Black brane

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 ${\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }}$.

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.

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]

• 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.