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I want to know what happens to the space a black hole crosses over as our galaxy travels through space.

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    $\begingroup$ What is your level in physics? $\endgroup$ – Mauricio Apr 2 '18 at 23:11
  • $\begingroup$ Since motion is relative, this is equivalent to asking what a black hole does to distant regions of space. The answer is that it does the same thing to distant regions of space as would have been done by the same mass in any other form, such as a normal star. $\endgroup$ – user4552 Apr 3 '18 at 3:07
  • $\begingroup$ Possible duplicate of Does a black hole move through space? What happens to other things around it? $\endgroup$ – Stéphane Rollandin Apr 3 '18 at 8:20
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Nothing. Or rather, a black hole is a (strong) curvature in space and time, so when it moves a fixed spot nearby (as defined relative to remote galaxies) experiences increasing curvature - which is experienced as gravitational attraction - followed by decreasing effects.

See also Does a black hole move through space? What happens to other things around it?

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  • $\begingroup$ So what you're saying is space can escape a blackhole? $\endgroup$ – Tim Lieberman Apr 3 '18 at 20:13
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    $\begingroup$ "Escape" is the wrong word. General relativity describes how spacetime curves in the presence of matter, fields and spacetime curvatures. It is not like space is some kind of fabric sitting on top of something more fundamental (although all the rubber sheet analogies we like to use as educational models may look like that). The behaviour of spacetime is also very different from the matter and fields residing in it - it can move faster than light (consider the early universe) and has no problems at all extending past event horizons. $\endgroup$ – Anders Sandberg Apr 3 '18 at 22:45

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