The Beautiful Destruction of the Graviton

Does Gravity Really Exist?--Dark Energy's Equivalence Principle

According to the equivalence principle, when you drop a pen to the floor, the pen can be considered at rest while the floor moves to the pen. Because the pen is allegedly at rest, no force is needed to act on it and it can be any mass. Now here's the tricky part, because the pen has mass, it has a tiny bit of gravity of its own. It attracts the earth, or rather, the earth is at rest and the pen moves to the earth, so the earth can be any mass and no force needs to act on it. So one has to wonder: Is the earth moving to the pen or vice versa? How can they both be at rest and be moving? Is gravity really a force? It has been argued that it is not. As we shall see, one could also argue that gravity does not exist, especially when one takes a closer look at dark energy.

Galaxies are accelerating apart as the universe expands. Allegedly they are not moving through space, but rather, they are moving with space--and because they are moving with space, they can be any mass. No force is needed to push them apart? Kind of reminds you of gravity and the equivalence principle, doesn't it? It seems that dark energy is a kind of negative gravity, or, maybe gravity is positive dark energy? So many questions! But these questions will be addressed in this post. First, we define the variables we need:

Both gravity and dark energy have something in common: spacetime. To understand why galaxies move apart and why pens fall to the floor, we need to examine spacetime. What is it exactly? On the surface, it's a vacuum with dimensions of space and time. If we were put in charge of building a universe, how would we go about creating this thing called spacetime?

We could start by taking all forces (strong, weak, Higgs, EM, and unkown), add them together to get an energy field (see equation 1 below). We use that energy to derive equation 5: the Planck length.

At equation 4, notice how the energy field cancels itself. It could be any magnitude and any type of energy or combination of energies. It doesn't matter. In any case we get the Planck length unit. We can use this unit to create an arbitrary length (r) at equation 6.

Normally when we think of dimensions of space we think of fixed Cartesian coordinates (x,y,z), but our universe is not static; it is expanding--so how fast does our coordinate system move? We derive the answer to this question below (see equation 12).

According to equation 12, our universe's coordinate system is expanding at light speed, but this seems to contradict equation 13 below, which shows the expansion velocity (vd) depends on the distance r. Then again, the following equations show the velocity depends on which clock you use. If you use Hubble's constant, you get a velocity of Hr. If you use time t, you get light speed.

At equation 15, note that velocity Hct can go to infinity. This does not violate the light-speed limit if we assume galaxies are moving with space instead of through space. Moving with space means they are at rest. If they are at rest then no force or pressure is moving them. The math below demonstrates that the galaxies move the way they do regardless of how much or how little pressure vacuum energy has. The implication is "dark energy" isn't energy or vacuum pressure.

At equations 20 and 21 we see that mass (m) cancels itself no matter how big or small it is. Whether spacetime has a little or a lot of energy (mass), it moves at the rate of Hr or light speed (c), depending on the clock used.

To prove that dark energy and gravity are the same phenomenon we set up a series of Minkowski diagrams. The first one below breaks velocity Hr into a time component (Hrt) and a space component (Hrs).

At 23 we integrate over all the varying velocities at each point in spacetime to get the Pythagorean relation between Hr and it's components.

In the next diagram we express the distances (r's) in terms of spacetime (ct) and make some substitutions:

In the next diagram we measure velocity using time (1/t) instead of Hubble's constant:

Time t cancels itself and that brings us to the familiar Minkowski diagram that can be used to calculate gravitational velocity (vo):

At 28 we once again integrate over all velocities at all points in spacetime and get the Pythagorean relation between time velocity (u), space velocity (vo) and light speed (c).

We extract dark-energy equation 29 below from equation 23. From 29 we derive Newton's gravity (see equation 32).

Taking a few more steps and making a few more substitutions we derive Einstein's gravity (see equation 39):

We can also take equation 31 (restated at 40 below) and derive the Lorentz factor (equation 45):

Bottom line? We can start with dark energy and derive gravity! Or vice versa!

Now, the following 1D diagrams show how the equivalence principle works in terms of dark-energy. We use time t rather than Hubble's constant. That means all masses are coupled with c^2--hence the famous equation: E = mc^2. However, our first diagram is a universe with pure spacetime, no masses, so we just have c^2 in all directions:

Notice points A and B. They are static, i.e., going nowhere. At equation 46 we see why. The c^2's are equal and opposite and sum to zero.

Suppose we add a mass at point B. Doing so causes time to slow at that point. As a result, velocity c is reduced to velocity u. Equation 47 confirms this causes point A to move to point B.

Let's add a mass (m') at point A. Time is reduced, reducing c to w. We can see from the equations and diagram below that point B moves to point A and vice versa. The masses, no matter how large or small, located at those points move at the same rates as those points. When we add masses m' and m to points A and B, c^2 becomes m'c^2 and mc^2 respectively.

