GR: Accelerated Observer's Local Speed of Light

[smoothoperator]

Hi guys,

in GR a free-falling observer will measure the local speed of light as c, like in SR.

My question is will an accelerated (non-inertial) observer locally measure a greater speed than c, or will he also measure the local speed of light as c. For instance, if there is an object that is at rest wrt to Earth, will the speed of light locally increase in that frame, since the object undergoes upward acceleration?

Thanks in advance.

 

[WannabeNewton]

My question is will an accelerated (non-inertial) observer locally measure a greater speed than c, or will he also measure the local speed of light as c.

The accelerated observer will also measure the local speed of light as $c$. This can be easily shown by going to an instantaneously comoving local inertial frame of the observer and writing down the speed of a passing light beam relative to this frame; I can show this explicitly if you would like.

 

[bcrowell]

For instance, if there is an object that is at rest wrt to Earth, will the speed of light locally increase in that frame, since the object undergoes upward acceleration?

By "increase," do you mean will it be greater than normal? Are are you asking whether it will show an increasing trend over time?

If I was going to expect an effect, I would expect it to be an anisotropy, not a trend or a general increase.

The answer to your question may depend on what you mean by the speed of light. We set c=1 in relativistic units, and 1 can't have a different value in an accelerated frame. On the other hand, the Sagnac effect can be interpreted as an anisotropy of the speed of light in a rotating frame.

 

[smoothoperator]

So is it true that locally, no matter what coordinate chart we use, the speed of light always is c in both inertial and non-inertial frames in GR? The coordinate speed of light at some distant point varies with the different choice of a coordinate system?

 

[stevendaryl]

So is it true that locally, no matter what coordinate chart we use, the speed of light always is c in both inertial and non-inertial frames in GR? The coordinate speed of light at some distant point varies with the different choice of a coordinate system?

The problem is that "speed" doesn't mean a lot in noninertial coordinates. Depending on how you define your coordinates, you can get the speed of light to be whatever numerical value you want.

But what is true is that no matter how curved spacetime becomes, you can choose a coordinate system that is approximately inertial in one particular, small region, and in that region, light will have speed c.

 

[WannabeNewton]

So is it true that locally, no matter what coordinate chart we use, the speed of light always is c in both inertial and non-inertial frames in GR? The coordinate speed of light at some distant point varies with the different choice of a coordinate system?

Yes precisely. That is exactly correct.