### 5.2 Kottler spacetime

The Kottler metric
is the unique spherically symmetric solution of Einstein’s vacuum field equation with a cosmological
constant . It has the form (69) with
It is also known as the Schwarzschild–deSitter metric for and as the Schwarzschild–anti-deSitter
metric for . The Kottler metric was found independently by Kottler [185] and by Weyl [348].
In the following we consider the Kottler metric with a constant and we ignore the
region for which the singularity at is naked, for any value of . For ,
there is one horizon at a radius with ; the staticity condition is
satisfied on the region . For , there are two horizons at radii
and with ; the staticity condition is satisfied on the
region . For there is no horizon and no static region. At the
horizon(s), the Kottler metric can be analytically extended into non-static regions. For ,
the resulting global structure is similar to the Schwarzschild case. For , the
resulting global structure is more complex (see [194]). The extreme case is discussed
in [279].

For any value of , the Kottler metric has a light sphere at . Escape cones and embedding
diagrams for the Fermat geometry (optical geometry) can be found in [313, 159] (cf. Figures 14 and 11 for
the Schwarzschild case). Similarly to the Schwarzschild spacetime, the Kottler spacetime can be joined to an
interior perfect-fluid metric with constant density. Embedding diagrams for the Fermat geometry (optical
geometry) of the exterior-plus-interior spacetime can be found in [314]. The dependence on
of the light bending is discussed in [193]. For the optical appearance of a Kottler white hole
see [195]. The shape of infinitesimally thin light bundles in the Kottler spacetime is determined
in [85].