nLab Schwarzschild-de Sitter spacetime

Idea

The Schwarzschild-de Sitter spacetime (SdS spacetime for short, $\Lambda\gt 0$) and Schwarzschild-anti de Sitter spacetime (SAdS spacetime for short, $\Lambda\lt 0$) are generalizations of the Schwarzschild spacetime ($\Lambda=0$) when additionally considering dark energy described by the cosmological constant $\Lambda$.

Formulation

Let $M$ be the mass, then the Schwarzschild-(anti-)de Sitter spacetime is given by:

$$\mathrm{d}^2s =\left( 1 -\frac{2M}{r} -\frac{\Lambda}{3}r^2 \right)\mathrm{d}^2t +\left( 1 -\frac{2M}{r} -\frac{\Lambda}{3}r^2 \right)^{-1}\mathrm{d}^2r +r^2\mathrm{d}\Omega.$$

Related concepts

black hole spacetimes	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild spacetime	Kerr spacetime
positive charge	Reissner-Nordstrom spacetime	Kerr-Newman spacetime

black hole spacetimes with dark energy	vanishing angular momentum	positive angular momentum
vanishing charge	Schwarzschild-de Sitter spacetime	Kerr-de Sitter spacetime
positive charge	Reissner-Nordström-de Sitter spacetime	Kerr-Newman-de Sitter spacetime

wormhole spacetimes	vanishing angular momentum
vanishing charge	Schwarzschild wormhole
positive charge	Reissner-Nordström wormhole

References

See also:

• Wikipedia, De Sitter–Schwarzschild metric