Rotating and expanding Bianchi type-IX model in $f(R,T)$ theory of gravity

Abstract

The spatially homogeneous shear-free, rotating and expanding Bianchi type-IX universe has been considered in the presence of perfect fluid in $f(R,T)$ theory of gravity. The exact solution of the field equations has been obtained and the functional form of $f(R,T)=R+2f(T)$ gravity has been reconstructed. The existence of such a solution suggests that the general relativistic shear-free perfect fluid conjecture which claims that a shear-free perfect fluid cannot rotate and expand at the same time, is not valid in this modified theory.

Appendices

Appendix A: Evolution and constraint equations of $f(R,T)$ gravity

1.1 A.1 Evolution equations

$$\begin{aligned} &{ \begin{aligned}[b] e_{0}\theta - e_{\alpha} \dot{u}^{\alpha} &= - \frac{1}{3}\theta^{2} + (\dot{u}_{\alpha} - 2a_{\alpha} )\dot{u}^{\alpha} - 2\sigma^{2} + 2 \omega^{2} \\ &\quad {}- \frac{1}{2}\bigl(\mu^{t} + 3p^{t}\bigr), \end{aligned}} \end{aligned}$$

(38)

$$\begin{aligned} &{ \begin{aligned}[b] e_{0}\omega^{\alpha} - \frac{1}{2} \varepsilon^{\alpha \beta \gamma} e_{\beta} \dot{u}_{\gamma} &= \varepsilon^{\alpha \beta \gamma} \biggl(\varOmega_{\beta} \omega_{\gamma} - \frac{1}{2}a_{\beta} \dot{u}_{\gamma} \biggr) - \frac{1}{2}n^{\alpha}_{\beta} \dot{u}^{\beta} \\ &\quad {}- \frac{2}{3}\theta \omega^{\alpha} + \sigma^{\alpha}_{\beta} \omega^{\beta}, \end{aligned}} \end{aligned}$$

(39)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{0}\sigma_{\alpha \beta} - e_{ \langle \alpha}\dot{u}_{\beta \rangle} \\ &\quad {}= \varepsilon_{\gamma \delta \langle \alpha }\bigl(2 \varOmega^{\gamma} {\sigma_{\beta \rangle}}^{\delta} - {n_{\beta \rangle}}^{\gamma} \dot{u}^{\delta} \bigr) + a_{ \langle \alpha } \dot{u}_{\beta \rangle} + \dot{u}_{ \langle \alpha }\dot{u}_{\beta \rangle} \\ &\qquad {}- \frac{2}{3}\theta \sigma_{\alpha \beta} - \sigma_{\gamma \langle \alpha }{\sigma_{\beta \rangle}}^{\gamma} - \omega_{ \langle \alpha } \omega_{\beta \rangle} \\ &\qquad {}- \biggl(E_{\alpha \beta} - \frac{1}{2} \pi_{\alpha \beta}^{t}\biggr), \end{aligned}} \end{aligned}$$

(40)

$$\begin{aligned} &{e_{0}\biggl(E_{\alpha \beta} + \frac{1}{2}\pi_{\alpha\beta}^{t}\biggr) - \varepsilon^{\gamma \delta}{}_{\langle \alpha}e_{\vert \gamma \vert }H_{\beta \rangle \delta} + \frac{1}{2}e_{\langle \alpha}q_{\beta\rangle}^{t}} \\ &{\quad {}= - 3n_{\gamma\langle \alpha }{H_{\beta \rangle}}^{\gamma} + \frac{1}{2}n^{\gamma}{}_{\gamma} H_{\alpha\beta} -\frac{1}{2}(a_{ \langle \alpha } + 2\dot{u}_{ \langle\alpha })q_{\beta \rangle}^{t}} \\ &{\qquad {} - \frac{1}{2}\bigl(\mu^{t} + p^{t}\bigr) \sigma_{\alpha \beta} - \theta \biggl(E_{\alpha \beta} + \frac{1}{6} \pi_{\alpha \beta}^{t}\biggr)} \\ &{\qquad {} + 3\sigma^{\gamma}{}_{ \langle \alpha} \biggl(E_{\beta \rangle \gamma} - \frac{1}{6}\pi_{\beta \rangle \gamma}^{t} \biggr)} \\ &{\qquad {}+ \varepsilon^{\gamma \delta}{}_{ \langle \alpha}\biggl[(2 \varOmega_{\gamma} + \omega_{\gamma} ) \biggl(E_{\beta \rangle \delta}+ \frac{1}{2}\pi_{\beta \rangle \delta}^{t}\biggr)} \\ &{\qquad {} + \frac{1}{2}n_{\beta \rangle \gamma} q^{t}_{\delta} + (2\dot{u}_{\gamma} - a_{\gamma} )H_{\beta \rangle \delta} \biggr],} \end{aligned}$$

