In this paper we have examined the validity of a proposed definition of gravitational entropy in the context of accelerating black hole solutions of the Einstein field equations, which represent the realistic black hole solutions. We have adopted a phenomenological approach proposed in [20] and expanded in [21], in which the Weyl curvature hypothesis is tested against the expressions for the gravitational entropy. Considering the C-metric for the accelerating black holes, we have evaluated the gravitational entropy and the corresponding entropy density for four different types of black holes, namely, non-rotating black hole, non-rotating charged black hole, rotating black hole and rotating charged black hole. We end up by discussing the merits of such an analysis and the possible reason of failure in the particular case of rotating charged black hole and comment on the possible resolution of the problem.

In this paper we study the evolution of the FRW universe filled with variable modified Chaplygin gas (VMCG). We begin with a thermodynamical treatment of VMCG described by the equation of state P = Aρ − Bρ −α , and obtain its temperature as a function of redshift z. We show that the results are consistent with similar works on other types of Chaplygin gas models. In addition to deriving the exact expression of temperature of the fluid in terms of the boundary conditions and redshift, we also used observational data to determine the redshift at the epoch of transition from the decelerated to the accelerated phase of expansion of the universe. The values of other relevant parameters like the Hubble parameter, the equation-of-state parameter and the speed of sound are obtained in terms of the redshift parameter, and these values are compared with the results obtained from previous works on MCG and other Chaplygin gas models for the various values of n permitted by thermodynamic stability. We assume the present value of temperature of the microwave background radiation to be given by T 0 = 2.7K, and the parameter A in the equation of state is taken as 1/3 since it corresponds to the radiation-dominated phase of the universe. The value of the parameter Ω x has been assumed to be 0.7 in our calculation. Since it is known that the redshift of photon decoupling is z 1100, we used this value to calculate the temperature of decoupling.

In this paper, we investigate the entropy of the free gravitational field for a given epoch for some well-known isotropic and anisotropic cosmologies. We use the definition of gravitational entropy proposed by Clifton, Ellis and Tavakol, where the 2-index square root of the 4-index Bel–Robinson tensor is taken to be the energy– momentum tensor for the free gravity. We examine whether in the vicinity of the initial singularity, the ratio of energy density of free gravity to that of matter density goes to zero, validating Penrose conjecture on Weyl curvature. Whenever this is true, the gravitational entropy increases monotonically with time, leading to structure formation. For the models considered by us, we identify the conditions for which the Weyl curvature hypothesis is valid, and the assumptions under which it is validated, or otherwise.

In this paper we have examined the validity of the generalized second law of thermodynamics (GSLT) in an expanding Friedmann Robertson Walker (FRW) universe filled with different variants of Chaplygin gases. Assuming that the universe is a closed system bounded by the cosmological horizon, we first present the general prescription for the rate of change of total entropy on the boundary. In the subsequent part we have analyzed the validity of the generalised second law of thermodynamics on the cosmological apparent horizon and the cosmological event horizon for different Chaplygin gas models of the universe. The analysis is supported with the help of suitable graphs to clarify the status of the GSLT on the cosmological horizons. In the case of the cosmological apparent horizon we have found that some of these models always obey the GSLT, whereas the validity of GSLT on the cosmological event horizon of all these models depend on the choice of free parameters in the respective models.

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