Actions for vacuum Einstein's equation with a killing symmetry

摘要

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 three-dimensional 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. [References: 9]