sZ/s400/S3.png"/></a></div>
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At 7 we set up another substitution. The final equations are 8 and 9 below:
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Equations 8 and 9 show why there's a least action principle and why energy is generally conserved. Suppose we have a conserved energy L. The divergent energy, delta-E, can be interpreted as energy borrowed from the vacuum. Because it's borrowed, it must vanish within time delta-t. The larger this energy, the shorter its lifespan. As a result, the energy L that you start with is the energy you end up with. It is conserved. Also, the action is the least action.
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At equation 10 we have a Lagrangian where there is no borrowed energy. Because no energy is borrowed, time delta-t is infinite. In other words, this scenario can last indefinitely and create the impression that energy is always conserved.
</p><p>
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</p><p>
At equation 11 we have the opposite extreme: a Lagrangian that diverges to infinity. The good news is delta-t is zero, which shows that infinite borrowed energy does not exist. We can also infer that large borrowed energies exist for too short of a time to be meaningfully observed and measured, so the energy we do observe and measure is small by comparison. Thus, renormalization works despite its ad hoc nature because nature wipes out divergences by means of the uncertainty principle and least action. The only time it is appropriate to keep the divergent terms is when divergent energy is added to the system and not borrowed from nothing. </p><p>
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</p><p>
Now, let's suppose L is a Lagrangian for vacuum energy (see equation 12). A Higgs boson (m-sub-H) pops into existence and has a lifespan of t-sub-H. A too-large Higgs mass would have a lifespan too short to provide a meaningful opportunity to observe it, so the mass we are most likely to observe is a smaller mass.
</p><p>
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</p><p>
More examples: Equation 13 below takes into account multiple particles. Equation 14 takes into account a Lagrangian or function with multiple terms and parameters.
</p><p>
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</p><p>
Since delta-s must be zero to minimize the action, then delta-s along D dimensions must also be zero. Further, both delta-s and s have units of momentum multiplied by position. If we integrate over position and/or momentum space, the following must be true:
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</p><p>
The uncertainty of knowing a particle's position is cancelled by knowing its momentum and vice versa. As a result, the particle's action is minimized along with its position path and momentum.
</p><p>
In conclusion, divergences are tamed if the least-action and uncertainty principles are applied. SUSY particles are not needed and ad hoc methods such as renormalization can be set aside.
</p><p>
References:
</p><p>
1. Lincoln, Don. 2013-05-21. What is Supersymmetry? Fermilab.
</p><p>
2. Martin, Stephen P. 1997. A Supersymmetry Primer. Perspectives on Supersymmetry. Advanced Series on Directions in High Energy Physics. Vol. 18.
</p><p>
3. Susskind, Leonard. 2012. Supersymmetry and Grand Unification Lectures. Stanford University </p><p>
4. McMahon, David. 2008. Quantum Field Theory Demystified. McGraw Hill
</p><p>
5. Baez, John. 11/14/2006. Renormalizability. math.ucr.edu
</p><p>
6. Renormalization. Wikipedia
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-33766885946849879132022-03-14T14:53:00.001-07:002022-03-15T13:04:54.047-07:00Giving Neutrinos Mass by Adjusting the Higgs and Electroweak Mathematics<p>
ABSTRACT:
</p><p>
This paper shows a new mathematical algorithm that allows weak-force bosons to have mass and leaves photons massless while giving mass to neutrinos and other leptons.
</p><p>
The right side of equation 1 below is the Higgs vector used to ensure that photons don't have mass and that the weak-force gauge bosons have mass. This same vector also ensures that leptons will have mass except for neutrinos.
</p><p>
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</p><p>
Neutrinos, however, are not massless. This fact indicates that the electroweak theory is not complete. To fix the theory we need something like the vector on the right side of equation 2:
</p><p>
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</p><p>
This new vector compensates for whatever contributes to neutrino mass. If we use this new vector when determining masses for leptons, and, use the old vector for determining masses for gauge bosons, we end up with the status quo and the extra bonus of slightly massive left-handed neutrinos.
</p><p>
To theoretically justify this new vector we'll examine how the old vector was derived from a Lagrange potential. We will make a minor adjustment to this Lagrange potential without changing its value and its gauge translation invariance. The minor adjustment will allow a derivation of the new vector as well as the old. Here is the Lagrange potential in its original form:
</p><p>
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</p><p>
Equations 3 through 6 demonstrate gauge translation invariance and lead to equation 7. Next, we take the derivative with respect to phi to acquire the minimum potentials:
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</p><p>
At 11 and 12 above we have the minimum potentials we find in the Higgs vector at equation 1. To derive the new vector at equation 2 we do the following:
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</p><p>
Two new terms are added to the potential that cancel each other, so the potential is the same. Once again gauge translation invariance is demonstrated. After all is said and done, we have two useful equations: 18 and 19. If we substitute the zero value at 18 into 19 we have the original Lagrange potential. Once again, we can take the derivative and derive the original Higgs vector. Or, we can take the derivative of equation 19 without the substitution:
</p><p>
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</p><p>
At 23 we end up with two non-zero solutions. If phi is small, we have the first approximate solution. If phi is larger we have the second approximate solution. This is consistent with, say, an electron being more massive than its family neutrino. We now have what we need to create the new vector (see equation 2).
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</p><p>
The next step is to show how this new vector is applied. When the mathematics is normally done, all terms containing h(x) are discarded at the end, but the solutions don't change if we discard h(x) early or leave it out of the vector. Doing so greatly streamlines the math. Thus the new vector becomes
</p><p>
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</p><p>
There are three families of leptons. Since the same mathematics applies to all three, let's just focus on the electron family. The normal interaction Lagrangian for the electron family is
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</p><p>
The problem with this Lagrangian is it assumes the left-handed neutrino has no mass, so we need to adjust the Yukawa coupling Ge to G.
</p><p>
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</p><p>
Since the right-handed electron doesn't have a corresponding right-handed neutrino, the Greek letter nu with an R subscript has a zero limit at equation 30. After performing the matrix operations we get
</p><p>
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</p><p>
Now let's set up some substitutions and define the modified Yukawa coupling G to include neutrino and electron masses:
</p><p>
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</p><p>
The final results are below. At 37 we have the left-handed neutrino's mass. At 38 we have the electron's mass (notice how the adjusted Yukawa coupling coupled with the new groundstate equals the standard Yukawa coupling coupled with the original groundstate. Both terms equal the electron's mass.)
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</p><p>
The forgoing exercise can be repeated for the other two families of leptons. To account for different masses, simply use different Yukawa couplings.
</p><p>
In conclusion, to give neutrinos mass requires a new Lagrange potential that can yield two field vectors: one for gauge bosons and one for leptons. Also, Yukawa couplings for massive neutrinos need to be added. When these requirements are met, the final solution shows that left-handed neutrinos do indeed have mass.
</p><p>
</p><p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-22707651501484459292022-02-07T18:56:00.026-08:002022-02-09T14:57:54.529-08:00Quantizing Gravity without the Graviton<p>Abstract:
</p><p>
This paper shows the connection between "dark energy" and gravity, the equivalence between matter and space, and how gravity works without the graviton or gravitational waves. It also suggests an alternate way to quantize gravity using smaller units--of mass, space and time--than the Planck units.
</p><p>
The same force acting on different masses will cause each mass to move at a different rate: F = ma, where m is mass and a is acceleration. However, the same "gravitational force" causes different masses to fall at the same rate. How can this be? Einstein proposed that when a body appears to be falling to earth, it is really at rest, and the earth is accelerating towards the body at a given rate. Thus the body's mass is irrelevant.
</p><p>Below is a diagram of Alice who is surrounded by four bodies, including Bob. Bob and the others appear to be either moving away from Alice or towards her, depending on how you follow the arrows. But if we go from left to right, we can think of Alice as the one who is moving away from the surrounding bodies. If we go from right to left, we can think of Alice as the one who is moving towards the surrounding bodies. As a consequence, the surrounding bodies can have any mass and the rate at which Alice and the surrounding bodies diverge or converge will be the same.
</p><p>
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</p><p>
Equations 1 and 2 below were derived from Einstein's field equations. Equation 3 was derived from a Friedmann equation where k is set to zero due to spacetime being flat at a large scale where the mass density (rho) is a small number and so is the curvature which is represented by the cosmological constant.
</p><p>
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</p><p>
At 4 and 5 we set up a couple of substitutions to be made at 7 and 8 below. Equation 6 shows how mass and space are equivalent. Divide any mass by the vacuum-mass density to get the equivalent volume. Equation 7 shows the universe must expand faster than light as the radius r tends to infinity; otherwise, light speed in a vacuum is not conserved! If both sides of the equation are multiplied by the universe's mass, then the universe's energy is conserved no matter how big or small the universe becomes.
</p><p>Also, the vacuum-mass density rho correlates with outward pressure, and that pressure is not diminished by an increase in distance r, so the outward pressure continues and so does the universe's expansion. At equation 8 we see that gravity looks similar to equation 7. As the speed of gravity increases, it is offset by the increase in spacetime curvature. As a result, the constant c and energy are conserved.
</p><p>
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</p><p>
Equation 9 below shows that gravitational waves are caused by gravity and angular frequency, so they cannot be the cause of gravity. In fact, if angular frequency is zero, there is still gravity but no gravitational waves.
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</p><p>
Now, let's imagine Alice and Bob are so far apart that Bob is moving away from Alice faster than light (see 10 below). Alice starts out with very little mass, but decides to go off her diet. As a result, she gains an enormous amount of mass. So much so that she becomes a black hole (see equation 11). The distance between her and Bob is the same but it is now less than Alice's Scharzschild radius. So are Alice and Bob still diverging or are they now converging?
</p><p>
We know that no signal, limited to light speed, ever reached Bob. This includes gravitons, gravitational waves, light, etc. At 12, Alice's mass is converted to it's equivalent space. Finally, the inequality at 13 shows that Bob is still moving faster than light if Alice is at rest, but Alice is no longer at rest--she's moving faster than Bob towards Bob. Thus Alice and Bob are converging as if a "gravitational force" is present.
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</p><p>
Take a proton and electron. Equation 14 below shows acceleration depends on both their respective charges, their masses and distance r. Clearly there is an information exchange between them. At equation 15 it's a different story: acceleration only depends on the mass of the proton and distance r. It is clear the two masses don't exchange information. It's as if the proton simply accelerates towards the electron.
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</p><p>
Now, let's consider Alice, Bob and Carl below. How fast Alice accelerates depends on the distance of the targets, the targets being Bob and Carl. Surely Alice needs to know the distance of each target so she can adjust her rate of acceleration accordingly?
</p><p>
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Consider the diagram below. The numbers in each section add up to the total volume of the 3X3X3 cube. At 17 the volume of each cube is divided by its square. If we multiply each term by the square of Hubble's parameter it becomes apparent that Alice is accelerating less than Bob and Bob is accelerating less than Carl. These three are diverging as if they are in an expanding universe.
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</p><p>
Now, let's add mass to the red section. Let's convert it to its equivalent volume which is 100 units. That brings the total to 101 units. Notice that the state of Bob's section (blue) and the state of Carl's section (green) do not change. Each have their previous volume. Also notice that the square areas do not change. This implies no information exchange between Alice, Bob and Carl. But now Alice is accelerating more than Bob and Bob is accelerating more than Carl. The three are converging is if they are in a gravitational field.
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</p><p>
The rate of acceleration depends on distance but, as demonstrated above, this does not imply an information exchange between the parties. To drive this point home, imagine we divide up our universe into volume cells, each cell expands at a given rate independently of every other cell. More cells cause a greater rate of expansion, but each cell has no clue how many cells an observer is looking at or what the other cells are doing. The cells exchange no information. Thus the rate of expansion is really up to the observer. Now, suppose you add more volume (cells) to the system without increasing the space. This was done when mass was added to Alice's section (red). Mass's equivalent volume doesn't change the distances between Alice, Bob, and Carl. The result is what we call gravitational acceleration (see equation 20 below). Again, each cell need not know the state of the others. The rate of acceleration depends on the distance the observer chooses to consider.
</p><p>
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</p><p>
Now let's turn to quantizing gravity. The popular choice is, of course, the graviton, but the graviton causes a major setback. If we assume the graviton is a real thing, surely we can come up with a reasonable estimate of how many gravitons are in the universe. For example, we could figure the total gravitational energy of the universe and divide that by the average energy of the graviton. At 21 below we plug in the entire mass of the universe to get the gravitational energy, but we don't get units of energy. Instead, we get velocity squared. One could argue that there is no gravitational energy, so there are no gravitons. Equation 22 is an attempt to counter this argument. It uses two masses which give an energy term, but how much gravitational energy there is depends on how much of the universe's total mass we assign to m and m'. Also, whatever gravitational-energy total we arrive at will be exceeded by a single black hole singularity and a single particle that have virtually zero distance between them.
