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Existence of stationary vacuum solutions of Einstein's equations in an exterior domain

Published online by Cambridge University Press:  17 February 2009

Jürgen Klenk
Jürgen Klenk
Affiliation:
IBM Zurich Research Laboratory, Säumerstrasse 4, 8803 Rüschlikon, Switzerland
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A proof is given for the existence and uniqueness of a stationary vacuum solution (M, g, ξ) of the boundary value problem consisting of Einstein's equations in an exterior domain M diffeomorphic to R × Σ (where Σ = R3\B(0, R)) and boundary data depending on the Killing field ξ on ∂Σ. The boundary data must be sufficiently close to that of a stationary, spatially conformally flat vacuum solution .

Type
Research Article
Information
The ANZIAM Journal , Volume 41 , Issue 2 , October 1999 , pp. 231 - 247
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Abraham, R., Marsden, J. and Ratiu, T., Manifolds, tensor analysis, and applications, Applied Mathematical Sciences vol. 75, Second ed. (Springer, 1988).Google Scholar
[2]Adams, R. A., Sobolev spaces (Academic Press, 1975).Google Scholar
[3]Bartnik, R., “The mass of an asymptotically flat manifold”, Comm. Pure Appl. Math. 34 (1986) 661–693.CrossRefGoogle Scholar
[4]Choquet-Bruhat, Y. and York, J. W., “The Cauchy problem”, in General relativity and gravitation 1 (ed. Held, A.), (Plenum Press, 1979) 99–172.Google Scholar
[5]Ehlers, J., “On limit relations between, and approximate explanations of, physical theories”, in Logic methodology and philosophy of science 7 (eds. Marcus, R. Barcan, Dorn, J. G. W. and Weingartner, P.), (North Holland, Amsterdam, 1986) 387–403.Google Scholar
[6]Eisenhart, L. P., Riemannian geometry, Second printing (Princeton University Press, 1949).Google Scholar
[7]Heilig, U., “On the existence of rotating stars in general relativity”, Comm. Math. Phys. 166 (1995) 457–493.CrossRefGoogle Scholar
[8]Lindblom, L., “Stationary stars are axisymmetric”, Astrophys. J. 208 (1976) 873–880.CrossRefGoogle Scholar
[9]Lindblom, L., “Fundamental properties of equilibrium stellar models”, Dissertation, University of Maryland, 1978.Google Scholar
[10]McOwen, R. C., “Boundary value problems for the Laplacian in an exterior domain”, Comm. Part. Diff. Eqns 6 (1981) 783–798.Google Scholar
[11]Pfister, H., “Zur Frage nach globalen Lösungen der Einsteinschen Feldgleichungen für rotierende Sterne [The quest for global solutions of Einstein's field equations for rotating stars]”, Wiss. Zs. Friedrich Schiller Univ. Jena Natur. Reihe 39 (1990) 152–160.Google Scholar
[12]Renardy, M. and Rogers, R. C., An introduction to partial differential equations (Springer, 1993).Google Scholar
[13]Reula, O., “On existence and behaviour of asymptotically flat solutions to the stationary Einstein equations”, Comm. Math. Phys. 122 (1989) 615–624.Google Scholar