Komar current from trace-reversed stress-energy ($J^{\mu }=R^{\mu }{}_{\nu }{\xi }^{\nu }$)

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 $8\pi G(T^{\mu }{}_{\nu }-\frac{1}{2}T{\delta }^{\mu }{}_{\nu }){\xi }^{\nu }$ 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.