Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T05:35:17.843Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baez, J. C., Segal, I. E. and Zhou, Z. F., The global Goursat problem and scattering for nonlinear wave equations, J. Funct. Analysis 93 (1990), 239–269.CrossRefGoogle Scholar
2.Christodoulou, D. and Klainerman, S., The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, Volume 41 (Princeton University Press, Princeton, NJ, 1993).Google Scholar
3.Chrusciel, P. T. and Delay, E., Existence of non trivial, asymptotically vacuum, asymptotically simple space-times, Class. Quant. Grav. 19 (2002), L71–L79 (erratum Class. Quant. Grav. 19 (2002), 3389).CrossRefGoogle Scholar
4.Chrusciel, P. T. and Delay, E., On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mem. Soc. Math. France 94 (2003), 1–10.Google Scholar
5.Corvino, J., Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys. 214 (2000), 137–189.CrossRefGoogle Scholar
6.Corvino, J. and Schoen, R. M., On the asymptotics for the vacuum Einstein constraint equations, J. Diff. Geom. 73(2) (2006), 185–218.Google Scholar
7.Dafermos, M. and Rodnianski, I., The red-shift effect and radiation decay on black hole space-times, preprint (available at http://arxiv.org/abs/gr-qc/0512119, 2006).Google Scholar
8.Friedlander, F. G., Radiation fields and hyperbolic scattering theory, Math. Proc. Camb. Phil. Soc. 88 (1980), 483–515.CrossRefGoogle Scholar
9.Friedlander, F. G., Notes on the wave equation on asymptotically Euclidean manifolds, J. Funct. Analysis 184 (2001), 1–18.CrossRefGoogle Scholar
10.Friedrich, H., Smoothness at null infinity and the structure of initial data, in The Einstein equations and the large scale behavior of gravitational fields (ed. Chrusciel, P. and Friedrich, H.), pp. 121–203 (Birkhäuser, Basel, 2004).CrossRefGoogle Scholar
11.Hörmander, L., A remark on the characteristic Cauchy problem, J. Funct. Analysis 93 (1990), 270–277.CrossRefGoogle Scholar
12.Klainerman, S. and Nicolò, F., On local and global aspects of the Cauchy problem in general relativity, Class. Quant. Grav. 16 (1999), R73–R157.CrossRefGoogle Scholar
13.Klainerman, S. and Nicolò, F., The evolution problem in general relativity, Progress in Mathematical Physics, Volume 25 (Birkhäuser, Basel, 2002).Google Scholar
14.Klainerman, S. and Nicolò, F., Peeling properties of asymptotically flat solutions to the Einstein vacuum equations, Class. Quant. Grav. 20 (2003), 3215–3257.CrossRefGoogle Scholar
15.Mason, L. J. and Nicolas, J.-P., Conformal scattering and the Goursat problem, J. Hyperbolic Diff. Eqns 1(2) (2004), 197–233.CrossRefGoogle Scholar
16.Morawetz, C. S., The decay of solutions of the exterior initial-boundary value problem for the wave equation, Commun. Pure Appl. Math. 14 (1961), 561–568.CrossRefGoogle Scholar
17.Penrose, R., Null hypersurface initial data for classical fields of arbitrary spin and for general relativity, Aerospace Research Laboratories Report 63-56 (P. G. Bergmann, 1963) (reprinted in Gen. Rel. Grav. 12 (1980), 225–64).CrossRefGoogle Scholar
18.Penrose, R., Zero rest-mass fields including gravitation: asymptotic behavior, Proc. R. Soc. Lond. A 284 (1965), 159–203.Google Scholar
19.Penrose, R. and Rindler, W., Spinors and space-time, Volumes 1 and 2 (Cambridge University Press, 1984/1986).CrossRefGoogle Scholar