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Could a neutron star turn into a blackhole if it is accelerated to near luminal velocities? And if it slows down after a while, does it become a neutron star again?

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In special relativity, changes of reference frame are symmetric between observer and observee. If moving a neutron star gave it enough kinetic energy to become a black hole, then you could effect this change by jumping into a moving reference frame yourself.

What matters is the neutron star's mass in its rest frame.

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