GR: Accelerated Observer's Local Speed of Light

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Discussion Overview

The discussion centers on the measurement of the local speed of light by accelerated (non-inertial) observers in the context of General Relativity (GR). Participants explore whether an accelerated observer measures a speed greater than c or if the speed remains c, similar to free-falling observers. The scope includes theoretical considerations and implications of coordinate systems in GR.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asserts that an accelerated observer will measure the local speed of light as c, similar to a free-falling observer, and offers to demonstrate this using a local inertial frame.
  • Another participant questions the interpretation of "increase" in the speed of light and suggests that any expected effect might be an anisotropy rather than a general increase or trend.
  • Concerns are raised about the meaning of "speed" in non-inertial coordinates, indicating that it can vary based on the chosen coordinate system.
  • Some participants agree that locally, the speed of light is c in both inertial and non-inertial frames, but note that the coordinate speed can differ based on the coordinate system used.
  • One participant highlights that while the speed of light is c in a small region of approximately inertial coordinates, the interpretation of speed can be complex in non-inertial frames.

Areas of Agreement / Disagreement

There is a mix of agreement and disagreement among participants. While some assert that the local speed of light remains c for both inertial and non-inertial observers, others raise questions about the implications of coordinate choices and the interpretation of speed in non-inertial frames. The discussion does not reach a consensus on the nuances of these interpretations.

Contextual Notes

Participants note that the definition of speed can vary significantly in non-inertial coordinates, and the discussion highlights the complexity of measuring light speed in different frames. There are unresolved aspects regarding the implications of the Sagnac effect and how it relates to anisotropy in the speed of light.

smoothoperator
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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.
 
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smoothoperator said:
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.
 
smoothoperator said:
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.
 
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?
 
smoothoperator said:
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.
 
smoothoperator said:
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.
 

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