# Asymptotically flat vacuum solution without a curvature singularity

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.

• I started writing an answer but i saw that you're interested in vaccum solutions without singularities. I do not think that this is possible as long as a metric describs a massive object, but i'm not sure. Commented Oct 22, 2023 at 6:49
• @Noone Thank you for the comment. Your answer didn't have vacuum solutions, but they were very neat metrics! Would you mind commenting the links you wrote (just for my own edification)? Commented Oct 22, 2023 at 6:51
• Yes, here are the papers: arxiv.org/abs/gr-qc/0009077, arxiv.org/abs/gr-qc/9911046. Commented Oct 22, 2023 at 7:01
• That textbook is free available, see Exact Solutions of Einstein's Field Equations (wikipedia).
– JanG
Commented Oct 22, 2023 at 14:18
• You are right, sorry. Try this one.
– JanG
Commented Oct 22, 2023 at 17:38

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.
• +1 This is a very nice book! I hadn't appreciated the Lichnoweritz theorem that "a static solution which is geodesically complete and flat at infinity on any space-like hypersurface must be flat." Commented Oct 31, 2023 at 17:49
• I have a somewhat naive question. Do the global smooth nontrivial solutions given in the book ensure that invariant quantities related to the curvature like $R_{\mu \nu \sigma \delta}R^{\mu \nu \sigma \delta}$ are bounded above as a function of the coordinates? Commented Oct 31, 2023 at 18:11
• @user196574 Yes, for each such solution scalar curvature invariants are bounded. Commented Nov 1, 2023 at 18:36
• @user196574 Also, in another thread you said There I'm nervous that there is some singularity still lurking in the geon solutions as some appropriate time coordinate goes to plus or minus infinity I share the concern. If we confine a system of gravitational waves in a finite region for a long time then nonlinear effects often lead to appearance of shorter and shorter wavelengths which would eventually produce tiny black holes and singularities. That is the mechanism for nonlinear instability of AdS space. Commented Nov 1, 2023 at 18:50

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.

• +1 This is very interesting, and I think this might be what I'm looking for. I'm worried that there might be a subtlety that rules them out, however. On some cursory searches, I've gotten a little nervous that the geon is not static in time. This isn't necessary to answer my question, of course, since I didn't require and didn't intend to ask for static solutions, but my fear is that there may be either a singularity or a source as some appropriate time coordinate goes to minus infinity. I'll sleep on it and I'll check this out in more detail in the coming days. Commented Oct 22, 2023 at 7:16
• To add on my previous comment, Wikipedia writes: "In 1997, Anderson and Brill gave a rigorous proof that geon solutions of the vacuum Einstein equation exist, though they are not given in a simple closed form." This Anderson+Brill paper arxiv.org/abs/gr-qc/9610074 has the following quote: Commented Oct 23, 2023 at 0:57
• "Gravitational geons are analogous to the original electromagnetic geons of Wheeler, which are virtual gravitationally bound states of electromagnetic energy. As such, geons have a finite lifetime and are not true non-radiative solutions. Gibbons and Stewart have shown that non-radiative geons, or other exactly periodic solutions, cannot exist in Einstein’s theory." To me, the key statement is that geons have a finite lifetime, which is worrying me for reasons discussed in my first comment. Commented Oct 23, 2023 at 1:01