Einstein metrics in higher dimensions

Einstein metrics

$S_D=\frac{1}{16\pi G}\int d^Dx \sqrt{-g} (R + \Lambda)$

Black holes with spherical horizon topology in higher dimensions

Dim	Metric	Authors (Age)	Symmetry	Parameters	Other features
D≥4	Schwarzschild-Tangherlini	Tangherlini (1963)	static, spherically symmetric	mass
D≥4	Myers-Perry	Myers, Perry (1986)	stationary, axially symmetric	mass, rotations

Schwarzschild-Tangherlini metric [F. R. Tangherlini, "Schwarzschild field in n dimensions and the dimensionality of space problem", Nuovo Cim. 27 (1963) 636-651]

The metric in $D$ dimensions is

$ds^2=-\left(1-\frac{\mu}{r^{D-3}}\right)dt^2+\left(1-\frac{\mu}{r^{D-3}}\right)^{-1}dr^2+r^2 d\Omega_{D-2}^2$

where $d\Omega_{D-2}^2$ is the standard metric on $S^{D-2}$ , given by

$d\Omega_{D-2}^2=d\theta_1^2+\sin^2\theta_1 d\theta_2^2+\cdots +(\sin^2\theta_1\cdots\sin^2\theta_{D-3})d\theta_{D-2}^2$

The mass is

$M = \frac{(D-2)\Omega_{D-2}}{16\pi G}\mu \,.$

Myers-Perry metric [R. C. Myers, M. J. Perry, Ann. of Phys. 172 (1986) 304]

The metric forms in odd and even dimensions are different. In odd dimensions $D=2n+1$ , the metric is

$ds^2=-dt^2+\frac{\Pi F}{\Pi-\mu r^2}dr^2+\sum_{i=1}^n(r^2+a_i^2)(d\mu_i^2+\mu_i^2d\phi_i^2)+\frac{\mu r^2}{\Pi F}\sum_{i=1}^n(dt-a_i\mu_i^2d\phi_i)^2$

where $\mu_i$ are constrained coordinates, that is,

$\sum_{i=1}^n\mu_i^2 = 1 \,.$

In even dimensions $D=2n$ , the metric is

$ds^2=-dt^2+\frac{\Pi F}{\Pi-\mu r}dr^2+\sum_{i=1}^n(r^2+a_i^2)(d\mu_i^2+\mu_i^2d\phi_i^2)+\frac{\mu r}{\Pi F}\sum_{i=1}^n(dt-a_i\mu_i^2d\phi_i)^2$

where $\mu_i$ satisfy

$\sum_{i=1}^n\mu_i^2 + \alpha^2 = 1 \qquad (-1\leq\alpha\leq 1) \,.$

In both cases, the functions contained in the metrics are given by

$F = 1-\sum_{i=1}^n\frac{a_i^2\mu_i^2}{r^2+a_i^2}\,, \qquad \Pi =\prod_{i=1}^n(r^2+a_i^2)\,.$

The mass and angular momenta for the i-th rotational plane are

$M = \frac{(D-2)\Omega_{D-2}}{16\pi G}\mu \,, \qquad J_i = \frac{\Omega_{D-2}}{16\pi G}\mu a_i\,.$

w/ cosmological constant

Dim	Metric	Authors (Age)	Symmetry	Parameters	Other features
5	Myers-Perry-(A)dS	Hawking, Hunter, Taylor-Robinson (1999)	stationary, axially symmetric	mass, rotations
D≥4	Myers-Perry-(A)dS	Gibbons, Lu, Page, Pope (2004)	stationary, axially symmetric	mass, rotations
D≥4	Myers-Perry-NUT-(A)dS	Chen, Lu, Pope (2006)	stationary, axially symmetric	mass, rotations, NUTs

Myers-Perry-NUT-(A)dS metric (Higher-dimensional Kerr-NUT-(A)dS metric) [W. Chen, H. Lu, C. N. Pope, Class. Quant. Grav. 23 (2006) 5323 (hep-th/0604125)]

