Inequalities Relating Area, Energy, Surface Gravity, and Charge of Black Holes

Abstract

The Penrose-Gibbons inequality for charged black holes is proved in spherical symmetry, assuming that outside the black hole there are no current sources, meaning that the charge e is constant, with the remaining fields satisfying the dominant energy condition. Specifically, for any achronal hypersurface which is asymptotically flat at spatial or null infinity and has an outermost marginal surface of areal radius r, the asymptotic mass m satisfies 2m $ r 1 e 2 ͞r. Replacing m by a local energy m, the inequality holds locally outside the black hole. A recent definition of dynamic surface gravity k also satisfies inequalities 2k # 1͞r 2 e 2 ͞r 3 and m $ m $ r 2 k 1 e 2 ͞r. All these inequalities are sharp in the sense that equality is attained for the Reissner-Nordström black hole. [S0031-9007(98) 07700-X] In general relativity, one expects that black holes will form if matter or energy is sufficiently concentrated, and conversely that a black hole of given size will require a certain minimal amount of energy. The latter expectation was formulated by Penrose [1] in the form of an inequalitywhere A is the area of the black hole and m is the mass or energy at spatial infinity, assuming an asymptotically flat space-time, and the Newtonian gravitational constant is unity. Penrose argued that this was required by his cosmic censorship hypothesis. Although the Penrose inequality remains a conjecture in general, it has recently been established for time-symmetric hypersurfaces [2] and in spherical symmetry [3]. The latter result also holds locally outside the black hole with m replaced by a local energy E, so that one need not assume asymptotic flatness. This proof and localization of the Penrose inequality in spherical symmetry was first reported as a special case of an inequality for the Hawking energy [4] and seems to be widely unknown. Moreover, there is another inequality relating the area and surface gravity k as recently defined for a dynamic, spherically symmetric black hole [5]:Physically this means that for a black hole of given size, the gravitational attraction (acceleration) cannot exceed a certain amount. The two inequalities combine to yield a third inequalitywhich also holds in Newtonian gravity. Physically this means that the total surface gravity (integrated over the surface) cannot exceed a value given by the asymptotic mass. Again this may be localized in terms of E. All these inequalities require the dominant energy condition. They are also sharp in the sense that equality is attained for the Schwarzschild black hole. However, equality is not attained for the Reissner-Nordström charged black hole. Gibbons [6] has therefore suggested that for charged black holes, there should be a sharpened version of the Penrose inequality, relating the area, mass, and charge. This Gibbons inequality is established here in spherical symmetry, along with sharpened versions of the other two inequalities and localized versions with m replaced (where it appears) by a local energy m.Geometry.-In spherical symmetry, the ...

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…The inequality has been proven for the outermost future apparent horizons outside the outermost past apparent horizon in maximal data sets in spherically-symmetric spacetimes [352] (see, also, [578, 250, 251]), for static black holes (using the Penrose mass, as mentioned in Section 7.2.5) [513, 514] and for the perturbed Reissner-Nordström spacetimes [301] (see, also, [302]). Although the original specific potential counterexample has been shown not to violate the Penrose inequality [214], the inequality has not been proven for a general data set.…”

Angular momentum conservation for dynamical black holes

“…It would be preferable to have independently motivated definitions of these quantities, but so far this has been done only in spherical symmetry [19,20,21], where there are natural definitions of E, κ and Φ = Q/R which can be applied anywhere in the space-time, coinciding with the above expressions on the outer horizons of a ReissnerNordström black hole. In the dynamical context, E ≥ M is generally not the ADM energy, since there may be matter or gravitational radiation outside the black hole.…”

“…So a black hole grows if something falls into it, otherwise staying the same size. Comprehensive treatments were subsequently given for spherical symmetry [19,20,21], cylindrical symmetry [22] and a quasi-spherical approximation [23,24,25,26]. In each case, definitions were found for all the non-zero physical quantities mentioned above, providing prototypes of all except J and Ω.…”

Kerr Black Holes in Horizon-Generating Form

“…However, vanishes at the bifurcation surface, leaving the possibility to include it in different foliations of the horizons. Another such foliation can be found simply by replacing a with ÿa in the definitions of t (5) and ' (8), which has the effect of changing the sign of d# in the metric, or of @ # in the inverse metric. The two sets of transverse surfaces have the same poles, but their equators are displaced relative to each other.…”