3.2 Area increase law
The qualitative result that the area
of cross-sections
increases monotonically on
follows immediately from the definition,
since
and
. Hence
increases monotonically in the direction of
. The non-trivial task is to obtain a quantitative
formula for the amount of area
increase.
To obtain this formula, one simply uses the
scalar and vector constraints satisfied by the Cauchy data
on
:
where
and
is the matter stress-energy tensor. The strategy is
entirely straightforward: One fixes two cross-sections
and
of
, multiplies
and
with appropriate lapse and shift fields
and integrates the result on a portion
which is
bounded by
and
. Somewhat surprisingly, if
the cosmological constant is non-negative, the resulting area
balance law also provides strong constraints on the topology of
cross sections
.
Specification of lapse
and shift
is equivalent to the specification of a vector field
with respect to which energy-flux
across
is defined. The definition of a DH provides a
preferred direction field, that along
. Hence it is
natural set
. We will begin with this choice and defer the
possibility of choosing more general vector fields until
Section 4.2.
The object of interest now is the flux of energy
associated with
across
. We denote the flux of matter energy across
by
:
By taking the appropriate combination of Equations (14) and (15) we obtain
Since
is foliated by compact 2-manifolds
, one can perform a 2 + 1 decomposition of various
quantities on
. In particular, one first uses the
Gauss-Codazzi equation to express
in terms of
,
, and a total divergence. Then, one uses
the identity
to simplify the expression. Finally one sets
(Note that
is just the shear tensor since the
expansion of
vanishes.) Then, Equation (18) reduces to
To simplify this expression further, we now make
a specific choice of the lapse
. We denote by
the area-radius function; thus
is constant on each
and satisfies
. Since we already know that area increases
monotonically,
is a good coordinate on
, and using it the 3-volume
on
can be decomposed as
, where
denotes the gradient
on
. Therefore calculations simplify if we choose
We will set
. Then, the integral on the
left side of Equation (21) becomes
where
and
are the (geometrical) radii of
and
, and
is the Gauss-Bonnet
topological invariant of the cross-sections
. Substituting back in Equation (21) one obtains
This is the general expression relating the change in area to
fluxes across
. Let us consider its ramifications in
the three cases,
being positive, zero, or negative:
- If
, the right side is positive
definite whence the Gauss-Bonnet invariant
is positive definite, and the topology of the
cross-sections
of the DH is necessarily that of
.
- If
, then
is either spherical or toroidal. The toroidal case
is exceptional: If it occurs, the matter and the gravitational
energy flux across
vanishes (see Section 3.3), the metric
is flat,
(so
can not be a FOTH), and
. In view of
these highly restrictive conditions, toroidal DHs appear to be
unrelated to the toroidal topology of cross-sections of the event
horizon discussed by Shapiro, Teukolsky, Winicour, and
others [121, 167, 139]. In the generic spherical case, the area
balance law (24) becomes
- If
, there is no control on the
sign of the right hand side of Equation (24). Hence, a priori any
topology is permissible. Stationary solutions with quite general
topologies are known for black holes which are asymptotically
locally anti-de Sitter. Event horizons of these solutions are the
potential asymptotic states of these DHs in the distant
future.
For simplicity, the
remainder of our discussion of DHs will be focused on the zero
cosmological constant case with
2-sphere topology.