5.2 Kottler spacetime

The Kottler metric
( 2) 2 2m-- Λr-- 2 -----dr------ 2( 2 2 2) g = − 1 − r − 3 dt + 1 − 2m--− Λr2 + r d 𝜗 + sin 𝜗dφ (108 ) r 3
is the unique spherically symmetric solution of Einstein’s vacuum field equation with a cosmological constant Λ. It has the form (69View Equation) with
2f(r) −1 2m-- Λr2- ------r------ e = S(r) = 1 − r − 3 , R (r) = 1 − 2m-− Λr2 . (109 ) r 3
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 [313159Jump 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 [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].


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