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

Abstract

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.