Penrose Singularity: Trapped & Anti-Trapped Surfaces Explained

[windy miller]

As I understand Penrose proved there must a be a singularity using certain assumption for black holes and something called a "trapped surface:. Can anyone give a lay person explanation of what this and "anti trapped surface" are? How were they used in the singularity theorems and what was new? Many thanks

 

[PeterDonis]

Can anyone give a lay person explanation of what this and "anti trapped surface" are?

A trapped surface is a 2-sphere on which radially outgoing light rays are not moving outward; or, to put it another way, the 2-sphere formed by the set of all radially outgoing null geodesics from the trapped surface has an area that is not increasing. Normally, we expect that light rays moving radially outward from a 2-sphere will form a 2-sphere whose area is increasing; a trapped surface violates this expectation.

I have not seen the term "anti trapped surface" in the literature. However, the application of trapped surfaces in the singularity theorems works in either direction of time; see further comments below.

How were they used in the singularity theorems

The singularity theorems say that if a spacetime contains a trapped surface and certain energy conditions are met (these conditions, heuristically, amount to gravity being always attractive), the same spacetime will also contain a singularity. More precisely, if a spacetime contains a trapped surface and meets certain energy conditions, the spacetime will also contain incomplete timelike or null geodesics (i.e., geodesics that cannot be extended past some finite value of their affine parameter).

When applied in the future direction of time (i.e., "outgoing" light rays are outgoing towards the future), the theorems tell us that there must be a singularity inside a black hole, since if the energy conditions are met there must be a trapped surface either at or inside the event horizon of a black hole.

When applied in the past direction of time (i.e., "outgoing" light rays are outgoing towards the past), the theorems tell us that there must be a singularity at the beginning of the universe in FRW spacetimes that satisfy the energy conditions.

 

[windy miller]

Thanks Peter , much appreciated

 

[PeterDonis]

You're welcome!