3.2 Extrinsic curvature of horizon, horizon as a minimal surface
The horizon surface
defined by the condition
in the metric (50) is a co-dimension 2 surface.
It has two normal vectors: a spacelike vector
with the only non-vanishing component
and a
timelike vector
with the non-vanishing component
. With respect to each normal vector one
defines an extrinsic curvature,
,
. The extrinsic curvature
identically
vanishes. It is a consequence of the fact that
is a Killing vector, which generates time translations.
Indeed, the extrinsic curvature can be also written as a Lie derivative,
, so that
it vanishes if
is a Killing vector. The extrinsic curvature associated to the vector
,
is vanishing when restricted to the surface defined by the condition
. It is due to the fact that the
term linear in
is absent in the
-expansion for
in the metric (50). This is required by the
regularity of the metric (50): in the presence of such a term the Ricci scalar would be singular at the
horizon,
.
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
-dimensional spacetime. On the other
hand, in the Lorentzian signature, the horizon
has the minimal area if considered on the hypersurface
of constant time
,
; thus, the latter has the topology of a wormhole.