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A rocket is traveling at a reasonable speed directly away from planet M at a large distance D. M is shaped like a coin and to the astronaut looking through a telescope directly out the back of the rocket, the front of M is spinning from the astronauts right hand to his left hand. Suddenly, the rocket pilot accelerates the rocket to a speed just under the speed of light ($c - w$, where $w$ is small) by supplying almost infinite energy. Distance D is large enough such that the rate of expansion of space over that distance is greater than $w$, hence the proper distance between the rocket and M is increasing at a rate greater than $c$.

  1. From the astronauts observation does M's position cross from time-like, through null, to space-like?

  2. What does the Astronaut observe through an ultrapowerful telescope? Nothing, M spinning from left hand to right, or something else?

  3. If the astronaut had a ultra sensitive equipment that could sense infinitesimal changes in the gravitational field coming from M, would they appear to be due to M spinning right to left, left to right, or something else?


Thank you for your answer and for the link.

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The velocities in cosmology that can exceed $c$ don't have much in common with the velocities in special relativity that can't. Even in special relativity, if you define a "recessional speed" similar to the way it's defined in cosmology, it can exceed $c$ even though the usual special-relativistic relative speed of the same objects is less than $c$. Here's a previous answer about this. (The question was "Can space expand with unlimited speed?")

In a toy special-relativistic cosmology, the relative speed of the rocket and the planet will be roughly what you'd get from the velocity-addition formula, so it will be less than $c$ even though the straight sum of the two speeds is larger than $c$. The rocket will appear highly redshifted, but there will be no null/spacelike worldlines or apparent reversal of time or anything of that sort. Since the distance to the rocket is small in cosmological terms, you can probably neglect spacetime curvature and use this model. If curvature can't be neglected, then there is no sensible way to define a sum of the cosmological and peculiar speeds, but the phenomenology is the same. Nothing will happen beyond a much increased redshift.

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