(41)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{0}H_{\alpha \beta} + {\varepsilon^{\gamma \delta}}_{ \langle \alpha }e_{\vert \gamma \vert }\biggl(E_{ \beta \rangle \delta} - \frac{1}{2}\pi_{\beta \rangle \delta}^{t}\biggr)\\ &\quad {}= {3n^{\gamma}}_{ \langle \alpha } \biggl(E_{\beta \rangle \gamma} - \frac{1}{2}\pi_{\beta \rangle \gamma}^{t} \biggr)- \frac{1}{2}{n^{\gamma}}_{\gamma} \biggl(E_{\alpha \beta} - \frac{1}{2}\pi_{\alpha \beta}^{t} \biggr) \\ &\qquad {} - \theta H_{\alpha \beta} + {3\sigma^{\gamma}}_{ \langle \alpha }H_{\beta \rangle \gamma} + \frac{3}{2}\omega_{ \langle \alpha }q_{\beta \rangle}^{t} \\ &\qquad {}+ {\varepsilon^{\gamma \delta}}_{ \langle \alpha}\biggl[(2 \varOmega_{\gamma} + \omega_{\gamma} )H_{\beta \rangle \delta} + a_{\gamma} \biggl(E_{\beta \rangle \delta} - \frac{1}{2}\pi_{\beta \rangle \delta}^{t} \biggr) \\ &\qquad {}+ \frac{1}{2}\sigma_{\beta \rangle \gamma} q_{\delta}^{t} - 2\dot{u}_{\gamma} E_{\beta \rangle \delta} \biggr], \end{aligned}} \end{aligned}$$

(42)

$$\begin{aligned} &{e_{0}\mu^{t} + \delta^{\alpha \beta} e_{\beta} q_{\alpha}^{t} + \bigl(\mu^{t} + p^{t}\bigr)\theta + \sigma^{\alpha \beta} \pi_{\alpha \beta}^{t}} \\ &{\quad {}+ 2(\dot{u}_{\alpha} - a_{\alpha} )q_{\alpha}^{t} = 0,} \end{aligned}$$

(43)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{0}q_{\alpha}^{t} + e_{\alpha} p^{t} + \delta^{\beta \gamma} e_{\gamma} \pi_{\alpha \beta}^{t} + \bigl(\mu^{t} + p^{t} \bigr)\dot{u}_{\alpha} + \frac{4}{3}\theta q_{\alpha}^{t} \\ &\quad {}+ \sigma^{\alpha \beta} q_{\beta}^{t} + \bigl(\dot{u}^{\beta} - 3a^{\beta} \bigr) \pi_{\alpha \beta}^{t} \\ &\quad {}- {\varepsilon_{\alpha}}^{\beta \gamma} \bigl[(\varOmega_{\beta} - \omega_{\beta} )q_{\gamma}^{t} + n_{\beta}^{\delta} \pi_{\delta \gamma}^{t}\bigr] = 0, \end{aligned}} \end{aligned}$$

(44)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{0}a_{\alpha} + \frac{1}{2}{\varepsilon_{\alpha \beta}}^{\gamma} (e_{\gamma} + \dot{u}_{\gamma} - 2a_{\gamma} )\varOmega^{\beta} + \frac{1}{2}\varepsilon_{\alpha \beta \gamma} \bigl(\dot{u}^{\beta} + a^{\beta} \bigr)\omega^{\gamma} \\ &\quad {}- \frac{1}{2}{\sigma_{\alpha}}^{\beta} ( \dot{u}_{\beta} + a_{\beta} ) + \frac{1}{3}( \dot{u}_{\alpha} + a_{\alpha} )\theta - \frac{1}{2}n_{\alpha \beta} \omega^{\beta}\\ &\quad {} + \frac{1}{2}\varepsilon_{\alpha \beta \lambda} {\sigma^{\beta}}_{\gamma} n^{\gamma \lambda} = 0, \end{aligned}} \end{aligned}$$

