Area increase law

3.2 Area increase law

The qualitative result that the area $aS$ of cross-sections $S$ increases monotonically on $H$ follows immediately from the definition,

since $Θ(ℓ) = 0$ and $Θ (n) < 0$ . Hence $aS$ increases monotonically in the direction of $^ra$ . 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 $(qab,Kab)$ on $H$ :

where

and $Tab$ is the matter stress-energy tensor. The strategy is entirely straightforward: One fixes two cross-sections $S1$ and $S2$ of $H$ , multiplies $HS$ and $HaV$ with appropriate lapse and shift fields and integrates the result on a portion $ΔH ⊂ H$ which is bounded by $S1$ and $S2$ . Somewhat surprisingly, if the cosmological constant is non-negative, the resulting area balance law also provides strong constraints on the topology of cross sections $S$ .

Specification of lapse $N$ and shift $N a$ is equivalent to the specification of a vector field $ξa = N ^τ a + N a$ with respect to which energy-flux across $H$ is defined. The definition of a DH provides a preferred direction field, that along $ℓa$ . Hence it is natural set $ξa = N ℓa ≡ N ^τa + N ^ra$ . 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 $ξa = N ℓa$ across $ΔH$ . We denote the flux of matter energy across $ΔH$ by $ℱ (mξa)tter$ :

By taking the appropriate combination of Equations (14

) and (15

) we obtain

∫ (ξ) --1--- a 3 ℱmatter = 16πG N (HS + 2^raH V) d V ∫ ΔH = --1--- N (ℛ + K2 − KabKab + 2^raDb (Kab − Kqab )) d3V. (18 ) 16πG ΔH

Since $H$ is foliated by compact 2-manifolds $S$ , one can perform a 2 + 1 decomposition of various quantities on $H$ . In particular, one first uses the Gauss–Codazzi equation to express $ℛ$ in terms of $^ ℛ$ , $K^ab$ , and a total divergence. Then, one uses the identity

to simplify the expression. Finally one sets

(Note that $σab$ is just the shear tensor since the expansion of $a ℓ$ vanishes.) Then, Equation (18

) reduces to

To simplify this expression further, we now make a specific choice of the lapse $N$ . We denote by $R$ the area-radius function; thus $R$ is constant on each $S$ and satisfies $2 aS = 4πR$ . Since we already know that area increases monotonically, $R$ is a good coordinate on $H$ , and using it the 3-volume $3 d V$ on $H$ can be decomposed as $d3V = |∂R |−1dRd2V$ , where $∂$ denotes the gradient on $H$ . Therefore calculations simplify if we choose

We will set $a a NR ℓ = ξ(R )$ . Then, the integral on the left side of Equation (21

) becomes

where $R1$ and $R2$ are the (geometrical) radii of $S1$ and $S2$ , and $ℐ$ is the Gauss–Bonnet topological invariant of the cross-sections $S$ . Substituting back in Equation (21

) one obtains

This is the general expression relating the change in area to fluxes across $ΔH$ . Let us consider its ramifications in the three cases, $Λ$ being positive, zero, or negative:

If $Λ > 0$ , the right side is positive definite whence the Gauss–Bonnet invariant $ℐ$ is positive definite, and the topology of the cross-sections $S$ of the DH is necessarily that of $S2$ .
If $Λ = 0$ , then $S$ is either spherical or toroidal. The toroidal case is exceptional: If it occurs, the matter and the gravitational energy flux across $H$ vanishes (see Section 3.3), the metric $^qab$ is flat, $ℒnΘ (ℓ) = 0$ (so $H$ can not be a FOTH), and $ℒ ℓΘ (ℓ) = 0$ . 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 , 140 ]. In the generic spherical case, the area balance law (24 ) becomes
If $Λ < 0$ , 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.