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Aside from the calculation, the specific scenario for which I have no sense of the solution is the following:

The equivalence principle proposes a parallel between the force experienced by an accelerating body compared to a body resisting the motion of a gravitational force. If a spectrometer motionless in space high above the earth (at a Lagrange point) measures the redshift of a monochromatic laser on earth, it will observe a specific value because the light leaving earth will be redshifted by earth's gravity.

Compare this to the 'equivalent' experiment where the light source and the the spectrometer are motionless relative to each other in deep space. Now the spectrometer is accelerated away from the light source at 1 g. The observed redshift should be increasing because the relative velocity of light source and detector is constantly increasing.

In both cases, there is a 1 g acceleration but the redshift observations should be different. Is my intuition correct? Is the 'equivalence' analogy breaking down here?

aquagremlin
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    You can't use the Equivalence Principle here. That only applies to local measurements, where the gravitational field strength is almost constant. – PM 2Ring Jul 26 '22 at 16:30
  • @PM2Ring Why would that be true? It seems like the equivalence principle to me. In Wheeler's book Exploring Black Holes he says , for a signal rising from Earth’s surface to be observed at the top of the tower, the period increases and the measured frequency decreases, an effect labeled gravitational red shift, which gives the higher observer the impression that clocks below him “run slow.” – foolishmuse Jul 26 '22 at 18:00
  • @foolishmuse Yes, that was famously demonstrated in 1959, in the Pound-Rebka experiment. The height difference in that experiment was 22.5 m, so the change in gravity from the bottom to the top was minute. A more recent experiment was done in Tokyo, in a much taller tower, with state-of-the art atomic clocks, so they could detect the effects due to the variation in gravity. https://physics.stackexchange.com/q/716323/123208 – PM 2Ring Jul 26 '22 at 19:04
  • The Equivalence Principle says that uniform gravitation is indistinguishable from uniform acceleration. However, gravitation is never perfectly uniform, but it's approximately uniform in a sufficiently small region of spacetime. So it doesn't apply to a scenario where the gravity at one end is significantly different to the gravity at the other end. Also, the observer in the moving spaceship can only perform local measurements. They're not allowed to observe some distant body that the ship's accelerating away from. – PM 2Ring Jul 26 '22 at 19:15
  • @PM2Ring So what you are saying is that for acceleration, we can only claim the equivalence principle if the guy in the elevator experiences a gravity like situation. - not an outside observer. – foolishmuse Jul 26 '22 at 19:23
  • @foolishmuse Something like that. ;) The EP (Equivalence Principle) is all about constant acceleration. The guy in the box can't tell if he's in deep space undergoing constant acceleration, or in a uniform gravitational field. The box has to be small enough that he can't measure a difference in gravitational strength between the top & bottom of the box, or measure that stuff doesn't fall down in parallel lines. And he can't make external observations, or communicate with an external observer. That would be cheating! – PM 2Ring Jul 26 '22 at 19:43
  • Thank you for all the comments. I did not anticipate the obvious effect of large distance. However can I modify the scenario? Pound at the bottom of Harvard steeple with a laser. Rebka at the top with a spectrometer. Can Rebka measure a redshift? The ‘equivalent situation is a putting pound and rebka in space separated by the same distance, and accelerating them together in the same direction at 1g. would the same redshift be measured? – aquagremlin Jul 27 '22 at 01:11
  • Yes, the same redshift would be measured in the steeple and in space. – PM 2Ring Jul 27 '22 at 16:27

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