Now, it is commonly believed that gravity and dark energy are two opposing forces. One pulls everything together and the other pushes them apart. The following diagrams show this is not the case. The universe will expand at the rate of c or Hr no matter how much mass there is. For comparison, let's start with a universe with no mass and measure its expansion rate:

In the above diagram we interpret one direction as positive and the opposite direction as negative, so the universe expands at +/- c or +/- Hr (taking the square root of equation 54). (If we used a 3D diagram, each velocity would have an equal and opposite velocity; however, we only need 1D to illustrate the point.)

In the next diagram we introduce a mass and this appears to reduce the overall expansion rate, but the following math suggests this is relative, it depends on where the observer is located. An observer some distance from the mass will sum up a total value of c^2. According to the above Minkowski diagram, if space expands less, time expands more or vice versa. The total expansion rate is always Hr.

Note that equation 57 agrees with the corresponding Minkowski diagram.

The final diagram shows how the thing we call "dark energy" causes the thing we call "gravity" when matter is present. The red arrows are pens falling on opposite sides (pens moving with spacetime). the blue and red curves are satellites or photons moving along geodesic paths. Objects not moving through space move with space at the same rate, since they are technically at rest (mc^2).

Now you might wonder why mc^2 is rest mass energy and not light-speed energy. Well, you might think you are at rest, but to an observer a cosmological-horizon distance away, you are moving away at light speed! In fact, every point in the universe is moving at light speed relative to such an observer. So "at rest" really means you are moving with space and not through it. Oddly enough, if time is measured in units of r/c instead of Hubble's constant's reciprocal (1/H), points near and far (and their respective masses) are always moving at the speed of light! So rest-mass energy truly is E = mc^2.

Below we derive the dark-energy Lagrangian and equations of motion:

At 64 we recognize that the Ricci tensor and scalar have the same units as quantum wave numbers (k). As we shall see, it doesn't seem to matter what kind of wave numbers we use.

At 72 we have the Lagrangian (L). At 73 we have the speed of mass-less particles in a gravitational field (light speed c or Hr at the cosmological horizon). At 74 we have the speed of mass particles in a gravitational field--or is it a dark-energy field?

Final thoughts: Assuming objects can move with space and need no force to act on them, does this imply that no force-carrying particle is needed? The above math and diagrams show that there is no attractive force, per se. Gravity appears to be a by-product of expanding spacetime in different reference frames. Could this be the reason the "graviton" has eluded particle physicists?

Debunking the Equivalence Principle Thought Experiments and Proving the Equivalence Principle Mathematically

According to the textbooks, the principle of equivalence asserts that gravity and inertia are one and the same--not similar, proportionate or related to each other, but the same. According to the principle, you could kidnap some scientists by smothering each of them with an ether rag, place them in a rocket ship with an acceleration of 9.8 meters per second per second. When they regain consciousness, they won't be able to tell if the gravity they feel is from the earth or the rocket ship--there are no windows for them to look outside.

As the video above says, they can drop a pen and it will fall to the floor like a pen on earth. They are free to do whatever experiment they wish--and they won't be able to distinguish earth's gravity from the rocket's acceleration? Hmmm ... let's take a closer look at how a rocket propels itself:

According to equation 1) above, rocket fuel and oxygen enter the nozzle at a certain mass-flow rate and velocity, then exit the nozzle at a higher velocity and final mass-flow rate. Add to that the pressure difference. The total is the thrust force. Recalling Newton's third law: this is the action. The equal and opposite reaction is the rocket accelerating.

Equation 2) shows that the perceived gravity inside the rocket is acceleration (g). Since action and reaction are equal and opposite, we can set the thrust force equal to the total mass of the rocket (passengers, supplies, fuel included) times the acceleration g:

Doing a bit of algebra gives us equation 4)--the function g. Let's compare equation 4) to equation 5) below:

Equation 5), of course, is Newton's law of gravitation. Let's assume the scientists aboard the rocket are aware of equations 4) and 5). They're also aware of equation 1) that shows fuel mass flowing out the rocket's nozzle. How hard would it be for them to figure out that they are in a rocket ship and not in a sealed box on earth?

Not hard at all! In fact, all they have to do is weigh themselves repeatedly over time. The rocket's total mass (m) is decreasing thanks to the burning fuel exploding out the rocket's back end. This would cause the rocket's acceleration (g) to increase. So when the scientists weigh themselves they notice over time they are gaining weight (scientists' mass * increased g). Either their Weight Watchers diet is failing or they are in an accelerating rocket.

Unlike the rocket, earth's mass is virtually constant. If the scientists weighed themselves there, they would notice little or no change (assuming they didn't quit their diet).

Now, take another look at equations 4) and 5). Notice how mass (m) is in equation 4's denominator, but equation 5) has a mass variable in its numerator. Any change in mass would yield opposite results when comparing the two equations. If mass increases, the acceleration (g) in equation 4) goes down, but equation 5's acceleration (g) increases--and vice versa!