In even dimensions $D=2n$ , the metric is

$ds^2=\frac{U}{X}dr^2+\sum_{\alpha=1}^{n-1}\frac{U_\alpha}{X_\alpha}dy_\alpha^2-\frac{X}{U}\left[W d\bar{t}-\sum_{i=1}^{n-1}\gamma_i d\bar{\phi}_i\right]^2$
$+\sum_{\alpha=1}^{n-1}\frac{X_\alpha}{U_\alpha}\left[\frac{(1+g^2r^2)W}{1-g^2y_\alpha^2}d\bar{t}-\sum_{i=1}^{n-1}\frac{(r^2+a_i^2)\gamma_i}{a_i^2-y_\alpha^2}d\bar{\phi}_i\right]^2$

In odd dimensions $D=2n+1$ , the metric is

$ds^2=\frac{U}{X}dr^2+\sum_{\alpha=1}^{n-1}\frac{U_\alpha}{X_\alpha}dy_\alpha^2-\frac{X}{U}\left[W d\bar{t}-\sum_{i=1}^na_i^2\gamma_i d\bar{\phi}_i\right]^2$
$+\sum_{\alpha=1}^{n-1}\frac{X_\alpha}{U_\alpha}\left[\frac{(1+g^2r^2)W}{1-g^2y_\alpha^2}d\bar{t}-\sum_{i=1}^n\frac{a_i^2(r^2+a_i^2)\gamma_i}{a_i^2-y_\alpha^2}d\bar{\phi}_i\right]^2$
$+\frac{\prod_{k=1}^na_k^2}{r^2\prod_{\alpha=1}^{n-1}y_\alpha^2}\left[(1+g^2r^2)Wd\bar{t}-\sum_{i=1}^n(r^2+a_i^2)\gamma_id\bar{\phi}_i\right]^2$

where

5d vacuum black hole solutions

In five dimensions, following the topology theorems, the event horizon topology must be either a sphere $S^3$ , a ring $S^1\times S^2$ , a lens space $S^3/\Gamma$ or their connected sums.

Black rings

Dim	Metric	Authors (Age)	Symmetry	Parameters	Λ	Other features
5	Singly spnning black ring	Emparan, Reall (2002)	stationary, axially symmetric	mass, 1 rotation	no	$S^1$ -rotation
5	Singly spnning black ring	Mishima, Iguchi (2006), Figueras (2005)	stationary, axially symmetric	mass, 1 rotation	no	$S^2$ -rotation, conical singularity
5	Doubly spnning black ring	Pomeransky, Sen'kov (2006)	stationary, axially symmetric	mass, 2 rotations	no

Black rings with $S^1$ -rotation [R. Emparan, H. S. Reall, PRL 88 (2002) 101101]

The metric is written in the C-metric coordinates in the form

$ds^2 = -\frac{F(y)}{F(x)}\left(dt-CR\frac{1+y}{F(y)}d\psi\right)^2+\frac{R^2F(x)}{(x-y)^2}\left[-\frac{G(y)}{F(y)}d\psi^2+\frac{G(x)}{G(y)}d\phi^2+\frac{dx^2}{G(x)}-\frac{dy^2}{G(y)}\right]$

where

$F(\xi)=1+\lambda \xi \,, \qquad G(\xi)=(1-\xi^2)(1+\nu \xi) \,, \qquad C= \sqrt{\lambda(\lambda-\nu)\frac{1+\lambda}{1-\lambda}}$

The regularity requires a balance condition, that is, $\lambda=2\nu/(1+\nu^2)$ . The mass and angular momentum are

$M = \frac{3\pi R^2}{4G}\frac{\lambda}{1-\nu}\,, \qquad J_\psi = \frac{\pi R^3}{2G}\frac{\sqrt{\lambda(\lambda-\nu)(1+\lambda)}}{(1-\nu)^2}$

Black rings with $S^2$ -rotation [T. Mishima, H. Iguchi, Phys. Rev. D 73 (2006) 044030; P. Figueras, JHEP 07 (2005) 039]

The metric is written in the C-metric coordinates as

$ds^2 = -\frac{H(y,x)}{H(x,y)}\left(dt-\frac{\lambda a y(1-x^2)}{H(y,x)}d\phi\right)^2+\frac{R^2H(x,y)}{(x-y)^2}\left[-\frac{(1-y^2)F(x)}{H(x,y)}d\psi^2+\frac{(1-x^2)F(y)}{H(y,x)}d\phi^2+\frac{dx^2}{(1-x^2)F(x)}-\frac{dy^2}{(1-y^2)F(y)}\right]$