Second law of gravitational thermodynamics in the locally rotationally symmetric Bianchi type-II universe

In this work, under the assumption that the universe is locally rotationally symmetric (LRS) Bianchi type-II and filled with anisotropic dark uid (dark energy and dark radiation), we investigate whether the generalized second law of thermodynamics (GSLT) is still valid. We also consider another important condition for LRS Bianchi type-II spacetimes whereby the model expansion is proportional to the shear, which leads to B = An (where n is a constant and A and B are the metric potentials). Based on this, we derive a general circumstance for the GSLT and find that its validity depends on the anisotropies of both the dark uid and the metric potentials. We conclude that the GSLT still holds at all times even if these two anisotropic conditions are removed from the results.