Abstract
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking's celebrated theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that i) it does not rely on approximate monotonicity identities and ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.
Highlights
In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface
For an open2 subset of regular initial data in high spatial dimensions that satisfy the assumptions of Hawking’s theorem, we show that the solution’s incompleteness is due to the formation of a Big Bang, that is, the formation of a curvature singularity along a spacelike hypersurface
Our work provides the first constructive proof of stable curvature blowup along a spacelike hypersurface as an effect of pure gravity for Einstein’s equations in more than one spatial dimension without symmetry assumptions
Summary
Hawking’s celebrated “singularity” theorem (see, for example, [59, Theorem 9.5.1]) shows that an interestingly large set of initial data for the Einstein-vacuum equations leads to geodesically incomplete solutions. Our work provides the first constructive proof of stable curvature blowup along a spacelike hypersurface as an effect of pure gravity (i.e., without the presence of matter) for Einstein’s equations in more than one spatial dimension without symmetry assumptions. For the Einstein-vacuum equations in low space dimensions, the only obstacle to the existence of such solutions is the Hamiltonian constraint equation (1.2a) Given such a background solution, the rest of our analysis is essentially dimensionally independent.. Our main theorem applies only when D ≥ 38, our approach here is of interest in itself since it is more robust compared to prior works on stable blowup for various Einstein-matter systems, and since it has further applications, for example in three spatial dimensions; see the end of Subsect. Given such a background solution, the rest of our analysis is essentially dimensionally independent. our main theorem applies only when D ≥ 38, our approach here is of interest in itself since it is more robust compared to prior works on stable blowup for various Einstein-matter systems, and since it has further applications, for example in three spatial dimensions; see the end of Subsect. 1.2 for further discussion on this point
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