AdS black brane

Template:Short description Template:Multiple issues An anti de Sitter black brane is a solution of the Einstein equations in the presence of a negative cosmological constant which possesses a planar event horizon. This is distinct from an anti de Sitter black hole solution which has a spherical event horizon. The negative cosmological constant implies that the spacetime will asymptote to an anti de Sitter spacetime at spatial infinity.

Math development

The Einstein equation is given by

$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=0,$

where

$R_{\mu \nu }$

is the Ricci curvature tensor, R is the Ricci scalar,

$\mathrm{\Lambda }$

is the cosmological constant and

$g_{\mu \nu }$

is the metric we are solving for.

We will work in d spacetime dimensions with coordinates $(t,r,x_1,...,x_{d-2})$ where $r\ge 0$ and $-\mathrm{\infty }<t,x_1,...,x_{d-2}<\mathrm{\infty }$. The line element for a spacetime that is stationary, time reversal invariant, space inversion invariant, rotationally invariant

and translationally invariant in the

$x_i$

directions is given by,

$d{s}^2=L^2(\frac{d{r}^2}{r^{2}h(r)}+r^2(-d{t}^{2}f(r)+d{\overrightarrow{x}}^2))$

.

Replacing the cosmological constant with a length scale L

$\mathrm{\Lambda }=-\frac{1}{2L^2}(d-1)(d-2)$

,

we find that,

$f(r)=a(1-\frac{b}{r^{d-1}})$

$h(r)=1-\frac{b}{r^{d-1}}$

with $a$ and $b$ integration constants, is a solution to the Einstein equation.

The integration constant $a$ is associated with a residual symmetry associated with a rescaling of the time coordinate. If we require that the line element takes the form,

$d{s}^2=L^2(\frac{d{r}^2}{r^2}+r^2(-d{t}^2+d\overrightarrow{x}))$, when r goes to infinity, then we must set $a=1$.

The point

$r=0$

represents a curvature singularity and the point

$r^{d-1}=b$

is a coordinate singularity when

$b>0$

. To see this, we switch to the coordinate system

$(v,r,x_1,...,x_{d-2})$

where

$v=t+r^{*}(r)$

and

$r^{*}(r)$

is defined by the differential equation,

$\frac{d{r}^{*}}{dr}=\frac{1}{r^{2}h(r)}$

.

The line element in this coordinate system is given by,

$d{s}^2=L^2(-r^{2}h(r)d{v}^2+2dvdr+r^{2}d{\overrightarrow{x}}^2)$

,

which is regular at

$r^{d-1}=b$

. The surface

$r^{d-1}=b$

is an event horizon.