3.2 Extrinsic curvature of horizon, horizon as a minimal surface

The horizon surface Σ defined by the condition ρ = 0 in the metric (50View Equation) is a co-dimension 2 surface. It has two normal vectors: a spacelike vector n1 with the only non-vanishing component n1ρ = 1 and a timelike vector n2 with the non-vanishing component n2 = 1 ∕ρ ϕ. With respect to each normal vector one defines an extrinsic curvature, a l p a kij = − γi γj ∇ln p, a = 1,2. The extrinsic curvature 2 kij identically vanishes. It is a consequence of the fact that 2 n is a Killing vector, which generates time translations. Indeed, the extrinsic curvature can be also written as a Lie derivative, kμν = − 12ℒng μν, so that it vanishes if n is a Killing vector. The extrinsic curvature associated to the vector n1,
1 1- l p k ij = − 2γ i γj ∂ργkn , (52 )
is vanishing when restricted to the surface defined by the condition ρ = 0. It is due to the fact that the term linear in ρ is absent in the ρ-expansion for γij(ρ, 𝜃) in the metric (50View Equation). This is required by the regularity of the metric (50View Equation): in the presence of such a term the Ricci scalar would be singular at the horizon, R ∼ 1∕ρ.

The vanishing of the extrinsic curvature of the horizon indicates that the horizon is necessarily a minimal surface. It has the minimal area considered as a surface in d-dimensional spacetime. On the other hand, in the Lorentzian signature, the horizon Σ has the minimal area if considered on the hypersurface of constant time t, ℋt; thus, the latter has the topology of a wormhole.


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