A study of universal thermodynamics in massive gravity: modified entropy on the horizons

Abstract

Universal thermodynamics for FRW model of the Universe bounded by apparent/event horizon has been considered for massive gravity theory. Assuming Hawking temperature and using the unified first law of thermodynamics on the horizon, modified entropy on the horizon has been determined. For simple perfect fluid with constant equation of state, generalized second law of thermodynamics and thermodynamical equilibrium have been examined on both the horizons.

Appendices

Appendix 1

Using Eq. (55) in Eq. (29), one obtains

$$\begin{aligned} S_A= & {} \frac{A_A}{4G}-\frac{\pi }{G} \int m_{g}^2 \frac{\dot{H}}{H^2}\frac{H}{H_c}\left[ -3\beta +2\gamma \frac{H}{H_c}-3\delta \frac{H^2}{H_{c}^{2}}\right] {\textit{HR}}_{A}^{4}dt \\= & {} \frac{A_A}{4G}-\frac{\pi m_{g}^2}{{\textit{GH}}_c} \int \left[ -3\beta +2\gamma \frac{H}{H_c}-3\delta \frac{H^2}{H_{c}^{2}}\right] \frac{\textit{dH}}{H^4} \\= & {} \frac{A_A}{4G}-\frac{\pi m_{g}^2}{{\textit{GH}}_c} \left[ \beta R_{A}^{3} -\frac{\gamma }{H_c}R_{A}^{2}+\frac{3\delta }{H_{c}^{2}}R_{A}\right] , \end{aligned}$$

which is Eq. (58).

Using Eq. (55) in Eq. (31), the entropy of the event horizon takes the form

$$\begin{aligned} S_E= & {} \frac{A_E}{4G}-\frac{\pi }{2G} \int \left( \frac{R_{A}^{2}R_E}{1-\epsilon }\right) \left( \frac{{\textit{HR}}_E+1}{{\textit{HR}}_E-1}\right) \\&\times \left[ m_{g}^2 \frac{\dot{H}}{H^2}\frac{H}{H_c} \left( -3\beta +2\gamma \frac{H}{H_c}-3\delta \frac{H^2}{H_{c}^{2}} \right) \right] {\textit{dR}}_E \\= & {} \frac{A_E}{4G}-\frac{\pi m_{g}^2}{2GH_c} \int R_E \left( \frac{{\textit{HR}}_E+1}{1-\epsilon } \right) \left[ -3\beta +2\gamma \frac{H}{H_c}-3\delta \frac{H^2}{H_{c}^{2}}\right] \frac{{\textit{dH}}}{H^3} \\= & {} \frac{A_E}{4G}-\frac{\pi m_{g}^2}{2GH_c} \int \left[ \frac{2R_E({\textit{HR}}_E+1)}{1-\dot{R_A}} \left\{ \frac{-3\beta }{H^3}+\frac{2\gamma }{H^2 H_c}-\frac{3\delta }{{\textit{HH}}_{c}^{2}}\right\} \right] {\textit{dH}}, \end{aligned}$$

which is Eq. (59).

Appendix 2

From Eq. (3),

$$\begin{aligned} T_{f} {\textit{dS}}_{fh}= & {} dE_f+pdV_h \\ or,\quad T_{f} \frac{{\textit{dS}}_{fh}}{dt}= & {} \dot{\rho }_m V_h+(\rho _m+p_m)\frac{{\textit{dV}}_h}{dt} \\ or,\quad T_{f} \dot{S}_{fh}= & {} -3H(\rho _m +p_m) \cdot \frac{4}{3}\pi R_{h}^{3}+4\pi (\rho _m+p_m)R_{h}^{2}\dot{R}_h \\ or,\quad \dot{S}_{fh}= & {} \frac{4\pi R_{h}^{2}}{T_f} (\rho _m+p_m)(\dot{R}_h-{\textit{HR}}_h), \end{aligned}$$

which is Eq. (60).