# General Relativity metrics

1. Aug 11, 2013

### Orion1

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

Kerr metric:
\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}

Reissner–Nordström metric:
$$ds^2 = \left( 1 - \frac{r_\mathrm{S}}{r} + \frac{r_Q^2}{r^2} \right) c^2\, dt^2 - \frac{1}{1 - r_\mathrm{S}/r + r_Q^2/r^2}\, dr^2 - r^2\, d\theta^2 - r^2 \sin^2 \theta \, d\phi^2,$$

Kerr–Newman metric:
$$c^{2} d\tau^{2} = -\left(\frac{dr^2}{\Delta} + d\theta^2 \right) \rho^2 + \left(c \, dt - \alpha \sin^2 \theta \, d\phi \right)^2 \frac{\Delta}{\rho^2} - \left(\left(r^2 + \alpha^2 \right) d\phi - \alpha c\, dt \right)^2 \frac{\sin^2 \theta}{\rho^2}$$

BTZ black hole metric:
$$ds^2 = -\frac{(r^2 - r_+^2)(r^2 - r_-^2)}{l^2 r^2}dt^2 + \frac{l^2 r^2 dr^2}{(r^2 - r_+^2)(r^2 - r_-^2)} + r^2 \left(d\phi - \frac{r_+ r_-}{l r^2} dt \right)^2$$

Alcubierre metric:
$$ds^2 = -\left(\alpha^2- \beta_i \beta^i\right)\,dt^2+2 \beta_i \,dx^i\, dt+ \gamma_{ij}\,dx^i\,dx^j$$

(traversable) wormhole metric:
$$ds^2= - c^2 dt^2 + dl^2 + (k^2 + l^2)(d \theta^2 + \sin^2 \theta \, d\phi^2)$$

Godel metric:
$$c^{2} d\tau^{2} = - \frac{1}{2 \Lambda} \left( -c^2 dt^2 + dr^2 - \frac{e^{2r} r^2}{2} \; d \theta^2 + r^2 \sin^2 \theta \; d\phi^2 - 2 r e^{r} \; c \; dt \; d\theta \right)$$

Friedmann-Lemaître-Robertson-Walker metric:
$$c^{2} d\tau^{2} = -c^{2} dt^2 + a(t)^2 \left(\frac{dr^2}{1 - k r^2} + r^2 d\theta^2 + r^2 \sin^2 \theta \; d\phi^2 \right)$$

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

Last edited: Aug 11, 2013
2. Aug 11, 2013

### yenchin

Last edited by a moderator: May 6, 2017
3. Aug 11, 2013

### PAllen

Without imposing plausibility constraints (e.g. energy conditions), any formally valid metric is a solution of EFE for the stress energy tensor taken as defined by the Einstein tensor G$\mu$$\nu$. That is, for any metric you can compute the Einstein tensor and then rename it the stress energy tensor and you formally have a solution. The problem is that most such solutions will have extreme violations of plausibility conditions (e.g. negative energy).

4. Aug 11, 2013

### George Jones

Staff Emeritus
Last edited by a moderator: May 6, 2017
5. Aug 11, 2013

### WannabeNewton

Are you asking for strictly physically relevant solutions by the way? For example, there are methods of taking a certain metric tensor with a special symmetry that solves the EFEs and generating an entire one-parameter family of solutions to the EFEs that retain the symmetry; they just might not be physically relevant.

The Taub-NUT space-times are especially interesting: http://en.wikipedia.org/wiki/Taub–NUT_space

6. Aug 11, 2013

### atyy

Minkowski (I suppose it's a special case of Schwarzschild, FLRW)
Vaidya metric http://en.wikipedia.org/wiki/Vaidya_metric
Mixmaster universe http://arxiv.org/abs/gr-qc/9612066
de Sitter space http://en.wikipedia.org/wiki/De_Sitter_space
Anti de Sitter space http://en.wikipedia.org/wiki/Anti-de_Sitter_space
Schwarzschild Anti de Sitter http://arxiv.org/abs/hep-th/0106112
Black Saturn (4+1D) http://arxiv.org/abs/hep-th/0701035

Incidentally, Ben Niehoff, who posts on PF, works on solutions. I would like to understand his work some day.
http://arxiv.org/abs/1103.0520
http://arxiv.org/abs/1203.1348
http://arxiv.org/abs/1204.1972
http://arxiv.org/abs/1303.5449

Last edited: Aug 11, 2013
7. Aug 11, 2013

### Orion1

metric physical properties...

Thanks for the excellent links, some more books for my library.

I have noticed some intrinsic patterns in General Relativity, specifically, metrics with both rotation and charge that are reducible to metrics that have no rotation and no charge:

$$A(J,Q) = \left( \begin{array}{ccccc} \; & J = 0 & J \neq 0 \\ Q = 0 & 0,0 & 1,0 \\ Q \neq 0 & 0,1 & 1,1 \end{array} \right)$$

For example:
$$A(0,0) = \text{Schwarzschild metric}$$
$$A(1,0) = \text{Kerr metric}$$
$$A(0,1) = \text{Reissner–Nordström metric}$$
$$A(1,1) = \text{Kerr–Newman metric}$$

I also noticed that the solution for a non-traversable wormhole metric is identical to the Schwarzschild metric for a black hole.

Therefore, it seems plausible that the remaining metric solutions are also solutions for non-traversable wormhole metrics with rotation and/or charge physical properties. So, that would be at least four black hole types and four wormhole types, all of which are reducible to $A(0,0)$, just by changing the inputs from the numerical integration of matrix $A(1,1)$ from some number greater than zero, to zero.

Also, each metric in General Relativity would be a solution at some position in this matrix example.

For example, the BTZ black hole metric has no rotation and charge $A(0,0)$, yet there should be at least three other solutions that have the remaining intrinsic physical properties, for example: $A(1,0),A(0,1),A(1,1)$. Also, would the BTZ black hole metric also be a solution for a non-traversable BTZ wormhole? are the solutions identical?

The same examples should also be true for the remaing metrics listed on post #1, such as the Alcubierre metric, the (traversable) wormhole metric, the Godel metric and the Friedmann-Lemaître-Robertson-Walker metric.

Although the exotic metrics may not have any modeling applications for objects that exist in the real universe, it would still be interesting to examine them in action on some supercomputer GR model simulator. I also still find it intriguing that a metric solution may exist in General Relativity for a rotating charged traversable wormhole.

Is there a tensor theorem to test a metric to determine if it has a negative energy condition?

I think that I have read that the Alcubierre metric has such a condition.

Last edited: Aug 11, 2013
8. Aug 11, 2013

### PAllen

There are several traditional tests for physical plausibility. Unfortunately, they are considered increasingly problematic because they are (at the same time) too permissive and too strict. That is, they allow solutions most physicists would consider implausible, while ruling out things the you might consider plausible (e.g. scalar fields). To the best of my knowledge, there is no accepted solution to this problem. One alternative is consider as plausible only evolution (using ADM initial value formalism) from some plausible starting point with a Lagrangian containing some desired laws of matter. To apply this to an arbitrary solution, you would have to show there exist initial conditions you consider plausible, and evolution consistent with laws of matter considered plausible.

For the traditional approach, on which most major theorems of GR are based, an introduction is:

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

For an evaluation of the problems with these:

http://arxiv.org/abs/gr-qc/0205066