In the previous works [RSR19, SR22] we have introduced a new type of self-similarity for the Einstein
vacuum equations characterized by the fact that the homothetic vector field may be spacelike on the past
light cone of the singularity. In this work we give a systematic treatment of this new self-similarity. In
particular, we provide geometric characterizations of spacetimes admitting the new symmetry and show
the existence and uniqueness of formal expansions around the past null cone of the singularity which
may be considered analogues of the well-known Fefferman–Graham expansions. In combination with
results from [RSR19] our analysis will show that the twisted self-similar solutions are sufficiently general
to describe all possible asymptotic behaviors for spacetimes in the small data regime which are selfsimilar and whose homothetic vector field is everywhere spacelike on an initial spacelike hypersurface. We
present an application of this later fact to the understanding of the global structure of Fefferman–Graham
spacetimes and the naked singularities of [RSR19, SR22]. Lastly, we observe that by an amalgamation of
the techniques from [RSR18, RSR19], one may associate true solutions to the Einstein vacuum equations
to each of our formal expansions in a suitable region of spacetime.
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This content will become publicly available on June 1, 2024
Naked Singularities in the Einstein-Euler System
Abstract In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of selfsimilar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a selfsimilar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to flatten the selfsimilar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.
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- NSF-PAR ID:
- 10419967
- Date Published:
- Journal Name:
- Annals of PDE
- Volume:
- 9
- Issue:
- 1
- ISSN:
- 2524-5317
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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