= Komar current from trace-reversed stress-energy
{c}
{title2=$J^\mu=R^\mu{}_\nu\xi^\nu$}
A <Killing vector field> gives an antisymmetric tensor $\nabla^\mu\xi^\nu$ whose divergence is this current, up to overall surface-orientation convention. The four-dimensional <Einstein field equations> express it as $8\pi G(T^\mu{}_\nu-\tfrac12T\delta^\mu{}_\nu)\xi^\nu$ at zero cosmological constant. The <contracted Bianchi identity> and Killing invariance of <scalar curvature> prove its divergence is zero. Stress conservation alone does not cancel the trace derivative without Killing invariance of the trace.
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