Null Geodesics and Improved Unique Continuation for Waves in Asymptotically Anti-de Sitter Spacetimes
| Autor: | Arick Shao, Alex McGill |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Physics
Physics and Astronomy (miscellaneous) Geodesic Null (mathematics) Boundary (topology) FOS: Physical sciences Conformal map General Relativity and Quantum Cosmology (gr-qc) General Relativity and Quantum Cosmology symbols.namesake Theory of relativity Mathematics - Analysis of PDEs 35A02 35Q75 83C30 35L05 symbols FOS: Mathematics Covariant transformation Anti-de Sitter space Klein–Gordon equation Mathematical physics Analysis of PDEs (math.AP) |
| Popis: | We consider the question of whether solutions of Klein--Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given by the second author with G. Holzegel, under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein--Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates---and hence new unique continuation results---for Klein--Gordon equations on a larger class of spacetimes, in particular with more general boundary geometries. Second, we argue for the optimality, in many respects, of our assumptions by connecting them to trajectories of null geodesics near the conformal boundary; these geodesics play a crucial role in the construction of counterexamples to unique continuation. Finally, we develop a new covariant formalism that will be useful---both presently and more generally beyond this article---for treating tensorial objects with asymptotic limits at the conformal boundary. 55 pages, 2 figures |
| Databáze: | OpenAIRE |
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