GM Jackson Physics and Mathematics: Using Quantum Physics to Find the Best Model for Gravity, Gravitational Waves, and the Vacuum

ABSTRACT: According to Einstein's theories of relativity, nothing is faster than light; yet, observations made by Newton, Laplace and Van Flandern led them to believe gravitational information is much much faster than light, virtually instantaneous. To thicken the plot further, LIGO observed gravitational waves propagating within the light-speed limit. Then there's the vacuum energy problem where the vacuum seems to have up to infinite energy! By making use of quantum physics and changing an initial assumption about the vacuum energy, it is possible to connect the dots between quantum physics and General Relativity. By exposing a fundamental flaw in the rubber-sheet model for curved spacetime, it is possible to create a superior model that reconciles the speed of gravitational waves with the illusion of faster-than-light gravitational information.

Quantum physics is probabilistic as opposed to deterministic. Given a vacuum that is composed of numerous (omega) energies it only makes sense to multiply each energy by a probability. The alternative is to simply add all the energies and get up to infinity! Assuming each energy has a wave function, each probability can be determined by squaring each wave function. Each energy is represented by the Hubble energy (Hubble's parameter * Planck's constant) multiplied by n. At equation 1 below, we calculate the vacuum mass density (rho). At equation 2 we determine the cosmological constant:

Now, let's introduce a particle with mass m (or it could be massless: m = E/c^2). It could be located anywhere and everywhere. Its location (x,y,z coordinates) is uncertain at best. We could add m multiple times to cover all its possible locations, but that would lead to a rediculously big number. Or, as we did with the vacuum, we could multiply each m location by a probability (see equations 3 and 4) and that will give us the expectation value for m which is really just m. Thus, multiple m's at multiple locations don't amount to more than just m. The velocity of m is also uncertain, but we can calculate its expected value at equation 5. At 6 we determine the expectation value for the wavelength of m.

At equation 7 we see how multiple locations of m impact its gravity:

The gravity of m is also at multiple locations along with m, but probabilties cut it all down to size. The size being the left side of equation 7: the gravity of the expectation value: m.

At equation 8 below, we redefine m as one or more particles (expected values). At 9 and 10, we calculate the final velocity and the final wavelength, respectively.

At 10 we have the De Broglie wavelength formula. At 11 we below can see that if mass m or final velocity v changes, the final wavelength lambda must instantaneously change to keep Planck's constant a constant. After a few algebraic steps, we derive the Schwarzshild radius at 16.

Note that any change of mass m causes an instantaneous change of its wavelength (lambda prime). This means that when a black hole's singularity mass changes, its Schwarzschild radius instantaneously changes! Equation 17 below confirms this. The light-speed constant c on the left side is not a constant unless mass m and Schwarzschild radius changes are synchronized. Thus, we don't have to wonder how the singularity sends information out as far as the Schwarzschild radius, assuming such information is limited to light speed. It doesn't need to. Every mass simply has, and is defined by, a corresponding wavelength and Schwarzschild radius.

But wait! It gets even better! The diagram below shows the total radius r (in black), the Schwarzschild radius (in red) and the remaining distance (in blue). Let's assume the circle below contains the volume (V) of the entire universe (or any volume you like). If mass m owns the volume as far out as the Schwarzschild radius, then the vacuum's claim along the total distance r is reduced. The rest of the universe owns only the volume along the remaining distance.

One can infer that the remaining distance (in blue) must change instantaneously in response to a change in the Schwarzschild radius (in red) which in turn changes instantaneously to a change in the average wavelength which responds instantaneously to a change in mass m.

Equation 18 shows that proper time, out to radius r, is also reduced and is proportional to the square root of the remaining distance over r. The reader may recognize a variation of the Lorentz factor on the left side. One may also infer that proper time is reduced instantaneously given its dependency on the remaining distance.

Using the diagram above, we can set up equation 19 below. From there we can navigate to Einstein's field equations at 21. At 22, we can further verify the instantaneous relationship between matter, spacetime, and gravity. Again, the constant c is not constant unless spacetime curvature responds instantaneously to a change in the stress-energy tensor.

