Improved existence for the characteristic initial value problem with the conformal Einstein field equations
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Improved existence for the characteristic initial value problem with the conformal Einstein field equations David Hilditch1 · Juan A. Valiente Kroon2
· Peng Zhao2
Received: 30 June 2020 / Accepted: 26 August 2020 © The Author(s) 2020
Abstract We adapt Luk’s analysis of the characteristic initial value problem in general relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall’s reduction strategy. In analysing the conformal Einstein equations we make use of the Newman–Penrose formalism and a gauge due to J. Stewart. Keywords Characteristic initial value problem · Conformal Einstein field equations · Newman–Penrose formalism · Asymptotically simple spacetimes
Contents 1 Introduction . . . . . . . . . . . . . . . . . . New insights . . . . . . . . . . . . . . . . . . Overview and main results . . . . . . . . Notation and conventions . . . . . . . . . 2 The vacuum conformal Einstein field equations 3 The geometry of the problem . . . . . . . . .
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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
123
85
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3.1 Basic setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Stewart’s gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The NP frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Spin connection coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Equations for the frame coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 The conformal gauge conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The formulation of the CIVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Specifiable
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