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Black brane

Generalization of a black hole to higher dimensions

Generalization of a black hole to higher dimensions

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.

In string theory, the term black brane describes a group of D1-branes that are surrounded by a horizon. 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. 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.

The metric for a black p-brane in a 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:

  • 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{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 (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 (generalization 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.

References

Bibliography

  • {{Cite book

References

  1. "black brane in nLab".
  2. Gubser, Steven Scott. (2010). "The Little Book of String Theory". [[Princeton University Press]].
  3. "String theory answers".
  4. Koji., Hashimoto. (2012). "D-brane : superstrings and new perspective of our world". Springer-Verlag Berlin Heidelberg.
  5. Obers (2009), p. 212–213
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