These horizons model black holes which are
themselves in equilibrium, but in possibly dynamical
space-times [13, 14, 26
, 16
]. For early
references with similar ideas, see [156, 106]. A useful example
is provided by the late stage of a gravitational collapse shown in
Figure 1
. In such physical
situations, one expects the back-scattered radiation falling into
the black hole to become negligible at late times so that the ‘end
portion’ of the event horizon (labelled by
in the figure) can be regarded as isolated to an excellent
approximation. This expectation is borne out in numerical
simulations where the backscattering effects typically become
smaller than the numerical errors rather quickly after the
formation of the black hole (see, e.g., [34
, 48
]).
The key idea is to extract from the notion of a
Killing horizon the minimal conditions which are necessary to
define physical quantities such as the mass and angular momentum of
the black hole and to establish the zeroth and the first laws of
black hole mechanics. Like Killing horizons, isolated horizons are
null, 3-dimensional sub-manifolds of space-time. Let us therefore
begin by recalling some essential features of such sub-manifolds,
which we will denote by . The intrinsic metric
on
has signature (0,+,+), and is simply
the pull-back of the space-time metric to
,
, where an underarrow indicates the
pullback to
. Since
is degenerate, it
does not have an inverse in the standard sense. However, it does
admit an inverse in a weaker sense:
will be said to be
an inverse of
if it satisfies
. As one would expect, the inverse is not unique: We
can always add to
a term of the type
, where
is a null normal to
and
any vector field tangential to
. All our constructions will be insensitive to this
ambiguity. Given a null normal
to
, the expansion
is defined as
Definition 1: A
sub-manifold of a
space-time
is
said to be a non-expanding horizon (NEH) if
The motivation behind this definition can be
summarized as follows. Condition 1 is imposed for definiteness; while
most geometric results are insensitive to topology, the case is physically the most relevant one.
Condition 3 is satisfied by all classical
matter fields of direct physical interest. The key condition in the
above definition is Condition 2 which is equivalent to requiring
that every cross-section of
be marginally trapped. (Note incidentally that
if
vanishes for one null normal
to
, it vanishes for all.)
Condition 2 is equivalent to requiring that
the infinitesimal area element is Lie
dragged by the null normal
. In particular, then,
Condition 2 implies that the horizon area is
‘constant in time’. We will denote the area of any cross section of
by
and define the horizon radius as
.
Because of the Raychaudhuri equation, Condition 2 also implies
whereThe zeroth and first laws of black hole mechanics
require an additional structure, which is provided by the concept
of a weakly isolated horizon. To arrive at this concept, let us
first introduce a derivative operator on
. Because
is degenerate, there is an
infinite number of (torsion-free) derivative operators which are
compatible with it. However, on an NEH, the property
implies that the space-time connection
induces a unique (torsion-free) derivative operator
on
which is compatible with
[26
, 136
]. Weakly isolated
horizons are characterized by the property that, in addition to the
metric
, the connection component
is also ‘time independent’.
Two null normals and
to an NEH
are said to belong to the
same equivalence class
if
for some positive constant
. Then, weakly isolated
horizons are defined as follows:
Definition 2: The
pair is said to
constitute a weakly isolated horizon (WIH) provided
is an NEH and each
null normal
in
satisfies
It is easy to verify that every NEH admits null
normals satisfying Equation (5), i.e., can be made a
WIH with a suitable choice of
. However the required
equivalence class is not unique, whence an NEH admits distinct WIH
structures [16
].
Compared to conditions required of a Killing horizon, conditions in this definition are very weak. Nonetheless, it turns out that they are strong enough to capture the notion of a black hole in equilibrium in applications ranging from black hole mechanics to numerical relativity. (In fact, many of the basic notions such as the mass and angular momentum are well-defined already on NEHs although intermediate steps in derivations use a WIH structure.) This is quite surprising at first because the laws of black hole mechanics were traditionally proved for globally stationary black holes [182], and the definitions of mass and angular momentum of a black hole first used in numerical relativity implicitly assumed that the near horizon geometry is isometric to Kerr [5].
Although the notion of a WIH is sufficient for
most applications, from a geometric viewpoint, a stronger notion of
isolation is more natural: The full
connection should be time-independent. This leads
to the notion of an isolated horizon.