</p><p>
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</p><p>
At equation 23 we switch to a smaller scale. We assume the gravitational energy produced by an electron and proton is nEg, where Eg is the energy of one graviton and n is the number of gravitons. Therefore two electron-proton pairs produce 2nEg or 2n gravitons? Not so fast. Since gravitons have energy, they also produce gravitons. At 24 they produce up to an infinite number of gravitons if the distance r limit is zero! So how many gravitons are in the universe? It depends on how you crunch the numbers.
</p><p>
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</p><p>
By contrast, it is much easier to estimate how many photons are in the universe, since photons don't produce photons. Electromagnetic energy is a function of charge and not energy or mass. So as long as photons don't have charge, they don't infinitely reproduce themselves. We can take the total luminous energy of the universe and divide it by a photon's average energy to get a reasonable estimate of how many photons there are.
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP89gtyReNYp8W0_vJTDCAJ7I89QIW8u6WFdyTWs3-zQfT6DaNN7aj6CxUwmWCJLsqFbDGcBi-mhp448dhkrYW-O66TFsjwml1LoESwi31Qxu7tlf0mdMUJ5XDlU7O_by_clfJy62ZkkpNh8xk97se8QoNWerueyvkOHNxidOd2w9mY3511Is7M33U/s478/TE10c.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="386" data-original-width="478" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP89gtyReNYp8W0_vJTDCAJ7I89QIW8u6WFdyTWs3-zQfT6DaNN7aj6CxUwmWCJLsqFbDGcBi-mhp448dhkrYW-O66TFsjwml1LoESwi31Qxu7tlf0mdMUJ5XDlU7O_by_clfJy62ZkkpNh8xk97se8QoNWerueyvkOHNxidOd2w9mY3511Is7M33U/s400/TE10c.png"/></a></div>
</p><p>
For the above reasons, the graviton is untenable. So then how should we quantize gravity? Gravity is a function of mass (defined as energy divided by light speed squared), space and time. Thus it would make sense to quantize these fundamental dimensions. We could ask, what is the shortest length or time, and, what is the smallest mass? The popular response is, the Planck length, the Planck time and the Planck mass, respectively. But are these really the smallest units? The Planck mass clearly is not the smallest mass. An electron mass is smaller. Also, the Scharzschild radius of an electron is much shorter than a Planck length. The time it takes for a photon to travel the shorter distance is less than the Planck time. So what are the smallest units? Here are the smallest units I have found so far. I call them the Hubble units:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNG8G43Adz5eO2cCEatcL4vBCqJcn9Ws0dAs9xqNQP_SMEZZbjM52xTwW0cVvwu_zW_1y2tK0dUIoo2YwvV_9O1vSu22PqRz06ToJ9xyHfqgtn9wIa7Jd0ntPNKr232g8s61UYNPYuLH-dDHSVoDk4RPqbWlhNy0mjQA6_CNPHv4wE0jN0BIw80rKa/s595/TE11.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="595" data-original-width="478" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNG8G43Adz5eO2cCEatcL4vBCqJcn9Ws0dAs9xqNQP_SMEZZbjM52xTwW0cVvwu_zW_1y2tK0dUIoo2YwvV_9O1vSu22PqRz06ToJ9xyHfqgtn9wIa7Jd0ntPNKr232g8s61UYNPYuLH-dDHSVoDk4RPqbWlhNy0mjQA6_CNPHv4wE0jN0BIw80rKa/s600/TE11.png"/></a></div>
</p><p>
We could take the Hubble length, for example, and quantize equation 13 as follows:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgcceAgLmwtC6UpGscWi022XhnDFZO2wRac_qcNdTWeXG8OseN6LA9l3f7AaaIOJjnO1K-Yj8Ykvr_CIzCXuD5oWcyf0QaYS1AxsQ5lLjiB1nbwxW-S7ZIXLRqSda0xULFawfmNR_IIfGtZmlS-tkSUHX6V1XV5ZJpgs3FIatPPT4s0fvou3gsZc52c=s453" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="219" data-original-width="453" src="https://blogger.googleusercontent.com/img/a/AVvXsEgcceAgLmwtC6UpGscWi022XhnDFZO2wRac_qcNdTWeXG8OseN6LA9l3f7AaaIOJjnO1K-Yj8Ykvr_CIzCXuD5oWcyf0QaYS1AxsQ5lLjiB1nbwxW-S7ZIXLRqSda0xULFawfmNR_IIfGtZmlS-tkSUHX6V1XV5ZJpgs3FIatPPT4s0fvou3gsZc52c=s400"/></a></div>
</p><p>
At equation 29 the first term has alpha Hubble lengths; the second term has beta Hubble lengths. The smallest rate of expansion is Hubble's parameter times one Hubble's length. Now, one probable objection to this scheme is space is chaotic and stochastic on the quantum scale, so how can we have such nice, neat units? The Hubble length, for example, could be an expectation value (average) of all the chaotic activity that may make up space. We can think of the Hubble units as perhaps the smallest average units that can be derived from fundamental constants and Hubble's parameter.
</p><p>
In conclusion, unlike the other fundamental interactions, gravity does not appear to have any means for bodies to communicate with each other, nor is communication necessary. Thus the graviton is unnecessary. Further, the graviton fails to conserve energy like its electromagnetic counterpart the photon. A better way to quantize gravity is to quantize space, time and/or mass.
</p><p>
Acknowledgments:
</p><p>
Amber Strunk. Education and Outreach Lead. LIGO Hanford Observatory.
</p><p>
Peter Laursen, Astrophysicist and science communicator at the Cosmic Dawn Center, University of Copenhagen.
</p><p>
References:
</p><p>
1. <a href="https://arxiv.org/pdf/2005.07211.pdf" target="blank">Parikh, Wilczek, Zahariade. 2020. The Noise of Gravitons. arxiv.org.</a>
</p><p>
2. <a href="https://blogs.umass.edu/grqft/files/2014/11/Feynman-gravitation.pdf" target="blank">Feynman, R.P. 07/03/1963. Quantum Theory of Gravitation. Acta Physica Polonica. Vol. XXIV.</a>
</p><p>
3. <a href="https://en.wikipedia.org/wiki/Graviton" target="blank">Graviton. Wikipedia.</a>
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4. <a arxiv.org="" gr-qc="" href="" https:="" pdf="" target="blank">Carlip, S. 12/1999. Aberration and the Speed of Gravity. arxiv.org.</a>
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5. <a href="https://arxiv.org/pdf/1005.3035.pdf" target="blank">Van Raamsdonk, M. 05/17/2010. Building up spacetime with quantum entanglement. arxiv.org.</a>
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6. <a href="https://arxiv.org/pdf/1404.4369.pdf" target="blank">Hanson, R.; Twitchen, D. J.; Markham, M.; Schouten, R. N.; Tiggelman, M. J.; Taminiau, T. H.; Blok, M. S.; Dam, S. B. van; Bernien, H. (2014-08-01). Unconditional quantum teleportation between distant solid-state quantum bits. Science. 345 (6196): 532–535.</a>
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7. <a href="https://en.wikipedia.org/wiki/Gravitational_wave" target="blank">Gravitational Wave. Wikipedia.</a>
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8. <a href="https://journals.aps.org/prd/pdf/10.1103/PhysRevD.101.063518" target="blank">de Rham, C., Tolley, A.J. 03/17/2020. Speed of Gravity. arxiv.org.</a>
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9. <a href="https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html" target="blank">Carroll, S.M. 12/1997. Lecture Notes on General Relativity. Enrico Fermi Institute.</a>
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10. <a href="https://www.gemarsh.com/wp-content/uploads/SpdGrav-3.pdf" target="blank">Marsh G.E., Nissim-Sabat. 3/18/1999. Comment on an article by Van Flandern on the speed of gravity. Physics Letters A Vol. 262, pp. 257-260 (1999)</a>
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11. <a href="https://www.mic.com/articles/19755/the-speed-of-gravity-why-einstein-was-wrong-and-newton-was-right" target="blank">Suede M. 11/29/2012. The Speed of Gravity: Why Einstein Was Wrong and Newton Was Right. Blog commentary re: Tom Van Flandern.</a>
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12. <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.161102" target="blank">Cornish N., Blas D., and Nardini, G. 10/18/2017. Bounding the Speed of Gravity with Gravitational Wave Observations. Phys. Rev. Lett. 119, 161102</a>
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13. <a href="https://www.gravitywarpdrive.com/Speed_of_Gravity.htm" target="blank">Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research
University of Maryland Physics Army Research Lab.</a>
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14. <a href="https://www.history.com/news/who-determined-the-speed-of-light" target="blank">Nix, E. 08/22/2018. Who Determined the Speed of Light. History.com.</a>
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15. <a href="https://en.wikipedia.org/wiki/Speed_of_gravity#:~:text=The%20effect%20of%20a%20finite,times%20the%20speed%20of%20light." target="blank">Speed of Gravity. Wikipedia.
</a></p><p>
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16. <a href="https://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury" target="blank">Tests of General Relativity. Wikipedia.
</a></p><p>
17. <a href="https://brilliant.org/wiki/gravitational-waves/#:~:text=These%20particles%20comprise%20gravitational%20waves%20in%20the%20same,Fourier%20analysis%2C%20and%20extensive%20knowledge%20of%20quantum%20mechanics." target="blank">Decross, M. et al. Gravitational Waves. Brilliant.com.</a>
</p><p>
18. Lawden, D.F. 1982. Introduction to Tensor Calculus, Relativity and Cosmology. Dover Publications, Inc.
</p><p>
19. <a href="https://arxiv.org/pdf/physics/0612019v6.pdf" target="blank">Stefanovich, E. V. 09/16/2018. A relativistic quantum theory of gravity. arxiv.org.</a>
</p><p>
20. <a href="https://en.wikipedia.org/wiki/Light-time_correction" target="blank">Light-time correction. Wikipedia.</a>
</p><p>
21. <a href="https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" target="blank">Liénard–Wiechert potential Wikipedia.</a>
</p><p>
22. <a href="https://arxiv.org/pdf/astro-ph/0311063.pdf" target="blank">Kopeikin, S. M. Fomalont, E. B. 03/27/2006. Aberration and the Fundamental Speed of Gravity in the Jovian Deflection
Experiment. arxiv.org. </a>
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23. <a href="https://arxiv.org/pdf/astro-ph/0303346.pdf" target="blank">Faber, J. A. 11/24/2018. The Speed of Gravity Has Not Been Measured From Time Delays. arxiv.org.</a>
</p><p>
24. <a href="https://arxiv.org/ftp/arxiv/papers/1108/1108.3761.pdf" target="blank"> Yin Zhu. 08/18/2011. Measurement of the Speed of Gravity. arxiv.org.</a>
</p><p>
25. <a href="https://www.macmillanlearning.com/studentresources/college/physics/tiplermodernphysics6e/more_sections/more_chapter_2_1-perihelion_of_mercurys_orbit.pdf" target="blank">Perihelion of Mercury’s Orbit. macmillanlearning.com.</a>
</p><p>
26. <a href="https://static1.squarespace.com/static/5852e579be659442a01f27b8/t/5cd46312eb393117fec080bb/1557422869040/Belenchia%5Bc.a.%5D_Wald_Giacomini_Castro-Ruiz_Brukner_Aspelmeyer_2019.pdf" target="blank">Belenchia A, Wald, R.M., Giacomini, F.,
Castro-Ruiz, E., Brukner, C., Aspelmeyer, M., 03/22/2019. Information Content of the Gravitational Field of a Quantum Superposition. Gravity Research Foundation.</a>
</p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-26106846908511999292022-01-21T15:02:00.240-08:002022-01-27T12:58:10.642-08:00The Faulty Premises of Black-hole Physics<p>ABSTRACT:</p>
<p>Re: the information paradox. When a theory contains a paradox, it is a clue that one or more premises the theory relies on are faulty. In this paper, we examine the premises, arguments and assumptions that are the foundation of black-hole physics.</p>
<p>All roads lead to Rome. Somewhere in Rome there is a particle trapped in a rotating potential well. Which road (or path) did the particle take to get to Rome? The Heisenberg Uncertainty Principle prevents us from simultaneously knowing the particle's position and momentum. That information could help us determine from whence the particle came. </p><p>Prior to being trapped in the potential well, the particle's momentum vector could have given us a sense of direction, but that vector is now rotated. The the information we need is lost. We can't determine the particle's previous positions and momenta, i.e., the road it took to Rome. This is one example of irreversibility that flies in the face of the claim that the state of a particle is always reversible. Yet, it is this claim or premise that leads to the black-hole information paradox.
