Tolman VII解的几何结构
The Geometrical Structure of the Tolman VII solution
摘要：
Tolman~VII解是具有理想流体源的球对称静态Einstein方程组的精确解析解，它具有许多特性，使它对模拟高密度物理天体很有意义。在这里，我们补充这些特征的几何张量，这个解决方案拥有，并发现外尔，黎曼和里奇张量组件显示出意想不到的数学行为，根据物理参数的变化，即使建模对象的质量是固定的。我们显示这些功能首先通过张量分量，然后通过标量的零四分体形式主义的纽曼和彭罗斯。分析的突出结论是Tolman~VII解与常密度Schwarzschild内部解之间的密切关系：前者是后者的直接推广，而避开了非物理的常密度。
英文摘要：
The Tolman~VII solution, an exact analytic solution to the spherically symmetric, static Einstein equations with a perfect fluid source, has many characteristics that make it interesting for modelling high density physical astronomical objects. Here we supplement those characteristics with the geometrical tensors that this solution possess, and find that the Weyl, Riemann, and Ricci tensor components show unexpected mathematical behaviour that change depending on physically motivated parameters, even though the mass of the modelled objects is fixed. We show these features firstly through tensor components, and then through the scalars in the null tetrad formalism of Newmann and Penrose. The salient conclusion of this analysis is the intimate relationship between the Tolman~VII solution and the constant density Schwarzschild interior solution: the former being a straight forward generalization of the latter while eschewing the unphysical constant density.
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