SchwarzschildDarstellung

Schwarzschild–de Sitter (SdS) metric is a solution to Einstein's field equations in general relativity that describes a non-rotating, spherically symmetric black hole embedded in a universe with a positive cosmological constant Λ. It extends the Schwarzschild solution by incorporating Λ, reflecting the presence of dark energy and the large-scale expansion of the universe. In standard static coordinates, the line element is ds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + r^2 dΩ^2, where f(r) = 1 - 2GM/(c^2 r) - Λ r^2/3. In geometric units (G=c=1), this becomes f(r) = 1 - 2M/r - Λ r^2/3.

The horizons of SdS spacetime are determined by the roots of f(r) = 0. For Λ > 0 and suitable M, f(r) has two positive real roots corresponding to the black hole event horizon and the cosmological horizon, with a static region between them. If M is too large for a given Λ, the horizons can merge or disappear, indicating the absence of a static region. The spacetime is not asymptotically flat; at large r it approaches de Sitter space rather than Minkowski space.

SdS is also known as the Kottler metric, after Friedrich Kottler who derived it in 1918. It reduces to Schwarzschild in the limit Λ → 0 and to de Sitter space when M → 0. The solution is employed in theoretical investigations of black holes in expanding universes, the causal structure of spacetimes with a positive cosmological constant, and the study of horizon thermodynamics and geodesic motion in such backgrounds.