To explain the evolution of ideas and provide
points of comparison, we will introduce the notion of dynamical
horizons following a chronological order. Readers who are not
familiar with causal structures can go directly to
Definition 5
of dynamical horizons (for which a more direct motivation can be
found in [31]).
As discussed in Section 1, while
the notion of an event horizon has proved to be very convenient in
mathematical relativity, it is too global and teleological to be
directly useful in a number of physical contexts ranging from
quantum gravity to numerical relativity to astrophysics. This
limitation was recognized early on (see, e.g., [113], page 319) and
alternate notions were introduced to capture the intuitive idea of
a black hole in a quasi-local manner. In particular, to make the
concept ‘local in time’, Hawking [111, 113] introduced the notions of a trapped
region and an apparent horizon, both of which are associated to a
space-like 3-surface
representing ‘an instant of time’. Let
us begin by recalling these ideas.
Hawking’s outer trapped
surface is a compact, space-like 2-dimensional
sub-manifold in
such that the expansion
of the outgoing null normal
to
is non-positive. Hawking then defined
the trapped region
in a surface
as the set of all points in
through which there passes an outer-trapped surface,
lying entirely in
. Finally, Hawking’s apparent horizon
is the
boundary of a connected component of
. The idea then
was to regard each apparent horizon as the instantaneous surface of
a black hole. One can calculate the expansion
of
knowing only the intrinsic 3-metric
and the extrinsic curvature
of
. Hence, to find outer trapped surfaces
and apparent horizons on
, one does not need to evolve
away from
even locally. In this sense
the notion is local to
. However, this locality is
achieved at the price of restricting
to lie in
. If we wiggle
even slightly, new outer
trapped surfaces can appear and older ones may disappear. In this
sense, the notion is still very global. Initially, it was hoped
that the laws of black hole mechanics can be extended to these
apparent horizons. However, this has not been possible because the
notion is so sensitive to the choice of
.
To improve on this situation, in the early
nineties Hayward proposed a novel modification of this
framework [117]. The main idea is
to free these notions from the complicated dependence on
. He began with Penrose’s notion of a trapped
surface. A trapped surface
a la Penrose is a compact, space-like 2-dimensional
sub-manifold of space-time on which
,
where
and
are the two null normals to
. We will focus on future trapped surfaces on which both
expansions are negative. Hayward then defined a space-time trapped region. A trapped region
a la Hayward is a
subset of space-time through each point of which there passes a
trapped surface. Finally, Hayward’s trapping
boundary
is a connected component of the
boundary of an inextendible trapped region. Under certain
assumptions (which appear to be natural intuitively but technically
are quite strong), he was able to show that the trapping boundary
is foliated by marginally trapped
surfaces (MTSs), i.e., compact, space-like 2-dimensional
sub-manifolds on which the expansion of one of the null normals,
say
, vanishes and that of the other, say
, is everywhere non-positive. Furthermore,
is also everywhere of one sign. These general
considerations led him to define a quasi-local analog of future
event horizons as follows:
Definition 4: A
future, outer, trapping horizon (FOTH) is a
smooth 3-dimensional sub-manifold of space-time,
foliated by closed 2-manifolds
, such that
In this definition, Condition 2 captures the
idea that is a future horizon (i.e., of black
hole rather than white hole type), and Condition 3 encodes the
idea that it is ‘outer’ since infinitesimal motions along the
‘inward’ normal
makes the 2-surface trapped.
(Condition 3 also serves to distinguish black
hole type horizons from certain cosmological ones [117
] which are not ruled
out by Condition 2). Using the Raychaudhuri equation,
it is easy to show that
is either space-like or
null, being null if and only if the shear
of
as well as the matter flux
across
vanishes. Thus, when
is null, it is a non-expanding horizon introduced in
Section 2.1. Intuitively,
is space-like in the dynamical region where
gravitational radiation and matter fields are pouring into it and
is null when it has reached equilibrium.
