Why Entanglement and Faster Than Light Speed Are Consistent with Relativity

ABSTRACT:

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.

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:

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.

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:

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:

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.

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:

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:

The speed of light also depends on the speed of our expanding universe--even if that speed is faster than light:

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.

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.

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.

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.

We bring back the Friedmann equation at 30 below. According to the WMAP spacecraft, space is nearly flat, so we set k to zero.

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.

Next, lets assume the vacuum and gravitational fields both exist everywhere. The equality is restored and the Friedmann equation is always valid:

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.

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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Conclusion:

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.

References:

1. Flanagan, Eanna. Hughes, Scott A. 2005. The Basics of Gravitational Wave Theory. New Journal of Physics.

2. Van Flandern, T. 1999. The Speed of Gravity What the Experiments Say. Meta Research University of Maryland Physics Army Research Lab.

3. Siegel, Ethan. August 30, 2018. Our Motion Through Space Isn't A Vortex, But Something Far More Interesting. Forbes

4. Tzortzakakis, Filippos, LIGO Analysis: Direct Detection of Gravitational Waves. Journal of Research Progress Vol. 1.

5. Roberts, Tom, Schleif, Siegmar. 2007. What is the experimental basis of Special Relativity?

6. Friedmann equations. Wikipedia

The Faulty Premises of Black-hole Physics

ABSTRACT:

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.

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.

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.

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:

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.

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:

The premise that irreversibility can't and should never happen seems untenable.

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:

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:

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).

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.

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:

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.

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:

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).

Adding a qubit of information to a black hole is done in the following manner:

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:

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!

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.

References:

1. Hossenfelder, Sabine (23 August 2019). "How do black holes destroy information and why is that a problem?". Back ReAction. Retrieved 23 November 2019.

2. Hawking, Stephen (1 August 1975). "Particle Creation by Black Holes" (PDF). Commun. Math. Phys. 43 (3): 199–220.

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.

4. Black hole information paradox. Wikipedia.

5. Mathur, Samir D. 03/21/2021. The Elastic Vacuum. Gravity Research Foundation.

6. Chaisson, Eric. Astronomy Today. Englewood, NJ: Prentice Hall, 1993: 503

Uncertainty Principle for Black Holes

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:

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:

We can imagine the electron being anywhere within a spherical cloud extending as far as the Bohr radius:

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:

Next, we add a pinch of algebra to get equation/inequality 5--an uncertainty principle for the black hole.

So far, so good, but we run into a problem when we reduce radius r to, say, the Planck length:

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).

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:

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.

Below we do some more algebra and derive 14:

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.

We can take what we have developed so far and apply it to an energy conservation technique used within a previous post titled "High Energy Quantum Gravity." At 15 below we take the total energy between two orbiting bodies and subtract the strong, weak and electromagnetic energies.

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.

Now, let's take what we now know and apply it to the singularity problems that crop up in the Schwarzshild metric below:

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:

From 19 to 23 we make some substitutions and simplify the metric at 24:

At 25 we factor in eta:

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).

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.

Special thanks to Cosmological {Prime} Causality for linking this post. Click here to read their blog.

Resolving the Black Hole Information Paradox: How Information is Lost and Conserved

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:

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.

Many hypotheses have been set forth that attempt to resolve this paradox. One popular one is the holographic principle (T'Hooft and Susskind). Unfortunately, it is easy to punch holes in this one. You can read about it by clicking here.

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.

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:

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.'

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.'

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.

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.

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.)

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.

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?

How to Make a Dent in Spacetime for Orbiting Satellites

The above video shows how marbles do elliptical orbits on lycra that is depressed by a massive steel ball. It's analogous to satellites orbiting stars and planets. In this post we show how real spacetime is dented or curved and work out the mathematics of orbiting satellites. So let's define some variables:

Below is diagram D-1. It's a bit crude, but it shall be our guide as we do the math.

Imagine two adjacent volumes of space. Both have volume V. Neither has any significant energy or mass--just empty space. That brings us to equation 1:

Suppose we add a star to one the volumes (see D-1). The star's matter and energy take up some of the space. That leaves a net volume V'. V' is only slightly less voluminous than V, since the star is made of atoms that are mostly space. Suppose the star collapses into a black hole. Surely a black hole takes up no space and V' should equal V. But the black hole and the star have common ground: both have the same energy density within volume V.

At the top of D-1 is the Compton wave formula. When energy (or mass) is added, wavelengths decrease. Shorter wavelengths take up less space and are equivalent to a smaller volume. At D-1, the squiggly lines represent the wavelengths. Note that the top portion (V) has longer wavelengths than the bottom portion (V') If we think of volume in terms of wavelengths, V' is definitely less than V.

The V minus V' average wavelength difference is equivalent to the volume taken up by the star. We can add that volume to V' to get V:

Thus the variables of equation 2 have the same values whether we have energy density in the form of a giant star or a black hole.

Now, let's divide by the z axis (see D-1) to get areas A and A'. Equation 4 gives us the net area (the broken-line rectangle at the center of D-1).

At 5 we divide the areas by the square of the average relative time it takes for a satellite to go along the x and y axes. At 6 through 8 we find the squared velocity of the satellite. It is small compared to light speed--due to gravity being weak ... and ... gravity is weak due to the minor difference in V and V'.

Equation 7 is illustrated at D-2 and D-3 below. Due to increasing energy density (shorter wavelengths), when particle-waves move toward the star, they gain momentum (or lose less momentum). Due to decreasing energy density (longer wavelengths), when particle-waves move away from the star, they lose momentum (or gain less momentum). Thus, on average, the particle waves that make up our satellite, spacetime, etc., are attracted to the star.

At D-3, the long arrows represent the increasing momentum of stuff coming in. The short arrows represent the decreasing momentum of stuff going out.

The converging arrows at D-4 represent the net momentum, the dent in the lycra--the gravitational field. Its intensity increases as the satellite falls due to increasing energy density.

At 10 through 12 we work the star's mass and momentum into the equation. Whenever mass is included in the field, so is it's inertia. Inertia cancels mass and thus different masses fall at the same rate, so we divide by mass as well as multiply.

At 13 through 16 we convert the mass and momentum into n units of Planck mass and Planck momentum.

At 17 we borrow from Heisenberg's uncertainty relations and write the Planck mass in terms of Planck's reduced constant. We make a substitution at 18. At 19 we convert the reciprocal of the Planck momentum squared and make another substitution at 20. Some variables cancel each other.

At 20 we have something close to what we need. All that's left to do is to put the variables back in that we took out earlier (for ease of computation):

At last! We have equation 24. It gives us the satellite's instant velocity at any point during its orbit. Equation 23 allows us to change the ellipse coefficient so we can have a variety of elliptical orbits (see D-5 below) on the dented lycra of spacetime.

Update: Below is a formal proof showing that equation 2 is a solution of Einstein's field equations.

Update: How much actual space does a black hole occupy? It has a singularity with a zero-limit radius (r), but a physical extent of radius (rs), the Scharzschild radius. The answer appears to be a volume with the Schwarzschild radius. Below is the mathematics showing how a black hole takes up space: