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In the special relativity it is postulated that the speed of light in vacuum is constant in all directions. This postulate is motivated by the Michelson-Morley experiment.

As far as I understand (am I wrong?) in general relativity it is also postulated that the speed of light is constant in all directions even in the presence of gravity (but no other matter). (This is my interpretation of the proof of formula (84.6) in Landau-Lifshitz, vol. 2. See $\S$ 84 there for more relevant details.)

Was this postulate confirmed experimentally at least in some situations? Is my interpretation correct?

Qmechanic
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MKO
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  • Hi, you should include the relevant formula and the relevant parts of the proof about which you are asking in your question (and preferably not as an image). Not everyone has a copy of LL open in another window :) –  Feb 23 '22 at 09:32
  • The speed of light is always locally equal to $c$ (in all directions) but away from the observer it can be greater or less than $c$ even in flat spacetime if the observer is accelerating. – John Rennie Feb 23 '22 at 09:38
  • @DvijD.C.: I think my question is clear without this formula. I referred to LL since that was just my motivation of the question. Including the formula to the post would require extra notation, would be too technical, and... not immediately relevant. – MKO Feb 23 '22 at 09:38
  • In my copy of L&L equation 84.6 is $d\ell^2 = \gamma_{\alpha\beta}dx^\alpha dx^\beta$. I don't see how this relates to the constancy of the speed of light. – John Rennie Feb 23 '22 at 09:44
  • @MKO Well, if one ignores your references to LL, the answer to your question is trivially a "yes". The speed of light is $1$ and is so in all directions (locally) in GR -- this follows directly from the local flatness theorem (or the equivalence principle). Since the highlighted part of your question asks if "[your] interpretation" is correct and the only "interpretation" that you have referenced to is "[your] interpretation of the proof of formula (84.6) in Landau-Lifshitz", I'd say the LL stuff is immediately relevant. –  Feb 23 '22 at 09:45
  • @DvijD.C.: Apparently you are right, and my question looks now like a tautology. Still, I am wondering if there were Michelson-Morley type experiments in presence of gravity, but I cannot formulate this question more precisely. – MKO Feb 23 '22 at 09:54
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    Does this answer your question? The Michelson-Morley experiment –  Feb 23 '22 at 10:03
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    You can't measure the one way speed of light. It was assumed in SR that speed of light is same in all direction. This assumption was then carried forward to GR. – KP99 Feb 23 '22 at 10:22
  • @DvijD.C.: Yes. Thank you. – MKO Feb 23 '22 at 10:23

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