In truly dynamical situations, then, is expected to be space-like. Furthermore, it turns
out that most of the key results of physical interest [30
, 31
], such as the area
increase law and generalization of black hole mechanics, do not
require the condition on the sign of
. It is
therefore convenient to introduce a simpler and at the same time
‘tighter’ notion, that of a dynamical horizon, which is better
suited to analyze how black holes grow in exact general
relativity [30, 31
]:
Definition 5: A smooth,
three-dimensional, space-like sub-manifold (possibly with
boundary) of space-time is said to be a dynamical horizon
(DH) if it can be foliated by a family of
closed 2-manifolds such that
Note first that, like FOTHs, dynamical horizons
are ‘space-time notions’, defined quasi-locally. They are not
defined relative to a space-like surface as was the case with
Hawking’s apparent horizons nor do they make any reference to
infinity as is the case with event horizons. In particular, they
are well-defined also in the spatially compact context. Being
quasi-local, they are not teleological. Next, let us spell out the
relation between FOTHs and DHs. A space-like FOTH is a DH on which the
additional condition
holds. Similarly, a DH satisfying
is a space-like FOTH. Thus, while
neither definition implies the other, the two are closely related.
The advantage of Definition 5 is that it refers only to the
intrinsic structure of
, without any conditions on
the evolution of fields in directions transverse to
. Therefore, it is easier to verify in numerical
simulations. More importantly, as we will see, this feature makes
it natural to analyze the structure of
using only the
constraint (or initial value) equations on it. This analysis will
lead to a wealth of information on black hole dynamics.
Reciprocally, Definition 4 has the advantage that, since it
permits
to be space-like or
null, it is better suited to analyze the transition to
equilibrium [31
].
A DH which is also a FOTH will be referred to as a space-like future outer horizon (SFOTH). To fully capture the physical notion of a dynamical black hole, one should require both sets of conditions, i.e., restrict oneself to SFOTHs. For, stationary black holes admit FOTHS and there exist space-times [166] which admit dynamical horizons but no trapped surfaces; neither can be regarded as containing a dynamical black hole. However, it is important to keep track of precisely which assumptions are needed to establish specific results. Most of the results reported in this review require only those conditions which are satisfied on DHs. This fact may well play a role in conceptual issues that arise while generalizing black hole thermodynamics to non-equilibrium situations3.
Let us begin with the simplest examples of
space-times admitting DHs (and SFOTHs). These are provided by the
spherically symmetric solution to Einstein’s equations with a null
fluid as source, the Vaidya metric [179, 137, 186]. (Further details and the inclusion of a
cosmological constant are discussed in [31
].) Just as the
Schwarzschild-Kruskal solution provides a great deal of intuition
for general static black holes, the Vaidya metric furnishes some of
the much needed intuition in the dynamical regime by bringing out
the key differences between the static and dynamical situations.
However, one should bear in mind that both Schwarzschild and Vaidya
black holes are the simplest examples and certain aspects of
geometry can be much more complicated in more general situations.
The 4-metric of the Vaidya space-time is given by
Let us focus our attention on the metric
2-spheres, which are all given by and
. It is easy to verify that the expansion of the
outgoing null normal
vanishes if and only if (
and)
. Thus, these are the only spherically symmetric marginally trapped
surfaces MTSs. On each of them, the expansion
of the ingoing normal
is negative. By
inspection, the 3-metric on the world tube
of these MTSs has signature
when
is non-zero and
if
is zero. Hence, in the left panel of
Figure 4
the surface
is the DH
. In the right panel of
Figure 4
the portion of this
surface
is the DH
, while the portion
is a non-expanding horizon. (The general issue of
transition of a DH to equilibrium is briefly discussed in
Section 5.) Finally, note that at these MTSs,
. Hence in both cases, the DH is an
SFOTH. Furthermore, in the case depicted in the right panel of
Figure 4
the entire surface
is a FOTH,
part of which is dynamical and part null.
|
What is the situation in a more general
gravitational collapse? As indicated in the beginning of this
section, the geometric structure can be much more subtle. Consider
3-manifolds which are foliated by marginally trapped
compact 2-surfaces
. We denote by
the normal whose expansion vanishes. If the
expansion of the other null normal
is negative,
will be called a marginally trapped
tube (MTT). If the tube
is space-like, it is a
dynamical horizon. If it is time-like, it will be called time-like membrane. Since future directed
causal curves can traverse time-like membranes in either direction,
they are not good candidates to represent surfaces of black holes;
therefore they are not referred to as horizons.
