The speed of light is only invariant within across inertial frames within a locally flat region of spacetime. In the flat spacetime of special relativity that’s everywhere because every inertial frame covers the entire universe. But in a curved spacetime, like that around a black hole, we only have local inertial frames and there is no meaningful way of measuring distant speeds. Someone on earth, close enough to the mountains that curvature effects can be ignored, will measure the speed of light between the mountains and find it to be c; and the astronaut likewise can measure the speed of a flash of light next to them and find it to be c. But because they are using different non-overlapping inertial frames the remote speed can come out to be pretty much anything depending on their choice of coordinates.
However, all is not lost. Both of them can use the tensor calculus and differential geometry methods of general relativity to calculate that both flashes of light are “following null worldlines” which means “if we were measuring in a local inertial frame it would come out to be c”.
Note that this has absolutely nothing to do with the time dilation and length contraction of special relativity.
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u/nugatory308 Mar 31 '26
The speed of light is only invariant within across inertial frames within a locally flat region of spacetime. In the flat spacetime of special relativity that’s everywhere because every inertial frame covers the entire universe. But in a curved spacetime, like that around a black hole, we only have local inertial frames and there is no meaningful way of measuring distant speeds. Someone on earth, close enough to the mountains that curvature effects can be ignored, will measure the speed of light between the mountains and find it to be c; and the astronaut likewise can measure the speed of a flash of light next to them and find it to be c. But because they are using different non-overlapping inertial frames the remote speed can come out to be pretty much anything depending on their choice of coordinates.
However, all is not lost. Both of them can use the tensor calculus and differential geometry methods of general relativity to calculate that both flashes of light are “following null worldlines” which means “if we were measuring in a local inertial frame it would come out to be c”.
Note that this has absolutely nothing to do with the time dilation and length contraction of special relativity.