To rule out a certain class of potential counterexamples to the (weak) cosmic censorship hypothesis [416], Penrose derived an inequality that any asymptotically flat initial data set with (outermost) apparent horizon must satisfy [418]: The ADM mass of the data set cannot be less than the irreducible mass of the horizon, (see, also, [213, 113, 354]). However, as stressed by Ben-Dov [75], the more careful formulation of the inequality, due to Horowitz [273], 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 two-surface , where has the minimum area among the two-surfaces enclosing the apparent horizon . In [75] 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 [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. (For the limitations of the proof of the Penrose inequality for the area of a trapped surface and the Bondi mass at past null infinity [345], see [82].) 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 [193]), and then the global existence of such a foliation between the apparent horizon and the two-sphere at infinity is proven. The first complete proof of the latter was given by Huisken and Ilmanen [278, 279]. (An alternative proof, using a conformal technique, was given by Bray [110, 111, 112].) A simple (but complete) proof of the Riemannian Penrose inequality is given in the special case of axisymmetric time-symmetric data sets by using Brill’s energy positivity proof [218].

A more general form of the conjecture, containing the electric charge parameter of the black hole,
was formulated by Gibbons [213]: The ADM mass is claimed not to be exceeded by .
Although the weaker form of the inequality, the Bogomolny inequality , has been proven
(under assumptions on the matter content, see, e.g., [219, 508, 344, 217, 371, 213]), Gibbons’ inequality
for the electric charge has been proven for special cases (for spherically-symmetric spacetimes see,
e.g., [251]), and for time-symmetric initial data sets using Geroch’s inverse mean curvature
flow [290]. As a consequence of the results of [278, 279] 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–Nordström black holes,
can be greater than the ADM mass, where is either the area of the
outermost marginally-trapped surface [546], or the sum of the areas of the individual black
hole horizons. On the other hand, the weaker inequality (13.1) below, derived from the cosmic
censorship assumption, does not seem to be violated, even in the presence of more than one black
hole.^{22}

Repeating Penrose’s argumentation (weak cosmic censorship hypothesis, the conjecture that the final state of black holes is described by some Kerr–Newman solution, Bondi’s mass-loss and the assumption that the Bondi mass is not greater than the ADM mass) in axisymmetric electrovacuum spacetime, and assuming that the angular momentum , measured at the future null infinity in the stationary stage (defined by the Komar integral using the Killing vector of axisymmetry) coincides with the ADM angular momentum , for the irreducible mass of the black hole we obtain the upper bound (see also [168])

Here the electric charge , measured at spatial infinity as well, is related to the charge parameter of the black hole final state as . If initially there are more, say , black holes, then in (13.1) is built from the irreducible masses of the individual black holes as . The inequality (13.1) implies that one of the following inequalities: holds: If (13.2) is violated, then, by (13.1), the inequality (13.3) holds (though one does not exclude the other). Both inequalities give positive lower bounds for the ADM mass in terms of the irreducible mass and other quantities measured also at spatial infinity. The Kerr–Newman solution saturates (13.3). However, while lower bounds for in terms of and can be given even on a general asymptotically flat data set, in lack of axisymmetry it does not seem to be possible to control in terms of , and hence, to derive lower bounds for in terms of , and .The structure of Eqs. (13.2) and (13.3) suggests another interpretation, too. In fact, since is a quasi-locally defined property of the black hole itself, it is natural to ask if the lower bound for the ADM mass can be given only in terms of quasi-locally defined quantities. In the absence of charges outside the horizon, is just the charge measured at , and if, in addition, the spacetime is axisymmetric and vacuum, then coincides with the Komar angular momentum also at . However, in general it is not clear what would have to be: The magnitude of some quasi-locally defined relativistic angular momentum, or only of the spatial part of the angular momentum, or even the Pauli–Lubanski spin?

Penrose-like inequalities are studied numerically in [295], while counter-examples to a new version, and to a generalized form (including charge) of the Penrose inequality are given in [129] and [166], respectively. Reviews of the Penrose inequality with an extended bibliography are [354, 355].

