The speed of light in a Schwarzschild space-time

[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.

 

[DW]

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.

 

[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?

 

[DW]

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.

 

[hellfire]

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.

 

[DW]

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.

 

[hellfire]

Excellent, this was of great help. Thanks.