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I am wondering as to how many metrics have been discovered that obey all the laws of General Relativity?

I am posting the ones that I have briefly studied, please post more if you have knowledge of them.

Schwarzschild metric:

[tex]c^2 {d \tau}^{2} =\left(1 - \frac{r_s}{r} \right) c^2 dt^2 - \left(1-\frac{r_s}{r}\right)^{-1} dr^2 - r^2 \left(d\theta^2 + \sin^2\theta \, d\phi^2\right),[/tex]

Kerr metric:

[tex]\begin{align} c^{2} d\tau^{2} = & \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Delta} dr^{2} - \rho^{2} d\theta^{2} \\ & - \left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha \sin^{2} \theta }{\rho^{2}} \, c \, dt \, d\phi \end{align}[/tex]

Reissner–Nordström metric:

[tex]ds^2 = \left( 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2} \right) c^2\, dt^2 - \frac{1}{1 - r_\mathrm{S}/r + r_Q^2/r^2}\, dr^2 - r^2\, d\theta^2 - r^2 \sin^2 \theta \, d\phi^2,[/tex]

Kerr–Newman metric:

[tex]c^{2} d\tau^{2} = -\left(\frac{dr^2}{\Delta} + d\theta^2 \right) \rho^2 + \left(c \, dt - \alpha \sin^2 \theta \, d\phi \right)^2 \frac{\Delta}{\rho^2} - \left(\left(r^2 + \alpha^2 \right) d\phi - \alpha c\, dt \right)^2 \frac{\sin^2 \theta}{\rho^2}[/tex]

BTZ black hole metric:

[tex]ds^2 = -\frac{(r^2 - r_+^2)(r^2 - r_-^2)}{l^2 r^2}dt^2 + \frac{l^2 r^2 dr^2}{(r^2 - r_+^2)(r^2 - r_-^2)} + r^2 \left(d\phi - \frac{r_+ r_-}{l r^2} dt \right)^2[/tex]

Alcubierre metric:

[tex]ds^2 = -\left(\alpha^2- \beta_i \beta^i\right)\,dt^2+2 \beta_i \,dx^i\, dt+ \gamma_{ij}\,dx^i\,dx^j[/tex]

(traversable) wormhole metric:

[tex]ds^2= - c^2 dt^2 + dl^2 + (k^2 + l^2)(d \theta^2 + \sin^2 \theta \, d\phi^2)[/tex]

Godel metric:

[tex]c^{2} d\tau^{2} = - \frac{1}{2 \Lambda} \left( -c^2 dt^2 + dr^2 - \frac{e^{2r} r^2}{2} \; d \theta^2 + r^2 \sin^2 \theta \; d\phi^2 - 2 r e^{r} \; c \; dt \; d\theta \right)[/tex]

Friedmann-Lemaître-Robertson-Walker metric:

[tex]c^{2} d\tau^{2} = -c^{2} dt^2 + a(t)^2 \left(\frac{dr^2}{1 - k r^2} + r^2 d\theta^2 + r^2 \sin^2 \theta \; d\phi^2 \right)[/tex]

Reference:

http://en.wikipedia.org/wiki/Schwarzschild_metric

http://en.wikipedia.org/wiki/Kerr_metric#Mathematical_form

http://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric

http://en.wikipedia.org/wiki/Kerr–Newman_metric

http://en.wikipedia.org/wiki/BTZ_black_hole#The_case_without_charge

http://en.wikipedia.org/wiki/Alcubierre_drive#Mathematics_of_the_Alcubierre_drive

http://en.wikipedia.org/wiki/Wormhole#Metrics

http://en.wikipedia.org/wiki/Gödel_metric#Definition

http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

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# General Relativity metrics

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