To claim the g in the rocket is indistinguishable from the g on earth requires us to ignore the physical processes involved. Prior to any mass loss, if one drops a pen in a rocket accelerating at 9.8 meters per second per second, the pen's inertia keeps it in place and allows the floor of the rocket to accelerate up to it. A casual observer aboard the rocket will most likely get the impression the pen dropped to the floor. Is this really sufficient evidence to support the claim that inertia and gravity are one and the same? If so, one could also claim that gravity is not a genuine force; it's a byproduct of another force such as the rocket's thrust force.

If gravity and inertia are the same, it should be possible to fool the kidnapped scientists aboard the rocket. Our first attempt failed--the rocket lost some of its mass causing g to increase. Let's try a different approach: We smother the scientists with ether rags and place them in a rotating, donut-shaped space station. When they regain consciousness, it is hoped they will fail to distinguish between the angular acceleration (g) and the earth's gravity.

The advantage of this new arrangement is the space station stays in one location, making it easy for a maintenance ship to top off the station's fuel tank. We avoid a mass decrease, due to fuel consumption. This should provide an artificial gravity indistinguishable from earth' gravity, right? But then something very routine happens: the scientists place their garbage and waste in the disposal chute. The waste is jettisoned into space. The scientists are not aware of this. For all they know, the garbage goes to a dumpster here on earth. But once again, when they weigh themselves, their respective weights increase.

Curses! It's that bloody conservation-of-momentum rule! When overall mass decreases due to garbage disposal, the velocity increases, so does the angular acceleration. If the scientists were on earth, jettisoning waste would still decrease the space station's mass, but the overall mass of earth would remain virtually the same, since the garbage ends up in a dumpster on earth. As a consequence, earth's gravity maintains its value.

Then again, if the garbage stays on the space station, the perceived gravity there should be the same as earth's, and, we should be able to say with confidence, "Inertia and gravity are one and the same." But before we get too cocky, let's put this claim to another test. Imagine the earth and a red meteor coming together. We can interpret this event in one of two ways: the meteor is falling to earth, or, the meteor is at rest and the earth is moving toward it.

If inertia and gravity are the same, then it should be possible for a blue meteor to be at rest on the other side of the earth. The earth should move toward both meteors:

No force acts on the meteors--they are at rest, floating in space. The force, whatever it may be, is moving the earth in opposite directions to meet the meteors. Then again, maybe it doesn't really work that way. Could it be that the meteors are really falling to earth?

Assuming the meteors are falling to earth, a force must be causing them to accelerate. We call that force gravity. It seems clear at this point that gravity and inertia are not one and the same but are only similar to a casual observer who doesn't ask too many questions.

There is another key difference between inertia and gravity that involves time ... the time it takes a boson, say, a graviton to notify matter of a change in status in other matter. To properly distinguish between gravity and inertia, we need to make Newton's equation and Einstein's field equations time dependent. The same goes for the rocket-thrust equation.

Take a look at equation 15) above. If the sun's mass (m) were to drastically change in the current time (t), it would take about eight minutes (r/c) for us here on earth to feel the gravitational effects. The change will be felt by us at time t+r/c. Equation 16) concurs. Any change in spacetime curvature won't happen instantaneously. It takes time for gravitons to move through space from the sun to the earth.

Equation 17) tells a different story. If the thrust force changes in the rocket ship we discussed earlier, the acceleration (or perceived gravity) changes instantaneously--there is no need for gravitons to notify distant objects of the change in status, and to tell them to accelerate. For it is the rocket floor that is accelerating to objects at rest. Thus, when inertia mimics gravity, it takes less time.

The diagrams below illustrate the time difference between gravity and inertia:

Notwithstanding the case presented above, it is possible to mathematically prove the equivalence principle. Imagine a mass m. It is equal to itself (see equation 18 below). To get m moving we add some energy.

If we add its rest-mass energy to its momentum we have a total energy of m'c^2 (See equation 19). We perform a little algebra, but when we get to equation 21, there's a problem.

Nothing can go faster than light, so we take the momentum term (mv). We reduce the velocity a little and increase the mass a little (see 22). We do some more steps. At equation 26 we get a velocity that is less than light speed (c). We derive equation 28 which shows mass m increasing to m' when it is going velocity v.

We can conclude that when mass m is in motion, its mass increases. When it is at rest, its mass stays the same. Now let's do the same exercise; only this time mass m is always at rest and never increases. However, we place mass m near mass M which has a strong gravitational field:

Mass m appears to be falling or orbiting but it is really at rest. It's as if mass M is moving instead, but it's at rest also. So what's going on? What's moving are the coordinates of spacetime. Mass m appears to be accelerating because its point of location is accelerating. Anything located at that point, regardless of its mass, will fall at the same rate.