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Positivity of energy in five-dimensional classical unified field theories. (English) Zbl 0574.53056
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.

MSC:
 53C80 Applications of global differential geometry to the sciences 83E15 Kaluza-Klein and other higher-dimensional theories
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References:
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