Astronomy:Black brane

In general relativity, a black brane is a solution of the equations that generalizes a black hole solution but it is also extended—and translationally symmetric—in p additional spatial dimensions. That type of solution would be called a black p-brane.

In string theory, the term black brane describes a group of D1-branes that are surrounded by a horizon. With the notion of a horizon in mind as well as identifying points as zero-branes, a generalization of a black hole is a black p-brane. However, many physicists tend to define a black brane separate from a black hole, making the distinction that the singularity of a black brane is not a point like a black hole, but instead a higher dimensional object.

A BPS black brane is similar to a BPS black hole. They both have electric charges. Some BPS black branes have magnetic charges.

The metric for a black p-brane in a n-dimensional spacetime is:

[math]\displaystyle{ {d s}^{2} = \left( \eta_{ab} + \frac{r_s^{n-p-3}}{r^{n-p-3}} u_a u_b \right) d \sigma^a d \sigma^b + \left(1-\frac{r_s^{n-p-3}}{r^{n-p-3}}\right)^{-1} dr^2 + r^2 d \Omega^2_{n-p-2} }[/math]

where:

• η is the (p + 1)-Minkowski metric with signature (−, +, +, +, ...),
• σ are the coordinates for the worldsheet of the black p-brane,
• u is its four-velocity,
• r is the radial coordinate and,
• Ω is the metric for a (n − p − 2)-sphere, surrounding the brane.

Curvatures

When [math]\displaystyle{ ds^2=g_{\mu\nu}dx^\mu dx^\nu + d\Omega_{n+1} }[/math].

The Ricci Tensor becomes [math]\displaystyle{ R_{\mu\nu}=R_{\mu\nu}^{(0)}+\frac{n+1}{r}\Gamma^r_{\mu\nu} }[/math], [math]\displaystyle{ R_{ij}=\delta_{ij} g_{ii} (\frac{n}{r^2}(1-g^{rr})-\frac{1}{r}(\partial_{\mu}+\Gamma^\nu_{\nu\mu})g^{\mu r}) }[/math].

The Ricci Scalar becomes [math]\displaystyle{ R=R^{(0)}+\frac{n+1}{r}g^{\mu\nu}\Gamma^r_{\mu\nu}+\frac{n(n+1)}{r^2}(1-g^{rr}) -\frac{n+1}{r}(\partial_\mu g^{\mu r}+\Gamma^\nu_{\nu\mu}g^{\mu r}) }[/math].

Where [math]\displaystyle{ R_{\mu\nu}^{(0)} }[/math], [math]\displaystyle{ R^{(0)} }[/math] are the Ricci Tensor and Ricci scalar of the metric [math]\displaystyle{ ds^2=g_{\mu\nu}dx^\mu dx^\nu }[/math].

Black string

A black string is a higher dimensional (D>4) generalization of a black hole in which the event horizon is topologically equivalent to S² × S¹ and spacetime is asymptotically M^d−1 × S¹.

Perturbations of black string solutions were found to be unstable for L (the length around S¹) greater than some threshold L′. The full non-linear evolution of a black string beyond this threshold might result in a black string breaking up into separate black holes which would coalesce into a single black hole. This scenario seems unlikely because it was realized a black string could not pinch off in finite time, shrinking S² to a point and then evolving to some Kaluza–Klein black hole. When perturbed, the black string would settle into a stable, static non-uniform black string state.

Kaluza–Klein black hole

A Kaluza–Klein black hole is a black brane (generalisation of a black hole) in asymptotically flat Kaluza–Klein space, i.e. higher-dimensional spacetime with compact dimensions. They may also be called KK black holes.

See also

• AdS black hole

Bibliography

• Obers, N.A. (2009). "Black Holes in Higher-Dimensional Gravity". Physics of Black Holes. Lecture Notes in Physics. 769. pp. 211–258. doi:10.1007/978-3-540-88460-6_6. ISBN 978-3-540-88459-0.