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5.2 Kottler spacetime

The Kottler metric
( 2) 2 2m-- /\r-- 2 -----dr------ 2( 2 2 2) g = - 1 - r - 3 dt + 1 - 2m-- /\r2+ r dh + sin h df (108) r 3
is the unique spherically symmetric solution of Einstein’s vacuum field equation with a cosmological constant /\. It has the form (69View Equation) with
2f(r) -1 2m-- /\r2- ------r------ e = S(r) = 1 - r - 3 , R(r) = 1- 2m-- /\r2 . (109) r 3
It is also known as the Schwarzschild-deSitter metric for /\ > 0 and as the Schwarzschild-anti-deSitter metric for /\ < 0. The Kottler metric was found independently by Kottler [186] 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 /\. For /\ < 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 < oo. For 0 < /\ < (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 /\ > (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 /\ < 0, the resulting global structure is similar to the Schwarzschild case. For 0 < /\ < (3m) -2, the resulting global structure is more complex (see [195]). The extreme case - 2 /\ = (3m) is discussed in [278].

For any value of /\, 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 [314160Jump To The Next Citation Point] (cf. Figures 14View Image and 11View Image 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 [315]. The dependence on /\ of the light bending is discussed in [194]. For the optical appearance of a Kottler white hole see [196]. The shape of infinitesimally thin light bundles in the Kottler spacetime is determined in [85].

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