Misner-Sharp Mass and the Unified First Law in Massive Gravity

We obtain the Misner-Sharp mass in the massive gravity for a four dimensional spacetime with a two dimensional maximally symmetric subspace via the inverse unified first law method. Significantly, the stress energy is conserved in this case with a widely used reference metric. Based on this property we confirm the derived Misner-Sharp mass by the conserved charge method. We find that the existence of the Misner-sharp mass in this case does not lead to extra constraint for the massive gravity, which is notable in modified gravities. In addition, as a special case, we also investigate the Misner-Sharp mass in the static spacetime. Especially, we take the FRW universe into account for investigating the thermodynamics of the massive gravity. The result shows that the massive gravity can be in thermodynamic equilibrium, which fills in the gap in the previous studies of thermodynamics in the massive gravity.

From thermodynamics to the solutions in gravity theory

In a recent work, we present a new point of view to the relation of gravity and thermodynamics, in which we derive the sch~solution through thermodynamic laws by the aid of the Misner-Sharp mass in an adiabatic system. In this paper we continue to investigate the relation between gravity and thermodynamics for obtaining solutions via thermodynamics. We generalize our studies on gravi-thermodynamics in Einstein gravity to modified gravity theories. By using the first law with the assumption that the Misner-Sharp mass is the mass for an adiabatic system, we reproduce the Boulware-Deser-Cai solution in Guass-Bonnet gravity. Using this gravi-thermodynamics thought, we obtain a NEW class of solution in $F(R)$ gravity in an $n$-dimensional (n$geq$3) spacetime which permits three-type $(n-2)$-dimensional maximally symmetric subspace, as an extension of our recent three-dimensional black hole solution, and four-dimensional Clifton-Barrow solution in $F(R)$ gravity.

Misner-Sharp Mass in $N$-dimensional $f(R)$ Gravity

We study the Misner-Sharp mass for the $f(R)$ gravity in an $n$-dimensional (n$geq$3) spacetime which permits three-type $(n-2)$-dimensional maximally symmetric subspace. We obtain the Misner-Sharp mass via two approaches. One is the inverse unified first law method, and the other is the conserved charge method by using a generalized Kodama vector. In the first approach, we assume the unified first still holds in the $n$-dimensional $f(R)$ gravity, which requires a quasi-local mass form (We define it as the generalized Misner-Sharp mass). In the second approach, the conserved charge corresponding to the generalized local Kodama vector is the generalized Misner-Sharp mass. The two approaches are equivalent, which are bridged by a constraint. This constraint determines the existence of a well-defined Misner-Sharp mass. As an important special case, we present the explicit form for the static space, and we calculate the Misner-Sharp mass for Clifton-Barrow solution as an example.

A quantitative criteria for the coincidence problem

The cosmic coincidence problem is a serious challenge to dark energy model. We suggest a quantitative criteria for judging the severity of the coincidence problem. Applying this criteria to three different interacting models, including the interacting quintessence, interacting phantom, and interacting Chaplygin gas models, we find that the interacting Chaplygin gas model has a better chance to solve the coincidence problem. Quantitatively, we find that the coincidence index C for the interacting Chaplygin gas model is smaller than that for the interacting quintessence and phantom models by six orders of magnitude.