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Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question:

Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there some instant of time in your development, namely $t \neq 0$, such that the slice $(M_t,g_t, k_t)$ is isomorphic to $(M,g,k)$, in the sense of that are isometric Riemannian manifolds and has the same second fundamental form?

Unfortunately, I do not know any answer for this question. I will be grateful with any answer or reference that can clarify this question.

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    $\begingroup$ Technically, the Minkowski space is a periodic solution for any period you want. In fact, any static solution (with or without matter field) also qualifies. Presumably you want a non-trivial solution to your problem, where it is periodic but not stationary. On top of that, you may want to specify what constraint you want for the matter model. Allowing arbitrary matter model allows you to cook up very arbitrary solutions. May be you want vacuum to start. In which case you should look at arxiv.org/abs/1504.04592 $\endgroup$ Jul 19, 2016 at 17:36
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    $\begingroup$ If you do not require some version of "finite energy" hypothesis, or are willing to look at spatially compact solutions,one easy way to get periodic solutions is to consider "plane wave" space-times. (Take the model pp-wave solution and set the free function $H$ to be periodic in $u$, then you get a time (and space) periodic solution that roughly represents monochromatic gravitational radiation, the analogous solution to the linear wave equation would be $\phi(t,\vec{x}) = \sin(t - x_1)$.) $\endgroup$ Jul 19, 2016 at 18:00
  • $\begingroup$ @WillieWong thank you for yours comments and references :-) $\endgroup$
    – asm
    Jul 19, 2016 at 18:44
  • $\begingroup$ Regarding your edit: most of the time additional conditions make the problem more restrictive. It makes more sense to ask about under what conditions there does not exist non-stationary periodic solutions; because even without any conditions there does exist non-stationary periodic solutions. $\endgroup$ Jul 19, 2016 at 18:49
  • $\begingroup$ Sorry for delay. Let me try to formulate better the question. $\endgroup$
    – asm
    Jul 21, 2016 at 2:38

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