Killing Reduction of 5-Dimensional Spacetimes

In a 5-dimensional spacetime (M, gab) with a Killing vector field ξ a which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of ξ gives a 4-dimensional space S. The reduction of (M, gab) is studied in the geometric language, which is a generalization of Geroch’s method for the reduction of 4-dimensional spacetime. A 4dimensional 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. Besides 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 . PACS number(s): 04.50.+h, 04.20.Fy