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arXiv 1301.0910gr-qcmath-phmath.MP
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广义导子与广义相对论

Generalized derivations and general relativity

M. Heller, T. Miller, L. Pysiak, W. Sasin

备注：
21 pages
发表信息：
Can. J. Phys., 2013, 91(10): 757-763
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摘要：

我们构造微分几何（联络，曲率等）基于代数${\cal A}$的广义导子。Bresar在1991年提出了这样一个导子，它由一个线性映射$u: {\cal A} \rightarrow {\cal A}$给出，使得对于所有$a,b \in \cal A$，存在一个满足广义Leibniz规则$u(a b) = u(a) b + a \, d(b)$的${\cal A}$的通常导子$d$.广义几何“测试”的情况下，光滑函数的代数流形。然后，我们应用这一机制来研究广义广义相对论。我们定义了爱因斯坦-希尔伯特作用量，并由此推出爱因斯坦场方程。我们表明，对于一类特殊的度量包含，除了通常的度量组件，只有一个非零项，行动减少到O 'Hanlon行动，这是潜在的Brans-Dicke行动和参数$ω$等于零。我们还表明，广义爱因斯坦方程（零能量应力张量）是等价的卡鲁扎-克莱因理论满足“修改的圆柱条件”，并有一个非紧额外的维度。这为考虑具有非紧额外维度的卡鲁扎-克莱因模型提供了一种可能性，这种额外维度对于宏观观察者来说是不可见的。在我们的方法中，这个额外的维度不是额外的物理时空维度，而是由于推导概念的推广而出现的。

英文摘要：

We construct differential geometry (connection, curvature, etc.) based on generalized derivations of an algebra ${\cal A}$. Such a derivation, introduced by Bresar in 1991, is given by a linear mapping $u: {\cal A} \rightarrow {\cal A}$ such that there exists a usual derivation $d$ of ${\cal A}$ satisfying the generalized Leibniz rule $u(a b) = u(a) b + a \, d(b)$ for all $a,b \in \cal A$. The generalized geometry "is tested" in the case of the algebra of smooth functions on a manifold. We then apply this machinery to study generalized general relativity. We define the Einstein-Hilbert action and deduce from it Einstein's field equations. We show that for a special class of metrics containing, besides the usual metric components, only one nonzero term, the action reduces to the O'Hanlon action that is the Brans-Dicke action with potential and with the parameter $ω$ equal to zero. We also show that the generalized Einstein equations (with zero energy-stress tensor) are equivalent to those of the Kaluza-Klein theory satisfying a "modified cylinder condition" and having a noncompact extra dimension. This opens a possibility to consider Kaluza-Klein models with a noncompact extra dimension that remains invisible for a macroscopic observer. In our approach, this extra dimension is not an additional physical space-time dimension but appears because of the generalization of the derivation concept.

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