3.1 The geometric setting of black-hole spacetimes

The notion of entanglement entropy is naturally applicable to a black hole. In fact, probably the only way to separate a system into two sub-systems is to place one of them inside a black-hole horizon. The important feature that, in fact, defines the black hole is the existence of a horizon. Many useful definitions of a horizon are known. In the present paper we shall consider only the case of the eternal black holes for which different definitions of the horizon coincide. The corresponding spacetime then admits a maximal analytic extension, which we shall use in our construction. The simplest example is the Schwarzschild black hole, the maximal extension of which is demonstrated on the well-known Penrose diagram. The horizon of the Schwarzschild black hole is an example of a Killing horizon. The spacetime in this case possesses a global Killing vector, ξt = ∂t, which generates the time translations. The Killing horizon is defined as a null hypersurface on which the Killing vector ξt is null, ξ2t = 0. The null surface in the maximal extension of an eternal black hole consists of two parts: the future horizon and the past horizon. The two intersect on a compact surface of co-dimension two, Σ, called the bifurcation surface. In the maximally extended spacetime a hypersurface ℋt of constant time t is a Cauchy surface. The bifurcation surface Σ naturally splits the Cauchy surface into two parts, ℋ − and ℋ+, respectively inside and outside the black hole. For asymptotically-flat spacetime, such as the Schwarzschild metric, the hypersurface ℋt has the topology of a wormhole. (In the case of the Schwarzschild metric it is called the Einstein–Rosen bridge.) The surface Σ is the surface of minimal area in ℋt. In fact the bifurcation surface Σ is a minimal surface not only in the (d − 1)-dimensional Euclidean space ℋt, but also in the d-dimensional spacetime. As a consequence, as we show below, the components of the extrinsic curvature defined for two vectors normal to Σ, vanish on Σ.

The spacetime in question admits a Euclidean version by analytic continuation t → iτ. It is a feature of regular metrics with a Killing horizon that the direction of Euclidean time τ is compact with period 2π βH, which is determined by the condition of regularity, i.e., the absence of a conical singularity. In a vicinity of the bifurcation surface Σ, the spacetime then is a product of a compact surface Σ and a two-dimensional disk, the time coordinate τ playing the role of the angular coordinate on the disk. The latter can be made more precise by introducing a new angular variable ϕ = β −H1τ, which varies from 0 to 2π. In this paper we consider the spacetime with Euclidean metric of the general type

ds2 = β2Hg(ρ)dϕ2 + dρ2 + γij(ρ,𝜃)d𝜃id𝜃j . (50 )
The radial coordinate ρ is such that the surface Σ is defined by the condition ρ = 0. Near this point the functions g(ρ) and γ (ρ,𝜃) ij can be expanded as
ρ2 4 (0) 2 g(ρ ) = -2-+ O (ρ ), γij(ρ,𝜃) = γij (𝜃) + O (ρ ) , (51 ) βH
where γ(ij0)(𝜃) is the metric on the bifurcation surface Σ equipped with coordinates { 𝜃i, i = 1,..,d − 2}. This metric describes what is called a non-degenerate horizon. The Hawking temperature of the horizon is finite in this case and equal to TH = 1∕(2πβH ).

It is important to note that the metric (50View Equation) does not have to satisfy any field equations. The entanglement entropy can be defined for any metric, which possesses a Killing-type horizon. In this sense the entanglement entropy is an off-shell quantity. It is useful to keep this in mind when one compares the entanglement entropy with some other approaches in which an entropy is assigned to a black hole horizon. Even though the metric (50View Equation) with (51View Equation) does not have to satisfy the Einstein equations we shall still call the complete space described by the Euclidean metric (50View Equation) the Euclidean black hole instanton and will denote it by E.


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