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Consider a vacuum solution to Einstein's equations that is asymptotically flat but is not merely Minkowski. Must it have a curvature singularity?

The reason I ask is because it seems that there are many vacuum solutions to the Einstein equations that suffer from curvature singularities. For example, the Schwarzschild metric is an asymptotically flat vacuum solution that suffers from a curvature singularity at $r=0$.

While a gravitational plane wave should give a vacuum solution without singularities, I don't think a plane wave that is infinite in extent will give an asymptotically flat metric.

I thank Mike for a useful comment giving lists of vacuum solutions, including wikipedia and a digital textbook. I don't currently have access to the textbook. My own comments in that thread are a little bit confused, and I hope this question will help resolve my confusions.

user196574
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2 Answers2

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A geon would fit your requirements.

I don't know if anyone has written down a simple metric for a geon, though it has been proved they must exist. A geon is non-singular and asymptotically flat.

John Rennie
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The result of Christodoulou and Klainerman on global nonlinear stability of Minkowski space provides a constructive proof of global smooth nontrivial solutions of vacuum Einstein equations which look in the large like the Minkowski space. In particular, such solutions are asymptotically flat and have no curvature singularities.

Note, that this global result appeared about 40 years after the result by Choquet-Bruhat on local existence for initial value problem for Einstein equations. Also, this result required a whole book:

  • D. Christodoulou and S. Klainerman. The global nonlinear stability of the Minkowski space, volume 41 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993.
A.V.S.
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