This is an important and natural question as the spacetime shear, inhomogeneity and tidal effects are all intertwined via the Einstein field equations. Though many solutions with these properties exist in the literature, in this paper we identify, via a geometrical analysis, the important physical reason behind these solutions. We show that such scenarios are possible for limited classes of equations of state that are solutions to a highly nonlinear and fourth order differential equation. To show this, we use a covariant semitetrad spacetime decomposition and present a novel geometrical classification of shear-free locally rotationally symmetric perfect fluid self-gravitating systems, in terms of the covariantly defined fluid acceleration and the fluid expansion. Noteworthily, we deduce the governing differential equation that gives the possible limited equations of state of matter.

这是一个重要且自然的问题，因为时空剪切、不均匀性和潮汐效应都通过爱因斯坦场方程交织在一起。尽管文献中存在许多具有这些属性的解决方案，但在本文中，我们通过几何分析确定了这些解决方案背后的重要物理原因。我们证明，对于作为高度非线性和四阶微分方程解的有限类状态方程来说，这种情况是可能的。为了证明这一点，我们使用协变半四分体时空分解，并根据协变定义的流体加速度和流体膨胀，提出了无剪切局部旋转对称完美流体自引力系统的新几何分类。值得注意的是，我们推导了控制微分方程，给出了可能的物质状态有限方程。