Revisiting the characteristic initial value problem for the vacuum Einstein field equations
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Revisiting the characteristic initial value problem for the vacuum Einstein field equations David Hilditch1 · Juan A. Valiente Kroon2
· Peng Zhao2
Received: 28 April 2020 / Accepted: 16 September 2020 © The Author(s) 2020
Abstract Using the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed. Keywords Characteristic problem · Newman-Penrose formalism · Initial value problem
Contents 1 2 3 4 5 6 7
Introduction . . . . . . . . . . . . . . . . . The geometry of the problem . . . . . . . . The initial data for the CIVP . . . . . . . . . Rendall’s local existence theory . . . . . . . Setting-up Luk’s strategy . . . . . . . . . . Main estimates . . . . . . . . . . . . . . . . Last slice argument and the end of the proof
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Juan A. Valiente Kroon [email protected] David Hilditch [email protected] Peng Zhao [email protected]
1
CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisbon, Portugal
2
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK 0123456789().: V,-vol
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A: The Einstein field equations in the NP formalism B: Inequalities . . . . . . . . . . . . . . . . . . . . C: Angular derivatives of a scalar function . . . . . D: Integration Identities . . . . . . . . . . . . . . . E: Details in Propositions 8 and 9 . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
D. Hilditch et al. . . . . . .
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1 Introduction The simplest setups of partial differential equations (PDEs) are of course the boundary value and Cauchy/initial value p
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