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Abstract We review recent mathematical results concerning the high-frequency solutions to the Einstein vacuum equations and the limits of these solutions. In particular, we focus on two conjectures of Burnett, which attempt to give an exact characterization of high-frequency limits of vacuum spacetimes as solutions to the Einstein–massless Vlasov system. Some open problems and future directions are discussed.
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Twisted Self-Similarity and the Einstein Vacuum Equations
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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- Award ID(s):
- 1900288
- PAR ID:
- 10439473
- Date Published:
- Journal Name:
- Communications in mathematical physics
- ISSN:
- 1432-0916
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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