On energy in 5-dimensional gravity and the mass of the Kaluza-Klein monopole

Abstract

We discuss the concept of energy in higher-dimensional gravity, with special attention given to the problem of the choice of a background. Three different approaches to the calculation of energy for solutions of the 5-dimensional Einstein equation are considered. They are then shown to be equivalent and applied to the calculation of the mass of the Kaluza-Klein monopole. The question of the reduction of a 5-dimensional theory to an effective 4-dimensional one in the presence of a Killing vector field along the compact dimension is then discussed, in particular with regard to the conformal ambiguity in the definition of the metric in the reduced theory. We show that there exists an essentially unique choice of reduction for which the 4-dimensional energy has the usual general relativistic form, and that for the Kaluza-Klein monopole this is the reduction that will make the “naive” 4-dimensional energy additive.