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In the proof that the speed of light is a constant we make the assumption that space at large scales is homogeneous, but there are patches of space where the density is higher and there are patches where it's lower. Does that mean that it is a little bit different in these patches? Maybe light could light travel faster or slower.

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The proofs are experimental. See the Experimental Basis of Special Relativity for details. In particular, you will be interested in the observations from cosmological sources. Perhaps the most interesting will be Brecher's observations of binary pulsars as that particular paper used high energy observations which are not subject to criticism based on optical extinction.

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  • $\begingroup$ How fast were the aircraft flying in the 1971 experiment? Do you know? $\endgroup$ Commented Jan 2 at 19:46
  • $\begingroup$ I do not know. They were commercial flights, so it would have been less than Mach 1. $\endgroup$
    – Dale
    Commented Jan 2 at 20:06
  • $\begingroup$ I did some digging. When compared to some universal starting point - perhaps the sun, the clocks travelled at: Westbound: 241 m/s Earthbound: 463 m/s Eastbound: 685 m/s So compared to their starting airport, they were about 220 m/s or 792 k/h. In reading the tests, it's funny that they say the westbound clock gained time, when this is only in comparison to the earthbound clocks. It would have been much better for them to say the westbound clocks were (nearly) stable and the earthbound clock lost some time and the eastbound clock lost even more time. $\endgroup$ Commented Jan 3 at 19:55
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You are correct that the speed of light "c" changes in different regions of space. This is because the speed of time changes in different regions. You have heard of warped spacetime. I am just referring to the warped time part of that.

First, keep in mind that if you are inside that region of space, you will never see the change in c because you, and any clock you hold, will experience the exact same change in the speed of time. But you can certainly notice it if you watch from a distant neutral point.

For example, imagine that there was a tunnel drilled right through Venus. And someone standing on Mercury pointed a laser at Earth. Before Venus crossed in between, the light would hit Earth in a certain time. But when Venus passed in between Mercury and Earth, it would take longer for light to hit Earth because of the time dilation caused by Venus. As you surmised, the density of mass in Venus caused a slowing of time and a slowing of the speed of light.

So in answer to your question, to a local observer the speed of light is constant. But to a distant observer, c can change because c simply reflects the speed of time itself.

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