The Lorentz transformations allow us to deduce how the acceleration of a material body changes when we move from one Galilean reference frame to another. But what if we consider not the acceleration of a material body but the acceleration of gravitation at a given point in the space of the gravitational field (which we could, for example, assume to be uniform)? Passing at this point from one Galilean reference frame to another with a constant relative velocity (e.g., the worldlines of two observers intersecting at a point with a given constant relative velocity), does the fact of having an acceleration of gravitational origin mean that it does not change during this passage? On the one hand, there could be no change, since the relative speed between the two reference frames is constant, or there could be a change, since there is a dilation of time and a contraction of lengths during the passage from one reference frame to another, which means that the gravitational acceleration, say g, would vary, going from g in reference frame R to g' in reference frame R', which is strictly different from g.
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2$\begingroup$ It is reference frame independent $\endgroup$– RitzthephysibeastCommented Sep 11 at 7:53
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$\begingroup$ What's the reason for this? $\endgroup$– RoyCommented Sep 11 at 22:16
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$\begingroup$ Hi Roy. Welcome to Phys.SE. How do you define a Galilean reference frame and change thereof in a curved spacetime? Or are you only asking about flat spacetime/special relativity? $\endgroup$– Qmechanic ♦Commented Sep 12 at 6:00
2 Answers
Let's say a catapult and a trebuchet are standing on a planet. These weapons happen to have similar specs in the planet frame.
They must have similar specs in other frames too, otherwise the winner of war becomes a frame dependent thing.
So the transformation laws of the specs of an weapon powered by potential energy of a spring are the same as the transformation laws of specs of a weapon powered by gravitational potential energy.
General relativity is reference frame independent because it's constructed using tensors, which means its laws take the same form in all coordinate systems. General relativity also doesn't contain any invariant geometric background structures, making it background independent.
You must remember that Einstein's motive for developing a new theory of gravity was because a frame independent theory was required and Newton's law of gravitation was based on mass (or energy via $E=mc^2$) which is not invariant (meaning the measurement is frame dependent), so it cannot form the basis of a physical theory. The result is now referred to as the General Theory of relativity and it is designed to be frame independent.
So, the value of g doesn't change in different reference frames.
I strongly recommend to see this- https://plato.stanford.edu/entries/spacetime-iframes/
Hope that answers your question.
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$\begingroup$ You mention the covariance of tensors with respect to changes of local charts of spacetime. The local diffeomorphisms of these chart changes do not involve relative velocities between observers; and this would be just as true in special relativity by non-parameterized and constant Lorentz group transformations on coordinate charts chosen on Minkowski space. But it turns out that some of these transformations are also parameterized by relative velocities between observers. And this is the situation I'm considering. Transferring the problem to general relativity doesn't change the situation. $\endgroup$– RoyCommented Sep 12 at 7:35
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$\begingroup$ Can you please elaborate what you want? The velocity change with respect doesn't seem to affect anything big in particular $\endgroup$ Commented Sep 12 at 9:54