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It is well understood that Einstein’s General Theory of Relativity explains how gravitational effects appear to occur instantaneously at a distance. Mass warps spacetime and so objects simply follow straight lines in this warped 4D spacetime. But at what “speed” does spacetime warp?

Gedankenexperiment: Suppose an electron is moving through space in a straight line according to some observer. At time t, exactly 1 light year away from the electron according to our observer, a massive object is dropped, warping spacetime. At what time after t does the electron’s path change to follow the new geodesic and ‘curve’ with respect to our observer?

You could reverse the thought experiment for removing mass which is perhaps more common in our universe at macroscopic scales.

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  • $\begingroup$ At what time after t does the electron’s path change” - Instantly - as soon as the moment of $t$ happens in the frame of the electron. Simultaneity is relative in relativity. Your question is not well defined, because $t$ in the frame of the mass is not the same as in the frame of the electron. $\endgroup$
    – safesphere
    May 10, 2020 at 13:37
  • $\begingroup$ Please note that my comment above does not contradict the answer of @S.McGrew. Both describe the same phenomenon from different angles. The key is that you need to be very clear whose clock you use to measure time of the change happening one light year away. Time is a local concept in general relativity. Measuring time remotely is not well defined. You can say ”here now”, but there is no such a thing in general relativity as “there now”. $\endgroup$
    – safesphere
    May 11, 2020 at 16:12

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Disturbances in a gravitational field propagate at the speed of light. That is, "warpage" of spacetime propagates at the speed of light.

The speed of light, measured locally in the vacuum, is always c. But if you work out a way to measure from here the speed of light somwhere else at a different gravitational potential, you will say that the speed there seems different from c. If you go over there and do the measurement, though, you will get precisely c. That is all due to - or at least tightly entwined with - gravitational time dilation.

The trajectory of a gravitational wave curves due to gravitation just the same way as the trajectory of a light wave; the two move at the same speed: always c when measured locally. A "warp" - i.e., a propagating change in the gravitational field - is tilted in 4-space in such a way that the change follows the local speed of light.

The only meaningful way to measure c is locally, since the relative value of c between here and there depends on gravitational potential at the places where it's measured. Light may take a billion years to get to you if it originates very close to a black hole's horizon just a million light years away. And of course it will be red shifted down to the microwave range.

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  • $\begingroup$ I've upvoted this Q&A, but, because of much recent consideration of black holes in cosmology, in combination with the fact that they're usually considered to be "causally separated" from the space containing objects whose visibility may be obscured by their presence, I have to point out the fact that Einstein, at least as early as the publication of his 1916 "popular science" book about General Relativity, specified that the speed of light may vary between localities, although not within the vacuum of our locality. $\endgroup$
    – Edouard
    May 9, 2020 at 20:04
  • $\begingroup$ For the benefit of the new participant, I've got to add that "locality" in physics means any place where the curvature of spacetime that manifests itself as "gravity" is negligible--in other words, where apparently-empty space appears to be "flat" in the simplest sense of "flat" (which means "not having curvature naturally and immediately perceptible, AS curvature, to organisms like ourselves"). $\endgroup$
    – Edouard
    May 9, 2020 at 20:33
  • $\begingroup$ @Edouard, please provide a link to a reference supporting "Einstein, at least as early as the publication of his 1916 "popular science" book about General Relativity, specified that the speed of light may vary between localities, although not within the vacuum of our locality. " $\endgroup$
    – S. McGrew
    May 9, 2020 at 21:06
  • $\begingroup$ Here's a link to a "free online" 1920 translation of Einstein's 1916 pop.sci. into English: The remark I'm referring to (which absolutely astonished me when I first read it, in a different "free online" format) was, unfortunately, somewhere in the middle, but the book's readable and requires very little math. I'll keep looking today, and let you know if I can cite chapter and verse: If you want to meanwhile post a PSE question about his assertion, you might get a response more quickly. The link to the book itself is en.wikisource.org/wiki/… $\endgroup$
    – Edouard
    May 9, 2020 at 21:39
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    $\begingroup$ @Edouard I am sorry, but I don’t know enough about the torsion theory to be comfortable adding a tag for it. Perhaps someone else with the relevant expertise? $\endgroup$
    – safesphere
    May 11, 2020 at 4:18
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The electron would feel the "warpage" of space and move on a new geodesic 1 year after the mass has been dropped. That is because as S. McGrew said, the "warpage" travels at the speed of light (since no information can travel faster than light speed). Similarly if the mass is removed, it would take 1 year for the electron's path to change (assuming that it's distance to the mass has not changed due to the geodesic it was taking.)

Response to comment: While GR suggests that the speed of light can decrease in a gravitational field, it can never increase. The SR postulate that information can never travel faster than light still holds. Gravity propagates at not the speed of light altered by the gravitational potential, but at the speed limit $c$ of our universe, which happens to be the speed of light when gravitational fields are absent. That is the difference. Also, the time I mentioned (1 year) is according to the electron. You could take into account time dilation, but come on: when the electron is 1 light year away and we are talking about time scales as large as a year, we can ignore it completely, to get a totally satisfactory answer.

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  • $\begingroup$ This answer is incorrect. See my comment above. You are not specifying by whose clock the time of events is measured. $\endgroup$
    – safesphere
    May 11, 2020 at 16:14

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