The 2 m ≤ r property of spherically symmetric static spacetimes

We prove that all spherically symmetric static spacetimes which are both regular at r = 0 and satisfying the single energy condition ρ + pr + 2pt ≥ 0 cannot contain any black hole region (equivalently, they must satisfy 2m/r ≤ 1 everywhere). This result holds even when the spacetime is allowed to contain a finite number of matching hypersurfaces. This theorem generalizes a result by Baumgarte and Rendall when the matter contents of the space-time is a perfect fluid and also complements their results in the general non-isotropic case. PACS Numbers: 04.20.Cv, 04.40.Dg, 04.40.Nr, 04.90.+e In a recent paper, Baumgarte and Rendall [1] have proven that all spherically symmetric static perfect-fluid solutions of the Einstein field equations with a regular centre cannot enter into a black hole region (up to where the pressure vanishes) provided that the energy-density ρ is continuous and non-negative and the central isotropic pressure is finite and positive. The method for obtaining their result was the analysis of the Tolman-Oppenheimer-Volkoff equation (we assume 8πG = c = 1 throughout) dp dr = − (ρ+ p) (2m(r) + pr) 2r (1− 2m(r)/r) ∗Also at Laboratori de F́ısica Matemàtica, Societat Catalana de F́ısica, IEC, Barcelona.