</p><p>
According to the current paradigm, quantum information is conserved. With perfect knowledge of a particle's current state, it should be possible to trace it backwards and forwards in time. This principle would be violated if information were lost. When information enters a black hole we might assume the information is inside, but then black holes evaporate due to Hawking radiation, and the black hole's temperature is as follows:
</p><p>
<a href="https://4.bp.blogspot.com/-L4Q1HGZJg8I/XfBXzvqcQvI/AAAAAAAAGWw/XOlNhbkRPZwWLTcrGwUs9eCa_M9HdjKcQCLcBGAsYHQ/s1600/1.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-L4Q1HGZJg8I/XfBXzvqcQvI/AAAAAAAAGWw/XOlNhbkRPZwWLTcrGwUs9eCa_M9HdjKcQCLcBGAsYHQ/s1600/1.png" data-original-width="433" data-original-height="548" /></a>
</p><p>
As the black hole evaporates, its mass shrinks and its temperature increases. Take note that equation 1 fails to tell us what information went into the black hole, so looking at the final information (remaining mass, momentum, charge) pursuant to the no-hair theorem we can't extrapolate that data backwards and determine what information went into the black hole. It's irreversible. But as shown earlier, irreversibility is not unique to black holes.
</p><p>
Here is yet another example of irreversibility: take two systems, each containing various particles with either positive or negative charge. Coarse grain both systems to get a final result of negative charge for each. The final result is identical for each system; yet, what went into each system varied widely. The final information (negative charge) fails to tell us what went in. A unitary operator would erroneously give the same previous state for each system:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj7ylknMPGAMWNz2O5USAvZKXFQ5upc3ubHv-mTl2OZc4oqHKjilUOzLlL6lw82umC8ptzZPpSpBpYKVeEnU9it1w0vIUys4MNdtWYe2Oe0y2eMDIKdZvETZduuIPpxEkezyCGm4DKoXbUyBFHP9a1v0iO_W56Hv25mE3WCF1FGVKkuX9YI-3Zp8ht5=s526" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="119" data-original-width="526" src="https://blogger.googleusercontent.com/img/a/AVvXsEj7ylknMPGAMWNz2O5USAvZKXFQ5upc3ubHv-mTl2OZc4oqHKjilUOzLlL6lw82umC8ptzZPpSpBpYKVeEnU9it1w0vIUys4MNdtWYe2Oe0y2eMDIKdZvETZduuIPpxEkezyCGm4DKoXbUyBFHP9a1v0iO_W56Hv25mE3WCF1FGVKkuX9YI-3Zp8ht5=s600"/></a></div>
</p><p>
The premise that irreversibility can't and should never happen seems untenable.
</p><p>
Now let's shift our focus to Hawking radiation. If Hawking radiation does not exist, life would be easy. The second law of thermodynamics would never be violated if the black hole maintains or gains mass:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiaeHUERMOomXnuybgzaLlJMxrCkQSDroiO73dbHp9bzJDjk5ykLmICMjU9jcdzaM30fG8u31GS4s4TyCb_YEVvt_NyqtZyqFRjtgdFNgvJDWGx7307wWTqbHOdIdtt3Zqis4gKC8Q0wN88G_Nzkb5LxWL32365qAiXioVfqC5MWK1HtKMnFNg-PmzU=s551" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="551" data-original-width="426" src="https://blogger.googleusercontent.com/img/a/AVvXsEiaeHUERMOomXnuybgzaLlJMxrCkQSDroiO73dbHp9bzJDjk5ykLmICMjU9jcdzaM30fG8u31GS4s4TyCb_YEVvt_NyqtZyqFRjtgdFNgvJDWGx7307wWTqbHOdIdtt3Zqis4gKC8Q0wN88G_Nzkb5LxWL32365qAiXioVfqC5MWK1HtKMnFNg-PmzU=s600"/></a></div>
</p><p>
One argument used to justify the existence of Hawking radiation is, "Black holes have temperature; therefore, they radiate." Unfortunately, temperature is not the only variable that determines how much a body radiates. The Stephan-Boltzmann equation below shows that emissivity also plays a role:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEjLpFUpHwqPkIJHrR9eqfLDn4IlSiNELxZkbekQIhUKF0o2EQv0wOqixOPw2c73HW54wPx9teu-Hz5WLkXWkZw0Djwc4clLlyg7JNCPHziMoyPUvEzy0nZ-Jc86tVK7lnRuBGyZd8YcovmQFpGCRJ2VtaFXBqnOpI-bbO5raOZTiDcyDeQmKzs8nIiv=s568" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="568" data-original-width="501" src="https://blogger.googleusercontent.com/img/a/AVvXsEjLpFUpHwqPkIJHrR9eqfLDn4IlSiNELxZkbekQIhUKF0o2EQv0wOqixOPw2c73HW54wPx9teu-Hz5WLkXWkZw0Djwc4clLlyg7JNCPHziMoyPUvEzy0nZ-Jc86tVK7lnRuBGyZd8YcovmQFpGCRJ2VtaFXBqnOpI-bbO5raOZTiDcyDeQmKzs8nIiv=s600"/></a></div>
</p><p>
The black hole's temperature is irrelevant if the emissivity is zero. And why would the emissivity be zero? Because a black hole's gravity is so strong ... nothing can escape--not even light. Of course, at the quantum scale, there are likely to be events that defy classical physics, but we don't observe them at the macro scale. Apparently, they cancel each other and the classical events are what we observe. So it is not a stretch to assert that a black hole's emissivity is zero (or negative if you count the stuff falling in). </p><p> Even if the emissivity is positive, large black holes have a lower temperature than the surrounding environment, so they won't be evaporating any time soon. Small black holes that have a higher temperature probably don't exist, since at least three solar masses are required to create a black hole. Thus, it is no surprise that Hawking radiation has not been observed.
</p><p>
Hawking radiation may also be untenable if the following axiom is true: a system's total mass and temperature emerge from smaller constituents. So the question arises: can a quantum particle pair emerge from parameters such as temperature and total mass? Take note of the following equations:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgQmcW4mLZq3NC1T9tOxPEvBTdWxxlOI2w7jX-Uiu4jTfGYKVvR8Cq8pxnnqvnl_WxSOMR66NxRH848ALNuxrYSZJbXCsKi55QFha_AEvUkwimx236uMCC0QrNs3w4oSvUN88d5uTiGSo2D5PQNyJO-GBqiO77hNJUeaYo6j_5k4oDG6q2qakjtNP1k=s821" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="821" data-original-width="664" src="https://blogger.googleusercontent.com/img/a/AVvXsEgQmcW4mLZq3NC1T9tOxPEvBTdWxxlOI2w7jX-Uiu4jTfGYKVvR8Cq8pxnnqvnl_WxSOMR66NxRH848ALNuxrYSZJbXCsKi55QFha_AEvUkwimx236uMCC0QrNs3w4oSvUN88d5uTiGSo2D5PQNyJO-GBqiO77hNJUeaYo6j_5k4oDG6q2qakjtNP1k=s600"/></a></div>
</p><p>
Equation 4 is consistent with the axiom: a sum of quantum masses make up the total black-hole mass. But at equation 5 we have a pair of radiation particles that depend on the black hole's average temperature which, in turn, depends on the black hole's mass. To sort this out, imagine a single photon at the sun's surface. It's frequency is independent of the sun's average temperature; but the average temperature depends on the photon's frequency along with countless other photons and their frequencies. </p><p>To assert that a photon's frequency depends on the temperature is to turn the axiom on its head. Put aside such an assertion and imagine each Hawking particle with its own frequency and other quantum parameters. Together they could be constituents of the black-hole temperature just like the information that entered the black hole. Thus one might be tempted to argue that, at least quantitatively, Hawking radiation preserves the information that entered the black hole. At the very minimum, if black holes evaporate, mass is conserved, so we can justify the following:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhCUVGHMb_EH8HUUj8XMLd2dwagtb_pQlrdIyWWRuJXDpyq9QldE0Okdgwk7nUJrcdaX-6JALx6Qp4acMAtYEbyryth5OtShHIOIAkFR6NHqNuUAkdkNHli5xHrIUWKymR-a8ByCgAo5UoHAZ14v-gjH0ksanRDBwvFlBfIgxYwL80EjCN8sxqjS6Kw=s810" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="410" data-original-width="810" src="https://blogger.googleusercontent.com/img/a/AVvXsEhCUVGHMb_EH8HUUj8XMLd2dwagtb_pQlrdIyWWRuJXDpyq9QldE0Okdgwk7nUJrcdaX-6JALx6Qp4acMAtYEbyryth5OtShHIOIAkFR6NHqNuUAkdkNHli5xHrIUWKymR-a8ByCgAo5UoHAZ14v-gjH0ksanRDBwvFlBfIgxYwL80EjCN8sxqjS6Kw=s600"/></a></div>
</p><p>
Equations 5b and 5c confirm that what leaves the black hole is equivalent to what went in. This may be the inspiration behind another premise: Information is conserved. Really? If it is proportionate to energy, yes. But it is not. It is proportionate to the imaginary surface area A of the event horizon (see equation 3).
</p><p>
Adding a qubit of information to a black hole is done in the following manner:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhS8stn3b0Mnzsp30w_YJt5lU-tgtocCurgioerz4YEx1_C5k6jO8N28vjqL7-3XkUdcolmKuKpM-tj7OFRUcX5NwjcKXQp3SA8gx8IJGtx3nQx3rSfss8dwqCbALFzT92A2sffywJgg4j9LG3zapYYSHCbkYoGAlJpZZWNziW_MlN1SJyZS6ZCSD-F=s599" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="477" data-original-width="599" src="https://blogger.googleusercontent.com/img/a/AVvXsEhS8stn3b0Mnzsp30w_YJt5lU-tgtocCurgioerz4YEx1_C5k6jO8N28vjqL7-3XkUdcolmKuKpM-tj7OFRUcX5NwjcKXQp3SA8gx8IJGtx3nQx3rSfss8dwqCbALFzT92A2sffywJgg4j9LG3zapYYSHCbkYoGAlJpZZWNziW_MlN1SJyZS6ZCSD-F=s600"/></a></div>
</p><p>
The result of equations 6 and 7 is one Planck area is added to the surface if a photon with the same wavelength as the Scharzschild radius falls into a black hole. The premise here is the Planck length is the shortest possible length; however, the change-of-Scharzschild radius is shorter than the Planck length if the Scharzschild radius is large (see equation 6). And then there's the sloppy math. Here's the math done properly:
</p><p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhOfV-L1AuzHRnk1zidk4R6TV7dyBVw5Dh8Bs8hv4HJWWZd46ErJeFwzOqziBpcRiyAEWFeJrfZENACs1sCMszVHbJrKW6TUsu704p3fu7jQEFtzJuvEu5CmSee7z_2nRzYyRA62xGSJfoXSIPBG1nZqATZv-n_W3CBYhKf1aHUSW-GSXVyoTc-XHvs=s592" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="548" data-original-width="592" src="https://blogger.googleusercontent.com/img/a/AVvXsEhOfV-L1AuzHRnk1zidk4R6TV7dyBVw5Dh8Bs8hv4HJWWZd46ErJeFwzOqziBpcRiyAEWFeJrfZENACs1sCMszVHbJrKW6TUsu704p3fu7jQEFtzJuvEu5CmSee7z_2nRzYyRA62xGSJfoXSIPBG1nZqATZv-n_W3CBYhKf1aHUSW-GSXVyoTc-XHvs=s600"/></a></div>
</p><p>
As you can see, at equation 8, there is an extra term added to the Planck-area term, and both terms are multiplied by 8pi. At 9 and 10, The change-of-radius variable is made independent of the Scharzschild radius, and, why not? What are the odds that a particle falling into a black hole will have a wavelength equal to the the Scharzschild radius? Equation 8 shows that the amount of area the particle contributes can vary depending on the size of the Scharzschild radius. If information is proportionate to area, then the amount of information contributed will vary as well. Also, at equation 3, entropy is a function of area. Since entropy must either remain the same or increase, so must information. Information (proportionate to entropy or area) is not conserved!
</p><p>
In conclusion, it is not surprising there is an information paradox, but that paradox is just the tip of the iceberg. It is a clue that one or more premises are flawed. Theorists need to re-examine them. And while they're at it, check the math.
</p><p>
References:
</p><p>
1. Hossenfelder, Sabine (23 August 2019). "How do black holes destroy information and why is that a problem?". Back ReAction. Retrieved 23 November 2019.
</p><p>
2. Hawking, Stephen (1 August 1975). "Particle Creation by Black Holes" (PDF). Commun. Math. Phys. 43 (3): 199–220.
</p><p>
3. Susskind, Leonard (2008-07-07). The Black Hole War: My Battle with Stephen Hawking to Make the World Safe for Quantum Mechanics. Little, Brown and Company.
