Go to previous page Go up Go to next page

13.2 Geometric inequalities for black holes

13.2.1 On the Penrose inequality

UpdateJump To The Next Update Information To rule out a certain class of potential counterexamples to the (weak) cosmic censorship hypothesis [303], Penrose derived an inequality that any asymptotically flat initial data set with (outermost) apparent horizon S must satisfy [305]: The ADM mass mADM of the data set cannot be less than the so-called irreducible mass of the horizon, V~ ---------------2-- M := Area(S)/(16pG ) (see also [156Jump To The Next Citation Point], and for a recent review of the problem and the relevant literature see [90]). However, as stressed by Ben-Dov [58Jump To The Next Citation Point], the more careful formulation of the inequality, due to Horowitz [202], is needed: Assuming that the dominant energy condition is satisfied, the ADM mass of the data set cannot be less than the irreducible mass of the 2-surface Smin, where Smin has the minimum area among the 2-surfaces enclosing the apparent horizon S. In [58] a spherically symmetric asymptotically flat data set with future apparent horizon is given which violates the first, but not the second version of the Penrose inequality.

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 [262] (see also [408185Jump To The Next Citation Point186Jump To The Next Citation Point]), for static black holes (using the Penrose mass, as we mentioned in Section 7.2.5[374375], and for the perturbed Reissner-Nordström spacetimes [225] (see also [226]). Although the original specific potential counterexample has been shown not to violate the Penrose inequality [157], the inequality has not been proven for a general data set. (For the limitations of the proof of the Penrose inequality for the area of trapped surface and the Bondi mass at past null infinity [258], see [64].) If the inequality were true, then this would be a strengthened version of the positive mass theorem, providing a positive lower bound for the ADM mass.

On the other hand, for time-symmetric data sets the Penrose inequality has been proven, even in the presence of more than one black hole. The proof is based on the use of some quasi-local energy expression, mostly of Geroch or of Hawking. First it is shown that these expressions are monotonic along the normal vector field of a special foliation of the time-symmetric initial hypersurface (see Sections 6.1.3 and 6.2, and also [143]), and then the global existence of such a foliation between the apparent horizon and the 2-sphere at infinity is proven. The first complete proof of the latter was given by Huisken and Ilmanen [207Jump To The Next Citation Point208Jump To The Next Citation Point]. (Recently Bray used a conformal technique to give an alternative proof [878889].)

A more general form of the conjecture, containing the electric charge e of the black hole, was formulated by Gibbons [156Jump To The Next Citation Point]: The ADM mass is claimed not to be exceeded by M + e2/(4G2M ). Although the weaker form of the inequality, the so-called Bogomolny inequality mADM > |e|/G, has been proven (under assumptions on the matter content, see for example [160369257159274156]), Gibbons’ inequality for the electric charge has been proven for special cases (for spherically symmetric spacetimes see for example [186Jump To The Next Citation Point]), and for time-symmetric initial data sets using Geroch’s inverse mean curvature flow [218]. As a consequence of the results of [207208] the latter has become a complete proof. However, this inequality does not seem to work in the presence of more than one black hole: For a time-symmetric data set describing n > 1 nearly extremal Reissner-Nodström black holes, 2 2 M + e /(4G M ) can be greater than the ADM mass, where 2 16pGM is either the area of the outermost marginally trapped surface [393], or the sum of the areas of the individual black hole horizons [124]. On the other hand, the weaker inequality derived from the cosmic censorship assumption, does not seem to be violated even in the presence of more than one black hole22. If in the final state of gravitational collapse the black hole has not only electric charge but angular momentum as well, then the geometry is described by the Kerr-Newman solution. Expressing the mass parameter m of the solution (which is just G times the ADM mass) in terms of the irreducible mass M of the horizon, the electric charge V~ -- q = e/ G and the angular momentum J = ma/G, we arrive at the more general form

V~ (------------)-----(------)-- -q2--- 2 -J---- 2 mADM > M + 4GM + 2GM (95)
of the Gibbons-Penrose inequality for asymptotically flat spacetimes with a single black hole. However, while mADM, M, and q are well-defined even in a general asymptotically flat data set, it is not clear what the angular momentum J should be: Is it the magnitude of the angular momentum defined at spatial infinity (which is already well-defined) or the (as yet not defined) quasi-local angular momentum of the horizon? Or should J be the length of the spatial angular momentum, or of the Pauli-Lubanski spin (or should the whole relativistic angular momentum contribute to J)? In the presence of more than one black hole only an inequality weaker than Equation (95View Equation) is expected to hold.

13.2.2 On the hoop conjecture

Update In connection with the formation of black holes and the weak cosmic censorship hypothesis another geometric inequality has also been formulated. This is the hoop conjecture of Thorne [367269], saying that ‘black holes with horizons form when and only when a mass m gets compacted into a region whose circumference C in every direction is C < 4pGm’ (see also [139Jump To The Next Citation Point391]). Mathematically, this conjecture is not precisely formulated: Neither the mass nor the notion of the circumference is well-defined. In certain situations the mass might be the ADM or the Bondi mass, but might be the integral of some locally defined ‘mass density’ as well [13935Jump To The Next Citation Point260Jump To The Next Citation Point240Jump To The Next Citation Point]. The most natural formulation of the hoop conjecture would be based on some spacelike 2-surface S and some reasonable notion of the quasi-local mass, and the trapped nature of the surface would be characterized by the mass and the ‘circumference’ of S. In fact, for round spheres outside the outermost trapped surface and the standard round sphere definition of the quasi-local energy (26View Equation) one has 4pGE = 2pr[1- exp(- 2a)] < 2pr = C, where we used the fact that r is an areal radius (see Section 4.2.1). If, however, S is not axi-symmetric then there is no natural definition (or, there are several inequivalent ‘natural’ definitions) of the circumference of S. Interesting necessary and also sufficient conditions for the existence of averaged trapped surfaces in non-spherically symmetric cases, both in special asymptotically flat and cosmological spacetimes, are found in [260240]. For the investigations of the hoop conjecture in the Gibbons-Penrose spacetime of the collapsing thin matter shell see [3635379298], and for colliding black holes see [407].
  Go to previous page Go up Go to next page