The speed of light in a Schwarzschild space-time

In summary, the conversation discusses the dependence of the speed of light on radial position in a Schwarzschild space-time and whether this effect can be measured in a local reference frame. It is explained that the remote observer variance in the vacuum speed of light has been observed, but a local experiment like the Michelson-Morley interferometer would not observe it. The conversation also touches on the appropriateness of using Schwarzschild coordinates for a remote observer's point of view.
  • #1
hellfire
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I have seen a derivation of the dependence of the speed of light inside a Schwarzschild space-time: c depends on the radial position (r), but a light ray which moves radially has a different dependence on r as a light ray which moves tangentially. My question is whether such an effect may be measurable somehow in a local reference frame and why did not the Michelson-Morley experiement record such an effect. Sorry if this question was already answered here, but after a short search I didn’t find any clear answer.

Thanks.
 
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  • #2
Originally posted by hellfire
I have seen a derivation of the dependence of the speed of light inside a Schwarzschild space-time: c depends on the radial position (r), but a light ray which moves radially has a different dependence on r as a light ray which moves tangentially. My question is whether such an effect may be measurable somehow in a local reference frame and why did not the Michelson-Morley experiement record such an effect. Sorry if this question was already answered here, but after a short search I didn’t find any clear answer.

Thanks.

Yes the remote observer variance in the vacuum speed of light has been observed from signals relayed from other planets, but this is a remote observer effect. The local vacuum speed of light is the invariant c and so a local experiment like their interferometer wouldn't observe it.
 
  • #3
Thanks for your answer, it seams to be a trivial point, but I am afraid I still do not get this. May be you can help me. Let’s take the derivation of for a radial light ray for example.

For light ds^2 = 0.

In a Schwarzschild space-time:
0 = (1-2m/r)(dt)^2 – (1-2m/r)^-1 (dr)^2
(the angular components vanish, since it moves only radially)

therefore:
(dr/dt)^2 = (1-2m/r)^2

and with:
dr/dt = c_r

one obtains:
c_r = 1 - 2m/r

Where is here the step with the assumption that this is for the remote observer only and not inside a local frame?
 
  • #4
Originally posted by hellfire
Thanks for your answer, it seams to be a trivial point, but I am afraid I still do not get this. May be you can help me. Let’s take the derivation of for a radial light ray for example.

For light ds^2 = 0.

In a Schwarzschild space-time:
0 = (1-2m/r)(dt)^2 – (1-2m/r)^-1 (dr)^2
(the angular components vanish, since it moves only radially)

therefore:
(dr/dt)^2 = (1-2m/r)^2

and with:
dr/dt = c_r

one obtains:
c_r = 1 - 2m/r

Where is here the step with the assumption that this is for the remote observer only and not inside a local frame?

You made it prior to your second equation when you chose to express ds^2 in terms of Schwarzschild coordinates. Those coordinates are appropriate for a remote observer's reconing.
 
  • #5
Originally posted by DW
You made it prior to your second equation when you chose to express ds^2 in terms of Schwarzschild coordinates. Those coordinates are appropriate for a remote observer's reconing.
I see. Are there other coordinates which are not appropiate for remote observers? Could you give me a hint or a link which explains which are the criteria to recognize that Schwarzschild coordinates are appropiate for remote observers? Regards.
 
  • #6
Originally posted by hellfire
I see. Are there other coordinates which are not appropiate for remote observers?

Yes infinitely many, take your pick. A well know class of coordinates that are not the remote observers coordinates are Kruskal-Szekeres coordinates for example.

Could you give me a hint or a link which explains which are the criteria to recognize that Schwarzschild coordinates are appropiate for remote observers? Regards.

Look at the limit as r goes to infinity and see that the metric approaches that of special relativity except transformed to spherical coordinates. That is what tells you that the coordinates are representative of a remote observer's appropriate choice.
 
  • #7
Excellent, this was of great help. Thanks.
 

1. How does the speed of light behave in a Schwarzschild space-time?

In a Schwarzschild space-time, the speed of light is constant and remains at its maximum value of 299,792,458 meters per second. This is because the speed of light is a fundamental constant and is not affected by the curvature of space-time.

2. Can the speed of light be exceeded in a Schwarzschild space-time?

No, the speed of light cannot be exceeded in a Schwarzschild space-time. According to Einstein's theory of general relativity, the speed of light is the maximum speed at which any object or information can travel in the universe.

3. How does the curvature of space-time affect the speed of light in a Schwarzschild space-time?

The curvature of space-time has no effect on the speed of light in a Schwarzschild space-time. The speed of light remains constant and at its maximum value, regardless of the curvature of space-time.

4. Does the speed of light change near a black hole in a Schwarzschild space-time?

No, the speed of light does not change near a black hole in a Schwarzschild space-time. The event horizon of a black hole does not affect the speed of light, as it is a property of space-time itself.

5. Can the speed of light be measured differently in a Schwarzschild space-time compared to normal space-time?

No, the speed of light cannot be measured differently in a Schwarzschild space-time compared to normal space-time. The speed of light is a constant and universal value, and it remains the same regardless of the space-time curvature.

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