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Popular expositions of general relativity start with a thought experiment showing that it is impossible to distinguish a constantly accelerating frame of reference in a free fall from a free floating frame of reference.

Thought Experiment:
Person A is a small closed box, free-falling towards earth. Person B is in a small closed box floating around in space. If they both do the same experiments, they should see the same results. For example, if they have a small ball and toss it inside their box, they would both see that ball travel in a straight line (not curving) towards the wall. They would also both feel themselves floating around as if there was no gravity.

The same thought experiment could be applied to a frame undergoing a constant jerk. Does that lead to a new theory of relativity?

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    $\begingroup$ In what thought experiment is it impossible to distinguish a constantly accelerating frame of reference from a free floating frame of reference? $\endgroup$ – md2perpe Sep 30 '18 at 16:21
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    $\begingroup$ I think you have either misspoken or misunderstood. Many popular expositions begin by talking about the path of a light beam in an accelerating frame of reference; showing that it is possible to distinguish accelerating frames from inertial ones. Then they talk about the equivalence between gravity and acceleration. $\endgroup$ – dmckee --- ex-moderator kitten Sep 30 '18 at 16:37
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    $\begingroup$ "free-falling towards earth" OK. That gets to the core issue. The notion of acceleration is defined differently in Newtonian and Einsteinian physics. I've written a little about the difference on the site before. The freely falling frame is inertial and the standing on the ground frame is not inertial. Wrapping your head around this requires starting from the principle of relativity and is a little easier if you have already had a detailed treatment of inertial pseudoforces in classical mechanics. $\endgroup$ – dmckee --- ex-moderator kitten Sep 30 '18 at 17:13
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    $\begingroup$ A frame of reference with a jerk will not be distinguishable from a gravity field varying in the same way. $\endgroup$ – md2perpe Sep 30 '18 at 17:20
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    $\begingroup$ @md2perpe "A frame of reference with a jerk will not be distinguishable from a gravity field varying in the same way." - According to what? It is not given and not obvious that the next derivative is relative. For example, speed is relative in the classical physics, but acceleration is not. So the fact that acceleration is relative in relativity does not automatically imply that jerk is also relative. $\endgroup$ – safesphere Sep 30 '18 at 17:53
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Gravity happens to measurably correspond to a constant acceleration, not jerk. So, no, this doesn't lead to a new theory of gravity. At least not immediately. On the other hand, strictly speaking, there is actually a slight jerk involved in free fall if an object falls long enough for the increasing gravitational field to become relevant. Whether a model for our world involving relativity of jerk could be built, is something I'd like to know myself...

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