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
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.