Investigations of f ( R)-gravity counterparts of the general relativistic shear-free conjecture by illustrative examples

By adopting a metric based approach and making use of f(R)-gravity extended tetrad equations, we have considered three spatially homogeneous metrics in order to investigate the existence of simultaneously rotating and expanding solutions of the f(R)-gravity field equations with shear-free perfect fluids as sources. We have shown that the Godel type expanding universe, as well as a rotating Bianchi-type II spacetime allow no such solutions of the field equations of this modified gravity. On the other hand, we have found that there exist two types of f(R) models in which a shear free Bianchi-type IX universe can expand and rotate at the same time. The matter content of this universe is described by a perfect fluid having positive or negative pressure, depending on the type of f(R) model and on the cosmological constant; in the particular case of a vanishing cosmological constant we have found that the universe is filled with a pure radiation. Whatsoever the cases, the universe exhibits always coasting anisotropic expansions along three spatial directions evolving like a flat Milne universe, and has a vorticity inversely proportional to cosmic time. A further result is that, due to the nonvanishing of the gravito-magnetic part of the Weyl tensor, this model allows for gravitational waves. Our solution constitutes one more example giving support to that in f(R)-gravity there is no counterpart of the general relativistic shear-free conjecture.