So far we have shown how matter interacts with spacetime. We should now take a look at how vacuum mass interacts with spacetime. All masses project a Schwarzshild radius regardless of how concentrated or diffuse they are. In each diagram below, the gray area represents the mass concentration. Note that each diagram has a mass of m and the same Schwarzschild radius.

Vacuum mass, like the star and black hole, also curves spacetime. The cosmological constant is that spacetime curvature caused by vacuum mass density. We add this to the field equations:

So a question arises: Why does vacuum mass density and its curvature cause the universe to expand? Below, term A implies term B, and, term C implies term D. Term B shows how matter (E) accelerates. When distance r is larger, the rate of acceleration is less. When r is smaller, the acceleration rate increases. This is because matter (E) is fixed. Contrast this with term D. At term D, the opposite happens: Acceleration increases as r increases and vice versa because vacuum mass is not fixed. It is proportional to volume. Both B and D contribute to spacetime curvature, but are seemingly opposite forces.

At equation 25 we set a scalar version of the Einstein tensor equal to the sum of the curvature caused by mass m and the curavature caused by vacuum mass. From there we derive equations 30 and 31 which show the tug of war between an expanding universe and gravity.

We can simulate the net acceleration rate at equation 31 with an accelerating rocket (see diagram A below). If we throw a shot put across, it will appear to fall along a curved path. Throwing the shot put gives it kinetic energy which may be lost and converted to gravitational waves (GWs, purple curved lines). At diagram B we have a fixed container with a magnetic field. Equations 31 and 32 demonstrate the time-delay difference between gravity and electromagnetism. Imagine t1 is the time it takes to initiate the rocket engine and the electromagnet. The magnetic field takes an additional r/c seconds to develop, since photons must propagate from the floor of B to the shot put (distance r) at speed c. After t1 + r/c seconds have passed, the shot put falls and takes time t2 to hit the floor. By contrast, the shot put at A immediately falls after t1 seconds.

Thus gravity's total time t is t1 + t2 seconds (equation 32). Electromagnetism's total time t is t1 + r/c + t2 (equation 33). Also note that the gravitational waves (GWs) do not cause the shot put to fall, but rather, it is the shot put's lost kinetic energy that causes the GWs.

Observations confirm that the GW strain (h) is consistent with 35 below and not 34. At 34 we have the curvature of the complete energy of the source; whereas, at 35, we just have the curvature of kinetic energy of the source. Notice at 36 and 37 electric and magnetic waves are proportionate and correlate with their respective fields. By contrast, GWs do not correlate with the full gravitational field.

The accelerating-rocket thought experiment above seems like a good approximation of gravity and gravitational waves; however, it seems to contradict the famous rubber-sheet model of gravity. Imagine placing the shot put on a rubber sheet. The shot put will depress the rubber sheet. Such depression, however, does not happen instantaneously. The depression curve takes time to form. As it's forming, one can imagine GWs propagating outward from the center of mass. But what happens if the shot put is moving fast (v > 0) and not at rest (v = 0)? One can imagine it not depressing the rubber sheet:

Imagine the earth is covered with a rubber sheet and the shot put has enough velocity v to orbit. It won't fall towards earth's center, so it won't depress the rubber sheet. We model this fact with equation 39:

One might erroneously conclude that if a mass moves fast enough, it won't curve spacetime! This could not be further from the truth. The truth is velocity enhances the curvature of space time:

Thus the rubber-sheet model has a fundamental flaw. The accelerating-rocket model is superior. It creates the illusion that gravity's speed is faster than light. This is consistent with observations made by Newton, Laplace and Van Flandern. I say "illusion" because nothing in the accelerating-rocket thought experiment exceeds the speed of light.

Given all the forgoing information, we can set up a timeline model for curved spacetime and gravitational waves (GW):

Equations 41 through 44 take into account the instantaneous interplay between matter and spacetime along with gravitational waves that don't exceed the light-speed limit. Equations 45 through 47 below confirm that 41 through 44 are correct; otherwise, the constant c would not be constant if the curvature of spacetime had to wait for gravitational waves or gravitons to propagate.