Is there a Penrose diagram for two black holes near each other. Perhaps they are colliding or circling each other? Or can this method only describe a single black hole.


1 Answer 1


The first step in drawing a Penrose diagram is to make an $n$-dimensional section or projection, where usually, but not always, $n = 2$. We want to take advantage of any symmetries, such as rotational symmetry, so that the final result will be informative, be representative of the whole spacetime, and accurately depict causal relationships in the original spacetime. Your spacetime with two black holes has a low degree of symmetry so this will not work for $n=2$. At this step we also need to make sure that lightlike geodesics in the original space correspond properly to lightlike geodesics in the submanifold.

The next step would be to apply a conformal transformation in order to make the diagram compact. If you want the diagram to be flat, then this requires the $n$-dimensional slice or projection to be conformally flat. All two-dimensional manifolds are conformally flat, but in your example we will have $n>2$, and I don't think you're going to get conformal flatness.

So no, I don't think you can draw a Penrose diagram for this spacetime.

  • $\begingroup$ Although the lack of symmetry makes a 2d section less informative (than in the spherical symmetric case), it will not be devoid of information, and one should be able to draw the corresponding penrose diagram. One must be very careful with its interpretation though. $\endgroup$
    – TimRias
    Dec 6, 2018 at 8:18
  • $\begingroup$ @mmeent: The main reason we want Penrose diagrams is so that we can read off causal relationships. If we can't do that, it's not very useful, and I wouldn't call it a Penrose diagram. I would just call it a section or a projection. $\endgroup$
    – user4552
    Dec 6, 2018 at 14:50

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