AdS black brane
Einstein equation solution with planar event horizon and negative cosmological constant.
Last updated
An anti de Sitter (AdS) black brane is a solution of the Einstein equations in the presence of a negative cosmological constant that possesses a planar event horizon, distinguishing it from an AdS black hole with a spherical event horizon. This solution is significant in theoretical physics for modeling spacetimes that asymptote to anti de Sitter spacetime at spatial infinity.
Last updated 2026-09-28 from the source article.
Quick Facts
- Known for
- Planar event horizon solution to Einstein equations with negative cosmological constant
- Type
- Spacetime solution
Facts from the source article.
Lore & Background
The AdS black brane is derived from the Einstein equation R_μν - 1/2 R g_μν + Λ g_μν = 0, where Λ is a negative cosmological constant. Working in d spacetime dimensions with coordinates (t, r, x₁, ..., x_{d-2}), the line element for a stationary, time reversal invariant, space inversion invariant, rotationally invariant, and translationally invariant spacetime is given by ds² = L² (dr²/(r² h(r)) + r²(-dt² f(r) + d→x²)). The cosmological constant is replaced with a length scale L via Λ = -1/(2L²) (d-1)(d-2).
The solution yields f(r) = a(1 - b/r^{d-1}) and h(r) = 1 - b/r^{d-1}, with a and b integration constants. The constant a is associated with a residual symmetry from rescaling time; requiring the line element to approach ds² = L² (dr²/r² + r²(-dt² + d→x²)) as r → ∞ forces a = 1. The point r = 0 is a curvature singularity, and r^{d-1} = b is a coordinate singularity when b > 0, as shown by switching to coordinates (v, r, x₁, ..., x_{d-2}) with v = t + r(r) and dr/dr = 1/(r² h(r)).
Reader's Guide
The AdS black brane is a key solution in the study of anti de Sitter spacetimes, particularly in the context of the AdS/CFT correspondence, where it serves as a gravitational dual to a finite-temperature field theory. Its planar horizon mimics a thermal bath in the boundary theory, making it essential for understanding holographic thermodynamics and transport phenomena. The solution's dependence on the cosmological constant Λ and the integration constant b, which sets the horizon location, allows for the exploration of black brane thermodynamics, including temperature and entropy. The coordinate singularity at r^{d-1} = b is removable via a coordinate transformation, revealing the event horizon.
The curvature singularity at r = 0 is a genuine singularity. The AdS black brane's translationally invariant horizon contrasts with spherical black holes, making it suitable for modeling systems with planar symmetry, such as those in condensed matter holography. Its legacy lies in providing a simple, exact solution that bridges gravity and quantum field theory, enabling calculations of transport coefficients and phase transitions in strongly coupled systems.
Frequently Asked Questions
What is an AdS black brane?
An AdS black brane is a solution to Einstein's field equations in a spacetime with a negative cosmological constant, characterized by a flat, planar event horizon. It describes a black object whose geometry extends uniformly in all spatial directions rather than wrapping into a sphere.
What are the defining mathematical ingredients of an AdS black brane?
The solution is constructed by solving Einstein's equations with a negative cosmological constant inserted, yielding a metric whose horizon is a flat plane. This contrasts with the spherical-horizon solution, where the horizon topology is that of a sphere.
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Sources
Compiled from Wikipedia and the sources listed below. Text from Wikipedia is available under CC BY-SA 4.0; this entry is adapted from it.
- Wikipedia: AdS black brane (CC BY-SA 4.0).
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