nLab Kerr-de Sitter spacetime

Idea

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

Formulation

Let $M$ be the mass and $a$ be the angular momentum, then the Kerr-(anti-)de Sitter spacetime is given by:

$$g_{\rm tt} =-\frac{3\left(a^2\sin^2\theta\left(a^2\Lambda\cos^2\theta+3\right)+a^2\left(\Lambda r^2-3\right)+\Lambda r^4-3r^2+6r\right)}{\left(a^2\Lambda+3\right)^2 \left(a^2\cos^2\theta+r^2\right)};$$

$$g_{\rm rr} =3\frac{a^2\cos^2\theta+r^2}{\left(a^2+r^2\right)\left(\Lambda r^2-3\right)+6r};$$

$$g_{\rm\theta\theta} =-\frac{3 \left(a^2\cos^2\theta+r^2\right)}{a^2\Lambda\cos^2\theta+3};$$

$$g_{\rm\phi\phi} =-\frac{3\left(\left(a^2+r^2\right)^2\sin^2\theta\left(a^2\Lambda\cos^2\theta+3\right) -a^2\sin^4\theta\left(\left(a^2+r^2\right) \left(3-\Lambda r^2\right)-6r\right)\right)}{\left(a^2\Lambda+3\right)^2\left(a^2\cos^2 \theta+r^2\right)};$$

$$g_{\rm t\phi} =\frac{3a\sin^2\theta\left(a^2\Lambda\left(a^2+r^2\right)\cos^2\theta +a^2\Lambda r^2+\Lambda r^4+6r\right)}{\left(a^2\Lambda+3\right)^2\left(a^2\cos^2\theta+r^2\right)}.$$

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