Prove that acceleration and gravity are really the same?

[HALON]

Here's how I entertained the old standard. I knew it of course.

In F=m_1a and F=Gm_1m_2/r^2

let m_1=1, let a=1, and let r=1. The mass of planet X is m_2 but let's call it m_X

Then to find m_X we get

m_X = 1/G

This particular answer is nice and neat. It's just a mnemonic.

 

[Nugatory]

This particular answer is nice and neat. It's just a mnemonic.

But rather unhelpful unless you know the value of $G$ in units in which $r$ and $a$ are both equal to one

 

[Wes Tausend]

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Interesting thread!
Equivalence is Thee Key to GR.

For the gentlemen that found difficulty in understanding the Equivalence principle, it may be helpful to imagine the situation of three clowns drifting in power-off-mode inside a free-floating spaceship in outer space, far from any major gravities.

Imagine two of them, Moe and Larry, are moving hand-to-hand along a longitudinal "fire" pole that runs dead-center the length of the spaceship just for such weightless ambulatory purposes. They have already traveled 100 feet along its length from the back towards the front of the ship. The ship is so massive that a couple of humans pulling on it to propel themselves has negligable inertial effect on the ship.

About this time, the third clown, Curly (always the darn fool) unfortunately started the powerful, poorly tuned spaceship engines in notch one throttle, and the back-firing spaceship suddenly accelerated with a bang up to 32 ft/sec/sec and held there. Moe and Larry lost their handhold and were left floating as the rear bulkhead of the ship approached rapidly. Moe happened to be carrying an accelerometer that had been faithfully reading zero, at least until the rear bulkhead crashed into them. The accelerometer needle bounced wildly high from the impact for a moment, and then settled down to the ship acceleration of 32 ft/sec/sec. The accelerometer lived... and Moe and Larry were livid too.

Curly had just caused them an accident equivalent to falling 100 feet on earth. When the engines fired, Moe and Larry didn't initially accelerate until they hit the floor (the bulkhead hit them), which is really a more correct Einsteinian-like Equivalence definition of an earthern episode of freefall. For the record, in spite of two broken legs, when he got the chance, Moe slapped Curly up and poked him in the eyes for the latest dumb stunt of course.
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Gravity is a force which causes acceleration. Gravity is not the same as acceleration.

If gravity were acceleration, then the radius of Earth would have to be expanding with an acceleration of 9.8 m/s^2, and we would have been very confused when we measured it from higher altitudes.

Actually, in both cases, Earth atomically expanding or gravitational attraction, higher altitudes would register less acceleration, just as one weighs less at the top of a mountain. It is a matching qualitive property only, and a little difficult to understand at first, but here is the reasoning.

Suppose Captain Kirk and Scotty always happened to weigh exactly the same as long as they weighed in side-by-side on the same bath scale on Earth. Then one day Kirk took the scale with and climbed a nearby mountain that was 1/2 mile high and weighed himself again. He called Scotty and told him he weighed slightly less at the top. Scotty, finding this interesting, asked Kirk to take a picture. Kirk took a selfie including the scale dial for proof and sure enough, later the picture showed he weighed less at the top. But back at the bottom, side-by-side he again weighed the same as Scotty.

The next day both men weigh in and they again weigh the same of course. But today is the start of their trip to a new quadrant of space, and they climb aboard the latest Enterprise spaceship. It is a big cigar-shaped craft, 1/2 mile long. Kirk goes to the head of the ship on the bridge. Scotty remains in the rear of the ship in the engine room, and they take off. For comfort with simulated gravity, they soon drop to 32 ft/sec/sec acceleration after initial take-off. They both end up standing on separate bulkheads behind them, the one nearest to themselves, Kirk in front and Scotty in the rear. They are 1/2 mile apart.

Under constant acceleration, Kirk will find that he weighs slightly less than Scotty again. The reasoning is that as the ship attains higher and higher speeds, it also begans to continuously foreshorten according to Special Relativity. So as the ship gets shorter, either Kirk is relatively accelerating slower than Scotty, or Scotty is relatively accelerating faster than Kirk. Either way Kirk experiences less acceleration than Scotty, proven if he checks himself against the trusty ol' spring-loaded bathroom scale.

Not only that, but Kirks precision watch will run faster than Scotty's identical timepiece, on the same ship!

Now just because both the mountain and ship are exactly the same half-mile length does not mean Kirks weight loss will necessarily be the same percentage in both cases. It is my understanding (according to a better mathematician on these forums than I) that the two effects, though similar, are not quantitively the same. They are, however qualitively very similar. And as well they should be... after all Equivalence is equivalent to a very great degree, good enough for Einstein to base his new theory of GR upon.

Wes
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[rcgldr]

Back to the original post, the equivalence principle is based on local observation, where the observer is either in a gravitational field or being accelerated. In the case of an "external" observer, such as an observer in an inertial frame where distant stars can be used as a frame of reference, then that observer can determine if an observed object is being affected by gravity and/or by acceleration.

 

[Wes Tausend]

Back to the original post, the equivalence principle is based on local observation, where the observer is either in a gravitational field or being accelerated. In the case of an "external" observer, such as an observer in an inertial frame where distant stars can be used as a frame of reference, then that observer can determine if an observed object is being affected by gravity and/or by acceleration.

Unless the acceleration effect was pervasive throughout the universe.

Is there a way by which we can prove that acceleration and gravity are really the same?

Yes, back to post #1 instead of post #2. When #2 immediately, and mistakenly, implied doubt, I thought it was quite important to point out to the OP that the Equivalence principle is much more equivalent than most folks think.

For the OP, it is not possible to "prove that acceleration and gravity are really the same", only point out that, under some conditions, if all is considered, it becomes impossible to tell the difference. They are at least identical twins. It has been merely our convenient choice to sometimes regard them as separate individuals. Einstein was not the first to realize the remarkable equivalence (Newton and others certainly did too), but he was the first to postulate that they are equivalent so that he could treat the "similarity" mathematically in the new light of SR.

To postulate Equivalence, Einstein imagined a couple of scientists being drawn up in an enclosed chest, and not being able to distinguish whether they were in a gravitational field or not. Although Einstein never stated so, one of the tests could have been that two objects, when dropped simultaneously, might fall parallel, or might fall in such a way as to move closer together on the way down. In a real gravitational system they would fall closer as they dropped (towards earthen center). An expanding globe would allow for this to be indistinguishable as long as the scientists, and chest, and space between them equally expanded also. The psuedo-gravities would be even more identical in this case over the plain chest example. The hapless scientists would not realize the wholesale change in size, only the resulting inertial effect they would surely assume is that of gravity.

This expansion principle brought up by post #2 arose before. Perhaps with great foresight, the great French mathematician, and philosopher of science, Poincaré, once imagined that the world could change dimension overnight, and no one could measure the difference the next day. Or perhaps the next moment. Measure no, but what about the inertia?

Wes
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[Nugatory]

Unless the acceleration effect was pervasive throughout the universe.

Which isn't possible (and I presume that's why you put the smiley in there).

An infinite charged sheet at constant potential would fill the entire universe with a constant electrical field, but there's no analogous gravitational solution.