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Let a black-hole moves linearly at a constant speed, $v$. We set $x$-axis parallel to $v$.

I suppose $v$ is fast enough to cannot ignore the Relativistic effects.

My question.
How is the mathematical formula Schwarzschild radius of the black-hole when it moves linearly at a constant speed such that the Relativistic effects cannot be ignored.

Note: My question is focused on the mathematical expression how the Schwarzkopf radius changes by constant-velocity linear motion using mathematical expressions.

In the 3436, there are some discussion about "Can an object near the speed of light become a black hole (due to an increase in mass)"? However, in this question, it does not matter whether or not a black hole is formed.

Furthermore, no one has answered how the Schwartz-Schuld radius specifically changes- This is the core of my question-.

To make the question simpler, I restricted the question if the star was originally a black hole.

【Comments】 I tried to calculate using a formula that could be used. The results are as follows:

Let ${M}_{0}$ be a Rest mass of the star, $G$ be a constant of gravitation, and $c$ be the speed of light.

According to the Wikipedia. the Relativistic mass, $M$ is represented by the following formula:

$$M=\frac{M_0}{\sqrt{1-\dfrac{v^2}{c^2}}}$$

The Schwarzschild radius is represented by the following formula:

$$r_s=\frac{2GM}{c^2}$$

On the other hand, an object of length ${L}_{0}$ shrinks to length $L$ by the Lorentz contraction as follows. $$L={L_0}\sqrt{1-\frac{v^2}{c^2}}$$

Therefore, it might be $$r_s=\frac{2GM}{c^2}\sqrt{1-\frac{v^2}{c^2}}\ =\frac{2G{M_0}}{c^2}.$$

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  • $\begingroup$ @Sebastiano Thank you for your editing. $\endgroup$ Dec 25, 2019 at 13:46
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    $\begingroup$ Possible duplicate: If a mass moves close to the speed of light, does it turn into a black hole? $\endgroup$
    – A.V.S.
    Dec 25, 2019 at 14:22
  • $\begingroup$ @AVS The question in your link is much more extensive than my question. However, it does not describe what happens in the case of constant velocity linear motion in mathematical formula.I'll make it more restrictive, to ask the mathematical formula. In addition, it does not matter whether it becomes a black hole. $\endgroup$ Dec 25, 2019 at 14:25
  • $\begingroup$ According to the @D. Halsey ’s comment in another thread, I'm in worrisome whether stars that are not black holes have a Schwarzschild radius.  Since the nature of the question is a change in the Schwarzschild radius, I reduce the question if the star definitely has a Schwarzschild radius. $\endgroup$ Dec 25, 2019 at 15:10

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This is not a meaningful question. We don't have global frames of reference in general relativity. There is no notion of a Lorentz boost in general relativity.

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  • $\begingroup$ Thank you for your answer. But isn't Lorentz contraction directly related to my question itself. "the relation between the Schwarz-Schild radius and velocity"? $\endgroup$ Dec 25, 2019 at 15:19

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