# Existence of a Trapped Surface to the Existence of a Black Hole

Does the existence of a trapped surface in a region of space (not necessarily either of the vacuum or symmetric spacetime) indicates (theoretically) the existence of a "black hole" there? And if yes, is it sufficient to describe the existence of a "black hole" (at least theoretically) by the existence of a trapped surface there?

Note: Penrose, in his famous paper in 1965 ["Gravitational collapse and space-time singularities". Phys. Rev. Lett. 14 (3): 57–59] claimed that [Source: Wikipedia],

A trapped surface is one where light is not moving away from the black hole.

But it is also well known that singularity can be avoided in the quantum treatment of such problem. Now the notion of singularity is eternally related to the concept of a trapped surface!

Along-with the previously cited references, I also had a look into the following references, but unfortunately my doubts remained uncleared (may be I've missed something there, which can clearly point out and answer my doubts):

So what's the way out? How the existence of a trapped surface and the existence of a black hole is (theoretically) connected to each other?

– SCh
Dec 20, 2022 at 15:18
• Without experiments, theory is vacuous speculation. Dec 20, 2022 at 15:25
• you don't have to test a definition
– Dale
Dec 20, 2022 at 15:36
• @JohnDoty We are dealing with the definitions of a trapped surface (which is mostly abstract mathematical) and the notion of a black hole and their relationship. So what is to be tested here? How can an abstract mathematical notion be tested experimentally in contrast to a "physical" theory?
– SCh
Dec 20, 2022 at 15:57
• The Hubble radius is called an anti trapped surface since you also can't escape to infinity, but it has the singularity in the past Dec 21, 2022 at 0:00

Does the existence of a trapped surface in a region of space (not necessarily either of the vacuum or symmetric spacetime) indicates (theoretically) the existence of a "black hole" there?

Not necessarily, without further assumptions on the spacetime geometry.

If we restrict ourselves to solutions of EFEs with matter satisfying weak or strong energy condition, then trapped surface ($$T$$) must be inside the black hole region of spacetime, $$T\subset B$$ (see Wald, proposition 12.2.2).

However if we drop constraints of EFEs then it is possible to have a trapped surface without a black hole (or alternatively a black hole without a trapped surface). For example, consider the following depiction of a “bouncing” spacetime (image taken from this paper):

This figure represents the lightcone structure (in $$t-r$$ plane of some reasonable coordinate system) of a spherically symmetric object (its outer surface is the thick line) that undergoes collapse and subsequent bounce. When the radius of the object decreases below its Schwarzschild radius, trapped surfaces appear (trapped surface would be a point in this figure). But nevertheless, there is no causally disconnected region in this spacetime, no event horizon and so no (true) black hole.

Of course, such a spacetime would not satisfy Einstein equations with a “normal” type of matter. But spacetimes with such features are considered as models of various quantum gravitational effects (such as black hole to white hole transitions). Often, the term “black hole” is still used for such spacetimes even though they no longer satisfy the classical definition.

• "...alternatively a black hole without a trapped surface". Is that actually possible? As in, can you have a spacetime with a black hole region and not a single spacelike foliation that has a trapped surface? Dec 21, 2022 at 17:28
• @TimRias: can you have a spacetime with a black hole region and not a single spacelike foliation that has a trapped surface Sure. Start with eternally growing black hole, for example Vaidya metric with mass asymptotically approachin some finite value. The expansion scalar of null geodesic congruence that contains horizon generators stays positive across horizon. Then simply excise the regions inside marginally trapped tube and declare that “spacetime ends here”. Since we do not bother with EFEs here we can construct true curvature singularity to cut away the trapping regions. Dec 22, 2022 at 7:15