Vol. 14, No. 8, 2021

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Wave equations with initial data on compact Cauchy horizons

Oliver Lindblad Petersen

Vol. 14 (2021), No. 8, 2363–2408

We study the following problem: given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be normalised to a nonzero constant. Examples include the Misner spacetime and the Taub-NUT spacetime. We prove an energy estimate close to the Cauchy horizon for wave equations acting on sections of vector bundles. Using this estimate we prove that if a linear wave equation can be solved up to any order at the Cauchy horizon, then there exists a unique solution on the globally hyperbolic region. As a consequence, we prove several existence and uniqueness results for linear and nonlinear wave equations without assuming analyticity or symmetry of the spacetime and without assuming that the generators close. We overcome in particular the essential remaining difficulty in proving that vacuum spacetimes with a compact Cauchy horizon, with constant nonzero surface gravity, necessarily admit a Killing vector field. This work is therefore related to the strong cosmic censorship conjecture.

compact Cauchy horizon, characteristic Cauchy problem, Misner spacetime, Taub-NUT spacetime, strong cosmic censorship
Mathematical Subject Classification 2010
Primary: 58J45
Secondary: 53C50
Received: 10 September 2018
Revised: 4 May 2020
Accepted: 31 July 2020
Published: 19 December 2021
Oliver Lindblad Petersen
Department of Mathematics
Uppsala University