List of spacetimes

These are spacetimes which admit the maximum number of isometries or Killing vector fields for a given dimension, and each of these can be formulated in an arbitrary number of dimensions.

• Minkowski spacetime

$${\displaystyle g=-dt^{2}+\sum _{i=1}^{n-1}dx_{i}^{2}}$$

• de-Sitter spacetime

$${\displaystyle g=-dt^{2}+\alpha ^{2}\sinh ^{2}\left({\frac {1}{\alpha }}t\right)dH_{n-1}^{2},}$$

where ${\displaystyle \alpha }$ is real and ${\displaystyle dH_{n-1}^{2}}$ is the standard hyperbolic metric.

• Anti de-Sitter spacetime

$${\displaystyle g={\frac {1}{y^{2}}}\left(-dt^{2}+dy^{2}+\sum _{i=1}^{n-2}dx_{i}^{2}\right)}$$

These spacetimes model black holes. The Schwarzschild and Reissner–Nordstrom black holes are spherically symmetric, while Schwarzschild and Kerr are electrically neutral.

• Schwarzschild spacetime

$${\displaystyle g=-\left(1-{\frac {2M}{r}}\right)dt^{2}+\left(1-{\frac {2M}{r}}\right)^{-1}dr^{2}+r^{2}d\Omega ^{2},}$$

where ${\displaystyle d\Omega ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}}$ is the round metric on the sphere, and ${\displaystyle M}$ is a positive, real parameter.

• Kruskal spacetime (Maximally extended Schwarzschild spacetime)

$${\displaystyle g=-{\frac {32M^{3}e^{-r(U,V)/2M}}{r(U,V)}}dUdV+r(U,V)^{2}d\Omega ^{2},}$$

where ${\displaystyle r(U,V)}$ is defined implicitly.

• Reissner–Nordstrom spacetime

$${\displaystyle g=-\left(1-{\frac {2M}{r}}+{\frac {e^{2}}{r^{2}}}\right)dt^{2}+\left(1-{\frac {2M}{r}}+{\frac {e^{2}}{r^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega ^{2}}$$

• Kerr spacetime
• Kerr–Newman spacetime

$${\displaystyle g=-{\frac {\Delta }{\rho ^{2}}}\left(dt-a\sin ^{2}\theta \,d\phi \right)^{2}+{\frac {\sin ^{2}\theta }{\rho ^{2}}}{\Big (}\left(r^{2}+a^{2}\right)\,d\phi -a\,dt{\Big )}^{2}+{\frac {\rho ^{2}}{\Delta }}dr^{2}+\rho ^{2}\,d\theta ^{2}.}$$

See Boyer–Lindquist coordinates for details on the terms appearing in this formula.

• FLRW spacetime

$${\displaystyle g=-dt^{2}+a(t)^{2}\left({\frac {dr^{2}}{1-kr^{2}}}+r^{2}d\Omega ^{2}\right)}$$

,

where ${\displaystyle k}$ is often restricted to take values in the set ${\displaystyle -1,0,1}$.

• Lemaître–Tolman spacetime

• pp-wave spacetime

• Alcubierre spacetime
• Einstein static universe spacetime
• Ellis wormhole spacetime
• Gödel spacetime
• Taub–NUT spacetime
• Kasner spacetime
• Mixmaster spacetime

• General Relativity, R. Wald, The University of Chicago Press, 1984, ISBN 0-226-87033-2
• Spacetime and Geometry, S. Carroll, Cambridge University Press, 2019, ISBN 9781108488396