The Raychaudhuri equations for the expansion, shear, and vorticity are generalized in a spacetime with torsion for timelike as well as null congruences. These equations are purely geometrical like the original Raychaudhuri equations and could be reduced to them when there is no torsion. Using the Einstein-Cartan-Sciama-Kibble field equations, the effective stress-energy tensor is derived. We also consider an Oppenheimer-Snyder model for the gravitational collapse of dust. It is shown that the null energy condition is violated before the density of the collapsing dust reaches the Planck density, hinting that the spacetime singularity may be avoided if there is a nonzero torsion, i.e., if the collapsing dust particles possess intrinsic spin.

中文翻译：

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爱因斯坦-嘉当理论中的瑞乔杜里方程和引力坍缩

膨胀、剪切和涡度的 Raychaudhuri 方程在具有扭转的时空中推广，用于类时和零同余。这些方程与原始 Raychaudhuri 方程一样是纯几何的，并且可以在没有扭转时简化为它们。使用Einstein-Cartan-Sciama-Kibble 场方程，可以推导出有效应力-能量张量。我们还考虑了尘埃引力坍塌的 Oppenheimer-Snyder 模型。结果表明，在坍缩尘埃的密度达到普朗克密度之前，零能量条件被破坏，这表明如果存在非零扭转，即如果坍缩尘埃粒子具有内在自旋，时空奇点可以避免。