### 2.11 An expression in terms of the determinant of the Laplacian on the surface

Even though the entanglement entropy is determined by the geometry of the surface , in general,
this can be not only its intrinsic geometry but also how the surface is embedded in the larger spacetime.
The embedding is determined by the extrinsic curvature. The curvature of the larger spacetime enters
through the Gauss–Cadazzi relations. But in some particularly simple cases the entropy can be given
a purely intrinsic interpretation. To see this for the case when is a plane we note that
the entropy (28) or (30) originates from the surface term in the trace of the heat kernel (27)
(or (31)). To leading order in , the surface term in the case of a massive scalar field
is
where
can
be interpreted as the trace of the heat kernel of operator , where is the intrinsic
Laplace operator defined on the -plane . The determinant of the operator is
determined by
Thus,
we obtain an interesting expression for the entanglement entropy
in terms of geometric objects defined intrinsically on the surface . A similar expression in the case of an
ultra-extreme black hole was obtained in [172] and for a generic black hole with horizon approximated by a
plane was obtained in [89].