In general relativity, a black brane is a solution of the 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 \( {\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }+d\Omega _{n+1}}. \)
The Ricci Tensor becomes \( {\displaystyle R_{\mu \nu }=R_{\mu \nu }^{(0)}+{\frac {n+1}{r}}\Gamma _{\mu \nu }^{r}} \), \( {\displaystyle R_{ij}=\delta _{ij}g_{ii}({\frac {n}{r^{2}}}(1-g^{rr})-{\frac {1}{r}}(\partial _{\mu }+\Gamma _{\nu \mu }^{\nu })g^{\mu r})}. \)
The Ricci Scalar becomes \( {\displaystyle R=R^{(0)}+{\frac {n+1}{r}}g^{\mu \nu }\Gamma _{\mu \nu }^{r}+{\frac {n(n+1)}{r^{2}}}(1-g^{rr})-{\frac {n+1}{r}}(\partial _{\mu }g^{\mu r}+\Gamma _{\nu \mu }^{\nu }g^{\mu r})}. \)
Where \( {\displaystyle R_{\mu \nu }^{(0)}} \), \( {\displaystyle R^{(0)}} \) are the Ricci Tensor and Ricci scalar of the metric \( {\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }}. \)
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 Md−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]
vte
String theory
Background
Strings History of string theory
First superstring revolution Second superstring revolution String theory landscape
Calabi-Yau-alternate
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
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 domain wall K-theory (physics) Twisted K-theory
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 Cleaver Curtright Dijkgraaf Distler Douglas Duff Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey 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 Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach
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. 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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