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At this point in Leonard Susskind's eighth lecture on general relativity, he begins a discussion about finding the metric of a black hole formed by an incoming, spherically symmetric shell of light. At this point, after explaining that flat spacetime is correct for the interior of the shell, while the Schwarzschild spacetime is correct for the exterior, he states that splicing the two correct regions of the diagrams together yields the answer.

Inspired by the discussion that follows, consider building a spherical mirror which can reflect the light back, with a radius smaller than the Schwarzschild radius. The two diagrams for the flat and Schwarzschild spacetimes would look like this

Minkowski spacetime with mirror

and this

Schwarzschild spacetime with mirror

respectively (the yellow lines are the paths of the light, and the green line is the event horizon).

Following the same principle of splicing diagrams, is it legitimate to conclude that this is the correct overall spacetime?

Combined spacetime

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The incoming light has a large mass, and therefore exerts a large impulse on the mirror when it's reflected. The impulse integrated over the whole mirror is zero, but force can't be transmitted through the mirror faster than light, so there is not enough time to avoid a local effect. The mirror has to be less massive than the light (otherwise it would already be a black hole) so it can't just tank the impulse.

In short, this setup unavoidably violates energy-momentum conservation, which is a purely geometric law in GR, so I think it's geometrically impossible to stitch these solutions together regardless of the details.

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  • $\begingroup$ "The impulse integrated over the whole mirror is zero" - Generally it is not. For example, in the Schwarzschild coordinates, the radial light moves inside the horizon all in the same spatial direction of decreasing $t$, so all its momentum adds up instead of canceling out. For an intuitive illustration: i.stack.imgur.com/P5QTF.jpg - or you can look at a plot of the radial null geodesics. $\endgroup$
    – safesphere
    Aug 13, 2022 at 8:18
  • $\begingroup$ Would it be possible to solve the energy-momentum violation problem by just dropping the assumption that the mirror does tank the impulse? $\endgroup$ Aug 13, 2022 at 12:54
  • $\begingroup$ Ah, I think I see the problem with that. If the light has energy $\omega$ and it perfectly reflects off the mirror with mass $M$, then the final energy of the mirror is $\frac{M^2+2M\omega+2\omega^2}{M+2\omega}$. Since we assume that the light has enough energy to form a black hole, and the mirror's final energy is at least $\omega$ at $M=0$, increasing with $M$, the mirror will always end up with enough energy to form a black hole. $\endgroup$ Aug 13, 2022 at 13:12

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