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  (see also [408, 185, 186]), for static black holes (using the Penrose mass, as we mentioned in Section 7.2.5) [374, 375], and for the perturbed Reissner-Nordström spacetimes  (see also ). Although the original specific potential counterexample has been shown not to violate the Penrose inequality , 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 , see .) 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 ), 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 [207, 208]. (Recently Bray used a conformal technique to give an alternative proof [87, 88, 89].)
A more general form of the conjecture, containing the electric charge of the black hole, was formulated by Gibbons : The ADM mass is claimed not to be exceeded by . Although the weaker form of the inequality, the so-called Bogomolny inequality , has been proven (under assumptions on the matter content, see for example [160, 369, 257, 159, 274, 156]), Gibbons’ inequality for the electric charge has been proven for special cases (for spherically symmetric spacetimes see for example ), and for time-symmetric initial data sets using Geroch’s inverse mean curvature flow . As a consequence of the results of [207, 208] 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 nearly extremal Reissner-Nodström black holes, can be greater than the ADM mass, where is either the area of the outermost marginally trapped surface , or the sum of the areas of the individual black hole horizons . 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 of the solution (which is just times the ADM mass) in terms of the irreducible mass of the horizon, the electric charge and the angular momentum , we arrive at the more general formsingle black hole. However, while , , and are well-defined even in a general asymptotically flat data set, it is not clear what the angular momentum 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 be the length of the spatial angular momentum, or of the Pauli-Lubanski spin (or should the whole relativistic angular momentum contribute to )? In the presence of more than one black hole only an inequality weaker than Equation (95) is expected to hold.
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