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Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal).

For Simplicity assume that there's no matter $T_{\mu\nu} =0$. And the spacetime satisfies Einstein's Field Equations.

Given a Cauchy hypersurface $S$ at initial time $t=t_0$,

the main question is:

Can there exist any solution where the evolution of at least a single point $p \in S$ is smooth BUT nowhere analytic in time?

The initial data is that of the Cauchy data and $S$ is supposed to be spacelike.

Assume $p \in U \subsetneq S$ where $U$ is an open subset of $S$ for the initial data set.

In case $U$ should not be even dense in $M$ or fulfill any other topological constraints, it must be clarified.

Physical motivation for the interested reader:

A smooth but no-where analytic evolution function for a Cauchy well-posed problem is my personal way to formulate indeterminism.

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  • $\begingroup$ The question is ill-posed as written. Evolution with respect to which PDE? When speaking of regularity, at least two things must be defined precisely: "regularity" of initial data and "regularity" of solutions. Only in relating the two can the "regularity" of evolution be meaningfully discussed. $\endgroup$
    – Igor Khavkine
    Commented Sep 30, 2023 at 10:32
  • $\begingroup$ Isn't it just too obvious that the PDE is Einstein's Field Equation for a generally globally hyperbolic spacetime as much as the OP is concerned with? @IgorKhavkine $\endgroup$
    – Bastam Tajik
    Commented Sep 30, 2023 at 11:35
  • $\begingroup$ Is there a coordinate-free definition of what it means for a solution of the EFE to be analytic? $\endgroup$
    – James E Hanson
    Commented Sep 30, 2023 at 17:27
  • $\begingroup$ I guess as much as there is smoothness, why not analyticity? Though for a deeper response I should honestly think more. @JamesHanson $\endgroup$
    – Bastam Tajik
    Commented Sep 30, 2023 at 17:32
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    $\begingroup$ @BastamTajik My point is just that by intentionally choosing your coordinate system poorly, you can make any metric look non-analytic. Conversely, it's not prima facie clear to me that there even is a pseudo-Riemannian manifold that doesn't admit a coordinate system in which the metric is analytic (although I suspect there is). $\endgroup$
    – James E Hanson
    Commented Sep 30, 2023 at 17:35

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If you're interested in chaos in the context of general relativity and quantum gravity, this Q&A may be helpful - https://physics.stackexchange.com/questions/369767/chaos-and-general-relativity

In particular, the first paper cited there, John D. Barrow Chaotic Behaviour in General Relativity Physics Reports 85(1) 1982 pp. 1-49 https://doi.org/10.1016/0370-1573(82)90171-5 may help in resolving your original question.

ETA: For further historical background on existence theory for the non-analytic case, it's useful to consult

Demetrios Christodoulou and Richard Kerner Editorial note to: Existence theorem for the Einsteinian gravitational field equations in the non-analytic case, by Yvonne Fourès-Bruhat,
Gen Relativ Gravit 54, 36 (2022). https://doi.org/10.1007/s10714-022-02916-5

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