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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

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

where ${\displaystyle R_{\mu \nu }}$ is the Ricci curvature tensor, R is the Ricci scalar, ${\displaystyle \Lambda }$ is the cosmological constant and ${\displaystyle g_{\mu \nu }}$ is the metric we are solving for.

We will work in d spacetime dimensions with coordinates ${\displaystyle (t,r,x_{1},...,x_{d-2})}$ where ${\displaystyle r\geq 0}$ and ${\displaystyle -\infty <t,x_{1},...,x_{d-2}<\infty }$. The line element for a spacetime that is stationary, time reversal invariant, space inversion invariant, rotationally invariant

and translationally invariant in the ${\displaystyle x_{i}}$ directions is given by,

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

Replacing the cosmological constant with a length scale L

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

we find that,

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

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

with ${\displaystyle a}$ and ${\displaystyle b}$ integration constants, is a solution to the Einstein equation.

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

${\displaystyle ds^{2}=L^{2}\left({\frac {dr^{2}}{r^{2}}}+r^{2}(-dt^{2}+d{\vec {x}})\right)}$, when r goes to infinity, then we must set ${\displaystyle a=1}$.

The point ${\displaystyle r=0}$ represents a curvature singularity and the point ${\displaystyle r^{d-1}=b}$ is a coordinate singularity when ${\displaystyle b>0}$. To see this, we switch to the coordinate system ${\displaystyle (v,r,x_{1},...,x_{d-2})}$ where ${\displaystyle v=t+r^{*}(r)}$ and ${\displaystyle r^{*}(r)}$ is defined by the differential equation,

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

The line element in this coordinate system is given by,

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

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

References

1. Witten, Edward (1998-04-07). "Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories". Advances in Theoretical and Mathematical Physics. 2 (3): 505–532. arXiv:hep-th/9803131. Bibcode:1998hep.th....3131W. doi:10.4310/ATMP.1998.v2.n3.a3.
2. ^ McGreevy, John (2010). "Holographic duality with a view toward many-body physics". Advances in High Energy Physics. 2010: 723105. arXiv:0909.0518. doi:10.1155/2010/723105. S2CID 16753864.

• Equations of physics
• Black holes