If you put a Black Hole inside of a Black hole, then you get a spherically symmetric vacuum outside the inner Black Hole and the Schwarzschild metric is the only spherically symmetric vacuum in GR. So this leads me to believe that the spacetime around a Black Hole inside a Black Hole is still a Schwarzschild spacetime, which means the Schwarzschild metric is the solution. But since time is finite inside a Black Hole, this leads me to believe that the metric outside of a Black Hole that is embedded inside a Black Hole should be different from the exterior Schwarzschild metric. So how do you get the metric around a Black Hole that is inside a Black Hole?
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3$\begingroup$ "If you put a Black Hole inside of a Black hole," makes no sense, it cannot happen, as when two black holes approach each other, they merge. space.com/what-happens-when-black-holes-merge . $\endgroup$– anna vCommented Nov 25 at 19:44
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$\begingroup$ But what if you had mass fall into a Black Hole which, while inside the Black Hole, concentrated and became a black hole. How would the vacuum around that Black Hole (which is inside the other Black Hole) be modelled? $\endgroup$– Chris LaforetCommented Nov 25 at 19:49
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$\begingroup$ Did you read the link I gave above? $\endgroup$– anna vCommented Nov 25 at 20:23
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$\begingroup$ @ChrisLaforet As observed from the outside, mass cannot fall “inside” a black hole. The event horizon cannot be crossed in finite coordinate time. $\endgroup$– controlgroupCommented Nov 25 at 20:25
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3$\begingroup$ The question is mainstream physics, although I doubt there is an analytical solution since this is a two body problem which is hard in general relativity. If the second black hole is so small that the local flat spacetime approximation of the first black hole holds at that scale you can add them up though, but if the second black hole's mass is not neglible compared to the first one's you need numerical relativity due to the backreaction. $\endgroup$– YukterezCommented Nov 26 at 5:51
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the Schwarzschild metric is the only spherically symmetric vacuum in GR. So this leads me to believe that the spacetime around a Black Hole inside a Black Hole is still a Schwarzschild spacetime, which means the Schwarzschild metric is the solution.
This is correct, provided spherical symmetry is maintained.
For example, suppose we had two concentric spherical shells of dust (no pressure), collapsing from rest. At some point the inner shell will collapse to the point that it is smaller than the Schwarzschild radius of its mass. Similarly, at some point the outer shell will collapse to the point that it is smaller than the Schwarzschild radius of its mass plus the mass of the inner shell.
At all times before the inner shell reaches the singularity, the shells divide spacetime into three regions. The inner region is Schwarzschild with $M=0$, which is flat spacetime. The middle region is Schwarzschild with $M$ equal to the inner shell mass. The outer region is Schwarzschild with $M$ equal to the sum of both shell masses. The shells themselves are not vacuum, so they are not Schwarzschild.
Regarding the horizons, there are two important events in the evolution. $A$ is when the inner shell reaches its Schwarzschild radius. $B$ is when the outer shell reaches the Schwarzschild radius of the total mass. From each of these events you can trace an outgoing light ray backwards (tracing an outgoing ray backwards is tracing inwards). The event where it traces all the way backwards to the center is the formation of the horizon. The outer horizon always forms first, regardless of whether $A$ or $B$ occurs first in some chosen coordinates. The formation events are timelike separated, even if $A$ and $B$ are not. I suspect that $A$ and $B$ must be spacelike separated, but I cannot give a solid justification. The outer horizon is an actual event horizon while the inner horizon is an apparent horizon.
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$\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$– Buzz ♦Commented Nov 27 at 17:59