</p><p>
4. Black hole information paradox. Wikipedia.
</p><p>
5. Mathur, Samir D. 03/21/2021. The Elastic Vacuum. Gravity Research Foundation.
</p><p>
6. Chaisson, Eric. Astronomy Today. Englewood, NJ: Prentice Hall, 1993: 503GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-18730647372434514502021-08-08T12:54:00.003-07:002021-08-08T13:24:17.993-07:00Is Math Reality?<p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-AE7xAMpzbzQ/YRA1dkHVr0I/AAAAAAAAG9w/mljyoVZ5I8kNM2XB9QF85VsHgl1j0q-yACLcBGAsYHQ/s687/B1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="391" data-original-width="687" src="https://1.bp.blogspot.com/-AE7xAMpzbzQ/YRA1dkHVr0I/AAAAAAAAG9w/mljyoVZ5I8kNM2XB9QF85VsHgl1j0q-yACLcBGAsYHQ/s320/B1.png"/></a></div>
</p><p>
Is math reality or just a proposed description of reality which may falsely describe, or not accurately or fully describe reality? According to some, the fact that 2 + 2 = 4 is proof that math is reality and vice versa. But what about 2 + 2 = 5? One could argue that this is wrong and fits no reality and isn't real. The counter argument is 2 + 2 = 5 is real (like many string theories) in some parallel universe inside a megaverse.
</p><p>
Unfortunately, there is no empirical evidence of parallel universes--only mathematical evidence. Yet, it is argued that mathematical evidence is every bit as valid as empirical evidence. When these two types of evidence disagree, we simply invent a new universe, i.e., a reality where they do agree. Problem solved?
</p><p>
Hmmmm ..., if math is always right somewhere within a megaverse, wouldn't every crackpot idea be right somewhere within a megaverse? For example, our earth isn't flat, but surely there is some parallel universe that contains a flat earth? It looks as though the scientific method becomes a joke if we are allowed to move the goalpost (invent new unobserved universes) when the math or theory is wrong in this universe.
</p><p>
To claim that math is reality is to ignore infinities--that have never been observed or verified--and negative probabilities (predicted by quantum physics math) and all the instances where the math is wrong, or, where the math is approximate. When calculating the area of a circle, who knows and uses the exact value of pi? No one to my knowledge. If we don't know and can't determine the exact value of pi, then how can we say with confidence that it is real? Our math in general is riddled with approximations and error margins. From an empirical standpoint, we have yet to make a perfect measurement of the circumference of an ellipse or calculate it with a perfect value of pi; yet, some argue that math is reality and reality is math.
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com6tag:blogger.com,1999:blog-3628542335168231161.post-53245484107305702732021-05-21T14:25:00.002-07:002021-05-21T21:16:35.642-07:00The Relativistic Nature of the Expanding Universe<p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/ytS7u8mCxEY" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</p><p>
Our expanding universe is not thought to be relativistic. Technically, the galaxies receding away from us are at rest. The space between them is expanding and creating the impression that the galaxies are in motion. However, each galaxy can be treated as a clock, and, it is not likely that any of these clocks tell time at the same rate.
</p><p>
Using the diagram below as a reference, consider Alice and Bob living in a static universe. Alice lives in galaxy A and Bob lives in galaxy B. Each has a photon gun and shoots one photon per second at the other. The vertical red arrows represent the photons fired from one galaxy to the other. When the first photon from Alice (after traveling light years) finally arrives at galaxy B, Bob intercepts it and records it as a unit of time. Since Alice fired photons at regular intervals, Bob only has to wait one second for the next photon and another second for the next one after that and so forth. He receives one photon per second from Alice, and, for similar reasons, Alice receives one photon per second from Bob. This is how Alice and Bob keep track of each other's time rate.
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-ZipZ3cjPYBY/YKf_zS1nrjI/AAAAAAAAG6A/E1Ei3LFcQ08unIQL8jixIczP4NLVI_0IwCLcBGAsYHQ/s813/1.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="813" data-original-width="624" src="https://1.bp.blogspot.com/-ZipZ3cjPYBY/YKf_zS1nrjI/AAAAAAAAG6A/E1Ei3LFcQ08unIQL8jixIczP4NLVI_0IwCLcBGAsYHQ/s600/1.png"/></a></div>
</p><p>
You will note there are some horizontal red arrows at the top of the above diagram. Also located at galaxy A is Gertrude. She also has a photon gun. She fires one photon per second at Alice. Alice therefore intercepts and records one photon per second from Gertrude. Bob also receives photons in the same manner from Norbert who is also lacated at galaxy B (see horizontal red arrows at the diagram's bottom). This is how Alice and Bob keep track of their own time rates.
</p><p>
When the universe is static, both Alice and Bob receive fired photons from all directions at a rate of one photon per second. All clocks seem to agree and equations 1 and 2 provide the relevant math. But suppose the galaxies move away from each other at velocity v due to expanding space. The next diagram represents this scenario:
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-ZGXZdNYnAuk/YKf_6xajZ8I/AAAAAAAAG6E/nGp3F-u0Qrwv8wgnsmIuDcTVRyC_boAjQCLcBGAsYHQ/s818/2.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="818" data-original-width="539" src="https://1.bp.blogspot.com/-ZGXZdNYnAuk/YKf_6xajZ8I/AAAAAAAAG6E/nGp3F-u0Qrwv8wgnsmIuDcTVRyC_boAjQCLcBGAsYHQ/s600/2.png"/></a></div>
</p><p>
The vertical red arrows show that the photons take longer to reach their respective destinations and fewer photons are received per second compared to photons represented by the horizontal arrows. Since each photon is counted as a unit of time, Alice and Bob have the impression that the other's time rate is slower. Since Gertrude is located in the same galaxy as Alice, she isn't receding from Alice the way Bob is, so Alice sees no change in her time rate, since she receives the same number of photons (time units) from Gertrude. Ditto for Bob and Norbert at galaxy B.
</p><p>
If we do a little algebra we can derive the Lorentz equation from equation 3 above. The final result is equation 10:
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-O-Y3H-17Trk/YKgABQcPD7I/AAAAAAAAG6I/bdbwc0Jrd8sx9AZE98JuNmsj_gB3o2DvACLcBGAsYHQ/s748/3.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="748" data-original-width="538" src="https://1.bp.blogspot.com/-O-Y3H-17Trk/YKgABQcPD7I/AAAAAAAAG6I/bdbwc0Jrd8sx9AZE98JuNmsj_gB3o2DvACLcBGAsYHQ/s600/3.png"/></a></div>
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-XFOd2P3C4vc/YKgAL4bvlcI/AAAAAAAAG6U/2MZR6vMiWwUjfzF6rQZhDCslFs9KCH5JwCLcBGAsYHQ/s711/4.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="600" data-original-height="483" data-original-width="711" src="https://1.bp.blogspot.com/-XFOd2P3C4vc/YKgAL4bvlcI/AAAAAAAAG6U/2MZR6vMiWwUjfzF6rQZhDCslFs9KCH5JwCLcBGAsYHQ/s600/4.png"/></a></div>
</p><p>
Equation 10 shows each galaxy's proper time (t') shrinking as they accelerate further apart. It does not matter if the galaxies are technically at rest, since the space inbetween and the photons are not at rest. If the space is at rest, here's the result after firing a photon gun for five seconds:
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-rYGGrSFlhzE/YKgcugRkJQI/AAAAAAAAG7I/SNrIU2NuMJge3VguZ-VVPuZSdojOnkp1ACLcBGAsYHQ/s354/a.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="290" data-original-width="354" src="https://1.bp.blogspot.com/-rYGGrSFlhzE/YKgcugRkJQI/AAAAAAAAG7I/SNrIU2NuMJge3VguZ-VVPuZSdojOnkp1ACLcBGAsYHQ/s320/a.png"/></a></div>
</p><p>
Notice the steady stream of photons. This is where t' = t. The next diagram shows what happens if space expands at a steady rate of v. The photons still have a steady rate but the interval between them has increased, so they are not counted as frequently. Time t' is less than t.
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-Fm-D2gyvAmA/YKgddsfm7UI/AAAAAAAAG7g/GWKN2CVGYXAWfGSeUQyHAlySFWrdptXGQCLcBGAsYHQ/s477/6.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="400" data-original-height="477" data-original-width="409" src="https://1.bp.blogspot.com/-Fm-D2gyvAmA/YKgddsfm7UI/AAAAAAAAG7g/GWKN2CVGYXAWfGSeUQyHAlySFWrdptXGQCLcBGAsYHQ/s400/6.png"/></a></div>
</p><p>
Finally, the next diagram is the most realistic, since the expansion of space is accelerating. Here the the interval between photons continues to grow and they are counted less and less frequently. Time t' is shrinking.
</p><p>
<div class="separator" style="clear: both;"><a href="https://1.bp.blogspot.com/-t4zx5VkmGCw/YKgd9tPrp0I/AAAAAAAAG7w/m8fdLtpb9zw71Ma4DMtr6DCowuU5Ng9oQCLcBGAsYHQ/s610/7.png" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" height="600" data-original-height="610" data-original-width="462" src="https://1.bp.blogspot.com/-t4zx5VkmGCw/YKgd9tPrp0I/AAAAAAAAG7w/m8fdLtpb9zw71Ma4DMtr6DCowuU5Ng9oQCLcBGAsYHQ/s600/7.png"/></a></div>
</p><p>
When Alice and Bob are far enough apart, neither will receive any photons from the other. Time t' will be zero. Thus the expanding universe is relativistic if one keeps track of the various time rates at different distances.
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-92205220356883985122020-08-14T16:16:00.042-07:002020-11-21T10:56:43.001-08:00The Beautiful Destruction of the Graviton
</div><br /><p><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/B4XzLDM3Py8" width="560"></iframe></p><p>
<b><!--Why k is constant:
<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-aJJI5e9qp5k/X1WPGEWpHPI/AAAAAAAAGp8/2M-fBaMy26c4svqHTifJpCAYZ540O0jzACLcBGAsYHQ/s1953/Why%2Bk%2Bis%2Bconstant.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="880" data-original-width="1953" src="https://1.bp.blogspot.com/-aJJI5e9qp5k/X1WPGEWpHPI/AAAAAAAAGp8/2M-fBaMy26c4svqHTifJpCAYZ540O0jzACLcBGAsYHQ/s320/Why%2Bk%2Bis%2Bconstant.png" width="320" /></a>
"It is well known that if a charged source moves at a constant velocity, the electric field experienced by a test particle points toward the source’s “instantaneous” position rather than its retarded position."--S. Carlip
"Why do photons from the Sun travel in directions that are not parallel to the direction of Earth’s gravitational acceleration toward the Sun?"--Tom Van Flandern
Caveat: The sun does not orbit the earth, so observers on the earth see the sun in its instant position 8.3 minutes ago. However, the earth's gravity on the sun would be a vector pointing to where the earth was, not where it is. Perhelian of mercury is caused by curvature, not light-time delays.
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<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-unejZS7eEtw/XzGdbyF6UOI/AAAAAAAAGn0/naGEpVtgWBsYy4sUzkRDpTYnCeAKmPXhwCLcBGAsYHQ/s1327/Waves2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="819" data-original-width="1327" src="https://1.bp.blogspot.com/-unejZS7eEtw/XzGdbyF6UOI/AAAAAAAAGn0/naGEpVtgWBsYy4sUzkRDpTYnCeAKmPXhwCLcBGAsYHQ/s640/Waves2.png" width="640" /></a></div><p>
</b><p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Frt1iavEuDM/XzNNQj85LwI/AAAAAAAAGoA/4cyJ5jjyw7Aa_QAxgixZLhOj-8dv35-2ACLcBGAsYHQ/s1091/Waves3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="855" data-original-width="1091" src="https://1.bp.blogspot.com/-Frt1iavEuDM/XzNNQj85LwI/AAAAAAAAGoA/4cyJ5jjyw7Aa_QAxgixZLhOj-8dv35-2ACLcBGAsYHQ/s640/Waves3.png" width="640" /></a></div>-->
</b></p><p>
ABSTRACT:
</p><p>
In his paper titled "Aberration and the Speed of Gravity," S. Carlip argues that gravity propagates at light speed, and, its "action at a distance" and the lack of observed aberration is canceled by velocity dependent interactions. However, the underlying assumption of his thesis is that gravity is caused by gravitational radiation propagating at light speed. Another assumption held by much of the physics community is the quantization of gravitational waves will lead to a spin-2 massless particle known as the graviton. In this paper, I show why gravitational waves and gravitons are not the root cause of gravity. Gravity emerges from an entangled relationship between spacetime and matter.
</p><p>
Modern physics has two conflicting ideas: 1. gravity propagates at light speed, and 2. the equivalence principle. Why are these two ideas in conflict? The first proposes that gravity works in the following manner: a person holds a pen in his hand and drops it. Before it hits the floor, however, the floor must emit gravitons that propagate at c to create a field of gravity, so the pen can receive the gravitational information; otherwise, the pen won't fall.
