具有非零Weyl张量的类Friedmann解
Friedmann-like solutions with a non-vanishing Weyl tensor
摘要：
我们求解了具有常空间曲率的一类度规的广义相对论的Einstein方程，并在具有各向异性压力分量的能量动量张量存在的情况下，找到了一个非零的Weyl张量.时空的时间演化由通常的弗里德曼方程引导，超曲面上的约束包括一个可以独立求解的独立方程组。与某些坐标选择引起的明显行为相反，我们得到的度规在任何地方都是完全正则的，并且没有奇点（除了在$t=0$的著名弗里德曼奇点）。该解的物理性质用准麦克斯韦方程（一组用Weyl张量描述引力场动力学的三阶微分方程）来阐明。测试粒子的运动也进行了分析，以确认所考虑的歧管的最大扩展。这些结果表明，各向异性压力可以模拟暗物质对某些测地线汇的影响，保持宇宙流不变。
英文摘要：
We have solved the Einstein equations of general relativity for a class of metrics with constant spatial curvature and found a non-vanishing Weyl tensor in the presence of an energy-momentum tensor with an anisotropic pressure component. The time evolution of the spacetime is guided by the usual Friedmann equations and the constraints on the hypersurface comprise a separated system of equations that can be independently solved. Contrary to the apparent behavior induced by some choices of coordinates, the metric we have obtained is completely regular everywhere and is free of singularities (except the well-known Friedmann singularity at $t=0$). The physical features of this solution are elucidated by using the Quasi-Maxwellian equations (a set of third order differential equations describing the dynamics of the gravitational field in terms of the Weyl tensor). The motion of test particles is also analyzed in order to confirm the maximal extension of the manifold under consideration. These results indicate that the anisotropic pressure could mimic dark matter effects on certain geodesic congruences keeping the cosmic flow unchanged.
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