# GR: Accelerated Observer's Local Speed of Light

• smoothoperator
In summary, the speed of light will always be measured as c by both an accelerated (non-inertial) observer and an inertial observer in GR. The coordinate speed of light may vary in different coordinate systems, but in a small region where spacetime is approximately inertial, light will always have a speed of c.
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?

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.

## 1. What is the "GR: Accelerated Observer's Local Speed of Light"?

The "GR: Accelerated Observer's Local Speed of Light" refers to the local speed of light as observed by an observer who is undergoing acceleration according to the principles of general relativity. This means that the speed of light will appear to be different for an observer who is accelerating compared to an observer who is at rest or moving with a constant velocity.

## 2. How does the "GR: Accelerated Observer's Local Speed of Light" differ from the speed of light in a vacuum?

The "GR: Accelerated Observer's Local Speed of Light" differs from the speed of light in a vacuum because it takes into account the effects of acceleration on the perception of the speed of light. In a vacuum, the speed of light is constant and is approximately 3 x 10^8 meters per second. However, for an observer who is accelerating, the local speed of light may appear to be different due to the warping of spacetime caused by acceleration.

## 3. What is the significance of the "GR: Accelerated Observer's Local Speed of Light" in general relativity?

The "GR: Accelerated Observer's Local Speed of Light" is significant in general relativity because it helps to explain the observed phenomena of time dilation and length contraction, which are predicted by Einstein's theory of relativity. The local speed of light is a crucial factor in understanding how acceleration affects the perception of time and space.

## 4. Can the "GR: Accelerated Observer's Local Speed of Light" be measured?

Yes, the "GR: Accelerated Observer's Local Speed of Light" can be measured using various experimental methods, such as the Michelson-Morley experiment or the Pound-Rebka experiment. These experiments involve measuring the speed of light in different frames of reference, including an accelerating frame of reference, to determine the effect of acceleration on the local speed of light.

## 5. How does the "GR: Accelerated Observer's Local Speed of Light" relate to the concept of spacetime?

The "GR: Accelerated Observer's Local Speed of Light" is closely related to the concept of spacetime. In general relativity, spacetime is a four-dimensional continuum that is affected by the presence of mass and energy. The local speed of light is a crucial factor in understanding the curvature of spacetime caused by acceleration, which in turn affects the perception of time and space for an observer undergoing acceleration.

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