= Papapetrou electromagnetic field
{c}
{title2=$F=dK^\flat$}
= Killing-generated Maxwell field
{c}
{synonym}
For a <Killing vector field> $K$, its metric-dual <differential one-form> produces the <electromagnetic field> $F_{ab}=2\nabla_{[a}K_{b]}=2\nabla_aK_b$. The <exterior derivative> gives $dF=0$, while the <second covariant derivative of a Killing vector> gives $\nabla^aF_{ab}=-2R_{bd}K^d$. Thus this construction solves the source-free <Maxwell equations> in a <vacuum spacetime> with zero <cosmological constant>. The field is a test field: adding its <stress-energy tensor> would generally change the original <metric tensor>.
Back to article page
