Actions for Vacuum Einstein's Equation with a Killing Symmetry

TL;DR

This paper proposes an alternative action principle for vacuum Einstein's equations in spacetimes with a Killing symmetry, leading to the same field equations as the traditional symmetry-reduced Einstein-Hilbert action.

Contribution

It introduces a new action formulation for vacuum Einstein's equations with a Killing symmetry, providing an equivalent but potentially more versatile approach.

Findings

The alternative action yields the same field equations as the symmetry-reduced Einstein-Hilbert action.

It offers a different perspective on the variational principle for spacetimes with Killing symmetries.

The approach could facilitate new methods in analyzing symmetric solutions in general relativity.

Abstract

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 $M$.