Penrose Diagram for Multi-Blackhole Solutions: Reference

[Roy_1981]

Can anybody recommend a reference (paper/textbook) where I can look up the Penrose diagram for multi-black hole spacetimes or multi-center solutions? Appreciate.

 

[bcrowell]

The answer is probably that no such thing exists. We had some previous discussion here about the conditions under which one can draw a Penrose diagram, and although we discussed and clarified some issues, I don't think we ever came to a precise and general conclusion:

https://www.physicsforums.com/threads/necessary-conditions-for-a-penrose-diagram.789802/

Normally we draw Penrose diagrams for spacetimes that satisfy two conditions: (1) they're conformally flat, and (2) they have some symmetry. The conformal flatness is necessary in order to be able to make the light cones look right, which is a defining characteristic of a Penrose diagram. The symmetry is needed so that we can represent a four-dimensional manifold on a two-dimensional piece of paper.

Your example is actually one that I have in my notes on this subject as a prototypical example of a spacetime for which I think you can't make a Penrose diagram. Say you have three black holes that are not all along the same line. Then your spacetime lacks any symmetry whatsoever.

The discussion we had of this was not totally conclusive, and some people did point out examples that lacked spherical symmetry, but for which it was still possible to draw a Penrose diagram for some two-dimensional slice through the spacetime. But in your example, it's not at all obvious that any such two-dimensional slice gives broad or representative information about the whole spacetime.

 

[Ben Niehoff]

You can still draw conformal diagrams, where the lightcones are all upright, but you might need to make 3-dimensional diagrams. Somewhere I have seen this done for the C-metric, but I forget where.

 

[Ben Niehoff]

Found it. I think I remembered it from Exact Space-Times in Einstein's General Relativity. It's also on the arXiv:

http://arxiv.org/abs/gr-qc/0609056

 

[bcrowell]

I think I finally understood what had been confusing me before. I posted about this in the earlier thread.

In the example with three noncollinear black holes, there is a low level of symmetry. There are two ways you could try to deal with this.

One way is to take a two-dimensional slice or projection. If you can do this in such a way that lightlike geodesics still look like lightlike geodesics, then you're all set, because a two-dimensional manifold is always conformally flat. But can you do the slice so that lightlike geodesics look lightlike? I don't know. Even if you can manage to do this, there is no guarantee that the 2-d version is sufficiently representative of the behavior of the whole spacetime to make it useful.

Another way, as suggested by Ben Niehoff, is to do a Penrose diagram in more than two dimensions. The problem here is that then you have a 3- or 4-dimensional manifold, and manifolds in that many dimensions may not be conformally flat. The original, 3+1-dimensional spacetime certainly isn't conformally flat in this example.

 

[Roy_1981]

Many thanks to Ben for the wonderful reference and to bcrowell for the inputs/thoughts. I was really struggling with this yesterday and got completely exhausted. Indeed more than three dimensional diagrams seem to be the way to go.

 

[bcrowell]

Many thanks to Ben for the wonderful reference and to bcrowell for the inputs/thoughts. I was really struggling with this yesterday and got completely exhausted. Indeed more than three dimensional diagrams seem to be the way to go.

Seems unlikely to me that this strategy will work, since I don't see how you're going to get a three-dimensional submanifold that is conformally flat. But if you do make it work, please post to tell us about it. I'd be interested to see your results.