# GM Jackson Physics and Mathematics

## Featured Post

### The Beautiful Destruction of the Graviton

ABSTRACT: In his paper titled "Aberration and the Speed of Gravity," S. Carlip argues that gravity propagates at light spee...

## Friday, August 14, 2020

### The Beautiful Destruction of the Graviton

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ABSTRACT:

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.

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.

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.

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:

From equation 2 we derive equation 3 which emphasizes that c is a component of rest-mass energy and not propagation speed.

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:

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.

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:

By contrast, if the gravitational field (gravitons) is removed, Newtonian gravity still exists:

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:

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.

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.

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.

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.

Acknowledgements:

Amber Strunk. Education and Outreach Lead. LIGO Hanford Observatory.

References:

1. Parikh, Wilczek, Zahariade. 2020. The Noise of Gravitons. arxiv.org.

2. Feynman, R.P. 07/03/1963. Quantum Theory of Gravitation. Acta Physica Polonica. Vol. XXIV.

4. Carlip, S. 12/1999. Aberration and the Speed of Gravity. arxiv.org.

5. Van Raamsdonk, M. 05/17/2010. Building up spacetime with quantum entanglement. arxiv.org.

7. Gravitational Wave. Wikipedia.

8. de Rham, C., Tolley, A.J. 03/17/2020. Speed of Gravity. arxiv.org.

9. Carroll, S.M. 12/1997. Lecture Notes on General Relativity. Enrico Fermi Institute.

14. Nix, E. 08/22/2018. Who Determined the Speed of Light. History.com.

15. Speed of Gravity. Wikipedia.

16. Tests of General Relativity. Wikipedia.

17. Decross, M. et al. Gravitational Waves. Brilliant.com.

18. Lawden, D.F. 1982. Introduction to Tensor Calculus, Relativity and Cosmology. Dover Publications, Inc.

19. Stefanovich, E. V. 09/16/2018. A relativistic quantum theory of gravity. arxiv.org.

20. Light-time correction. Wikipedia.

21. LiĆ©nard–Wiechert potential Wikipedia.

23. Faber, J. A. 11/24/2018. The Speed of Gravity Has Not Been Measured From Time Delays. arxiv.org.

24. Yin Zhu. 08/18/2011. Measurement of the Speed of Gravity. arxiv.org.

25. Perihelion of Mercury’s Orbit. macmillanlearning.com.

## Monday, May 25, 2020

### Why Time is More Than Real

"Reality is merely an illusion, albeit a very persistent one."--Albert Einstein

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.

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.

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"?

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.

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:

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:

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.

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.

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):

"How long do particles exist without time?"

## Monday, May 11, 2020

### 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.

## Friday, March 27, 2020

### High Energy Quantum Gravity

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:

Let's start things off by taking two arbitrary masses (m',m) and creating a reduced-mass Schwarzschild radius:

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.

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):

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.

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.

From 4 we derive Heisenberg's uncertainty principle for the singularity (reduced by the alpha factor):

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.

## Tuesday, December 10, 2019

### 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? 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?

## Thursday, December 5, 2019

### Introducing Stochastic Trigonometry for Quantum Physics and Statistical Mechanics

In the field of quantum physics, each eigenvalue has an eigenvector, and, when the eigenvector is normalized and squared, we get the probability for the eigenvalue. The normalized eigenvector is sometimes referred to as the probability amplitude.

When all the probability amplitudes are squared and added, the total should be 1. We can represent this with the Pythagorean theorem and the right triangle below:

The above diagram consists of two probability amplitudes: 'a' and 'b.' One is a wave function cos(theta) and the other is a wave function sin(theta).

Now, suppose there are more than two eigenvalues/eigenvectors? The diagram below shows that a and b can be broken up into smaller pieces or smaller and more numerous probability amplitudes. As before, when they are all squared and summed, they give us a total of 1.

It is possible to break up 'a' and 'b' into as many pieces as we like. Below we focus on amplitude 'a':

We can imagine breaking up amplitude 'a' into as many as an infinite number of sub-amplitudes. This can be done in both Euclidean and curved space. Equation 10 below shows how amplitude 'a' and its sub-amplitudes are invariant within flat or curved space.

