Killing reduction of 5-dimensional spacetimes

In a 5-dimensional spacetime ${(M,g}_{\mathrm{ab}})$ with a Killing vector field ${\xi }^a$ which is either everywhere time like or everywhere space like, the collection of all trajectories of ${\xi }^a$ gives a 4-dimensional space S. The reduction of ${(M,g}_{\mathrm{ab}})$ is studied in the geometric language, which is a generalization of Geroch’s method for the reduction of 4-dimensional spacetime. A 4-dimensional gravity coupled to a vector field and a scalar field on S is obtained by the reduction of vacuum Einstein’s equations on M, which gives also an alternative description of the 5-dimensional Kaluza-Klein theory. In addition to the symmetry-reduced action from the Hilbert action on M, an alternative action of the fields on S is also obtained, the variations of which lead to the same fields equations as those reduced from the vacuum Einstein equation on M.