Positivity of energy in five-dimensional classical unified field theories

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 V ₅ with metric tensor y _;; which admits a space-like Killing vector ξ^α. It is assumed that: (1) V ₅ has the topology of V ₄×S ¹, S ¹ is a circle and V ₄ is a four-dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor Γ_αβ of V ₅ satisfies , where u ^α and v ^β are future oriented time-like vectors with . 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 ^;=Γ_αβ=0 then V ₅ must admit a time-like Killing vector. Lichnerowicz's results then imply that V ₅ must be flat. A lower bound for P ⁴ (the mass) similar to that found by Gibbons and Hull is obtained.