Black brane#Black string

{{Short description|Generalization of a black hole to higher dimensions}}

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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 {{mvar|p}} additional spatial dimensions. That type of solution would be called a black {{mvar|p}}-brane.{{Cite web|url=http://ncatlab.org/nlab/show/black+brane|title=black brane in nLab|website=ncatlab.org|access-date=2017-07-18}}

In string theory, the term black brane describes a group of D1-branes that are surrounded by a horizon.{{Cite book|title=The Little Book of String Theory|last=Gubser|first=Steven Scott|date=2010|publisher=Princeton University Press|isbn=9780691142890|location=Princeton|pages=[https://archive.org/details/littlebookofstri0000gubs/page/93 93]|oclc=647880066|url=https://archive.org/details/littlebookofstri0000gubs/page/93}} 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.{{Cite web|url=http://superstringtheory.com/blackh/blackh5.html|title=String theory answers|website=superstringtheory.com|access-date=2017-07-18|archive-url=https://web.archive.org/web/20180111135120/http://www.superstringtheory.com/blackh/blackh5.html|archive-date=2018-01-11|url-status=dead}} 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.{{Cite book|title=D-brane : superstrings and new perspective of our world|last=Koji.|first=Hashimoto|date=2012|publisher=Springer-Verlag Berlin Heidelberg|isbn=9783642235740|location=Berlin, Heidelberg|oclc=773812736}}

The metric for a black {{mvar|p}}-brane in a {{mvar|n}}-dimensional spacetime is:

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

where:

{{mvar|η}} is the {{math|(p + 1)}}-Minkowski metric with signature {{math|(−, +, +, +, ...)}},
{{mvar|σ}} are the coordinates for the worldsheet of the black {{mvar|p}}-brane,
{{mvar|u}} is its four-velocity,
{{mvar|r}} is the radial coordinate,
{{math|Ω}} is the metric for a {{math|(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

$\begin{align}
R_{\mu\nu} &= R_{\mu\nu}^{(0)} + \frac{n+1}{r}\Gamma^r_{\mu\nu}, \\
R_{ij} &= \delta_{ij} g_{ii} \left(\frac{n}{r^2}(1-g^{rr}) - \frac{1}{r}(\partial_{\mu} + \Gamma^\nu_{\nu\mu})g^{\mu r}\right),
\end{align}$

and the Ricci Scalar becomes

$R = R^{(0)} + \frac{n+1}{r}g^{\mu\nu}\Gamma^r_{\mu\nu} + \frac{n(n+1)}{r^2}(1-g^{rr}) - \frac{n+1}{r}(\partial_\mu g^{\mu r} + \Gamma^\nu_{\nu\mu}g^{\mu r}),$

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 ({{math|D > 4}}) generalization of a black hole in which the event horizon is topologically equivalent to {{math|S² × S¹}} and spacetime is asymptotically {{math|M^d−1 × S¹}}.

Perturbations of black string solutions were found to be unstable for {{mvar|L}} (the length around {{math|S¹}}) greater than some threshold {{mvar|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 {{math|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.Obers (2009), p. 212–213

References

Bibliography

{{Cite book

| pages = 211–258

| last = Obers

| first = N.A.

| title = Physics of Black Holes

| volume = 769

| chapter = Black Holes in Higher-Dimensional Gravity

| year = 2009

| doi = 10.1007/978-3-540-88460-6_6

| series = Lecture Notes in Physics

| isbn = 978-3-540-88459-0

| arxiv = 0802.0519

| s2cid = 14911870

}}

Category:Black holes