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. 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 = ( η a b + r s n − p − 3 r n − p − 3 u a u b ) d σ a d σ b + ( 1 − r s n − p − 3 r n − p − 3 ) − 1 d r 2 + r 2 d Ω n − p − 2 2 {\displaystyle {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

d s 2 = g μ ν d x μ d x ν + d Ω n + 1 , {\displaystyle ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }+d\Omega _{n+1},}

the Ricci Tensor becomes

R μ ν = R μ ν ( 0 ) + n + 1 r Γ μ ν r , R i j = δ i j g i i ( n r 2 ( 1 − g r r ) − 1 r ( ∂ μ + Γ ν μ ν ) g μ r ) , {\displaystyle {\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 ) + n + 1 r g μ ν Γ μ ν r + n ( n + 1 ) r 2 ( 1 − g r r ) − n + 1 r ( ∂ μ g μ r + Γ ν μ ν g μ r ) , {\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 R μ ν ( 0 ) {\displaystyle R_{\mu \nu }^{(0)}} , R ( 0 ) {\displaystyle R^{(0)}} are the Ricci Tensor and Ricci scalar of the metric d s 2 = g μ ν d x μ d x ν . {\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 (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.

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