Abstract
Following earlier work of Masood-ul-Alam, we consider a uniqueness problem for non-rotating stellar models. Given a static, asymptotically flat perfectfluid spacetime with barotropic equation of state θ(p), and given another such spacetime which is spherically symmetric and has the same θ(p) and the same surface potential: we prove that both are identical provided θ(p) satisfies a certain differential inequality. This inequality is more natural and less restrictive than the conditions required by Masood-ul-Alam.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Masood-ul-Alam, A.K.M.: A proof of uniqueness of static stellar models with smalldч/dp. Class. Quantum Grav.5, 409–421 (1988)
Künzle, H.P., Savage, J.R.: A global analysis approach to the general relativistic fluid ball problem. Gen. Rel. Grav.12, 155–174 (1980)
Schmidt, B.G.: Kugelsymmetrische statische Materielösungen der Einsteinschen Feldgleichungen. Diploma thesis. Hamburg 1966
Shapiro, S.L., Teukolsky, S.A.: Black holes white dwarfs, and neutron stars. New York, Chichester, Brisbane, Toronto, Singapore: Wiley 1983
Harrison, B.K., Thorne, K.S., Wakano, M., Wheeler, J.A.: Gravitation theory and gravitational collapse. Chicago, London: The University of Chicago Press 1965
Schmid, W.: Numerische Integration der Tolman-Oppenheimer-Volkov Gleichungen. University of Vienna 1972 (unpublished)
Schoen, R., Yau, S.-T.: Proof of the possitive mass theorem. Commun. Math. Phys.65, 45–76 (1979)
Bunting, G.L., Masood-ul-Alam, A.K.M.: Nonexistence of multiple black holes in asymptotically Euclidean static vacuum space-time. Gen. Rel Grav.19, 147–154 (1987)
Beig, R., Simon, W.: On the spherical symmetry of static perfect fluids in general relativity. Lett. Math. Phys.21, 245–250 (1991)
Buchdahl, H.A.: A relativistic fluid sphere resembling the Emden polytrope of index 5. Astrophys. J.140, 1512–1516 (1964)
Israel, W.: Event horizons in static vacuum space-times. Phys. Rev.164, 1776–1779 (1967)
Robinson, D.C.: A simple proof of the generalization of Israel's theorem. Gen., Rel. Grav.8, 695–698 (1977)
Lindblom, L.: Some properties of static general relativistic stellar models. J., Math. Phys.21, 1455–1459 (1980)
Lindblom, L.: Static uniform-density stars must be spherical in general relativity. J. Math. Phys.29, 436–439 (1988)
Oppenheimer, J.R., Volkoff, G.M.: On massive neutron cores. Phys. Rev.55, 374–381 (1939)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1983
Künzle, H.P.: On the spherical symmetry of a static perfect fluid. Commun. Math. Phys.20, 85–100 (1971)
Avez, A.: Leds 2 de Schwarzschild parmi lesds 2 stationnaires. Ann. Inst. Henri Poincaré1, 291–300 (1964)
Lichnerowicz, A.: Théories relativistes de la gravitation et de l'électromagnétisme. Paris: Masson 1955
Beig, R., Simon, W.: Proof of a multipole conjecture due, to Geroch. Commun. Math. Phys.78, 75–82 (1980)
Beig, R.: Conformal properties of static spacetimes. Class. Quantum Grav.8, 263–271 (1991)
Simon, W., Beig, R.: The multipole structure of stationary space-times. J. Math. Phys.24, 1163–1171 (1984)
Beig, R.: A remarkable property of spherical harmonics. J. Math. Phys.26, 769–770 (1985)
Müller zum Hagen, H.: On the analyticity of static solutions of Einstein's equations. Proc. Camb. Phil. Soc.67, 415–421 (1970)
Bartnik, R.: The mass of an asymptotically flat manifold Commun. Pure Appl. Math.49, 661–693 (1986)
Müller zum Hagen, H.: The static two body problem. Proc. Camb. Phil. Soc.75, 249–260 (1974)
Lichtenstein, L.: Über einige Eigenschaften der Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Math. Z.28, 635–640 (1928)
Gidas, B., Ni, W.-M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations inR n. Adv. Math. Suppl. Stud.7A, 369–402 (1981)
Sperb, R.P.: Maximum principles and their applications. New York, London, Toronto, Sydney, San Francisco: Academic Press 1981
Weinberger, H.F.: Remark on the preceding paper of Serrin. Arch. Rational Mech. Anal.43, 319–320 (1971)
Author information
Authors and Affiliations
Additional information
Communicated by S.-T. Yau
Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, project P-7197
Rights and permissions
About this article
Cite this article
Beig, R., Simon, W. On the uniqueness of static perfect-fluid solutions in general relativity. Commun.Math. Phys. 144, 373–390 (1992). https://doi.org/10.1007/BF02101098
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02101098