(45)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{0}n^{\alpha \beta} + \frac{1}{3}n^{\alpha \beta} \theta + \delta^{\alpha \beta} \bigl[(e_{\gamma} + \dot{u}_{\gamma} ) \bigl(\varOmega^{\gamma} + \omega^{\gamma} \bigr)\bigr] \\ &\quad {}- \delta^{\gamma ( \beta }\bigl[(e_{\gamma} + \dot{u}_{\gamma} ) \bigl(\varOmega^{\alpha )} + \omega^{\alpha )} \bigr)\bigr] + \varepsilon^{\gamma \delta ( \alpha }(e_{\gamma} + \dot{u}_{\gamma} ){\sigma^{\beta )}}_{\delta} \\ &\quad {}- {2\sigma^{ ( \alpha }}_{\gamma} n^{\beta )\gamma} + 2\varepsilon^{\gamma \lambda ( \alpha }{n^{\beta )}}_{\gamma} ( \varOmega_{\lambda} + \omega_{\lambda} ) = 0. \end{aligned}} \end{aligned}$$

(46)

1.2 A.2 Constraint equations

$$\begin{aligned} &{ \begin{aligned}[b] &e_{\beta} {\sigma_{\alpha}}^{\beta} - \frac{2}{3}e_{\alpha} \theta + {\varepsilon_{\alpha}}^{\beta \gamma} e_{\beta} \omega_{\gamma} - 3a_{\beta} {\sigma_{\alpha}}^{\beta} - n_{\alpha \beta} \omega^{\beta} \\ &\quad {}- {\varepsilon_{\alpha}}^{\beta \gamma} \bigl[n_{\beta \delta} {\sigma^{\delta}}_{\gamma} + (a_{\beta} - 2\dot{u}_{\beta} )\omega_{\gamma} \bigr] + q_{\alpha}^{t} = 0, \end{aligned}} \end{aligned}$$

(47)

$$\begin{aligned} &{e_{\alpha} \omega^{\alpha} - ( 2a_{\alpha} + \dot{u}_{\alpha} )\omega^{\alpha} = 0,} \end{aligned}$$

(48)

$$\begin{aligned} &{ \begin{aligned}[b] &H_{\alpha \beta} + e_{ \langle \alpha } \omega_{ \beta \rangle} - {\varepsilon^{\gamma \delta}}_{ \langle \alpha }e_{\vert \gamma \vert } \sigma_{ \beta \rangle \delta} + (2\dot{u}_{ \langle \alpha } + a_{ \langle \alpha }) \omega_{ \beta \rangle} \\ &\quad {}+ 3n_{\gamma \langle \alpha }{\sigma_{ \beta \rangle}}^{\gamma} - \frac{1}{2}{n^{\gamma}}_{\gamma} \sigma_{\alpha \beta} \\ &\quad {} - {\varepsilon^{\gamma \delta}}_{ \langle \alpha }(n_{ \beta \rangle \gamma} \omega_{\delta} - a_{\gamma} \sigma_{ \beta \rangle \delta} )= 0, \end{aligned}} \end{aligned}$$

(49)

$$\begin{aligned} &{\delta^{\beta \gamma} e_{\gamma} \biggl(E_{\alpha \beta} + \frac{1}{2}\pi_{\alpha \beta}^{t} \biggr) - \frac{1}{3}e_{\alpha} \mu^{t} - 3a^{\beta} \biggl(E_{\alpha \beta} + \frac{1}{2} \pi_{\alpha \beta}^{t}\biggr)} \\ &{\quad {} + \frac{1}{3}\theta q_{\alpha}^{t} - \frac{1}{2}\sigma_{\alpha}^{\beta} q_{\beta}^{t}- 3\omega^{\beta} H_{\alpha \beta}- {\varepsilon_{\alpha}}^{\beta \gamma} \biggl[\sigma_{\beta}^{\delta} H_{\delta \gamma} - \frac{3}{2} \omega_{\beta} q_{\gamma}^{t}} \\ &{\quad {} + {n_{\beta}}^{\delta} \biggl(E_{\delta \gamma} + \frac{1}{2}\pi_{\delta \gamma}^{t} \biggr)\biggr] = 0,} \end{aligned}$$