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 [506, 366], saying that ‘black holes with horizons form when and only when a mass gets compacted into a region whose circumference in every direction is ’ (see, also, [188, 538]). 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 [188, 50, 350, 320]. The most natural formulation of the hoop conjecture would be based on some spacelike two-surface 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 . In fact, for round spheres outside the outermost trapped surface and the standard round-sphere definition of the quasi-local energy (4.7) one has , where we use the fact that is an areal radius (see Section 4.2.1).

Another formulation of the hoop conjecture, also for the spherically symmetric configurations, was given by Ó Murchadha, Tung, Xie and Malec in [402] using the Brown–York energy. They showed that a spherical 2-surface, which is embedded in a spherically symmetric asymptotically flat 3-slice with a regular center and which satisfies , is trapped. Moreover, if holds for all embeddings, then the surface is not trapped. The root of the deviation of the numerical coefficient in front of the quasi-local energy here (viz. ) from the one in Thorne’s original formulation (i.e., ) is the fact that on the event horizon of a Schwarzschild black hole is , rather than the expected . It is also shown in [402] that no analogous statement can be proven in terms of the Kijowski–Liu–Yau or the Wang–Yau energies.

If, however, is not axisymmetric, then there is no natural definition (or, there are several inequivalent ‘natural’ definitions) for the circumference of . 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 [350, 320]. For the investigations of the hoop conjecture in the Gibbons–Penrose spacetime of the collapsing thin matter shell see [51, 50, 518, 411], and for colliding black holes see [574]. One reformulation of the hoop conjecture, using the new concept of the ‘trapped circle’ instead of the ill-defined circumference, is suggested by Senovilla [450]. Another version of the hoop conjecture was suggested by Gibbons in terms of the ADM mass and the Birkhoff invariant of horizon of spherical topology, and this form of the conjecture was proved in a number a special cases [215, 159].

The Kerr–Newman solution describes a black hole precisely when the mass parameter dominates the angular momentum and the charge parameters: . Thus, it is natural to ask whether or not an analogous inequality holds for more general, dynamic black holes. As Dain has proven, in the axisymmetric, vacuum case there is an analogous inequality, a consequence of an extremality property of Brill’s form of the ADM mass. Namely, it is shown in [165], that the unique absolute minimum of the ADM mass functional on the set of the vacuum Brill data sets with fixed ADM angular momentum is the extreme Kerr data set. Here a Brill data set is an axisymmetric, asymptotically flat, maximal, vacuum data set, which, in addition, satisfies certain global conditions (viz. the form of the metric is given globally, and nontrivial boundary conditions are imposed) [218, 165]. The key tool is a manifestly positive definite expression of the ADM energy in the form of a three-dimensional integral, given in globally defined coordinates. If the angular momentum is nonzero, then by the assumption of axisymmetry and vacuum, the data set contains a black hole (or black holes), and hence, the extremality property of the ADM energy implies that the ADM mass of this (in general, nonstationary) black hole cannot be less than its ADM angular momentum. For further discussion of this inequality, in particular its role analogous to that of the Penrose inequality, see [164]; and for earlier versions of the extremality result above, see [163, 162, 161].

Since in the above result the spacetime is axisymmetric and vacuum, the ADM angular momentum could be written as the Komar integral built from the Killing vector of axisymmetry on any closed spacelike spherical two-surface homologous to the large sphere near the actual infinity. Thus, the angular momentum in Dain’s inequality can be considered as a quasi-local expression. Hence, it is natural to ask if the whole inequality is a condition on quasi-locally defined quantities or not. However, as already noted in Section 12.1, in the stationary axisymmetric but nonvacuum case it is possible to arrange the matter outside the horizon in such a way that the Komar angular momentum on the horizon is greater than the Komar energy there, or the latter can even be negative [20, 21]. Therefore, if a mass–angular momentum inequality is expected to hold quasi-locally at the horizon, then it is not obvious which definitions for the quasi-local mass and angular momentum should be used. In the stationary axisymmetric case, the angular momentum could still be the Komar expression, but the mass is the area of the event horizon [266]: . For the extremal case (even in the presence of Maxwell fields), see [22]. (For the extremality of black holes formulated in terms of isolated and dynamic horizons, see [99] and Section 13.3.2.)

For a recent, very well-readable and comprehensive review of the Dain inequality with the extended bibliography, where both the old and the recent results are summarized, see the topical review of Dain himself in [167].

Living Rev. Relativity 12, (2009), 4
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