</p><p>
The second idea is often set forth using a thought experiment where the person holding the pen is in a spaceship. The thrust of the engines cause the floor to accelerate toward the pen when the pen is dropped; otherwise, the pen would float freely in space and never make contact with the floor. In this scenario, no gravitons or gravitational field are needed. The floor is on a collision course with the pen and does not need to send a signal to the pen to let it know it's coming. According to Einstein, this is indistinguishable from gravity. Therefore, this great idea and the one that precedes it create a paradox: the first idea implies a force is causing the pen to fall, so a force-carrying particle is necessary. The second idea implies there is no force. <br />
</p><p>
That begs the question: does gravity require gravitons? Let's examine what may be a source of gravitons and strong evidence that gravity's velocity is c: gravitational waves. Equation 2 below is a gravitational-wave equation:
</p><p>
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<p></p><p>
From equation 2 we derive equation 3 which emphasizes that c is a component of rest-mass energy and not propagation speed.
</p><p>
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<p></p><p>
Does gravity exist if there are no gravitational waves? To find out, we take angular frequency to zero. The time (t') it takes for no waves to propagate a distance r is also zero. The final result is equation 5:
</p><p>
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<p></p><p>
Where there are no gravitational waves there is zero angular frequency and zero strain measured at distance r, but on the left side of equation 5 we see Newtonian gravity is not zero. Thus, the magnitude of gravitational acceleration does not depend on the magnitude of gravitational waves nor their quanta. Further, a zero time delay (t') implies action at a distance.
</p><p>
A comparison between electric waves and gravitational waves reveals why photons are observable and gravitons are not. The wave equation i below represents an electric field (photons) propagating at c. Equations ii through iv demonstrate how the removal of the electric field (photons) leads to no electromagnetic force:
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<p></p><p>
By contrast, if the gravitational field (gravitons) is removed, Newtonian gravity still exists:
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<p></p><p>
What's been shown so far is not surprising when you consider problems surrounding the hypothetical graviton:</p><p>
1. Since electromagnetic force is around 10<sup>37</sup> times greater than gravity, one might imagine a graviton with 10<sup>37</sup> times the Compton wavelength of an electron. The graviton's wavelength would span much of the known universe! Not exactly quantum scale.</p><p>
2. Or, one could imagine one graviton (with the same Compton wavelength as an electron and same speed as a photon) per 10<sup>37</sup> photons. If it takes 10<sup>37</sup> photons one second to interact with X number of atoms, the graviton would take 10<sup>37</sup> seconds to interact with X number of atoms--many orders of magnitude longer than the age of the universe! Further, each graviton interaction would have the same strength as a photon interaction or electromagnetic force.</p><p>
3. One expects gravitons to spread out to form a field. It is not clear how the gravitons of a black hole can escape each other (if they have a light-speed limit) and not clump together due to mutual attraction (caused by their spin, angular momentum, mass-energy equivalence, etc.).</p><p>
4. Gravitational wave wavelengths are inconsistent with hypothetical graviton wavelengths.</p><p>
5. There's a renormalization problem.</p><p>
6. The graviton has never been observed.</p><p>
Assuming gravitons are not the root cause of gravity, what exactly is? If we begin with the spacetime metric (equation 6), we can derive equations 9 and 10 below:
</p><p>
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<p></p><p>
Imagine, for the sake of argument, there is a graviton propagating at velocity c. Equation 9 shows if the graviton's energy (E) changes, the spacetime must also change instantaneously; otherwise the constant c would have a different value during the time it takes the graviton to emit another graviton which then transports information to surrounding spacetime. In other words, if the speed of gravity is limited to c, there would be a time lag where c is no longer c! The same holds for Planck's reduced constant at equation 10. The very constants physics relies on would fail to be constant if gravity is required to propagate an information-carrying particle no faster than c.
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A careful examination of equation 9 reveals the graviton's energy, when divided by Planck's constant, has the same dimension as frequency, and the spacetime has the same dimension as wavelength. Frequency and wavelength have an entangled relationship. If you measure the value of one, you instantaneously know the value of the other.
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In the case of our graviton, a change in its energy instantaneously updates its surrounding spacetime. Our graviton does not need to emit a graviton--and neither does any particle, planet, star, or black hole.
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Thus, if the graviton is ever discovered, it is not the root cause of gravity. Gravity is the result of an entangled relationship between matter and spacetime. Matter has a certain energy and moves in certain ways because of a certain configuration of spacetime, and spacetime has a certain configuration because matter has a certain energy and moves in certain ways. Like frequency and wavelength, one does not exist without the other. This new hypothesis is consistent with "action at a distance" observations but inconsistent with the highly contraversial Jovian deflection experiment (see endnote 22).
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<!--But what about gravitational waves? Surely they are strong evidence of gravitons. Not really. The Compton wavelength for gravitational waves is many orders of magnitude shorter than the hypothetical graviton's.
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Then there's the wavelength problem. If a typical graviton is 10^40 times weaker than a photon that can create a positron-electron pair, its momentum would be 10^40 times weaker, so its Compton wavelength would be 10^40 times the Bohr radius or many of orders of magnitude longer than the visible universe! Not exactly quantum scale.
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Of course the wavelength problem is not a problem if the graviton had the momentum of say, an electron. That would put its wavelength on the Bohr-radius scale. Unfortunately, one such graviton would only be produced per 10^40 photons or 10^40 electron-positron pairs or matter of equivalent energy. It would no doubt be a very busy graviton, since it has to interact with 10^40 other particles to ensure a gravitational field between them.
"The most amazing thing I was taught as a graduate student of celestial mechanics at Yale in the 1960s was that all gravitational interactions between bodies in all dynamical systems had to be taken as instantaneous."--Van Flandern.
Astronomical observers use light-time corrections, so gravity must be instantaneous for all such observers to agree that the constants are constant. It's the true state of an object versus its apparent state. If time delays were used, Newton's formula and the field equations would include r/c.
I show that aberration in general relativity is almost exactly canceled by
velocity-dependent interactions, permitting cg = c. This cancellation is dictated by conservation laws and the quadrupole nature of gravitational radiation.--S. Carlip
The assumption here is gravity is caused by gravitational radiation. Let's assume a graviton is propagating at velocity c. A veocity c is frequency X wavelength. The frequency does not cause the wavelength and vice versa--one simply can't exist without the other. By the same token, certain particles moving certain ways can't do so without a certain spacetime and a certain spacetime cannot exist without certain particles moving certain ways. So the graviton need not send a signal to spacetime or vice versa. The same can be said for all other particles. Thus, the graviton is unnessary since there is no cause and effect and no information that needs to be exchanged between matter and spacetime. If gravity appears instantaneous, it's because it is. Carlip fails to address the black hole concern of Van Flandern or a change in mass.
"It is certainly true, although perhaps not widely enough appreciated, that observations
are incompatible with Newtonian gravity with a light-speed propagation delay added in [3,4]."--Carlip
Although gravity propagates at the speed of light in
general relativity, the expected aberration is almost exactly canceled by velocity-dependent terms in the interaction.--Carlip
But photons propagate at the speed of light and Carlip fails to address why their aberrations are not cancelled by velocity dependant terms.
"Around 1676, Danish astronomer Ole Roemer became the first person to prove that light travels at a finite speed. He studied Jupiter’s moons and noted that their eclipses took place sooner than predicted when Earth was nearer to Jupiter and happened later when Earth was farther away from Jupiter. Roemer reasoned this was the result of light moving at a finite speed; it took longer to make it to Earth when Jupiter was a greater distance away."--Elizabeth Nix
We make observations thet lead to the conclusion that light has finite speed. But what observations lead to gravity has finite speed apart from gravitational waves?
Intantaneous gravity does not violate causality in that frequency does not cause wavelength or vice versa--one value can't be had without the other. A perceived violation of causality is caused by light-time delay (the time it takes light to reach each observer). If all observers make light-time adjustments, then all will agree that causality is not violated.
How can black holes have gravity when nothing can get out because escape speed is greater than the speed of light?--Van Flandern
There is no cause to doubt that photons arriving now from the Sun left 8.3 minutes ago, and arrive at Earth from the direction against the sky that the Sun occupied that long ago. But the analogous situation for gravity is less obvious, and we must always be careful not to mix in the consequences of light propagation delays. Another way, besides aberration, to represent what gravity is doing is to measure the acceleration vector for the Earth’s motion, and ask if it is parallel to the direction of the arriving photons. If it is, that would argue that gravity propagated to Earth with the same speed as light; and conversely.--Van Flandern-->
<!--Abstract: Excellent! Today I derived an equation that totally explains why gravity is instantaneous and gravitational waves propagate at the speed of light. It turns out there are no gravitons. Gravitational acceleration, at the quantum level, is a component of the wavelength of a gravitational wave, so a change in energy instantaneously changes the wavelength and its gravity component. No information is transferred, the system is entangled. Thus gravity really does work like the equivalence principle and no gravitons are necessary. What's really really cool is my equation's solution agrees with the LIGO data!-->
<!--Dont forget to add that gravitons should have no energy like photons have no charge and should produce particle pairs.-->
<!--"One could again
try to formulate an alternative theory in which gravity propagated instantaneously, but, as
in electromagnetism, only at the expense of “deunifying” the field equations and treating
gravity and gravitational radiation as independent phenomena."--S. Carlip-->
<!--In 1998 Van Flandern wrote a paper[32] asserting that astronomical observations imply that gravity propagates at least twenty billion times faster than light, or even infinitely fast.-->
<!--"One could again
try to formulate an alternative theory in which gravity propagated instantaneously, but, as
in electromagnetism, only at the expense of “deunifying” the field equations and treating
gravity and gravitational radiation as independent phenomena."--S. Carlip"
Carlip demonstrates that velocity-dependant interactions cancel aberation in general relativity, but also electromagnetism (EM). Unfortunately he fails to address the following: "As viewed from the Earth’s frame, light from the Sun has aberration. Light requires about 8.3 minutes to arrive from the Sun, during which time the Sun seems to move through an angle of 20 arc seconds. The arriving sunlight shows us where the Sun was 8.3 minutes ago."--T. Van Flandern. Caveat: light is visible, g-waves are not, so the two are incomparable to an observer dependent on light. If we could not see light (and used a different medium), would all motion appear instantaneously driven?
The first attempts to measure the speed of light were made in the 17th century. During this time, no one knew if light was composed of waves, as Christiaan Huygens thought, or particles, as Isaac Newton believed, and no one knew if the speed of light was infinite.
Here's a direct observation that caused Roemer to suspect the speed of light was finite. Were there any direct observations that made an early scientist wonder if the speed of gravity was finite?
"Laplace's c must be very large. As is now known, it may be considered to be infinite in the limit of straight-line motion, since as a static influence it is instantaneous at distance when seen by observers at constant transverse velocity. For orbits in which velocity (direction of speed) changes slowly, it is almost infinite."--Wikipedia
"In September 2002, Sergei Kopeikin and Edward Fomalont announced that they had measured the speed of gravity indirectly, using their data from VLBI measurement of the retarded position of Jupiter on its orbit during Jupiter's transit across the line-of-sight of the bright radio source quasar QSO J0842+1835. Kopeikin and Fomalont concluded that the speed of gravity is between 0.8 and 1.2 times the speed of light, which would be fully consistent with the theoretical prediction of general relativity that the speed of gravity is exactly the same as the speed of light.[22]"--Wikipedia
Several physicists, including Clifford M. Will and Steve Carlip, have criticized these claims on the grounds that they have allegedly misinterpreted the results of their measurements. Notably, prior to the actual transit, Hideki Asada in a paper to the Astrophysical Journal Letters theorized that the proposed experiment was essentially a roundabout confirmation of the speed of light instead of the speed of gravity.[23]
Light speed confirmation bias suspected. Prior to Einstein, did anyone suspect gravity speed was not infinite?
"Isaac Newton's formulation of a gravitational force law requires that each particle with mass respond instantaneously to every other particle with mass irrespective of the distance between them. In modern terms, Newtonian gravitation is described by the Poisson equation, according to which, when the mass distribution of a system changes, its gravitational field instantaneously adjusts. Therefore, the theory assumes the speed of gravity to be infinite. This assumption was adequate to account for all phenomena with the observational accuracy of that time. It was not until the 19th century that an anomaly in astronomical observations which could not be reconciled with the Newtonian gravitational model of instantaneous action was noted: the French astronomer Urbain Le Verrier determined in 1859 that the elliptical orbit of Mercury precesses at a significantly different rate from that predicted by Newtonian theory.[4]"--wikipedia
In general relativity, this remaining precession, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime. Einstein showed that general relativity[3] agrees closely with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity.--wikipedia
Carlip addresses why gravity appears to be instantaneous, but fails to adequately address why "velocity dependant terms" don't also cancel aberation caused by light. According to his paper, velocity dependant interactions cancel EM aberation, but this fails to address the following:-->
<!--1st Postulate of Relativity
The laws of physics are the same for all uniformly moving observers. ... Any uniformly moving observer can consider themselves to be "at rest".
