Stress-energy current from a Killing vector ($J^a=T^a{}_b{K}^b$)

If $T^{ab}$ is symmetric and covariantly conserved and $K$ is a Killing vector field, then $J^a=T^a{}_b{K}^b$ is conserved. Its flux through a spacelike hypersurface is the energy or momentum charge generated by the isometry.