Black brane

In general relativity, a black brane is a solution of the Einstein field equations that generalizes a black hole solution but it is also extended—and translationally symmetric—in Template:Mvar additional spatial dimensions. That type of solution would be called a black Template:Mvar-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 Template:Mvar-brane in a Template:Mvar-dimensional spacetime is: $${ds}^2=({\eta }_{ab}+\frac{r_s^{n-p-3}}{r^{n-p-3}}{u}_a{u}_b)d{\sigma }^{a}d{\sigma }^b+{(1-\frac{r_s^{n-p-3}}{r^{n-p-3}})}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}_{n-p-2}^2$$ where:

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

Curvatures

When $$d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{\mathrm{\Omega }}_{n+1},$$ the Ricci Tensor becomes $$\begin{array}{rl}{R}_{\mu \nu }& =R_{\mu \nu }^{(0)}+\frac{n+1}{r}{\mathrm{\Gamma }}_{\mu \nu }^r,\\ R_{ij}& ={\delta }_{ij}{g}_{ii}(\frac{n}{r^2}(1-g^{rr})-\frac{1}{r}({\partial }_{\mu }+{\mathrm{\Gamma }}_{\nu \mu }^{\nu })g^{\mu r}),\end{array}$$ and the Ricci Scalar becomes $$R=R^{(0)}+\frac{n+1}{r}{g}^{\mu \nu }{\mathrm{\Gamma }}_{\mu \nu }^r+\frac{n(n+1)}{r^2}(1-g^{rr})-\frac{n+1}{r}({\partial }_{\mu }{g}^{\mu r}+{\mathrm{\Gamma }}_{\nu \mu }^{\nu }{g}^{\mu r}),$$ where $R_{\mu \nu }^{(0)}$, $R^{(0)}$ are the Ricci Tensor and Ricci scalar of the metric $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }.$

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 Template:Mvar (the length around S¹) greater than some threshold Template:Mvar. 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