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How does a graviton escape a black hole? </p><p>A graviton is a spin-2 particle, so it has angular momentum and energy. A black hole also has energy, and, since energy seems to attract energy, it is not clear how a graviton escapes a black hole and carries its information to the far reaches of the universe. </p><p>To make matters worse, the average photon has around 10^40 times the energy of a graviton but can't escape the black hole. Therefore, any graviton hypothesis would need to explain why the graviton is exempt from the black hole's gravitational attraction, or why gravitons don't attract gravitons. What prevents gravitons from clumping together? To make a consistently isotropic gravitational field, they need to diverge evenly in accordance to the inverse square law.
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It is no surprise to this author the graviton has never been observed, discovered or pinned down. It probably doesn't exist. But what about gravitational waves? Surely they are strong evidence of gravitons. Not really. The Compton wavelength for gravitational waves is many orders of magnitude shorter than the hypothetical graviton's. </p><p>
<!--correct this:-->
<!--Gravitational waves are produced by any object accelerating in space-time, including the Earth’s rotation around the Sun.
There is also an anti-correlation between a gravitational wave's strength and gravity's strength. The strongest gravitational waves come from black holes so far away the average person is not aware of their gravitational impact. By contrast, the average person can easily detect Earth's gravity, but Earth's gravitational waves are grossly insignificant. So gravitons, if they exist, are not the quanta of gravitational waves.
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Given the problems I've outline above, things aren't looking good for the graviton. But surely there must be such a particle. How can gravity work on the quantum scale or any scale without it? That's what we shall cover below. First, let's define some variables:
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<a href="https://1.bp.blogspot.com/-eduJwrBzuk4/XxOPaoJfaoI/AAAAAAAAGmQ/C_e_ViW3puUDoyTxbUUjTLQH1ptnGcPSACLcBGAsYHQ/s1600/Vars.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-eduJwrBzuk4/XxOPaoJfaoI/AAAAAAAAGmQ/C_e_ViW3puUDoyTxbUUjTLQH1ptnGcPSACLcBGAsYHQ/s1600/Vars.png" data-original-width="353" data-original-height="287" /></a>
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<a href="https://2.bp.blogspot.com/-b5MKK-eYvSU/XxOU_wyIZwI/AAAAAAAAGmc/5SCoejkmhjEGuHClAFG3ZZJfbHXThyQDgCLcBGAsYHQ/s1600/1.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-b5MKK-eYvSU/XxOU_wyIZwI/AAAAAAAAGmc/5SCoejkmhjEGuHClAFG3ZZJfbHXThyQDgCLcBGAsYHQ/s1600/1.png" data-original-width="320" data-original-height="459" /></a>
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<a href="https://4.bp.blogspot.com/--MzzuGLjKi0/XxOVDNfzmRI/AAAAAAAAGmg/bHTA1KGoQUIoEQMXUOSh-YtVSsFbYaSqgCLcBGAsYHQ/s1600/2.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/--MzzuGLjKi0/XxOVDNfzmRI/AAAAAAAAGmg/bHTA1KGoQUIoEQMXUOSh-YtVSsFbYaSqgCLcBGAsYHQ/s1600/2.png" data-original-width="387" data-original-height="307" /></a>
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<a href="https://1.bp.blogspot.com/-yZjtfwqjBRM/XxOVGgvGFlI/AAAAAAAAGmk/o5bfHLo60FosgPe0GVxhZfelQK0ELRKfACLcBGAsYHQ/s1600/3.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-yZjtfwqjBRM/XxOVGgvGFlI/AAAAAAAAGmk/o5bfHLo60FosgPe0GVxhZfelQK0ELRKfACLcBGAsYHQ/s1600/3.png" data-original-width="795" data-original-height="817" /></a>
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<a href="https://4.bp.blogspot.com/-y0me3gYZHz8/XxOVJ30eawI/AAAAAAAAGmo/dJad7bSsuJwtyKTDKg7QbTfaEP8a1FG1gCLcBGAsYHQ/s1600/4.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-y0me3gYZHz8/XxOVJ30eawI/AAAAAAAAGmo/dJad7bSsuJwtyKTDKg7QbTfaEP8a1FG1gCLcBGAsYHQ/s1600/4.png" data-original-width="687" data-original-height="794" /></a>
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<a href="https://2.bp.blogspot.com/-IQKNUB3i6zQ/XxOVNPqTnAI/AAAAAAAAGms/IdkWaayyVKkP3TXxILMfrQDIfCZNzInDQCLcBGAsYHQ/s1600/5.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-IQKNUB3i6zQ/XxOVNPqTnAI/AAAAAAAAGms/IdkWaayyVKkP3TXxILMfrQDIfCZNzInDQCLcBGAsYHQ/s1600/5.png" data-original-width="457" data-original-height="372" /></a>
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<!--Address Gaussian noise
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<a href="https://3.bp.blogspot.com/-e6CLCyqfax0/XxOVQDX47dI/AAAAAAAAGm0/zP3GTILDeC0iGoiqa_PUAMOXcOPcWhOuwCLcBGAsYHQ/s1600/6.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-e6CLCyqfax0/XxOVQDX47dI/AAAAAAAAGm0/zP3GTILDeC0iGoiqa_PUAMOXcOPcWhOuwCLcBGAsYHQ/s1600/6.png" data-original-width="457" data-original-height="343" /></a>
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<a href="https://3.bp.blogspot.com/-13eJNjf4b7A/XxOVUiOqdqI/AAAAAAAAGm4/INyVoBkcDQoi7Ayfl60Rsq7NzZKfDeZOwCLcBGAsYHQ/s1600/7.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-13eJNjf4b7A/XxOVUiOqdqI/AAAAAAAAGm4/INyVoBkcDQoi7Ayfl60Rsq7NzZKfDeZOwCLcBGAsYHQ/s1600/7.png" data-original-width="639" data-original-height="313" /></a>$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$START:-->
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Acknowledgements:
</p><p>
Amber Strunk. Education and Outreach Lead. LIGO Hanford Observatory.
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References:
</p><p>
1. <a href="https://arxiv.org/pdf/2005.07211.pdf" target="blank">Parikh, Wilczek, Zahariade. 2020. The Noise of Gravitons. arxiv.org.</a>
</p><p>
2. <a href="https://blogs.umass.edu/grqft/files/2014/11/Feynman-gravitation.pdf" target="blank">Feynman, R.P. 07/03/1963. Quantum Theory of Gravitation. Acta Physica Polonica. Vol. XXIV.</a>
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3. <a href="https://en.wikipedia.org/wiki/Graviton" target="blank">Graviton. Wikipedia.</a>
</p><p>
4. <a arxiv.org="" gr-qc="" href="" https:="" pdf="" target="blank">Carlip, S. 12/1999. Aberration and the Speed of Gravity. arxiv.org.</a>
</p><p>
5. <a href="https://arxiv.org/pdf/1005.3035.pdf" target="blank">Van Raamsdonk, M. 05/17/2010. Building up spacetime with quantum entanglement. arxiv.org.</a>
</p><p>
6. <a href="https://arxiv.org/pdf/1404.4369.pdf" target="blank">Hanson, R.; Twitchen, D. J.; Markham, M.; Schouten, R. N.; Tiggelman, M. J.; Taminiau, T. H.; Blok, M. S.; Dam, S. B. van; Bernien, H. (2014-08-01). Unconditional quantum teleportation between distant solid-state quantum bits. Science. 345 (6196): 532–535.</a>
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7. <a href="https://en.wikipedia.org/wiki/Gravitational_wave" target="blank">Gravitational Wave. Wikipedia.</a>
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8. <a href="https://journals.aps.org/prd/pdf/10.1103/PhysRevD.101.063518" target="blank">de Rham, C., Tolley, A.J. 03/17/2020. Speed of Gravity. arxiv.org.</a>
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9. <a href="https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html" target="blank">Carroll, S.M. 12/1997. Lecture Notes on General Relativity. Enrico Fermi Institute.</a>
</p><p>
10. <a href="https://www.gemarsh.com/wp-content/uploads/SpdGrav-3.pdf" target="blank">Marsh G.E., Nissim-Sabat. 3/18/1999. Comment on an article by Van Flandern on the speed of gravity. Physics Letters A Vol. 262, pp. 257-260 (1999)</a>
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11. <a href="https://www.mic.com/articles/19755/the-speed-of-gravity-why-einstein-was-wrong-and-newton-was-right" target="blank">Suede M. 11/29/2012. The Speed of Gravity: Why Einstein Was Wrong and Newton Was Right. Blog commentary re: Tom Van Flandern.</a>
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12. <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.161102" target="blank">Cornish N., Blas D., and Nardini, G. 10/18/2017. Bounding the Speed of Gravity with Gravitational Wave Observations. Phys. Rev. Lett. 119, 161102</a>
</p><p>
13. <a href="https://www.gravitywarpdrive.com/Speed_of_Gravity.htm" target="blank">Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research
University of Maryland Physics Army Research Lab.</a>
</p><p>
14. <a href="https://www.history.com/news/who-determined-the-speed-of-light" target="blank">Nix, E. 08/22/2018. Who Determined the Speed of Light. History.com.</a>
</p><p>
15. <a href="https://en.wikipedia.org/wiki/Speed_of_gravity#:~:text=The%20effect%20of%20a%20finite,times%20the%20speed%20of%20light." target="blank">Speed of Gravity. Wikipedia.
</a></p><p>
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16. <a href="https://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury" target="blank">Tests of General Relativity. Wikipedia.
</a></p><p>
17. <a href="https://brilliant.org/wiki/gravitational-waves/#:~:text=These%20particles%20comprise%20gravitational%20waves%20in%20the%20same,Fourier%20analysis%2C%20and%20extensive%20knowledge%20of%20quantum%20mechanics." target="blank">Decross, M. et al. Gravitational Waves. Brilliant.com.</a>
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18. Lawden, D.F. 1982. Introduction to Tensor Calculus, Relativity and Cosmology. Dover Publications, Inc.
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19. <a href="https://arxiv.org/pdf/physics/0612019v6.pdf" target="blank">Stefanovich, E. V. 09/16/2018. A relativistic quantum theory of gravity. arxiv.org.</a>
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20. <a href="https://en.wikipedia.org/wiki/Light-time_correction" target="blank">Light-time correction. Wikipedia.</a>
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21. <a href="https://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" target="blank">Liénard–Wiechert potential Wikipedia.</a>
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22. <a href="https://arxiv.org/pdf/astro-ph/0311063.pdf" target="blank">Kopeikin, S. M. Fomalont, E. B. 03/27/2006. Aberration and the Fundamental Speed of Gravity in the Jovian Deflection
Experiment. arxiv.org. </a>
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23. <a href="https://arxiv.org/pdf/astro-ph/0303346.pdf" target="blank">Faber, J. A. 11/24/2018. The Speed of Gravity Has Not Been Measured From Time Delays. arxiv.org.</a>
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24. <a href="https://arxiv.org/ftp/arxiv/papers/1108/1108.3761.pdf" target="blank"> Yin Zhu. 08/18/2011. Measurement of the Speed of Gravity. arxiv.org.</a>
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25. <a href="https://www.macmillanlearning.com/studentresources/college/physics/tiplermodernphysics6e/more_sections/more_chapter_2_1-perihelion_of_mercurys_orbit.pdf" target="blank">Perihelion of Mercury’s Orbit. macmillanlearning.com.</a>
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26. <a href="https://static1.squarespace.com/static/5852e579be659442a01f27b8/t/5cd46312eb393117fec080bb/1557422869040/Belenchia%5Bc.a.%5D_Wald_Giacomini_Castro-Ruiz_Brukner_Aspelmeyer_2019.pdf" target="blank">Belenchia A, Wald, R.M., Giacomini, F.,
Castro-Ruiz, E., Brukner, C., Aspelmeyer, M., 03/22/2019. Information Content of the Gravitational Field of a Quantum Superposition. Gravity Research Foundation.</a>
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</p><p></p><p></p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com3tag:blogger.com,1999:blog-3628542335168231161.post-49200362058805513102020-05-25T17:54:00.001-07:002020-05-25T18:20:22.900-07:00Why Time is More Than Real<p><iframe width="560" height="315" src="https://www.youtube.com/embed/CJAXU7L5F0Y" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
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"Reality is merely an illusion, albeit a very persistent one."--Albert Einstein
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It is apparent from the above quote that reality distinguishes itself from ordinary illusions by its persistent nature. Reality is true even if you choose not to believe in it. Thus, if we are trying to settle the question whether time is real, we should examine time to see, if like reality, it too is a persistent illusion.
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The time variable is very persistent and ubiquitous in so many physics equations. On that basis we can claim it's real, but just how real is it compared to things like matter, energy, mass, distance, force, your neighbor's barking dog? It's not like we can grab time out of the air, hold in hand and look at it like a hunk of clay. However, like clay, time can be stretched or compressed depending on how close to the speed of light you are traveling. How is that possible if time is just a product of human imagination? Surely any relative differences in time would also be limited to the human imagination and not an empirical reality.
