MTW, p. 924, defines a caustic as a point where a null geodesic originating from the external universe enters a black hole's event horizon, remaining in the horizon afterward for some finite affine interval. (A null geodesic of this type is called a generator of the horizon.) They introduce this as a preliminary to stating and proving a theorem by Penrose, 1968. The theorem says that: (1) after a generator enters the horizon, it stays there forever, and never intersects another generator except at a caustic; (2) any point on the horizon (if not a caustic point) lies on exactly one generator.(adsbygoogle = window.adsbygoogle || []).push({});

In the case of a black hole that forms by gravitational collapse, is there a single event in all of spacetime that is a caustic, or does the set of caustics form a two-dimensional caustic surface?

If I look at the Penrose diagram for an astrophysical black hole, it looks to me like there is one point on the diagram that represents a caustic, and this point lies on the axis of symmetry, so it really is a single point, not a 2-sphere. (To see this on my diagram, you need to imagine the diagram as being rotated about the symmetry axis at its left edge.)

However, this seems odd, because I would then imagine this event as the one at which the black hole first formed, i.e., the earliest moment at which any infalling matter crossed the newly formed horizon. ("Earliest" would mean that it's in the causal past of all other events at which infalling matter crossed the horizon.) So it's trapped behind a horizon that is going to be formed by other matter that isn't there yet.

Based on Newtonian intuition, it's also tempting to imagine this as the event at which the singularity formed, but on the Penrose diagram I think it's clear that the caustic is separated from the singularity by some finite proper time, which I guess equals the mass of the hole.

If the caustic was going to be associated with a surface rather than a point, I guess it would have to be the first trapped null surface...? But where does that lie on the Penrose diagram?

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# Is caustic surface of an astrophysical black hole one event?

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