A stationary spacetime is defined to be one for which there exists a timelike Killing vector field such that the Lie derivative of the metric wrt this vector field vanishes, i.e.
$\mathcal{L}_X g_{\mu\nu}=0$.
I would like to know what this implies for the properties of other quantities related to this spacetime.
Let's for example assume that we have some given action with a stationary spacetime acting as the background metric. The stress-energy tensor is then defined to be the variation of the action wrt the metric, i.e.
$T^{\mu\nu} = \frac{\sqrt{-g}}{2} \frac{\delta S}{\delta g_{\mu\nu}} $.
Will this energy-momentum tensor have properties linked to the stationarity?
All the best,
