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.

Acknowledgements:

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

References:

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Resolving the Cosmological Constant Problem

Imagine you're a quantum physicist and you just got through calculating the energy density of the vacuum. To your horror you find that your calculation is around a hundred orders of magnitude greater than vacuum energy density measurements! This, in a nutshell, is the cosmological constant problem. According to your numbers, the cosmological constant should be huge. According to WMAP measuremments, the cosmological constant is too small to tweet about. In this post we shed some light on this problem. First, here are the variables we will be using:

Below we derive the cosmological constant value based on the WMAP vacuum energy density measurement of approximately 10^-10 Joules per cubic meter (see equations 3 and 4):

The next order of business is to consider the limitations caused by the Compton wavelength formula and the Heisenberg uncertainty principle--so we derive equation 8:

There are two ways we can interpret equation 8: 1. If delta-x grows smaller, energy (delta-E) grows bigger. This makes perfect sense when you consider that smaller spaces contain shorter wavelengths, and shorter wavelengths correspond to higher energies. 2. If delta-x grows smaller, delta-E becomes more uncertain. In any case, we use equation 8 as our means to make sense of the cosmological constant problem.

Equation 9 below is equation 8 modified. On the left side we have the vacuum energy measurement (E) and one meter (x). Of course the product of these values is far greater than Planck's constant, so the right side has the scale factor n:

OK, here's what we need to do: we need to reduce the vacuum's volume while maintaining a constant energy density. This can be done, but only up to a point. The question becomes, up to what point? Or, if you prefer, down to what point? How small can we make the energy and space before things get weird? Here's the answer:

Equation 10 above is the formula for finding the cutoff, or, the Heisenberg uncertainty limit. The product of the energy and space can't go below h-bar/2 without violating the uncertainty principle.

At 11 and 12 below, we plug-n-chug:

At the cutoff, the distance is around 10^-4 meters. The volume is 10^-12 cubic meters. The energy is approximately 10^-22 Joules. If the space is reduced further, the energy value will explode! With an upper limit of infinity! Obviously, if the energy skyrockets, or becomes more uncertain, the energy density won't be the same--and any calculation done at that level will yield ridiculous results.

The diagram below shows vacuum energy density matches observations at or above the cutoff. Any meaningful measurement is taken above that point. But if you prefer infinities and uncertainties, feel free to go smaller than the Heisenberg limit.

Thus, the cosmological constant problem appears to be a natural consequence of doing calculations and measurements below the cutoff limit.

Dark Energy In - Dark Energy Out = Gravity

In the diagram below there is a body falling to earth. After a distance of dr, what is its instant velocity?

Let's derive a formula that can give us the instant velocity:

Equations 8 and 9 give us the square of the instant velocity. Equation 7 shows that the squared velocity is also equal to gdr. Equation 8 shows how gdr is a function of the different time rates. At the top of dr (see diagram above), time is t; at the bottom, time is t'. Such time differences can be tested with atomic clocks.

At equation 9, notice how gdr or the instant velocity squared is the difference between light speed squared (c^2) and a lesser velocity squared (u^2). We can derive this difference again, starting with Einstein's field equations:

Equation 20 shows us again the instant velocity is the square root of the difference between two squared velocities. What exactly are these velocities? Starting with Einstein's energy equation, we can derive them yet again:

Equations 30 and 31 are all too familiar. One interesting property they seem to have is momentum is not conserved. It looks as though we could couple the velocities with any mass and they would remain constant?

Momentum not being conserved is consistent with any mass falling in a gravitational field--all masses fall at the same rate. By contrast, momentum is conserved when an electromagnetic force is applied to different masses. (See equations below.)

Equations 33 and 34 are classical force equations. Normally, when mass (m) increases, acceleration (a) decreases--force is conserved. The time derivative of conserved force is conserved momentum, so the instant velocity decreases as well. When mass is decreased, the acceleration and velocity increase. Equation 35 fits the norm; however, equation 36 does not. Note that electromagnetism (EM, equation 35) is not enhanced by mass, but charges (q' and q). By contrast, gravity (equation 36) has mass (m) in both the numerator and denominator, so it cancels itself.

The mathematical proof below confirms the squared velocities we derived above are unaffected by different masses. We start with equation 27:

At equation 38, momentum is not conserved. If we increase the mass (ps/c) and assume velocity (v) drops (making the term constant), ps on the equation's right also increases, but there is no v there to offset it. A similar problem occurs if we reduce ps. Armed with this information we derive equation 41 below:

Equation 41 works if we hold the velocities constant, but allow any mass. What we end up with is a momentum squared equals the difference between two squared momenta. Now look at what we can derive from this:

Equation 46 shows that curved spacetime is the difference between wave numbers (wave cycles per distance). The larger wave number (ks) corresponds to more wave cycles per distance, more space expansion pressure, and faster time. The smaller wave number (ke) corresponds to fewer wave cycles per distance, less space expansion pressure, and slower time (see proof at the update below). The net wave number (k) is the net spacetime expansion or gravitational field.

Now, take a close look at equation 22:

The intent of equation 22 was to isolate the momentum energy (pc). That's easy enough. It's just the total energy (E) minus the rest-mass energy (mc^2), all squared, of course. But something anti-gravity happens when you assume the total energy is conserved and increase the rest-mass energy: the momentum energy decreases! If equation 22 is a gravity equation, the momentum should increase when the rest-mass increases.

