Einstein metrics in four dimensions

Einstein metrics

Einstein metrics are vacuum solutions of the Einstein equation with cosmological constant. In particular, if the cosmological constant vanishes, the solutions are Ricci-flat,

$R_{\mu\nu} = 0$ .

Black hole solutions

Metric	Author (Age)	Symmetry	Parameters	Λ	Other features
Schwarzschild	Schwarzschild (1916)	static, spherically symmetric	mass	no
Schwarzschild-(A)dS	Kottler (1918)	static, spherically symmetric	mass	yes
Taub-NUT	Taub (1951), Newman, Tamburino, Unti (1963)	stationary, axially symmetric	mass, NUT	no
Kerr	Kerr (1963)	stationary, axially symmetric	mass, rotation	no
Kerr-NUT	Demianski (1966), Kramer, Neugebauer (1968), Robinson, Robinson, Zund (1969)	stationary, axially symmetric	mass, rotation, NUT	no
Schwarzschild-NUT-(A)dS	Demianski (1972)	static, spheircally symmetric	mass, NUT	yes
Kerr-(A)dS	Demianski (1973)	stationary, axially symmetric	mass, rotation	yes
Kerr-NUT-(A)dS	Frolov (1973)	stationary, axially symmetric	mass, rotation, NUT	yes

Schwarzschild metric [Schwarzschild (1916), Sitzungsber.Preuss.Akad.Wiss.Berlin (Math.Phys.) 1916, 189-196 (arXiv:physics/9905030)]

$ds^2 = -\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)$

Kerr metric [Kerr (1963), Physical Review Letters 11 (5) 237–238]

$ds^2 = -\frac{\Delta}{\Sigma}(dt-a \sin^2\theta d\phi)^2+\frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2+\frac{\sin^2\theta}{\Sigma}(adt-(r^2+a^2)d\phi)^2$