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 $p$ 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,
$Ω$ 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 ${\begin{aligned}R_{\mu \nu }&=R_{\mu \nu }^{(0)}+{\frac {n+1}{r}}\Gamma _{\mu \nu }^{r},\\R_{ij}&=\delta _{ij}g_{ii}\left({\frac {n}{r^{2}}}(1-g^{rr})-{\frac {1}{r}}(\partial _{\mu }+\Gamma _{\nu \mu }^{\nu })g^{\mu r}\right),\end{aligned}}$ and 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 $S 2 \times S 1$ and spacetime is asymptotically $M d -1 \times S 1$ .

Perturbations of black string solutions were found to be unstable for $L$ (the length around $S 1$ ) 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 $S 2$ 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

^ "black brane in nLab". ncatlab.org. Retrieved 2017-07-18.
^ Gubser, Steven Scott (2010). The Little Book of String Theory. Princeton: Princeton University Press. pp. 93. ISBN 9780691142890. OCLC 647880066.
^ "String theory answers". superstringtheory.com. Archived from the original on 2018-01-11. Retrieved 2017-07-18.
^ Koji., Hashimoto (2012). D-brane : superstrings and new perspective of our world. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 9783642235740. OCLC 773812736.
^ Obers (2009), p. 212–213

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.

[1] "black brane in nLab". ncatlab.org. Retrieved 2017-07-18.

[2] Gubser, Steven Scott (2010). The Little Book of String Theory. Princeton: Princeton University Press. pp. 93. ISBN 9780691142890. OCLC 647880066.

[3] "String theory answers". superstringtheory.com. Archived from the original on 2018-01-11. Retrieved 2017-07-18.

[4] Koji., Hashimoto (2012). D-brane : superstrings and new perspective of our world. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg. ISBN 9783642235740. OCLC 773812736.

[5] Obers (2009), p. 212–213

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v t e String theory
Background	Strings Cosmic strings History of string theory First superstring revolution Second superstring revolution String theory landscape
Theory	Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string Type IIA string Type IIB string Heterotic string N=2 superstring F-theory String field theory Matrix string theory Non-critical string theory Non-linear sigma model Tachyon condensation RNS formalism GS formalism
String duality	T-duality S-duality U-duality Montonen–Olive duality
Particles and fields	Graviton Dilaton Tachyon Ramond–Ramond field Kalb–Ramond field Magnetic monopole Dual graviton Dual photon
Branes	D-brane NS5-brane M2-brane M5-brane S-brane Black brane Black holes Black string Brane cosmology Quiver diagram Hanany–Witten transition
Conformal field theory	Virasoro algebra Mirror symmetry Conformal anomaly Conformal algebra Superconformal algebra Vertex operator algebra Loop algebra Kac–Moody algebra Wess–Zumino–Witten model
Gauge theory	Anomalies Instantons Chern–Simons form Bogomol'nyi–Prasad–Sommerfield bound Exceptional Lie groups (G₂, F₄, E₆, E₇, E₈) ADE classification Dirac string p-form electrodynamics
Geometry	Worldsheet Kaluza–Klein theory Compactification Why 10 dimensions? Kähler manifold Ricci-flat manifold Calabi–Yau manifold Hyperkähler manifold K3 surface G₂ manifold Spin(7)-manifold Generalized complex manifold Orbifold Conifold Orientifold Moduli space Hořava–Witten theory K-theory (physics) Twisted K-theory
Supersymmetry	Supergravity Eleven-dimensional supergravity Type I supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup
Holography	Holographic principle AdS/CFT correspondence
M-theory	Matrix theory Introduction to M-theory
String theorists	Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Curtright Dijkgraaf Distler Douglas Duff Dvali Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey 't Hooft Hořava Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Sagnotti Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach

Curvatures

Black string

Kaluza–Klein black hole

See also

References

Bibliography