AdS black brane

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

$R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=0,$

where $R_{\mu \nu }$ is the Ricci curvature tensor, R is the Ricci scalar, $\Lambda$ 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 $r\geq 0$ and $-\infty <t,x_{1},...,x_{d-2}<\infty$. 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,

$ds^{2}=L^{2}\left({\frac {dr^{2}}{r^{2}h(r)}}+r^{2}(-dt^{2}f(r)+d{\vec {x}}^{2})\right)$.

Replacing the cosmological constant with a length scale L

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

we find that,

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

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

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,

$ds^{2}=L^{2}\left({\frac {dr^{2}}{r^{2}}}+r^{2}(-dt^{2}+d{\vec {x}})\right)$, 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,

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

The line element in this coordinate system is given by,

$ds^{2}=L^{2}(-r^{2}h(r)dv^{2}+2dvdr+r^{2}d{\vec {x}}^{2})$,

which is regular at $r^{d-1}=b$. The surface $r^{d-1}=b$ is an event horizon.