With a little algebra, we can derive equation 14:

Equation 14 shows amplitude 'a' consists of an infinite number of eigenvalues (eta), each with its own probability (P(eta)). Without the probabilities, the etas would add up to infinity, and that would necessitate some sort of re-normalization technique. If we assume, however, that all quantum numbers have a probability, we will not get infinity; rather, we get the expectation value, i.e., the value actually observed.

What kind of probability values yield a finite result when eta increases linearly to infinity? Probability values that decrease exponentially. Below we derive such a probability function by using the natural-log function and converting eta to 'n':

At 16.2 we have a probability function that will reduce the probability exponentially. It gives us a number between 0 and 1, but we can derive a better function that gives us a number between 0 and 1, and, we can make a substitution. The end game is equation 16.9:

Equation 16.9 claims that if n = Q, the probability of Q (P(Q)) equals the definite integral of the probability function over a range from Q-1 to Q. We can further justify this claim with the diagram below which shows the relation between discrete values (in red) with continuous values (blue line).

Note how the area under the blue line, say, from Q-1 to Q is the same as the area of the red squares from Q-1 to Q. Equation 17 models the fact the the area under the blue line is the same as the area of the red squares over the entire range.

Now, to get a finite expectation value (amplitude 'a') we could combine equations 16.9 and 17, but the math would be more complicated than need be. To simply the math we will encounter later, let's first stretch the above diagram vertically:

Next, we draw a yellow line from zero N+1. This new line is going to make our lives easier and has the same area beneath it as the red line. Wouldn't it be nice if we could nix the red and substitute the yellow? Sure! But first we have to rotate the diagram:

Ah ... now we're in business! Below is the adjusted diagram and equation 18 with a new slope of N/(N+1):

The integral has a new range of zero to N+1, so we give the probability integral the same range:

Let's combine equations 18 and 19 to get 20:

If the limit of N is infinity, equation 20 will always give us the finite probability amplitude 'a.' No re-normalization required.

Using the diagram below and equations 21 and 22, we can derive a formula that finds probability densities:

What we've covered so far allows to find probabilities for integer values. This works fine if the value is, for example, the number of vertices in a Feynman diagram. Albeit, energy can have values of n+.5. Below is the math for that circumstance:

Notice if we divide both sides of equation 26 by Q+.5, and use summation signs, we arrive at equation 23, the formula for finding probability densities.

Now that we have the math the way we want it, let's put it to a test. Let's say we want to add up an infinite number of quantum numbers to get a finite value. Let's assume that the principle of least action applies: the most probable value will be the least action (e.g. least energy, least time, least distance, least resources required, etc.). The least probable value will be the action or event that requires the most resources, time, energy, etc. So we expect the probability to drop exponentially as the value of 'n' increases linearly--this will ensure a finite result.

Let's also assume that experiments confirm that probabilities change according to equation 27:

OK, now we only have to do some complicated math to find the expectation value 'a,' right? Wrong! At 28 and 29 below we convert the right side of 27 to a natural exponent function. If we look at equation 20, it becomes obvious that we can solve this problem by mere inspection. Looking at the exponent, everything to the right of -n is 1/a. Thus equation 31 is our final result.

Here's another test: What is the probability that a particle will travel a distance 'Q' along a pathway 'omega'? Equation 34 below can answer that. At 32 we assume that each pathway has the same probability if the distance traveled is constant, since the action is the same along each pathway (except for the direction, angles, curves, twists, turns, etc.).

Equation 35 gives us a definite answer if we want to know the probability density of a range of distances and pathways the particle can travel:

As you can see, stochastic trigonometry simplifies mathematics that can turn into a complicated, ugly, and infinite mess. It can also improve statistical mechanic's coarse-graining techinique:

Why use squares when you can use triangles?

Update: The following math shows both a convergent series and divergent series can yield a finite number 'q.' First, we start with a divergent series and make it convergent by using the probability function we derived above.Next, we take a divergent series and assume the coefficients (the c's) don't add up to 1. Each could be any finite size; they could be a random series. The strategy is to factor out 'c' from the coefficients and use one of Ramanujan's techniques:

Another update: The following math generalizes the idea that a finite value can result from any arbitrary convergent or divergent series: