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.[1][2] 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

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The Einstein equation is given by

 

where   is the Ricci curvature tensor, R is the Ricci scalar,   is the cosmological constant and   is the metric we are solving for.

We will work in d spacetime dimensions with coordinates   where   and  . The line element for a spacetime that is stationary, time reversal invariant, space inversion invariant, rotationally invariant

and translationally invariant in the   directions is given by,

 .

Replacing the cosmological constant with a length scale L

 ,

we find that,

 

 

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

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

 , when r goes to infinity, then we must set  .

The point   represents a curvature singularity and the point   is a coordinate singularity when  . To see this, we switch to the coordinate system   where   and   is defined by the differential equation,

 .

The line element in this coordinate system is given by,

 ,

which is regular at  . The surface   is an event horizon.[2]

References

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  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. ^ a b 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.