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Tuesday, April 4, 2017

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.

2 comments:

  1. These examples seem pedantic to me.

    In your first example we could vary the fuel output to keep the acceleration constant to within the accuracy of the set of scales. I admit that it is probably impossible to damp the noise of the engines though.

    The second seems slightly ridiculous because they could easily tell that they were in a spinning spaceship via the procession of a gyroscope. Incidentally if you expelled the waste parallel to the angular momentum vector then you wouldn't change the rate of rotation of the disc.

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    1. See equations 3 and 4 for the first example. The acceleration is a function of mass, so to maintain it requires maintaining constant mass. For the second, I hadn't thought about the gyroscope, so good one! I don't see your last point, since the rate of rotation would increase, given a fixed amount of force, regardless of the angle you expel the waste. Less mass is less mass in any case.

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