I am trying to prove a result regarding the Lie derivative of the Dirac distribution, whose support would be an integral curve of the Killing vector. I expect it to be zero, because this delta distribution would be an object completely characterised by the Killing vector itself.
In particular, let $k^a$ be a Killing vector on a given manifold, $x^\alpha$ a chart of coordinates and let $\mathcal{C}$ be an integral curve of $k^a$, which we will parametrise by a set of four equations $x^\alpha=z^\alpha(\tau)$, with $\tau$ the propertime along $\mathcal{C}$. Let now $\delta_4(x^\alpha-z^\alpha(\tau))$ be the covariant Dirac delta whose support is thus $\mathcal{C}$. Do we have $\mathcal{L}_k \delta_4$ = 0 ? And if so, any idea as to how it could be proven ?
Many thanks
Hypaulite