Positivity of energy in five-dimensional classical unified field theories (Q1063888)

A positive energy theorem valid in five-dimensional classical unified field theories as well as in Yang-Mills theory with gauge group U(1), is described in terms of a Lorentzian five-dimensional space \(V_ 5\) with metric tensor \(\gamma_{\alpha \beta}\) which admits a space-like Killing vector \(\xi^{\alpha}\). It is assumed that: (1) \(V_ 5\) has the topology of \(V_ 4\times S^ 1\), \(S^ 1\) is a circle and \(V_ 4\) is a four-dimensional Lorentzian space that is asymptotically flat and (2) the Einstein tensor \(\Gamma_{\alpha \beta}\) of \(V_ 5\) satisfies \(\Gamma_{\alpha \beta} u^{\alpha} v^{\beta}\leq 0\), where \(u^{\alpha}\) and \(v^{\beta}\) are future oriented time-like vectors with \(\gamma_{\alpha \beta} v^{\alpha} \xi^{\beta}=0\). The spinor approach of Witten, Nester, and Moreschi and Sparling is used to show that the conserved five-dimensional energy momentum vector \({\mathcal P}^{\alpha}\) is nonspace-like. If \({\mathcal P}^{\alpha}=\Gamma_{\alpha \beta}=0\) 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 \({\mathcal P}^ 4\) (the mass) similar to that found by Gibbons and Hull is obtained.