Proposition: Current divergence

For any symmetric tensor distribution $s_{\mu \nu }$,

Consequently the currents $j_{X,\mu }=X^{\nu }{t}_{\mu \nu }$ are conserved under Assumption ass:improvement.

Proof. Apply the product rule. Symmetry of $s$ annihilates the antisymmetric part of ${\partial }^{\mu }{X}^{\nu }$, and the conformal Killing equation contracts the remaining part to ${\sigma }_X{s}^{\mu }{}_{\mu }$.