Actions for Vacuum Einstein's Equation with a Killing Symmetry

In a space-time $M$ with a Killing vector field ${\xi }^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ${\xi }^a$ gives a 3-dimension space $S$. Besides the symmetry-reduced action from that of Einstein-Hilbert, an alternative action of the fields on $S$ is also proposed, which gives the same fields equations as those reduced from the vacuum Einstein equation on $M$.