Speed of light one more time

[Rick16]

TL;DR

the equivalence principle and the speed of light

I have now several times come across the situation where the equivalence principle is used to explain why a gravitational field bends light. Here is a very clear presentation of the situation from Hartle:

Every time I read about this, I wonder why does nobody ever mention what happens when the light ray moves parallel to the laboratory. When the light ray enters the laboratory at the top and leaves it at the bottom and the laboratory accelarates, then the observer inside the laboratory will measure a shorter time for the light ray to traverse the laboratory compared to the non-accelarating case, whereas the distance that the light ray covers is still the same for the observer. This means that the observer should measure a speed of light > c. Since the frame is accelarating, this does not violate the postulate of special relativity. Then, by the equivalence principle, the same should happen in a gravitational field, i.e. speed of light > c in a gravitational field. What is wrong about this reasoning?

 

[Ibix]

It's the two-way speed that's invariant - the time for the light pulse to travel from the bottom to the top and back. The one-way speed is coordinate dependent and can deviate a long way from $c$, even moving inertially in flat spacetime. Even the two-way speed probably does deviate from $c$ in this case, though, yes.

Also, speed over long distances in curved spacetime isn't well defined, so "the speed of light" over long distances can vary. This is one way to interpret Shapiro delay when (e.g.) radar measuring the distance to Venus.

 

[PeroK]

TL;DR Summary: the equivalence principle and the speed of light

I have now several times come across the situation where the equivalence principle is used to explain why a gravitational field bends light. Here is a very clear presentation of the situation from Hartle:
View attachment 354515
Every time I read about this, I wonder why does nobody ever mention what happens when the light ray moves parallel to the laboratory. When the light ray enters the laboratory at the top and leaves it at the bottom and the laboratory accelarates, then the observer inside the laboratory will measure a shorter time for the light ray to traverse the laboratory compared to the non-accelarating case, whereas the distance that the light ray covers is still the same for the observer. This means that the observer should measure a speed of light > c. Since the frame is accelarating, this does not violate the postulate of special relativity. Then, by the equivalence principle, the same should happen in a gravitational field, i.e. speed of light > c in a gravitational field. What is wrong about this reasoning?

Here's a simple thought experiment. A light pulse is sent from point O to a mirror, and reflected back to point O. Point O and the mirror are at relative inertial rest and are a distance $D$ apart.

One observer remains at point O and measures a time $t_1$ for the round trip. And, hopefully, finds that $t_1 = \frac {2D}{c}$.

Meanwhile, a second observer takes a quick break, speeds off somewhere and back to point O, returning before the light pulse returns. Their clock will measure a time $t_2 < t_1$ (due to the differential ageing associated with the short non-inertial detour). This observer measures the average speed of the light pulse to be $v = \frac{2D}{t_2} > c$.

This $v$, as @Ibix says, is the coordinate speed of a non-inertial observer and exceeds the speed of light in vacuum.

 

[Dale]

This depends entirely on how you define the coordinates in the accelerated frame. If you use radar coordinates then the one way speed of light is c, as usual. In other words, this is not about the acceleration.

 

[FactChecker]

Your example gets very complicated. The top and bottom of the ship is accelerating on different time schedules according to an outside, non-accelerating observer and the length of the ship is shrinking.

For the inside, accelerating observer, he can feel the acceleration so why would he think that he is measuring the speed of light correctly?

 

[Nugatory]

Then, by the equivalence principle, the same should happen in a gravitational field, i.e. speed of light > c in a gravitational field. What is wrong about this reasoning?

Calculate the magnitude of the effect that you are considering.

To make things concrete, take the acceleration to be the earth's surface gravity and the distance between source and receiver to be 22.5 meters - we now have an idealization of the Pound-Rebka experiment from three-quarters of a century ago. Your choice of coordinates determines whether you explain your observations as the light not moving at speed $c$ or as redshift caused by gravitational time dilaion.