Definition 3: A WIH
is said to constitute
an isolated horizon (IH) if
While an NEH can always be given a WIH structure
simply by making a suitable choice of the null normal, not every
WIH admits an IH structure. Thus, the passage from a WIH to an IH
is a genuine restriction [16]. However, even for
this stronger notion of isolation, conditions in the definition are
local to
. Furthermore, the
definition only uses quantities intrinsic to
; there are no
restrictions on components of any fields transverse to
. (Even the full 4-metric
need not be time
independent on the horizon.) Robinson-Trautman solutions provide
explicit examples of isolated horizons which do not admit a
stationary Killing field even in an arbitrarily small neighborhood
of the horizon [68
]. In this sense, the
conditions in this definition are also rather weak. One expects
them to be met to an excellent degree of approximation in a wide
variety of situations representing late stages of gravitational
collapse and black hole mergers2.
The class of space-times admitting NEHs, WIHs,
and IHs is quite rich. First, it is trivial to verify that any
Killing horizon which is topologically is also an
isolated horizon. This in particular implies that the event
horizons of all globally stationary black holes, such as the
Kerr-Newman solutions (including a possible cosmological constant),
are isolated horizons. (For more exotic examples, see [155].)
|
More generally, using the characteristic initial
value formulation [92, 161], Lewandowski [140] has constructed an infinite dimensional
set of local examples. Here, one considers two null surfaces and
intersecting in a 2-sphere
(see Figure 3
). One can freely
specify certain data on these two surfaces which then determines a
solution to the vacuum Einstein equations in a neighborhood of
bounded by
and
, in which
is an isolated horizon.
On IHs, by contrast, the situation is
dramatically different. Given an IH , generically
the Condition (6
) in Definition 3 can not be
satisfied by a distinct equivalence
class of null normals
. Thus on a generic IH, the
only freedom in the choice of the null normal is that of a
rescaling by a positive constant [16
]. This freedom
mimics the properties of a Killing horizon since one can also
rescale the Killing vector by an arbitrary constant. The triplet
is said to constitute the geometry of the isolated
horizon.
Next, let us consider the Ricci-tensor
components. On any NEH we have:
,
.
In the Einstein-Maxwell theory, one further has: On
,
and
.
Let be a spherical cross section
of
. The degenerate metric
naturally projects
to a Riemannian metric
on
, and similarly the 1-form
of
Equation (7
) projects to a 1-form
on
. If the vacuum equations hold on
, then given
on
, there is, up to diffeomorphisms, a unique non-extremal isolated horizon geometry
such that
is the projection of
,
is the projection of the
constructed from
, and
. (If the vacuum equations do not hold,
the additional data required is the projection on
of the space-time Ricci tensor.)
The underlying reason behind this result can be
sketched as follows. First, since is degenerate
along
, its non-trivial part is just its projection
. Second,
fixes the connection on
; it is only the quantity
that is
not constrained by
, where
is a 1-form on
orthogonal to
, normalized so that
. It is easy to show that
is symmetric and
the contraction of one of its indices with
gives
:
.
Furthermore, it turns out that if
, the field
equations completely determine the angular part of
in terms of
and
. Finally, recall that the surface gravity is not
fixed on
because of the rescaling freedom in
; thus the
-component of
is not part of the free data. Putting all these
facts together, we see that the pair
enables us
to reconstruct the isolated horizon geometry uniquely up to
diffeomorphisms.
On any non-extremal NEH, the 1-form can be used to construct preferred foliations of
. Let us first examine the simpler, non-rotating case
in which
. Then Equation (9
) implies that
is curl-free and
therefore hypersurface orthogonal. The 2-surfaces orthogonal to
must be topologically
because, on any
non-extremal horizon,
. Thus, in the non-rotating
case, every isolated horizon comes equipped with a preferred family
of cross-sections which defines the rest frame [26
]. Note that the
projection
of
on any leaf of this
foliation vanishes identically.
The rotating case is a little more complicated
since is then no longer curl-free. Now the idea is to
exploit the fact that the divergence
of the projection
of
on a cross-section is
sensitive to the choice of the cross-section, and to select a
preferred family of cross-sections by imposing a suitable condition
on this divergence [16
]. A mathematically
natural choice is to ask that this divergence vanish. However, (in
the case when the angular momentum is non-zero) this condition does
not pick out the
cuts of the Kerr horizon
where
is the (Carter generalization of the)
Eddington-Finkelstein coordinate. Pawlowski has provided another
condition that also selects a preferred foliation and reduces to
the
cuts of the Kerr horizon:
In the asymptotically flat context, boundary conditions select a universal symmetry group at spatial infinity, e.g., the Poincaré group, because the space-time metric approaches a fixed Minkowskian one. The situation is completely different in the strong field region near a black hole. Because the geometry at the horizon can vary from one space-time to another, the symmetry group is not universal. However, the above result shows that the symmetry group can be one of only three universality classes.