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Suppose that there is a black hole with such a mass (e.g. one thousand ton), so that it would evaporate (at rest, and no inflow of material or radiation) in ~80 seconds. If it is somehow accelerated up to (or the conditions under which it appears forces it to move at) 99.9% of light-speed, so that the Lorentz factor is ~22.37, would the black hole evaporate the same time (80 sec), or multiplied by Lorentz factor (~1790 sec) (for someone who is not moving)?

In other words, would the ultrarelativistic speed of the micro-black hole make its decay time longer than usual (for an observer who is at rest)? Or the Hawking radiation time is independent on the speed of the object?

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Time dilation is universal, it does not depend on the method used to measure time. So yes, a black hole moving at 99.9% of light speed would need much more time to evaporate.

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