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When examining time, one has to make the distinction between how we measure time and time itself. One popular argument claims that if all particles in the universe stopped changing their states and came to rest, time would stop and cease to exist. This seems reasonable. If nothing happens, how would we experience the "flow of time"? </p><p>But what if the "flow of time" is just our experience when we measure time? If your watch stops, you don't assume that time has stopped. You only assume your ability to measure and experience "the flow of time" has stopped. So it seems reasonable to assume that time continues even if every particle comes to a grinding halt. Think of a stalled universe as one big watch that stopped.
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So what exactly is time if not a flowing, evolving, ever-changing environment of entropy? The following equation, for me, is a real eye-opener and has forced me to rethink time:
</p><p>
<a href="https://1.bp.blogspot.com/-X82IMl_YyNo/Xsxf7BKKjCI/AAAAAAAAGjs/PpWPuBGX9iEgQ2yF2AjOWluzd41x9WZLQCLcBGAsYHQ/s1600/time1.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-X82IMl_YyNo/Xsxf7BKKjCI/AAAAAAAAGjs/PpWPuBGX9iEgQ2yF2AjOWluzd41x9WZLQCLcBGAsYHQ/s1600/time1.png" data-original-width="459" data-original-height="259" /></a>
</p><p>
E is energy and psi is the wave function, tp is the Planck time, G, c, and h-bar are the gravitational constant, light speed and Planck's constant, respectively. The above equation shows that it doesn't matter how much or little energy there is, or whether states change frequently or not at all, whether they go forward or backward. No matter what values you plug in for E and psi, you get forward time, specifically, the Planck time. Imagine having zero energy, zero change and still having a Planck time. How is that possible? Thought experiment time:
</p><p>
Imagine a universe with no energy, no distance or space, no charges, no masses, no momentum, no oscillators--just a single zero-dimensional point, a singularity. According to the above equation, time still exists. Why? Because the singularity is persistent--it is real. What exactly is this singularity? It's literally nothing ... except time at a single reference frame, at a single point. No clocks, no observers, just pure time. </p><p>Time is so essential to reality, that no "persistent illusion" can persist without it. Time can persist without anything else we would deem real, but nothing we deem real can persist without time. The words "reality," "existence," "persistence," "presence" all imply the passage of time. At this juncture, one could argue that time is not only real, but reality's most essential component. And, when we perform the above thought experiment, we witness time in its purest form.
</p><p>
So if you ever encounter a skeptic who believes time isn't real, that particles exist without time, ask the following question (but don't hold your breath):</p><p>"How long do particles exist without time?"
</p><p>GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com1tag:blogger.com,1999:blog-3628542335168231161.post-62650764099816341012020-05-11T18:53:00.000-07:002020-05-12T18:44:28.029-07:00Uncertainty Principle for Black Holes<p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/-q7EvLhOK08" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</p><p>
The above video discusses black-hole mathematical singularity problems. The current laws of physics seem to break down once a particle crosses a black hole's event horizon. One mathematical singularity occurs at the Schwarzschild radius; another occurs at the black hole's center. That being said, we will show if Heisenberg's uncertainty principle is employed, the singularity problems vanish and the laws of physics are restored. First, we define the variables we will use:
</p><p>
<a href="https://1.bp.blogspot.com/-NdkATU8kOJE/Xrh4x9mvJGI/AAAAAAAAGfg/h8oj2N70Cp4wwQiKQ9Z_mMKGnX862wB2wCLcBGAsYHQ/s1600/0.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-NdkATU8kOJE/Xrh4x9mvJGI/AAAAAAAAGfg/h8oj2N70Cp4wwQiKQ9Z_mMKGnX862wB2wCLcBGAsYHQ/s1600/0.png" data-original-width="526" data-original-height="565" /></a>
</p><p>
Before we examine a black hole, let's look at an electron orbiting a hydrogen nucleus. If we know the electron's mass and its approximate velocity (close to light speed c),i.e., its momentum, then we don't know its exact position. Its position could be anywhere within the Bohr radius. The product of its uncertain position and momentum gives us a number close to Planck's reduced constant:
</p><p>
<a href="https://3.bp.blogspot.com/-gdiujBzUWNs/Xrh42Pbt2mI/AAAAAAAAGfk/4DWZ7YUbnO4tME-JTkeYRmUQ3z916wRiQCLcBGAsYHQ/s1600/1.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-gdiujBzUWNs/Xrh42Pbt2mI/AAAAAAAAGfk/4DWZ7YUbnO4tME-JTkeYRmUQ3z916wRiQCLcBGAsYHQ/s1600/1.png" data-original-width="295" data-original-height="106" /></a>
</p><p>
We can imagine the electron being anywhere within a spherical cloud extending as far as the Bohr radius:
</p><p>
<a href="https://3.bp.blogspot.com/-t2s-hZdGX6U/Xrh46vUXN3I/AAAAAAAAGfo/7kV_QaLQBf46iXD3qbYKQGJXRTJTHbIKgCLcBGAsYHQ/s1600/1a.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-t2s-hZdGX6U/Xrh46vUXN3I/AAAAAAAAGfo/7kV_QaLQBf46iXD3qbYKQGJXRTJTHbIKgCLcBGAsYHQ/s1600/1a.png" data-original-width="482" data-original-height="333" /></a>
</p><p>
Now, let's take the mass of a black hole. Let's assume it is greater than the Planck mass. At the black hole's Schwarzschild radius, equation 3 is true:
</p><p>
<a href="https://4.bp.blogspot.com/--Kjnorhij-A/Xrh5AhisEcI/AAAAAAAAGfs/XaIvpcFv7j0xomp6hqv5frz9Nny4CzlwACLcBGAsYHQ/s1600/2.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/--Kjnorhij-A/Xrh5AhisEcI/AAAAAAAAGfs/XaIvpcFv7j0xomp6hqv5frz9Nny4CzlwACLcBGAsYHQ/s1600/2.png" data-original-width="452" data-original-height="218" /></a>
</p><p>
Next, we add a pinch of algebra to get equation/inequality 5--an uncertainty principle for the black hole.
</p><p>
<a href="https://3.bp.blogspot.com/-YIs4Vszg-PQ/XrnicgfSqNI/AAAAAAAAGh8/LTAeTc0WcII-CIzdtZ9VNHiFNQA2CnJZgCLcBGAsYHQ/s1600/3.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-YIs4Vszg-PQ/XrnicgfSqNI/AAAAAAAAGh8/LTAeTc0WcII-CIzdtZ9VNHiFNQA2CnJZgCLcBGAsYHQ/s1600/3.png" data-original-width="519" data-original-height="259" /></a>
</p><p>
So far, so good, but we run into a problem when we reduce radius r to, say, the Planck length:
</p><p>
<a href="https://2.bp.blogspot.com/-1mNKxDU1sLw/Xrh5HzBlnRI/AAAAAAAAGf4/wtuS5APp7sMrEsRPP8v81zHU--MaXRx_QCLcBGAsYHQ/s1600/4.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-1mNKxDU1sLw/Xrh5HzBlnRI/AAAAAAAAGf4/wtuS5APp7sMrEsRPP8v81zHU--MaXRx_QCLcBGAsYHQ/s1600/4.png" data-original-width="366" data-original-height="142" /></a>
</p><p>
The inequality at 6 clearly violates the uncertainty principle. The left side is required to be greater or equal to the right side--not less! The problem is caused by the momentum term containing nothing but constants (the Planck mass, c, and m).
</p><p>
If we are more certain about the position or size of the black hole's physical singularity, we need to be more uncertain about its momentum, so we need a momentum uncertainty factor represented by the Greek letter eta:
</p><p>
<a href="https://3.bp.blogspot.com/-ju-LfXwXzew/XrtQ8FcQ-RI/AAAAAAAAGig/XH-is7b3qa4nTG532tkd4fgGKjYB7E6CwCLcBGAsYHQ/s1600/5.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-ju-LfXwXzew/XrtQ8FcQ-RI/AAAAAAAAGig/XH-is7b3qa4nTG532tkd4fgGKjYB7E6CwCLcBGAsYHQ/s1600/5.png" data-original-width="509" data-original-height="997" /></a>
</p><p>
At 8 we see the uncertainty principle is restored. When radius r shrinks to a Planck or even a zero limit, eta blows up as it should.
</p><p>
Below we do some more algebra and derive 14:
</p><p>
<a href="https://3.bp.blogspot.com/-obRKBHJ6bUY/XrmUu9aQIlI/AAAAAAAAGhE/Az-UNz6QJqMtpE2BC5gb3A9BoISvvzKEACLcBGAsYHQ/s1600/6.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-obRKBHJ6bUY/XrmUu9aQIlI/AAAAAAAAGhE/Az-UNz6QJqMtpE2BC5gb3A9BoISvvzKEACLcBGAsYHQ/s1600/6.png" data-original-width="417" data-original-height="765" /></a>
</p><p>
At 14 we see the total energy on the right side never exceeds the total finite energy on the left side. A large momentum uncertainty (eta) cancels position certainty due to a small or zero radius. The inequality/equation at 14 also implies the black hole's singularity position is uncertain if the momentum is known. It could be located anywhere within a sphere bounded by the Schwarzschild radius. The most probable location being the center.
</p><p>
<a href="https://4.bp.blogspot.com/-XLVDr2cAwaw/Xrh5Y_FjhOI/AAAAAAAAGgM/b3bBXTd6VdM4hgTIOTM4qIXWfNwP-ruIQCLcBGAsYHQ/s1600/7.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-XLVDr2cAwaw/Xrh5Y_FjhOI/AAAAAAAAGgM/b3bBXTd6VdM4hgTIOTM4qIXWfNwP-ruIQCLcBGAsYHQ/s1600/7.png" data-original-width="515" data-original-height="427" /></a>
</p><p>
We can take what we have developed so far and apply it to an energy conservation technique used within a previous post titled <a href="http://gmjacksonphysics.blogspot.com/2020/03/high-energy-quantum-gravity.html" target="blank">"High Energy Quantum Gravity."</a> At 15 below we take the total energy between two orbiting bodies and subtract the strong, weak and electromagnetic energies. </p><p>The gravitational energy that remains will have a radius (ro) independent of radius r. The total gravitational energy remains constant no matter the distance r. However, we've factored in eta to conserve the Heisenberg uncertainty principle if r shrinks below the Scharzschild limit. Equation 15 reveals that a small force over a large area is equal to a large force over a small area.
</p><p>
<a href="https://2.bp.blogspot.com/-UgOToXkSq1k/XroOjtkEh5I/AAAAAAAAGiU/pKnyA7m9tCk-FeJKkAe35m5Ffytlt2bZgCLcBGAsYHQ/s1600/8.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-UgOToXkSq1k/XroOjtkEh5I/AAAAAAAAGiU/pKnyA7m9tCk-FeJKkAe35m5Ffytlt2bZgCLcBGAsYHQ/s1600/8.png" data-original-width="708" data-original-height="349" /></a>
</p><p>
Now, let's take what we now know and apply it to the singularity problems that crop up in the Schwarzshild metric below:
</p><p>
<a href="https://1.bp.blogspot.com/-rNXcu_6U5f4/XrmXmtx5EWI/AAAAAAAAGhQ/nCHa9XoGt3Q8xvTgM0AjfFFhXRbiRw__ACLcBGAsYHQ/s1600/9.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-rNXcu_6U5f4/XrmXmtx5EWI/AAAAAAAAGhQ/nCHa9XoGt3Q8xvTgM0AjfFFhXRbiRw__ACLcBGAsYHQ/s1600/9.png" data-original-width="540" data-original-height="432" /></a>
</p><p>
At 16, the right side's first term is infinity if r = rs. This implies the spacetime interval (ds) is infinite at the Schwarzschild radius--which is ridiculous. If r = 0, the last term, proper time, is infinite--also ridiculous. But of course, we have the tools to vanquish these mathematical singularities. We know the following Lorenz equations are true:
</p><p>
<a href="https://2.bp.blogspot.com/-twW3IAsEPt4/XrmXsEy76eI/AAAAAAAAGhU/TehFF0oQ6XY5piBbgugKfg-ceJAqMpwbQCLcBGAsYHQ/s1600/10.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-twW3IAsEPt4/XrmXsEy76eI/AAAAAAAAGhU/TehFF0oQ6XY5piBbgugKfg-ceJAqMpwbQCLcBGAsYHQ/s1600/10.png" data-original-width="681" data-original-height="410" /></a>
</p><p>
From 19 to 23 we make some substitutions and simplify the metric at 24:
</p><p>
<a href="https://1.bp.blogspot.com/-MsXmJifuoXE/XrmXyigHqJI/AAAAAAAAGhY/R0srII8bh0U60mnHmjBRBRM-SrLHynv5wCLcBGAsYHQ/s1600/11.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-MsXmJifuoXE/XrmXyigHqJI/AAAAAAAAGhY/R0srII8bh0U60mnHmjBRBRM-SrLHynv5wCLcBGAsYHQ/s1600/11.png" data-original-width="626" data-original-height="763" /></a>
</p><p>
At 25 we factor in eta:
</p><p>
<a href="https://4.bp.blogspot.com/-yikOM7TupSk/XrmX5AYlJpI/AAAAAAAAGhg/ENswEjUxYi0EQQmw0WrwqFvPoRgMADObgCLcBGAsYHQ/s1600/12.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-yikOM7TupSk/XrmX5AYlJpI/AAAAAAAAGhg/ENswEjUxYi0EQQmw0WrwqFvPoRgMADObgCLcBGAsYHQ/s1600/12.png" data-original-width="527" data-original-height="146" /></a>
</p><p>
Now, the only time we get infinity is when r is infinity and kappa is greater than zero. This makes sense if you stop and think about it (see results below).
