### 3.3 The wave function of a black hole

Although the entanglement entropy can be defined for any co-dimension two surface, when the surface is
a horizon particular care is required. In order to apply the general prescription outlined in Section 2.1, we
first of all need to specify the corresponding wave function. Here we will follow the prescription proposed by
Barvinsky, Frolov and Zelnikov [15]. This prescription is a natural generalization of the one in flat
spacetime discussed in Section 3.8. On the other hand, it is similar to the “no-boundary” wave
function of the universe introduced in [127]. We define the wave function of a black hole by
the Euclidean path integral over field configurations on the half-period Euclidean instanton
defined by the metric (50) with angular coordinate changing in the interval from to
. This half-period instanton has Cauchy surface (on which we can choose coordinates
) as a boundary where we specify the boundary conditions in the path integral,
where is the action of the quantum field . The functions and are
the boundary values defined on the part of the hypersurface , which is respectively inside () and
outside () the horizon . As was shown in [15], the wave function (53) corresponds to the
Hartle–Hawking vacuum state [126].