Constancy of speed of light in GR

Question

The speed of light is the same constant $c$ as seen by any inertial frame in SR. But, is the speed of light same in all frames (inertial or otherwise) in GR?

Answer 1

There is a difference between saying the speed of light is the same in any frame, and saying it is the same in all frames - because certain unexamined assumptions creep into the second statement, about the comparability of speeds.

It is like asserting that there is the same ratio of boys and girls in any school classroom. Equivalently, Is the ratio the same in all classrooms? Yes.

But, from this, do we infer that every classroom must contains the same number of girls and boys? No - in fact a classroom may contain no children, or be stuffed like the black hole of Calcutta, and still the asserted ratio is maintained so long as every boy that is present has a counterpart girl that is present.

So what people fail to readily appreciate is that, whilst the ratio may be constant, the underlying number of boys and girls may differ freely.

In other words, what does not need to stay constant in relativity, is the scales of time and distance in which the measurement of speed is expressed.

So the speeds are not the same in all frames, when the person asking the question implicitly expects that the speeds in each frame that are being compared, are being measured on the same scales.

The frame, in relativity, is the thing that defines those scales against which speed is expressed.

I am of course saying nothing yet about why this approach is valid - I am not an educator, so I may not be entirely clear, and may raise more questions than I answer, but here goes.

The reason is it valid is because we have no known way of measuring time without measuring distance. That is, clocks don't measure "time", they measure speed - the rate of a physical process that involves regular changes in spatial position.

The pendulum of the grandfather clock moves back and forth in space - and it is only by devising a much more complex mechanism, that any apparent movement forward in "time" is counted and seen.

Indeed, it is not entirely clear what "time" actually is - it is that "other thing" inside a timepiece that is not the position of the mechanism, but something else that has to be postulated to account for how the positions of various parts of the mechanism move. Nor is it clear that a perfect pendulum moves forward in time - since it simply repeats the same positional relations, once forwards and once backwards.

Other clock mechanisms exist besides the pendulum, but they are all predicated on a reciprocating action across a measure of space, and some assumption about the constancy and regularity of the time it takes to traverse that space.

It follows therefore that if that reciprocating action takes a different period of time to complete a cycle across a region of space, the time measured will change.

That is how, in relativity, the local speed of light is always constant - that is, a distance traversed reciprocally, always involves a constant period of time on a clock, because the clock itself measures that.

Therefore, if it takes longer for a fixed distance to be traversed, the clock will slow down, and therefore the "time" taken to traverse that distance always remains a constant on the clock, because it is the speed of that journey which the clock was measuring in the first place - it was not measuring "time".

In other words again, clocks assume the speed of light to be constant. If it is not constant (in objective terms), then the clock cannot tell, because the reading on the clock face is always in a fixed relationship to the local speed of light.

That is why it is a local constant in relativity, but relativity does not say that the speed of light in one frame is equivalent to the speed in another (because each frame, by definition, is using a different scale of time, which is based on a local measurement of the speed of light by a clock that is stationary, and which is not equivalent between frames).

And what relativity does say, is that the differing speeds of light for different objects, will only be detected by clocks that are moving relative to one another. It will not be detected by co-moving, local, clocks.