## Sunday, February 7, 2016

### The Weyl tensor of the five-dimensional Myers-Perry metric

In this post I write the Weyl tensor of the Myers-Perry black hole as a sum of squares.
The metric of the Myers-Perry black hole is $$\label{eq:20160203a} ds^2 = -dt^2 + \frac{\rho^2}{R^2} dr^2 + \rho^2 d\theta^2 + \rho^2 \left( \sin^2\theta d\phi^2 + \cos^2\theta d\psi^2\right) + \frac{2m}{\rho^2} \left( dt - a \sin^2\theta d\phi- a \cos^2\theta d\psi\right)^2$$ with $\rho^2 = r^2+a^2$ and $R^2 = \dfrac{(r^2+a^2)^2}{r^2} - 2 m$. This metric describes a rotating black hole with two equal angular frequencies.

The Weyl tensor $C_{\mu \nu\kappa \lambda}$ of the metric \eqref{eq:20160203a} can be decomposed as $$\label{eq:20160203b} C = - m \rho^2 \left( 2\ F \otimes F + \frac{1}{2} \ F~\wedge\!\!\!\!\!\!\bigcirc~F + \frac{1}{8} F_{\alpha\beta}F^{\alpha\beta} g ~\wedge\!\!\!\!\!\!\bigcirc~ g + \ F^2 ~\wedge\!\!\!\!\!\!\bigcirc~ g\right)$$ with $F$ the anti-symmetric matrix defined by $F = dA$ with \begin{equation*} A = \dfrac{1}{\rho^2} \left( - dt + a \sin^2\theta d\phi + a \cos^2\theta d\psi\right) \end{equation*} Furthermore, \begin{align*} \otimes & = \text{the tensor product}\\ \wedge\!\!\!\!\!\!\bigcirc&= \text{the Kulkarni – Nomizu product}\\ F^2 &= \text{the symmetric matrix with components } (F^2)_{\mu \nu} = F_{\mu \lambda} F^{\lambda}_{\quad\nu} \end{align*} Remarks
• The Wikipedia article about the Kulkarni – Nomizu product defines this product for symmetric matrices only, but I also use it in \eqref{eq:20160203b} for the anti-symmetric matrix $F_{\mu \nu}$.
• The form $A$ has similarities to the potential of a charged rotating sphere. I do not know the reason for this resemblance.
• Myers and Perry have written down the metric of a rotating black hole in $D$ dimensions. I have not analyzed if a decomposition as \eqref{eq:20160203b} holds in general $D$.
• A review of the Myers-Perry black holes can be found in arXiv:1111.1903
• In a previous post, I decomposed the Weyl tensor of the Reissner–NordstrÃ¶m black hole.