</p><p>
<a href="https://3.bp.blogspot.com/-8mplzyfE3N8/XrmX9ahPsBI/AAAAAAAAGho/g-ERNSsk9zYI6KIWXtqTpAobYbMX5jU8wCLcBGAsYHQ/s1600/13.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-8mplzyfE3N8/XrmX9ahPsBI/AAAAAAAAGho/g-ERNSsk9zYI6KIWXtqTpAobYbMX5jU8wCLcBGAsYHQ/s1600/13.png" data-original-width="738" data-original-height="505" /></a>
</p><p>
When the radius is equal to the Schwarzschild radius, the spacetime interval is finite and the proper time is zero. When the radius is zero, the spacetime interval is only the outside observer's time, which makes sense, since nothing can move through zero space (a single point). The proper time is also zero, which makes sense, since it implies that time began after the universe expanded beyond a single point. Thus the current laws of physics that previously broke down are now at least partially fixed.
</p><p>
<a href="https://churchofentropy.wordpress.com/2020/05/04/cosmological-prime-causality/" target="blank">Special thanks to Cosmological {Prime} Causality for linking this post. Click here to read their blog.</a>
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-13676490932653215722020-03-27T13:49:00.000-07:002020-03-27T20:59:42.309-07:00High Energy Quantum Gravity<p><iframe width="560" height="315" src="https://www.youtube.com/embed/5foUTeRdqII" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</p><p>
In the above video, Sabine Hossenfelder discusses one of the shortcomings of quantizing gravity. At high energies or short distances things go haywire and you get crazy big numbers or infinities. In this post I present one possible solution to this problem. It is not the only solution, and, only experiments will reveal which solution is correct, or, reveal that none are correct. Here is a list of variables we will be working with:
</p><p>
<a href="https://2.bp.blogspot.com/-ssL3B7JtDFA/Xn5aesyQmjI/AAAAAAAAGbw/UsqLW0E20mAdzV793txtNqoGRnhTUaLNQCLcBGAsYHQ/s1600/1.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-ssL3B7JtDFA/Xn5aesyQmjI/AAAAAAAAGbw/UsqLW0E20mAdzV793txtNqoGRnhTUaLNQCLcBGAsYHQ/s1600/1.png" data-original-width="495" data-original-height="550" /></a>
</p><p>
Let's start things off by taking two arbitrary masses (m',m) and creating a reduced-mass Schwarzschild radius:
</p><p>
<a href="https://2.bp.blogspot.com/-0yZAooZ796c/Xn5bbqH6nBI/AAAAAAAAGb8/J_-RGtPLRnUdr7mEgj4YjwCvl6QM2eWkgCLcBGAsYHQ/s1600/2.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-0yZAooZ796c/Xn5bbqH6nBI/AAAAAAAAGb8/J_-RGtPLRnUdr7mEgj4YjwCvl6QM2eWkgCLcBGAsYHQ/s1600/2.png" data-original-width="529" data-original-height="157" /></a>
</p><p>
At equation 2 below, we express the maximum energy of the gravitational field between the two masses. Notice at equation 3, if the radius between the two masses is taken to the zero limit, you don't end up with an infinity. Instead the maximum gravitational energy is conserved, and, said energy never exceeds the maximum energy available--which is always finite.
</p><p>
<a href="https://1.bp.blogspot.com/-1WHwwb1__BY/Xn5bgMM8vPI/AAAAAAAAGcA/S8p7E2jylII5vutnoYUL1jzFE5Zg3aNjwCLcBGAsYHQ/s1600/3.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-1WHwwb1__BY/Xn5bgMM8vPI/AAAAAAAAGcA/S8p7E2jylII5vutnoYUL1jzFE5Zg3aNjwCLcBGAsYHQ/s1600/3.png" data-original-width="526" data-original-height="250" /></a>
</p><p>
The equation at lines 2 and 3 can be written in a more familiar form of work (energy) equals force times distance (see 3a below):
</p><p>
<a href="https://3.bp.blogspot.com/-XQ3TVgUrN7M/Xn5g-14k_9I/AAAAAAAAGcc/tISb7JHWjO05s34s7kelNJnopVJc4kDZQCLcBGAsYHQ/s1600/3a.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-XQ3TVgUrN7M/Xn5g-14k_9I/AAAAAAAAGcc/tISb7JHWjO05s34s7kelNJnopVJc4kDZQCLcBGAsYHQ/s1600/3a.png" data-original-width="382" data-original-height="431" /></a>
</p><p>
If the distance (carrot r) is great, the gravitational force (mg) is small, but if the distance goes to zero, the force blows up to infinity, but ... the energy is conserved, since the infinite force is only along a zero distance.
</p><p>
The next step is to quantize what we have so far. Let's take equation 3a and use a scale factor (alpha). At equation 4, we multiply alpha by a time derivative of h-bar. That takes care of energy (E). On the right side of 4 we have alpha times the time derivative of momentum (p) times the distance. The time derivative of momentum is, of course, the force.
</p><p>
<a href="https://3.bp.blogspot.com/-dfmpq6GdHtI/Xn5bjaUNHKI/AAAAAAAAGcE/idGlMxuM-JEjffB_fq7-YREBHKnOEL3cwCLcBGAsYHQ/s1600/4.png" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-dfmpq6GdHtI/Xn5bjaUNHKI/AAAAAAAAGcE/idGlMxuM-JEjffB_fq7-YREBHKnOEL3cwCLcBGAsYHQ/s1600/4.png" data-original-width="526" data-original-height="119" /></a>
</p><p>
From 4 we derive Heisenberg's uncertainty principle for the singularity (reduced by the alpha factor):
</p><p>
<a href="https://2.bp.blogspot.com/-ygU2x018y40/Xn5bnNxplUI/AAAAAAAAGcI/vzQC4jHBbBIsLZzwhuPmvZXuZVcC3CaHwCLcBGAsYHQ/s1600/5.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-ygU2x018y40/Xn5bnNxplUI/AAAAAAAAGcI/vzQC4jHBbBIsLZzwhuPmvZXuZVcC3CaHwCLcBGAsYHQ/s1600/5.png" data-original-width="531" data-original-height="353" /></a>
</p><p>
What does equation 7 tell us? It indicates that if we know the exact position of a singularity, we are completely uncertain about its momentum, and vice versa.
</p>
GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0tag:blogger.com,1999:blog-3628542335168231161.post-40520067671412937022019-12-10T20:41:00.005-08:002020-11-18T19:26:19.004-08:00Resolving the Black Hole Information Paradox: How Information is Lost and Conserved<p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/9XkHBmE-N34" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
</p><p>
According to the current paradigm, quantum information is conserved. With perfect knowledge of the current universe it should be possible to trace the universe backwards and forwards in time. This principle would be violated if information were lost. When information enters a black hole we might assume the information is inside, but then black holes evaporate due to Hawking radiation, and the black hole's temperature is as follows:
</p><p>
<a href="https://4.bp.blogspot.com/-L4Q1HGZJg8I/XfBXzvqcQvI/AAAAAAAAGWw/XOlNhbkRPZwWLTcrGwUs9eCa_M9HdjKcQCLcBGAsYHQ/s1600/1.png" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-L4Q1HGZJg8I/XfBXzvqcQvI/AAAAAAAAGWw/XOlNhbkRPZwWLTcrGwUs9eCa_M9HdjKcQCLcBGAsYHQ/s1600/1.png" data-original-width="433" data-original-height="548" /></a>
</p><p>
As the black hole evaporates, its mass shrinks and its temperature increases. Take note that equation 1 fails to tell us what information went into the black hole, so looking at the final information (remaining mass, momentum, charge) pursuant to the no-hair theorem we can't extrapolate that data backwards and determine what information went into the black hole. This is known as the Black Hole Information Paradox.
</p><p>
Many hypotheses have been set forth that attempt to resolve this paradox. One popular one is the holographic principle (T'Hooft and Susskind). <a href="https://gmjacksonphysics.blogspot.com/2019/05/how-to-falsify-holographic-principle.html" target="blank"> Unfortunately, it is easy to punch holes in this one. You can read about it by clicking here.</a> </p><p>Another common proposal is the information goes inside the black hole, then through a wormhole into another universe. Personally, I don't care for this one, since it requires the establishment of another universe (good luck!). Then there's the explanation that begins with a shrug and ends with a sigh: the information is lost. </p><p>Of course I'm not without a brainstorm of my own, which is why I'm now scribbling. It occurred to me that maybe there's at least two kinds of information: information that is conserved and information that is not. This random thought popped into my head when I was working on the following math proof:
</p><p>
<a href="https://1.bp.blogspot.com/-FAq4rEOS0ts/XfBX4DHX_8I/AAAAAAAAGW0/JVCctBXRiYIhrpRQ-kc5K4xor3pDWr4lwCLcBGAsYHQ/s1600/2.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-FAq4rEOS0ts/XfBX4DHX_8I/AAAAAAAAGW0/JVCctBXRiYIhrpRQ-kc5K4xor3pDWr4lwCLcBGAsYHQ/s1600/2.png" data-original-width="552" data-original-height="655" /></a>
</p><p>
The proof starts with the absurd claim that if 'a' doesn't equal 'b' then 'a' is equal to 'b.' Let's suppose 'a' is information. At equation 2 it is defined. However, by the time we get to equation 4, 'a' becomes undefined. Zero times infinity can equal any number, so the definite information we started with appears to be lost. Although, unlike black-hole information, by the time we get to equation 7, 'a' is defined again, but this time it is defined as 'b.'
</p><p>
What we can take away from the proof above is specific information is not conserved, but information overall is conserved. The information changed from 'a' to undefined to 'b.' Unfortunately, even though the information is conserved, we can't tell by looking at 'b,' that it was once 'a.'
</p><p>
Below is another example of what I'm scribbling about. Start with two distinct binary numbers. Let's pretend they enter a fictitious binary black hole and come out identical (zeros on the left, ones on the right). Now we put them into a cosmic hat. You reach in and pull one out. Can you tell whether it used to be 1010 or 0101? I don't see how. The information is conserved however--there's still the same number of ones and zeros.
</p><p>
<a href="https://1.bp.blogspot.com/-x9eIuY_-Gdw/XfBX8cb1F6I/AAAAAAAAGW4/g-F85gXVf90l7f6x8YBgVRqB1RZuNR9yACLcBGAsYHQ/s1600/3.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-x9eIuY_-Gdw/XfBX8cb1F6I/AAAAAAAAGW4/g-F85gXVf90l7f6x8YBgVRqB1RZuNR9yACLcBGAsYHQ/s1600/3.png" data-original-width="457" data-original-height="245" /></a>
</p><p>
Rather than say information is lost, perhaps it is more prudent to say it is undefined. In the case of a black hole, most of the information becomes undefined. Having perfect knowledge of it doesn't help us trace it back to its defined state prior to entering a black hole.
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There are many examples in everyday life where we can observe information evolving from defined to undefined. Write a message on a blackboard. Erase the message. The chock that made up the message is now smeared onto the eraser. Give the eraser to a physicist and see if he/she can tell you what your unique message was. At this point, the chalk has mass, for instance, but chances are excellent that physicist won't be able to know your unique message. That information is lost. It was defined, now it is undefined (except for some basic properties like mass, etc.)
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The no-hair theorem reminds me of brown paint. Imagine some masterpiece paintings, each with a unique set of information. The paint on each painting is scraped off the canvass and mixed in a bucket of paint thinner. At the end you have several buckets of brown paint. If they are mixed up and you choose one at random, can you tell which painting it came from? Probably not. It's another case where defined information becomes less defined--so it may also be true even at the quantum scale. For example, according to quantum field theory, particles and their unique, well-defined properties are excitations of fields where the information is kind of blurry or undefined.
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Imagine an electron-positron pair popping into existence. The electron is spin up, the positron is spin down. They annihilate. Is it possible to look at the resulting photon and know it was previously a spin-up electron and a spin-down positron? For all you know the electron was spin-down and the positron was spin up. Yet another case of information evolving into something where you can't know its previous state. So why should we be surprised there's an information paradox if we believe perfect knowledge of the current state of information allows us to trace it backwards and forwards in time?
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GM Jacksonhttp://www.blogger.com/profile/02363192260461368016noreply@blogger.com0