Black brane

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In

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:

${\displaystyle {ds}^{2}=\left(\eta _{ab}+{\frac {r_{s}^{n-p-3}}{r^{n-p-3}}}u_{a}u_{b}\right)d\sigma ^{a}d\sigma ^{b}+\left(1-{\frac {r_{s}^{n-p-3}}{r^{n-p-3}}}\right)^{-1}dr^{2}+r^{2}d\Omega _{n-p-2}^{2}}$

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 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

• AdS black hole

References