Kottler spacetime

5.2 Kottler spacetime

The Kottler metric

is the unique spherically symmetric solution of Einstein’s vacuum field equation with a cosmological constant $Λ$ . It has the form (69

) with

It is also known as the Schwarzschild–deSitter metric for $Λ > 0$ and as the Schwarzschild–anti-deSitter metric for $Λ < 0$ . The Kottler metric was found independently by Kottler [185 ] and by Weyl [348 ].

In the following we consider the Kottler metric with a constant $m > 0$ and we ignore the region $r < 0$ for which the singularity at $r = 0$ is naked, for any value of $Λ$ . For $Λ < 0$ , there is one horizon at a radius $rH$ with $0 < rH < 2m$ ; the staticity condition $ef(r) > 0$ is satisfied on the region $rH < r < ∞$ . For $0 < Λ < (3m )− 2$ , there are two horizons at radii $rH1$ and $rH2$ with $2m < rH1 < 3m < rH2$ ; the staticity condition $ef(r) > 0$ is satisfied on the region $rH1 < r < rH2$ . For $− 2 Λ > (3m )$ there is no horizon and no static region. At the horizon(s), the Kottler metric can be analytically extended into non-static regions. For $Λ < 0$ , the resulting global structure is similar to the Schwarzschild case. For $0 < Λ < (3m )−2$ , the resulting global structure is more complex (see [194 ]). The extreme case $− 2 Λ = (3m )$ is discussed in [279 ].

For any value of $Λ$ , the Kottler metric has a light sphere at $r = 3m$ . 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 $Λ$ 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 ].