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After doing much more digging than I thought I had to do, I found out that the speed of light is NOT invariant in an accelerated reference frame. Has anyone done any experiments to confirm this? In particular a Michelson-Morley experiment in an accelerated reference frame? I figured light being invariant in any constant speed frame would automatically imply being invariant in any frame whatsoever.

I have to credit Richard Mould's Basic Relativity with informing me about this fact.

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You say:

I found out that the speed of light is NOT invariant in an accelerated reference frame

but things are more complicated than this. The local speed of light measured by an observer is always equal to $c$, and this remains the case whether the observer is stationary, moving, accelerating or anything else you might think of. So if your Michelson interferometer is small enough to make a local measurement then it will report the speed of light is the same in all directions.

However it is certainly true that an accelerated observer would observe the speed of light to be different at distant locations. This is because the coordinate system used by an accelerating observer doesn't match up with the coordinate system of a non-accelerating observer. Specifically, non-accelerating observers find spacetime to be described by the Minkowski metric, while accelerating observers find spacetime to be described by the Rindler metric.

We can make this quantitative. We start with the Rindler metric:

$$ \mathrm ds^2 = -\left(1 + \frac{g}{c^2}z \right)^2 c^2 ~\mathrm dt^2 + \mathrm dz^2 \tag{1} $$

and note that for light the line interval $\mathrm ds$ is always zero. Setting $\mathrm ds = 0$ in equation (1) and rearranging gives us the equation for the speed of light as observed by an accelerating observer:

$$ \frac{\mathrm dz}{\mathrm dt} = c\left(1 + \frac{g}{c^2}z\right) \tag{2} $$

where positive $z$ is in the direction of acceleration and negative $z$ is opposite to the direction of acceleration.

We can now get some idea of what distances we need to make measurements over to get a change in the velocity of light. For example, to get a 1% reduction in the velocity of light we need:

$$ 1 + \frac{g}{c^2}z = 0.99 $$

or:

$$ z = -0.01 \frac{c^2}{g} $$

For an acceleration of 1G we get:

$$ z = -0.01 \frac{c^2}{10} \approx 9 \times 10^{13} \text{m} \approx 0.01 \space\text{light years} $$

dmckee mentions the Pound-Rebka experiment in his answer, but this is really measuring the curvature of spacetime. Actually it's not a bad approximation to analyse the Pound-Rebka experiment by taking spacetime to be flat and using the Rindler metric. The reason the PR experiment managed to measure anything over its 20m distance is because it is exquisitely sensitive.

A last note, since we're having such fun, looking at equation (2) you might notice that the speed of light goes to zero at $z = -c^2/g$. This is the Rindler horizon and it's closely analogous to the event horizon of a black hole. The speed of light does indeed go to zero at the Rindler horizon just as it does at a black hole event horizon.

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The first to do something equivalent to that were Pound and Rebka who first measured the gravitational redshift in 1959. I'm not aware of anyone who actually used a Michelson interferometer in a upright orientation.

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even though John puts it down quite nicely, I don't think that was the answer you sought?

Yes, the Michelson-Morley experiment has, to my knowledge, only been done in accelerating reference frames, because of the rotation and gravity of the earth, some with precision high enough to measure both gravity or the rotation velocity. Unfortunately I can't name-drop who, but rotating Michelson-interferometer experiments there has also been conducted.

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