Physics: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.[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
The Einstein equation is given by
[math]\displaystyle{ R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}+\Lambda g_{\mu\nu}=0, }[/math]
where [math]\displaystyle{ R_{\mu\nu} }[/math] is the Ricci curvature tensor, R is the Ricci scalar, [math]\displaystyle{ \Lambda }[/math] is the cosmological constant and [math]\displaystyle{ g_{\mu\nu} }[/math] is the metric we are solving for.
We will work in d spacetime dimensions with coordinates [math]\displaystyle{ (t,r,x_1,...,x_{d-2}) }[/math] where [math]\displaystyle{ r\geq0 }[/math] and [math]\displaystyle{ -\infin\lt t,x_1,...,x_{d-2}\lt \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]\displaystyle{ x_i }[/math] directions is given by,
[math]\displaystyle{ 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]\displaystyle{ \Lambda=-\frac{1}{2L^2}(d-1)(d-2) }[/math],
we find that,
[math]\displaystyle{ f(r)=a\left(1-\frac{b}{r^{d-1}}\right) }[/math]
[math]\displaystyle{ h(r)=1-\frac{b}{r^{d-1}} }[/math]
with [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] integration constants, is a solution to the Einstein equation.
The integration constant [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ a=1 }[/math].
The point [math]\displaystyle{ r=0 }[/math] represents a curvature singularity and the point [math]\displaystyle{ r^{d-1}=b }[/math] is a coordinate singularity when [math]\displaystyle{ b\gt 0 }[/math]. To see this, we switch to the coordinate system [math]\displaystyle{ (v,r,x_1,...,x_{d-2}) }[/math] where [math]\displaystyle{ v=t+r^*(r) }[/math] and [math]\displaystyle{ r^*(r) }[/math] is defined by the differential equation,
[math]\displaystyle{ \frac{dr^*}{dr}=\frac{1}{r^2h(r)} }[/math].
The line element in this coordinate system is given by,
[math]\displaystyle{ ds^2=L^2(-r^2h(r)dv^2+2dvdr+r^2d\vec{x}^2) }[/math],
which is regular at [math]\displaystyle{ r^{d-1}=b }[/math]. The surface [math]\displaystyle{ 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". Advances in Theoretical and Mathematical Physics 2 (3): 505–532. doi:10.4310/ATMP.1998.v2.n3.a3. Bibcode: 1998hep.th....3131W.
- ↑ 2.0 2.1 McGreevy, John (2010). "Holographic duality with a view toward many-body physics". Advances in High Energy Physics 2010: 723105. doi:10.1155/2010/723105.
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