Positivity of energy in five-dimensional classical unified field theories

Abstract

Five-dimensional classical unified field theories as well as in Yang-Mills theory with gauge group U(1), are described in terms of a Lorentzian five-dimensional space V5 with metric tensor y;; which admits a space-like Killing vector ξα. It is assumed that: (1) V5 has the topology of V4×S1, S1 is a circle and V4 is a four-dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γαβ of V5 satisfies \(\Gamma _{\alpha \beta } u^\alpha v^\beta \leqslant 0\), where uα and vβ are future oriented time-like vectors with \(\gamma _{\alpha \beta } \upsilon ^\alpha \xi ^\beta = 0\). The spinor approach of Witten, Nester, and Moreschi and Sparling is used to show that the conserved five-dimensional energy momentum vector P; is nonspace-like. If P;=Γαβ=0 then V5 must admit a time-like Killing vector. Lichnerowicz's results then imply that V5 must be flat. A lower bound for P4 (the mass) similar to that found by Gibbons and Hull is obtained.