Schwarzschildde

Schwarzschild–de Sitter, sometimes written Schwarzschild-de Sitter and known as the Kottler metric, is a static, spherically symmetric solution to Einstein’s field equations with a nonzero cosmological constant Λ. It generalizes the Schwarzschild solution by including Λ, and describes a non-rotating black hole embedded in a universe with a cosmological constant. When Λ = 0 it reduces to the Schwarzschild solution, and when M = 0 it reduces to de Sitter space.

The spacetime is described by the line element in static coordinates: ds^2 = -f(r) c^2 dt^2 + f(r)^{-1} dr^2 + r^2(dθ^2 + sin^2 θ dφ^2), where f(r) = 1 - 2GM/(rc^2) - (Λ r^2)/3. Here M is the mass parameter of the central object, G is Newton’s gravitational constant, and c is the speed of light. The geometry is determined by the function f(r); horizons occur at the roots of f(r) = 0. For Λ > 0, there can be a black hole event horizon and a cosmological horizon; depending on M and Λ, there may be two positive roots or a configuration with a third (negative) root. In the special Nariai limit, the two positive horizons coincide.

Schwarzschild–de Sitter is used to study black holes in cosmological contexts, particularly the influence of a positive cosmological constant on horizon structure, thermodynamics, and particle trajectories. In regimes where the cosmological constant is very small, deviations from Schwarzschild are negligible near astrophysical black holes, but the solution provides a useful exact background in theoretical investigations of gravity with Λ.