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])
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].
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Living Rev. Relativity 12, (2009), 4
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