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This question of mine started shaping in my head first while I was looking for the most fundamental answer for the speed of light's value and its property of being the limit.

I have convinced myself that it's due to the structure of spacetime, or in other words because the spacetime interval is expressed as $ds^2=dx^2+dy^2+dz^2-c^2dt^2$ which is not Euclidean (for the limit property part) and has the value $9\times10^{10}$ in front of the $dt^2$ term (for the value part).

So from this conclusion, and by knowing the fact that the coefficients of those coordinate terms can change with a "curvature" in spacetime, I wonder if the speed of light can have some other value in different curved regions in spacetime.

I don't mean it as differing between different observers in that same curved region but as differing between different observers in different regions with different curvatures.

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  • $\begingroup$ The value of $c$ is $3\times10^8\,$m/s so $c^2=9\times10^{16}\,$(m/s)$^2$ i.e. you are missing 6 orders of magnitude or something. But it really does not matter. The actual thing that matters is that if you wish to understand the Special Theory of Relativity, you should be measuring time in light-metres, and if you do so, then all 4 measurements are using the same units, and it no longer makes sense to ask what happens if different regions of spacetime have different speeds of light. Even if it is different, no experiment can tell that they are different. $\endgroup$
    – naturallyInconsistent
    Commented Jul 24, 2023 at 2:53
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    $\begingroup$ The local speed of light is always c for any observer, but see en.wikipedia.org/wiki/Shapiro_time_delay $\endgroup$
    – PM 2Ring
    Commented Jul 24, 2023 at 2:56
  • $\begingroup$ @naturallyInconsistent I meant that in km/s but yeah guess I forgot the units. Thank you for your answer. $\endgroup$
    – gnyszbr21
    Commented Jul 24, 2023 at 3:13

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It is not a change in curvature that precipitates a change in the coefficients of the metric, but a change of coordinates. Indeed, at any event in any spacetime (regardless of how strongly curved it is), there are coordinate systems where the coefficients are all $\pm 1$.

What curvature does is prevent extending such coordinate systems. These inertial coordinates are only valid locally.

Similarly, even in flat spacetime the metric coefficients can be changed arbitrarily at a given event by simply changing the coordinates appropriately. This can change the speed of light to something other than $c$. Note: this does not violate the second postulate because the second postulate applies only to inertial frames.

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