(50)

$$\begin{aligned} &{ \begin{aligned}[b] &\delta^{\beta \gamma} e_{\gamma} H_{\alpha \beta} + \frac{1}{2}{\varepsilon_{\alpha}}^{\beta \gamma} e_{\beta} q_{\gamma}^{t} - 3a_{\beta} {H_{\alpha}}^{\beta} + \bigl(\mu^{t} + p^{t} \bigr)\omega_{\alpha} \\ &\quad {}- \frac{1}{2}{n_{\alpha}}^{\beta} q_{\beta}^{t} + 3\omega^{\beta} \biggl(E_{\alpha \beta} - \frac{1}{6} \pi_{\alpha \beta}^{t}\biggr) \\ &\quad {}+ {\varepsilon_{\alpha}}^{\beta \gamma} \biggl[{\sigma_{\beta}}^{\delta} \biggl(E_{\delta \gamma} + \frac{1}{2}\pi_{\delta \gamma}^{t}\biggr) - \frac{1}{2}a_{\beta} q_{\gamma}^{t} - {n_{\beta}}^{\delta} H_{\delta \gamma} \biggr] = 0, \end{aligned}} \\ \end{aligned}$$

(51)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{\beta} n^{\alpha \beta} + \varepsilon^{\alpha \beta \gamma} e_{\beta} a_{\gamma} - 2\varepsilon^{\alpha}{}_{\beta\gamma} \omega^{\beta} \varOmega^{\gamma} + 2\sigma^{\alpha}{}_{\beta} \omega^{\beta} \\ &\quad {}+ \frac{2}{3}\theta \omega^{\alpha} - 2n^{\alpha \beta} a_{\beta} = 0, \end{aligned}} \end{aligned}$$

(52)

$$\begin{aligned} &{ \begin{aligned}[b] &e_{ \langle \alpha }a_{ \beta \rangle} + b_{ \langle \alpha \beta \rangle} + (e_{\gamma} - 2a_{\gamma} ){n^{\lambda}}_{ \langle \alpha } {\varepsilon^{\gamma}}_{ \beta \rangle \lambda} \\ &\quad {}+ \frac{1}{3}\theta \sigma_{\alpha \beta} - {\sigma^{\gamma}}_{ \langle \alpha }\sigma_{ \beta \rangle \gamma} + 2\omega_{ \langle \alpha } \varOmega_{ \beta \rangle} - \omega_{ \langle \alpha }\omega_{ \beta \rangle} \\ &\quad {}- \biggl(E_{\alpha \beta} + \frac{1}{2}\pi_{\alpha \beta}^{t} \biggr) = 0, \end{aligned}} \end{aligned}$$

(53)

$$\begin{aligned} &{ \begin{aligned}[b] &4e_{\alpha} a^{\alpha} - 6a_{\alpha} a^{\alpha} - n^{\alpha \gamma} n_{\alpha \gamma} + \frac{1}{2}{n^{\gamma}}_{\gamma} {n^{\alpha}}_{\alpha} - 4\omega^{\lambda} \varOmega_{\lambda} - 2\mu^{t}\\ &\quad {} + \frac{2}{3}\theta^{2} + 2 \sigma^{2} + 2\omega^{2} = 0, \end{aligned}} \end{aligned}$$

(54)

where $E_{ab}$ and $H_{ab}$ are the electric ($E_{ab} = C_{aebd}u^{e}u^{d}$) and the magnetic ($H_{ab} = (1/2){\varepsilon_{a}}^{cd}C_{cdbe}u^{e}$) parts of the conformal Weyl curvature tensor $C_{abcd}$.

Appendix B: Dynamic quantities of the rotating Bianchi type-IX model

$$\begin{aligned} &{ \begin{aligned}[b] \mu^{t} &= - \frac{v_{1}^{2}}{k_{1}^{2}}\biggl(2\frac{\ddot{a}}{a} + \frac{\dot{a}^{2}}{a^{2}}\biggr) + 3\frac{\dot{a}^{2}}{a^{2}} - \frac{1}{4}\biggl[ \frac{k_{1}^{2} - 3v_{1}^{2}}{k_{2}^{2}k_{3}^{2}}+ \frac{k_{2}^{2}}{k_{1}^{2}k_{3}^{2}}\\ &\quad {} + \frac{k_{3}^{2}}{k_{1}^{2}k_{2}^{2}} - \frac{2}{k_{1}^{2}} - \frac{2}{k_{2}^{2}} - \frac{2}{k_{3}^{2}})\biggr]\frac{1}{a^{2}}, \end{aligned}} \end{aligned}$$

