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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,
1 1 ^K = &tidle;qabDa^rb = --&tidle;qab∇a (ℓb − nb) = − -Θ (n) > 0, (13 ) 2 2
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:

H := ℛ + K2 − KabK = 16πG ¯T ^τa^τ b, (14 ) Sa ( ab ab)ab bcab a H V := Db K − Kq = 8πG T¯ ^τcq b, (15 )
where
1 T¯ab = Tab − -----Λgab, (16 ) 8πG
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:

∫ (ξ) a b 3 ℱ matter := ΔHTab^τ ξ d V. (17 )
By taking the appropriate combination of Equations (14View Equation) and (15View Equation) 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
q^ab(Kab + K^ab ) = Θ(ℓ) = 0 (19 )
to simplify the expression. Finally one sets
σab = ^qac^qbd∇a ℓb, ζa = ^qab^rc∇c ℓb. (20 )
(Note that σab is just the shear tensor since the expansion of a ℓ vanishes.) Then, Equation (18View Equation) reduces to
∫ ∫ ∫ ( ) N ℛ^d3V = 16πG T¯ab^τ aξb d3V + N |σ|2 + 2 |ζ|2 d3V. (21 ) ΔH ΔH ΔH

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

N = |∂R | ≡ NR. (22 )
We will set a a NR ℓ = ξ(R ). Then, the integral on the left side of Equation (21View Equation) becomes
∫ ∫ ∮ 3 R2 2 NR ℛ^ d V = dR ℛ^d V = ℐ (R2 − R1), (23 ) ΔH R1
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 (21View Equation) one obtains
∫ ( Λ ) ∫ ( ) ℐ (R2 − R1) = 16πG Tab − ----gab ^τaξb(R)d3V + NR |σ |2 + 2|ζ|2 d3V. (24 ) ΔH 8πG ΔH
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:

For simplicity, the remainder of our discussion of DHs will be focused on the zero cosmological constant case with 2-sphere topology.


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