Black brane explained

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:

{d s} ^{2} = \left( η _{ab} + (r ^{n - p -3} _{s})/(r ^{n-p-3}) u _{a} u _{b} \right) d \sigma ^{a} d \sigma ^{b} + \left( 1- (r ^{n - p -3} _{s})/(r ^{n-p-3}) \right) ^{-1} dr ^{2} + r ^{2} d \Omega ^{2} _{n - p -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 ^{( 0 )} _{\mu\nu} + (n +1)/(r) \Gamma ^{r} _{\mu\nu} , R _{ij} = \delta _{ij} g _{ii} ( (n)/(r ^{2}) ( 1- g ^{rr} ) - (1)/(r) (\partial _{\mu} + \Gamma ^{\nu} _{\nu\mu} )g ^{\mu} ) .

The Ricci Scalar becomes

R = R ^{(0)} + (n +1)/(r) g ^{\mu\nu} \Gamma ^{r} _{\mu\nu} + (n(n +1 ))/(r ^{2}) ( 1- g ^{rr} ) - (n +1)/(r) (\partial _{\mu} g ^{\mu} + \Gamma ^{\nu} _{\nu\mu} g ^{\mu} ) .

Where

R ^{( 0 )} _{\mu\nu} , 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]

See also

• AdS black hole

Bibliography

• Book: Obers , N.A. . 211–258. Physics of Black Holes. 769. Black Holes in Higher-Dimensional Gravity. 2009. 10.1007/978-3-540-88460-6_6. Lecture Notes in Physics. 978-3-540-88459-0. 0802.0519. 14911870.

Notes and References

1. Web site: black brane in nLab. ncatlab.org. 2017-07-18.
2. Book: Gubser, Steven Scott. The Little Book of String Theory. 2010. Princeton University Press. 9780691142890. Princeton. 93. 647880066.
3. Web site: String theory answers. superstringtheory.com. 2017-07-18. https://web.archive.org/web/20180111135120/http://www.superstringtheory.com/blackh/blackh5.html. 2018-01-11. dead.
4. Book: Koji., Hashimoto. D-brane : superstrings and new perspective of our world. 2012. Springer-Verlag Berlin Heidelberg. 9783642235740. Berlin, Heidelberg. 773812736.
5. Obers (2009), p. 212–213