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Regularity at space-like and null infinity

Part of: Partial differential equations Equations of mathematical physics and other areas of application General relativity Hyperbolic equations and systems Classical differential geometry

Published online by Cambridge University Press:  24 October 2008

Lionel J. Mason  and
Jean-Philippe Nicolas
Lionel J. Mason
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles', Oxford OX1 3LB, UK ([email protected])
Jean-Philippe Nicolas
Affiliation:
M.A.B., Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France ([email protected])
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Abstract

We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino–Schoen/Chrusciel–Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ‘Morawetz vector field’ to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino–Schoen/Chrusciel–Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino–Schoen/Chrusciel–Delay space-times.

Keywords

conformal compactificationSchwarzschild space-timewave equationpeelingvector field methodsweighted estimates

MSC classification

Secondary: 35B40: Asymptotic behavior of solutions 35B65: Smoothness and regularity of solutions 35L05: Wave equation 35Q75: PDEs in connection with relativity and gravitational theory 53A30: Conformal differential geometry 83C57: Black holes
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 1 , January 2009 , pp. 179 - 208
Copyright
Copyright © Cambridge University Press 2008

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