In Vaidya metrics, there is precisely one MTT to
which all three rotational Killing fields are tangential and this
is the DH . In the Oppenheimer-Volkoff dust
collapse, however, the situation is just the opposite; the unique
MTT on which each MTS
is spherical is time-like [180, 47
]. Thus we have a
time-like membrane rather than a dynamical horizon. However, in
this case the metric does not satisfy the smoothness conditions
spelled out at the end of Section 1 and the
global time-like character of
is an artifact of the lack of
this smoothness. In the general perfect fluid spherical collapse,
if the solution is smooth, one can show analytically that the
spherical MTT is space-like at sufficiently late times, i.e., in a
neighborhood of its intersection with the event horizon [102
]. For the spherical
scalar field collapse, numerical simulations show that, as in the
Vaidya solutions, the spherical MTT is space-like
everywhere [102
]. Finally, the
geometry of the numerically evolved MTTs has been examined in two
types of non-spherical situations: the axi-symmetric collapse of a
neutron star to a Kerr black hole and in the head-on collision of
two non-rotating black holes [48
]. In both cases, in
the initial phase the MTT is neither space-like nor time-like all
the way around its cross-sections
. However, it quickly
becomes space-like and has a long space-like portion which
approaches the event horizon. This portion is then a dynamical
horizon. There are no hard results on what would happen in general,
physically interesting situations. The current expectation is that
the MTT of a numerically evolved black hole space-time which
asymptotically approaches the event horizon will become space-like
rather soon after its formation. Therefore most of the ongoing
detailed work focuses on this portion, although basic analytical
results are available also on how the time-like membranes evolve
(see Appendix A of [31
]).
Even in the simplest, Vaidya example discussed
above, our explicit calculations were restricted to spherically
symmetric marginally trapped surfaces. Indeed, already in the case
of the Schwarzschild space-time, very little is known analytically
about non-spherically symmetric marginally trapped surfaces. It is
then natural to ask if the Vaidya metric admits other,
non-spherical dynamical horizons which also asymptote to the
non-expanding one. Indeed, even if we restrict ourselves to the
3-manifold , can we find another foliation by
non-spherical, marginally trapped surfaces which endows it with
another dynamical horizon structure? These considerations
illustrate that in general there are two uniqueness issues that
must be addressed.
First, in a general space-time , can a space-like 3-manifold
be foliated by two distinct families of marginally
trapped surfaces, each endowing it with the structure of a
dynamical horizon? Using the maximum principle, one can show that
this is not possible [93]. Thus, if
admits a dynamical
horizon structure, it is unique.
Second, we can ask the following question: How
many DHs can a space-time admit? Since a space-time may contain
several distinct black holes, there may well be several distinct
DHs. The relevant question is if distinct DHs can exist within each
connected component of the (space-time) trapped region. On this
issue there are several technically different uniqueness
results [27]. It is simplest to
summarize them in terms of SFOTHs. First, if two non-intersecting
SFOTHs
and
become tangential to the same
non-expanding horizon at a finite time (see the right panel in
Figure 4
), then they coincide
(or one is contained in the other). Physically, a more interesting
possibility, associated with the late stages of collapse or
mergers, is that
and
become asymptotic to the
event horizon. Again, they must coincide in this case. At present,
one can not rule out the existence of more than one SFOTHs which
asymptote to the event horizon if they intersect each other
repeatedly. However, even if this were to occur, the two horizon
geometries would be non-trivially constrained. In particular, none
of the marginally trapped surfaces on
can lie entirely to
the past of
.
A better control on uniqueness is perhaps the most important open issue in the basic framework for dynamical horizons and there is ongoing work to improve the existing results. Note however that all results of Sections 3 and 5, including the area increase law and the generalization of black hole mechanics, apply to all DHs (including the ‘transient ones’ which may not asymptote to the event horizon). This makes the framework much more useful in practice.
The existing results also provide some new
insights for numerical relativity [27]. First, suppose that a MTT
is generated by a foliation of a region of space-time by partial
Cauchy surfaces
such that each MTS
is the outermost MTS in
. Then
can not be a time-like membrane. Note however that this does not
imply that
is necessarily a dynamical horizon
because
may be partially time-like and partially space-like
on each of its marginally trapped surfaces
. The requirement that
be space-like - i.e.,
be a dynamical horizon - would restrict the choice of the foliation
of space-time and reduce the unruly freedom in the
choice of gauge conditions that numerical simulations currently
face. A second result of interest to numerical relativity is the
following. Let a space-time
admit a DH
which asymptotes to the event horizon. Let
be any partial Cauchy
surface in
which intersects
in one of the marginally trapped surfaces, say
. Then,
is the outermost marginally
trapped surface - i.e., apparent horizon in the numerical
relativity terminology - on
.