If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations
\begin{equation*}
\nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0
\end{equation*}
and there is a spinor $\psi$ such that
\begin{equation}\label{eq:20161115b}
(\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0
\end{equation}
and $i \bar \psi \gamma^{\mu} \psi$ is time-like
then the Einstein equations are satisfied as well:
\begin{equation*}
R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu}
\end{equation*}
Monday, December 5, 2016
On Killing spinors in general dimensions
The following property is true in four spacetime dimensions [1] [2]
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