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Sharp bounds on \(2m/r\) of general spherically symmetric static objects. (English) Zbl 1151.83014
Summary: In 1959 H. A. Buchdahl [Phys. Rev., II. Ser. 116, 1027–1034 (1959; Zbl 0092.20802)] obtained the inequality \(2M/R\leq 8/9\) under the assumptions that the energy density is non-increasing outwards and that the pressure is isotropic. Here \(M\) is the ADM mass and \(R\) the area radius of the boundary of the static body. The assumptions used to derive the Buchdahl inequality are very restrictive and for instance neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider any static solution of the spherically symmetric Einstein equations for which the energy density \(\rho\geq 0\), and the radial and tangential pressures \(p\geq 0\) and \(p_T\) satisfy \(p+2p_T\leq \Omega\rho\), \(\Omega>0\), and we show that
\[ \sup_{r>0} \frac{2m(r)}{r}\leq \frac{(1+2\Omega)^2-1} {(1+2\Omega)^2}. \]
where \(m\) is the quasi-local mass, so that in particular \(M=m(R)\). We also show that the inequality is sharp under these assumptions. Note that when \(\Omega=1\) the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with \(p\geq 0\) which satisfies the dominant energy condition satisfies the hypotheses with \(\Omega=3\).

MSC:
83C15 Exact solutions to problems in general relativity and gravitational theory
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
83E05 Geometrodynamics and the holographic principle
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References:
[1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover Publications New York · Zbl 0515.33001
[2] Aikawa, H.; Essén, M., Potential theory – selected topics, Lecture notes in math., vol. 1633, (1996), Springer-Verlag Berlin, Heidelberg · Zbl 0865.31001
[3] Andréasson, H., On the buchdahl inequality for spherically symmetric static shells, Comm. math. phys., 274, 399-408, (2007) · Zbl 1121.83007
[4] Andréasson, H., On static shells and the buchdahl inequality for the spherically symmetric einstein – vlasov system, Comm. math. phys., 274, 409-425, (2007) · Zbl 1121.83008
[5] Andréasson, H., The einstein – vlasov system/kinetic theory, Living rev. relativ., 8, (2005)
[6] Andréasson, H.; Rein, G., A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric einstein – vlasov system, Classical quantum gravity, 23, 3659-3677, (2006) · Zbl 1096.83035
[7] Andréasson, H.; Rein, G., On the steady states of the spherically symmetric einstein – vlasov system, Classical quantum gravity, 24, 1809-1832, (2007) · Zbl 1112.83014
[8] Baumgarte, T.W.; Rendall, A.D., Regularity of spherically symmetric static solutions of the Einstein equations, Classical quantum gravity, 10, 327-332, (1993)
[9] Binney, J.; Tremaine, S., Galactic dynamics, (1987), Princeton Univ. Press · Zbl 1130.85301
[10] Böhmer, C.G.; Harko, T., Bounds on the basic physical parameters for anisotropic compact general relativistic objects, Classical quantum gravity, 23, 6479-6491, (2006) · Zbl 1117.83058
[11] Bondi, H., Massive spheres in general relativity, Proc. R. soc. lond. ser. A, 282, 303-317, (1964) · Zbl 0125.21003
[12] Bondi, H., Anisotropic spheres in general relativity, Mon. not. R. astron. soc., 259, 365-368, (1992)
[13] Bowers, R.L.; Liang, E.P.T., Anisotropic spheres in general relativity, Astrophys. J., 188, 657-665, (1974)
[14] Buchdahl, H.A., General relativistic fluid spheres, Phys. rev., 116, 1027-1034, (1959) · Zbl 0092.20802
[15] Christodoulou, D., The formation of black holes and singularities in spherically symmetric gravitational collapse, Comm. pure appl. math., 44, 339-373, (1991) · Zbl 0728.53061
[16] Christodoulou, D., The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of math., 149, 183-217, (1999) · Zbl 1126.83305
[17] Fraundiener, J.; Hoenselaers, C.; Konrad, W., A shell around a black hole, Classical quantum gravity, 7, 585-587, (1990)
[18] Guven, J.; Ó Murchadha, N., Bounds on \(2 m / R\) for static spherical objects, Phys. rev. D, 60, 084020, (1999)
[19] Heintzmann, H.; Hillebrandt, W., Neutron stars with an anisotropic equation of state: mass, redshift and stability, Astronom. astrophys., 38, 51-55, (1975)
[20] Herrera, L.; Santos, N.O., Local anisotropy in self-gravitating systems, Phys. rep., 286, 53, (1997)
[21] Lemaitre, G., L’univers en expansion, Ann. soc. sci. brux. A, 53, 51, (1993) · Zbl 0007.33104
[22] Lieb, E.; Loss, M., Analysis, Grad. stud. math., vol. 14, (1997), Amer. Math. Soc.
[23] Mars, M.; Martín-Prats, M. Mercè; Senovilla, J.M.M., The \(2 m \leqslant r\) property of spherically symmetric static spacetimes, Phys. lett. A, 218, 147, (1996)
[24] Rendall, A.D., An introduction to the einstein – vlasov system, (), 35-68 · Zbl 0892.35148
[25] Schwarzschild, K., Über das gravitationsfeld einer kugel aus inkompressibler flussigkeit nach der einsteinschen theorie, Sitz. preuss. akad. wiss. Berlin kl. math.-phys. tech., 424-434, (1916) · JFM 46.1297.01
[26] Wald, R.M., General relativity, (1984), The University of Chicago Press · Zbl 0549.53001
[27] Weinberg, S., Gravitation and cosmology, (1972), John Wiley and Sons, Inc.
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