# AdS black brane

An **anti de Sitter black brane** is a solution of the Einstein field 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 space spacetime at spatial infinity.

## Math development

The Einstein equation is given by

<math> R_{\mu\nu}-\frac{1}{2} g_{\mu\nu}+\Lambda g_{\mu\nu}=0,</math>

where <math>R_{\mu\nu}</math> is the Ricci curvature tensor, R is the Ricci scalar, <math>\Lambda</math> is the cosmological constant and <math>g_{\mu\nu}</math> us the metric we are solving for.

We will work in d spacetime dimensions with coordinates <math>(t,r,x_1,...,x_{d-2})</math> where <math>r\geq0</math> and <math>-\infin<t,x_1,...,x_{d-2}<\infin</math>. The line element for a spacetime that is stationary, time reversal invariant, space inversion invariant, rotationally invariant

and translationally invariant in the <math>x_i</math> directions is given by,

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

Replacing the cosmological constant with a length scale L

<math>\Lambda=\frac{1}{2L^2}(d-1)(d-2)</math>,

we find that,

<math>f(r)=a\left(1-\frac{b}{r^{d-1}}\right)</math>

<math>h(r)=1-\frac{b}{r^{d-1}}</math>

with <math>a</math> and <math>b</math> integration constants, is a solution to the Einstein equation.

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

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

The point <math>r=0</math> represents a curvature singularity and the point <math>r^{d-1}=b</math> is a coordinate singularity when <math>b>0</math>. To see this, we switch to the coordinate system <math>(v,r,x_1,...,x_{d-2})</math> where <math>v=t+r^*(r)</math> and <math>r^*(r)</math> is defined by the differential equation,

<math>\frac{dr^*}{dr}=\frac{1}{r^2h(r)}</math>.

The line element in this coordinate system is given by,

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

which is regular at <math>r^{d-1}=b</math>. The surface <math>r^{d-1}=b</math> is an event horizon.^{[2]}

## References

- ↑ Witten, Edward (1998-04-07). "Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories".
*arXiv:hep-th/9803131*. - ↑
^{2.0}^{2.1}McGreevy, John (2010-05-08). "Holographic duality with a view toward many-body physics".*arXiv:0909.0518 [cond-mat, physics:hep-th]*. doi:10.1155/2010/723105.

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