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 μ ν − 1 2 R g μ ν + Λ g μ ν = 0 , {\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=0,} where R μ ν {\displaystyle R_{\mu \nu }} is the Ricci curvature tensor, R is the Ricci scalar, Λ {\displaystyle \Lambda } is the cosmological constant and g μ ν {\displaystyle 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 ) {\displaystyle (t,r,x_{1},...,x_{d-2})} where r ≥ 0 {\displaystyle r\geq 0} and − ∞ < t , x 1 , . . . , x d − 2 < ∞ {\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 x i {\displaystyle x_{i}} directions is given by,
d s 2 = L 2 ( d r 2 r 2 h ( r ) + r 2 ( − d t 2 f ( r ) + d x → 2 ) ) {\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
Λ = − 1 2 L 2 ( d − 1 ) ( d − 2 ) {\displaystyle \Lambda =-{\frac {1}{2L^{2}}}(d-1)(d-2)} , we find that,
f ( r ) = a ( 1 − b r d − 1 ) {\displaystyle f(r)=a\left(1-{\frac {b}{r^{d-1}}}\right)}
h ( r ) = 1 − b r d − 1 {\displaystyle h(r)=1-{\frac {b}{r^{d-1}}}}
with a {\displaystyle a} and b {\displaystyle b} integration constants, is a solution to the Einstein equation. The integration constant a {\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,
d s 2 = L 2 ( d r 2 r 2 + r 2 ( − d t 2 + d x → ) ) {\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 a = 1 {\displaystyle a=1} .
The point r = 0 {\displaystyle r=0} represents a curvature singularity and the point r d − 1 = b {\displaystyle r^{d-1}=b} is a coordinate singularity when b > 0 {\displaystyle b>0} . To see this, we switch to the coordinate system ( v , r , x 1 , . . . , x d − 2 ) {\displaystyle (v,r,x_{1},...,x_{d-2})} where v = t + r ∗ ( r ) {\displaystyle v=t+r^{*}(r)} and r ∗ ( r ) {\displaystyle r^{*}(r)} is defined by the differential equation, d r ∗ d r = 1 r 2 h ( r ) {\displaystyle {\frac {dr^{*}}{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 + 2 d v d r + r 2 d x → 2 ) {\displaystyle 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 {\displaystyle r^{d-1}=b} . The surface r d − 1 = b {\displaystyle r^{d-1}=b} is an event horizon.
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