Black hole metric of reflected shell of incoming light

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

and this

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?

• "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. Aug 13, 2022 at 8:18
• 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. Aug 13, 2022 at 13:12