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, , which generates the time translations. The Killing
horizon is defined as a null hypersurface on which the Killing vector is null, .
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, , called the bifurcation surface. In the maximally extended spacetime a hypersurface
of constant time is a Cauchy surface. The bifurcation surface naturally splits
the Cauchy surface into two parts, and , respectively inside and outside the black
hole. For asymptotically-flat spacetime, such as the Schwarzschild metric, the hypersurface
has the topology of a wormhole. (In the case of the Schwarzschild metric it is called the
Einstein–Rosen bridge.) The surface is the surface of minimal area in . In fact the
bifurcation surface is a minimal surface not only in the -dimensional Euclidean
space , but also in the -dimensional spacetime. As a consequence, as we show below,
the components of the extrinsic curvature defined for two vectors normal to , vanish on
The spacetime in question admits a Euclidean version by analytic continuation . It is a feature
of regular metrics with a Killing horizon that the direction of Euclidean time is compact with period
, which is determined by the condition of regularity, i.e., the absence of a conical singularity. In a
vicinity of the bifurcation surface , the spacetime then is a product of a compact surface and a
two-dimensional disk, the time coordinate playing the role of the angular coordinate on the disk. The
latter can be made more precise by introducing a new angular variable , which varies from
to . In this paper we consider the spacetime with Euclidean metric of the general type
The radial coordinate is such that the surface is defined by the condition . Near this point
the functions and can be expanded as
where is the metric on the bifurcation surface equipped with coordinates .
This metric describes what is called a non-degenerate horizon. The Hawking temperature of the horizon is
finite in this case and equal to .
It is important to note that the metric (50) 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 (50) with (51) does not have to satisfy the Einstein equations we shall still call the
complete space described by the Euclidean metric (50) the Euclidean black hole instanton and will denote
it by .