# Equivalence of gravity and acceleration

Neil
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Topic 192263

I was doing a (special) relativistic problem the other day and to help started reading some of my old books. I rediscovered this idea in one of them as being (one of) the things which leads to the general theory. I asked myself this question but couldn't answer it...

If I create two clocks and accelerate one (A) continuously using, what would appear to be in its rest frame, a constant force. It would get faster and faster, and its ticking compared to the B clock would get slower and slower.

If I create two clocks and put one (A) in a gravitational field, does its ticking not only slow down, but gradually slow down more and more in comparison with the B clock?

Or if I take two clocks to different places on the earth, say one next to a very heavy goldmine in a very thick piece of crust, and one high over the ocean on a very thin piece of crust, where the gravitational field has different strength. Would they tick at different rates and would the difference between the ticking rates increase over time?

Neil

Brian O'Reilly
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### Equivalence of gravity and acceleration

There are two effects here. One is the time dilation effect from the object moving
faster and faster (assuming constant acceleration). The other is not a cumulative
effect since the clock's condition does not change as a function of time i.e. it
experiences a constant acceleration due to gravity and will tick slower. There's a
good tutorial on several aspects of relativity at:

http://www.astro.ucla.edu/~wright/relatvty.htm

Brian

Neil
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### RE: It experiences a

Message 58598 in response to message 58597

Quote:

It experiences a constant acceleration due to gravity and will tick slower.

Thanks. I sort of knew that was the case, but I think I have been confused by the use of the word "equivalence".

It's quite clear that the two situations are not equivalent. In the first there is relative motion and in the second none. But I'm struggling to think of a case where one can substitute continuous acceleration in the place of a static gravitational field, and not end up with relative motion. (That is other than the case of a single particle). Take my example 2b where identical, non-moving clocks placed at different points in a non-uniform but static gravitational field tick at different rates. How can I substitute accelerations, without any field at all in place of the field and end up with an "equivalent" situation? Is the best I can do in this equivalence game to accelerate both clocks equally, and get rid of the field at one of the locations, leaving some residual field at the other?

Mike Hewson
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### RE: Thanks. I sort of knew

Message 58599 in response to message 58598

Quote:

Thanks. I sort of knew that was the case, but I think I have been confused by the use of the word "equivalence".

It's quite clear that the two situations are not equivalent. In the first there is relative motion and in the second none. But I'm struggling to think of a case where one can substitute continuous acceleration in the place of a static gravitational field, and not end up with relative motion. (That is other than the case of a single particle). Take my example 2b where identical, non-moving clocks placed at different points in a non-uniform but static gravitational field tick at different rates. How can I substitute accelerations, without any field at all in place of the field and end up with an "equivalent" situation? Is the best I can do in this equivalence game to accelerate both clocks equally, and get rid of the field at one of the locations, leaving some residual field at the other?

What resolves this is, I think, is that the Equivalence Principle is only locally true - and probably only exactly true when taken as a limit. By 'locally' I don't just mean in space but in time also ie. local in spacetime. Thus you cannot apply it globally in space or time, so thus it will fail to some level of accuracy by awaiting sufficient extension in space ( travelling ) or time ( passing ).

To exhibit the differential behaviour of the clocks you would separately integrate each's experience, ie. a limiting argument : with an 'infinite' number of 'infinitesimal' steps, along their respective histories. An asymmetry would then appear between the descriptions of the two scenarios - basically because there is going to finally be quite a physical separation between the two clocks in the first instance that does not occur in the second example. I guess that this would devolve to Brian's description...

I think we should wait for 4D glasses to be invented, then it would all be much clearer! :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal