## Wednesday, January 13, 2016

### The Weyl tensor of the D-dimensional Reissner–Nordström metric

In this post I write the Weyl tensor of the Reissner–Nordström black hole as a sum of squares of the electromagnetic field tensor.
The Reissner–Nordström black hole has the metric $$\label{eq:20160113a} ds^2 = g_{\mu \nu} dx^{\mu} dx^{\nu} = -f dt^2 + \frac{1}{f} dr^2 + r^2 d\Omega^2$$ with \begin{equation*} f(r) = 1 - \frac{2 m}{r^{D-3}} + \frac{q^2}{r^{2 D - 6}} \end{equation*} and $d\Omega^2$ the metric of the $D-2$ dimensional sphere. The electromagnetic potential is \begin{equation*} A = \frac{Q}{r^{D-3}} dt\quad\text{with}\quad Q = \left( \frac{1}{8 \pi}\ \frac{D-2}{D-3} \right)^{1/2} q \end{equation*} I observed that the Weyl tensor $C_{\mu \nu\kappa \lambda}$ of the metric \eqref{eq:20160113a} can be decomposed as $$\label{eq:20160113b} C =h \left( F_{\alpha\beta}F^{\alpha\beta} g \odot g + 2 (D-1) (F^2 \odot g) + (D-1) ( D-2) F \odot F \right)$$ with \begin{align*} h &= \frac{2\pi}{(D - 2) (D-3)} \left(\frac{2(2 D -5)}{D-1} - \frac{M r^{D-3}}{q^2} \right)\\ \odot&= \text{the Kulkarni – Nomizu product}\\ F^2 &= \text{the symmetric matrix with components } (F^2)_{\mu \nu} = F_{\mu \lambda} F^{\lambda}_{\ \nu} \end{align*} The Ricci tensor is a sum of squares of the electromagnetic field tensor as a consequence of Einstein's equations; expression \eqref{eq:20160113b} shows that the curvature tensor of the Reissner–Nordström black hole is a sum of such squares as well. I find expression \eqref{eq:20160113b} interesting because it implies that the Reissner–Nordström metric is algebraically special in $D = 5$, see for example equation 30 in arXiv:1008.2955.
Remarks