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Imagine you could delete the sun from its existence at an instance. Information cannot travel faster than the speed of light so the earth will keep orbiting the sun for about 8 minutes without knowing the sun is already gone (the distance between the earth and sun is ~ 8 lightminutes). After these 8 minutes the information that the sun is missing will reach the earth and the earth will fly away on a straight path.

So what will happen if we did this to a black hole? The curvature near a black hole is so extreme that information cannot travel outwards inside the event horizon. So if we removed the mass of the black hole in the center does the outside world know that we removed the mass?

And a related question: If we have a black hole with charge $+Q$ and we add to it a charge $-Q$, does the outside world know that the black hole is now uncharged? Again this addition is done instantaneously, without this charge ever crossing the event horizon. The difference with the previous case is that the changing the charge doesn't cause the curvature to disappear so this might result in a different outcome.

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To address the issue concerning 'the impossibility' of this setup: this process is impossible in real life, but at least for the case of the earth-sun system there should be no issues mathematically. You have some matter distribution $\rho(x^\mu)$ which, by the Einstein field equations, determine how the curvature of the spacetime surrounding it evolves. Setting $\rho$ to zero still allows a solution albeit with discontinuities. Which you can circumvent too.

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  • $\begingroup$ From math.ucr.edu/home/baez/physics/Relativity/BlackHoles/… "In this sense the black hole is a kind of "frozen star": the gravitational field is a fossil field. The same is true of the electromagnetic field that a black hole may possess." $\endgroup$ – PM 2Ring Apr 3 '20 at 10:59
  • $\begingroup$ Related, possible duplicate: physics.stackexchange.com/q/12169/123208 $\endgroup$ – PM 2Ring Apr 3 '20 at 11:03
  • $\begingroup$ If you are able to "delete" a star, all bets are off. $\endgroup$ – m4r35n357 Apr 3 '20 at 11:19
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    $\begingroup$ Good point, @m4r35n357. If a hypothetical scenario involves doing things that GR flat-out says are impossible, like magically deleting or teleporting mass, then you can't turn around and use GR to preduct what happens next. $\endgroup$ – PM 2Ring Apr 3 '20 at 12:29
  • $\begingroup$ Where does the energy go ? It can't just vanish. But if the energy inside the black hole goes away (magically !), you've got nothing to distort space, so no black hole because no event horizon. Maybe think of it like instant black hole evaporation. $\endgroup$ – StephenG Apr 3 '20 at 12:40
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This setup is impossible in general relativity. The Einstein field equations require that the energy-momentum acting as a source be divergence free, i.e. its energy and momentum must be conserved.

Clearly, "deleting" a mass at some instant violates energy-conservation, and there simply will be no valid solution to the GR equations.

For black holes, there is an additional complication in that the "source" is not part of the actual manifold. Black holes are solutions to the vacuum (sourceless) Einstein (-Maxwell) equations. From this perspective, it is even mathematically unclear what you would mean by "deleting" the mass.

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