### 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 [121167140]. 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.