(55)

$$\begin{aligned} &{ \begin{aligned}[b] p^{t} &= \frac{1}{3}\frac{v_{1}^{2}}{k_{1}^{2}}\biggl(4 \frac{\ddot{a}}{a} + 5\frac{\dot{a}^{2}}{a^{2}}\biggr) - 2\frac{\ddot{a}}{a} - \frac{\dot{a}^{2}}{a^{2}} + \frac{1}{12}\biggl[\frac{k_{1}^{2} + v_{1}^{2}}{k_{2}^{2}k_{3}^{2}} + \frac{k_{2}^{2}}{k_{1}^{2}k_{3}^{2}} \\ &\quad {}+ \frac{k_{3}^{2}}{k_{1}^{2}k_{2}^{2}} - \frac{2}{k_{1}^{2}} - \frac{2}{k_{2}^{2}} - \frac{2}{k_{3}^{2}})\biggr]\frac{1}{a^{2}}, \end{aligned}} \end{aligned}$$

(56)

$$\begin{aligned} &{q_{1}^{t} = 2\frac{v_{1}}{k_{1}}\biggl( \frac{\ddot{a}}{a} - \frac{\dot{a}^{2}}{a^{2}}\biggr) - \frac{k_{1}v_{1}}{2k_{2}^{2}k_{3}^{2}} \frac{1}{a^{2}},} \end{aligned}$$

(57)

$$\begin{aligned} &{ \begin{aligned}[b] \pi_{11}^{t} &= - \frac{4}{3} \frac{v_{1}^{2}}{k_{1}^{2}}\biggl(\frac{\ddot{a}}{a} - \frac{\dot{a}^{2}}{a^{2}}\biggr) + \frac{1}{3}\biggl(\frac{2k_{1}^{2} - v_{1}^{2}}{k_{2}^{2}k_{3}^{2}} - \frac{k_{2}^{2}}{k_{3}^{2}k_{1}^{2}} -\frac{k_{3}^{2}}{k_{1}^{2}k_{2}^{2}}\\ &\quad {} + \frac{2}{k_{1}^{2}} - \frac{1}{k_{2}^{2}} - \frac{1}{k_{3}^{2}} \biggr)\frac{1}{a^{2}}, \end{aligned}} \end{aligned}$$

(58)

$$\begin{aligned} &{ \begin{aligned}[b] \pi_{22}^{t} &= \frac{2}{3} \frac{v_{1}^{2}}{k_{1}^{2}}\biggl(\frac{\ddot{a}}{a} - \frac{\dot{a}^{2}}{a^{2}}\biggr) + \frac{1}{3}\biggl(\frac{2k_{2}^{2}}{k_{3}^{2}k_{1}^{2}} - \frac{k_{3}^{2}}{k_{1}^{2}k_{2}^{2}} - \frac{2k_{1}^{2} - v_{1}^{2}}{2k_{2}^{2}k_{3}^{2}} \\ &\quad {}+ \frac{2}{k_{2}^{2}} - \frac{1}{k_{3}^{2}} - \frac{1}{k_{1}^{2}} \biggr)\frac{1}{a^{2}}, \end{aligned}} \end{aligned}$$

(59)

$$\begin{aligned} &{ \begin{aligned}[b] \pi_{33}^{t} &= \frac{2}{3} \frac{v_{1}^{2}}{k_{1}^{2}}\biggl(\frac{\ddot{a}}{a} - \frac{\dot{a}^{2}}{a^{2}}\biggr) + \frac{1}{3}\biggl(\frac{2k_{3}^{2}}{k_{1}^{2}k_{2}^{2}} - \frac{2k_{1}^{2} - v_{1}^{2}}{2k_{2}^{2}k_{3}^{2}} - \frac{k_{2}^{2}}{k_{3}^{2}k_{1}^{2}}\\ &\quad {} + \frac{2}{k_{3}^{2}} - \frac{1}{k_{1}^{2}} - \frac{1}{k_{2}^{2}} \biggr)\frac{1}{a^{2}}, \end{aligned}} \end{aligned}$$

(60)

$$\begin{aligned} &{\pi_{23}^{t} = \frac{v_{1}(k_{2}^{2} - k_{3}^{2})}{k_{2}k_{3}k_{1}^{2}} \frac{\dot{a}}{a^{2}}.} \end{aligned}$$

(61)