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$$x = R\cosh\eta, \qquad t = R\sinh\eta$$

AFAIK based on Equivalence Principle (EP) all experiments executed inside the "unidimensional elevator" can be explained introducing an uniform gravitational field inside it. The Equivalence here is not just local but we can extend it for the entire region of

*flat*spacetime involved.

Now suppose you leave a particle at rest at quote ##R_{0}## in the elevator. At that moment I believe it shares the proper acceleration of the elevator point in which it is at rest and then, during its free-fall, it follows a geodesic described as just a straight line in ##(x,t)## coordinates. Basically its initial proper acceleration has been "converted" into the elevator ##(R,\eta)## coordinate acceleration.

If what said is correct, now each elevator point (with spatial ##R_{x}## coordinate inside the elevator itself) has a

*different*proper acceleration (according Rindler acceleration profile) thus the coordinate acceleration for objects released from rest at different elevator quotas is not a constant; in other words my conclusion is that uniform gravitational field inside the elevator is

*not*the case !

Can you help me in understanding if the argument is right or where is wrong ? Thanks.