Suppose we assume there are two energy fields instead of one? Each has its own momentum and rest-mass energies. Spacetime would have a low rest-mass energy compared to, say, earth, but its momentum energy (dark energy?) could be higher than earth's energy field's. So imagine high-momentum spacetime expansion pressing against the earth and lower-momentum spacetime expansion pressing away from earth. The difference is a net inward force we call gravity. Equations 23 through 25 below concur:

If we take the above three equations together, we get a gravitational relationship between the net momentum (p) and earth's rest-mass energy (me*c^2). The greater the potential energy or rest-mass energy, the greater the net momentum towards earth. This is consistent with greater energy density causing more spacetime curvature, i.e., gravity. Where there's more energy density, there is bound to be more rest-mass energy.

Now, let's take a look at dark energy. Let's define it as vacuum energy. According to the Wilkinson Microwave Anisotropy probe, the energy density of the vacuum is around e-10 Joules per cubic meter. That would give it a mass or mass-equivalent density of approximately e-27 kilograms per cubic meter. Energy density, of course, is equivalent to pressure, so the vacuum has a pressure of e-27c^2 Pascals. Equations 47 and 48 below reveal a key difference between typical pressure and vacuum pressure.

At equation 47, expanding volume (V) relieves pressure, and, one would guess that the pressure eventually drops to zero and expansion stops. Equation 48 shows this is not the case. An increase in volume increases the vacuum energy, so the pressure remains constant and expansion continues. Equation 50, using the vacuum mass density, calculates the net pressure we interpret as gravity. At equation 51 we can increase the net pressure by adding a falling mass (m) to the vacuum mass. Equation 52 shows that the instant falling velocity is unaffected by the additional mass, since the mass's inertia cancels the mass's pressure contribution.

The diagram below is a one-dimensional representation of how gravity emerges from the expanding vacuum:

The red rectangle shows the difference between the incoming pressure (m'c^2/V) and the outgoing pressure (m'u^2/V). Notice how the falling mass (m') slows vacuum expansion to velocity w. One would think this would affect velocity v, but the velocity w goes in all directions and cancels itself. Equation 53 confirms this and confirms that mass m' inertia cancels its pressure contribution: m'/m'.

The next diagram demonstrates how mass m' causes a tidal effect:

In the above diagram, gravity is weaker above mass m and stronger below. The next diagram demonstrates the gravity of mass m'. (Notice in all the diagrams how the spacetime vacuum expands at the outskirts at a maximum rate of c^2. This appears as acceleration when Hubble's constant is used. For more details on the rate of expansion, click here.)

Of course we know when masses fall, they accelerate, so velocity v must increase, but how does this work exactly? Let's zoom in and have a look:

As a body falls closer to the planet's surface, overall energy density increases (more rest-mass energy in the given space). The vacuum expansion is slower and slower (progressively shorter double arrows). As a result, velocity v grows. Equations 56 and 56b sum all the little changes in velocity during a body's descent.

Using equation 57 below we can calculate the velocity a mass will accelerate to in a gravitational field. Massless particles, such as photons, don't accelerate, they gain frequency (see equation 58).

On a cosmological scale, galaxies are moving apart but they also have gravity. The diagram below shows how this possible:

In the above diagram, if we sum up all the vector arrows within the red rectangles, we get a net gravitational velocity (v) of zero. As a result, spacetime is free to expand. If, however, we zoom in closer to one of the galaxies (red dots), we encounter gravity (blue rectangles: c^2 - u^2 = v^2). Ironically, this gravity has no impact on the overall expansion. If two galaxies (blue dots below) are sufficiently close together, they can slow the expansion enough to create a gravitational attraction between them, but spacetime on the outskirts will still expand (see diagram below).

At the quantum scale, it seems highly probable that gravity boson(s) originate from the vacuum. The mysterious "dark energy" and graviton could be one and the same. It's also possible that any boson or combination of bosons can do the job.

Imagine a black hole with gravity so strong that not even gravitons or dark energy can escape. In other words, the huge mass of the black hole prevents an outflow of expanding spacetime. How does a particle floating in space know there's a black hole nearby? It knows from the massive inflow of vacuum pressure! In general, particles know how to move due to the net flow of the vacuum pressure--and whatever boson(s) it carries with it.

Update: Here's a mathematical proof showing the correlation between wave number and time rate:

Update: Below are dark energy-gravity equations that take into account Hubble's observations. First we define the variables:

The first equation below is power per area at any location in spacetime. Check out what can be derived from this equation.

Equation 3 above is the orbital velocity of a satellite. Equation 4 is the familiar Newtonian g-force. These equations were derived by dividing both sides of equation 1 by the dark energy velocity (Hr) and the mass density (ps). Below we derive the dark energy velocity by dividing both sides of equation 1 by the gravitational velocity and mass density. (See equation 7.)

We use the reciprocal of Hubble's constant as a benchmark time. Relative or proper time is 1/H'. If we take the limit of H' to infinity, we get the dark energy velocity at half the Schwarzschild radius. (See equation 8.) This shows that there is always some expansion velocity even when gravity is as powerful as a black hole's.

Update: When in a gravitational or dark-energy field, if objects move with space instead of through space, terms like pressure, energy and force only give us units. Read the new post entitled, "Does Gravity Really Exist?--Dark Energy's Equivalence Principle."