Black brane

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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 p additional spatial dimensions. That type of solution would be called a black p-brane.^[1]

In string theory, the term black brane describes a group of D1-branes that are surrounded by a horizon.^[2] 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.^[3] 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.^[4]

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

{ds}^{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 _{n-p-2}^{2}

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 $ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }+d\Omega _{n+1}$ .

The Ricci Tensor becomes $R_{\mu \nu }=R_{\mu \nu }^{(0)}+{\frac {n+1}{r}}\Gamma _{\mu \nu }^{r}$ , $R_{ij}=\delta _{ij}g_{ii}({\frac {n}{r^{2}}}(1-g^{rr})-{\frac {1}{r}}(\partial _{\mu }+\Gamma _{\nu \mu }^{\nu })g^{\mu r})$ .

The Ricci Scalar becomes $R=R^{(0)}+{\frac {n+1}{r}}g^{\mu \nu }\Gamma _{\mu \nu }^{r}+{\frac {n(n+1)}{r^{2}}}(1-g^{rr})-{\frac {n+1}{r}}(\partial _{\mu }g^{\mu r}+\Gamma _{\nu \mu }^{\nu }g^{\mu r})$ .

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

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 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.^[5]

References

Bibliography

Obers, N.A. (2009). "Black Holes in Higher-Dimensional Gravity". Physics of Black Holes. Lecture Notes in Physics. Vol. 769. pp. 211–258. arXiv:0802.0519. doi:10.1007/978-3-540-88460-6_6. ISBN 978-3-540-88459-0. S2CID 14911870.

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