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This is an important and natural question as the spacetime shear,inhomogeneity and tidal effects are all intertwined via the Einstein fieldequations. However, as we show in this paper, such scenarios are possible forlimited classes of equations of state that are solutions to a highly non-linearand fourth order differential equation. To show this, we use a covariantsemitetrad spacetime decomposition and present a novel geometricalclassification of shear-free Locally Rotationally Symmetric (LRS-II) perfectfluid self-gravitating systems, in terms of the covariantly defined fluidacceleration and the fluid expansion. Noteworthily, we deduce the governingdifferential equation that gives the possible limited equations of state ofmatter.

What makes a shear-free spherical perfect fluid be inhomogeneous with tidal effects?