Evolution of axially and reflection symmetric source in energy–momentum squared gravity

Abstract

This paper aims to investigate the effects of $f(R, T^2)$ gravity on the dynamical evolution of axially and reflection symmetric anisotropic and dissipative fluid, where R is the scalar curvature and $T^2 =T_{\omega \lambda }T^{\omega \lambda }$, with $T_{\omega \lambda }$ being the stress–energy tensor. With a systematic structure, we investigate modified gravitational field and dynamical equations. In addition, we use the heat transport equation to analyze the thermodynamics of our problem. The Weyl tensor is utilized to evaluate a generalized version of scalar variables from the Riemann tensor. We investigate the influence of $f(R, T^2)$ corrections on the dynamics of the gravitating source. We conclude that modified scalar variables bearing the effects of electric and magnetic components of the Weyl tensor play a vital role in the evolution of compact objects. We analyze that in geodesic and vorticity-free evolution, $\varepsilon _{KL}=\frac{\kappa '}{2}\Pi _{KL}$, where $\kappa '=\frac{\kappa }{f_R}(1-2\mu f_{T^2})$. This reveals that the physics is the same but the couplings of both theories (GR and $f(R, T^2)$) are different.

Appendix

The quantities appearing in Eqs. (61) and (62) are

$$\begin{aligned} \chi _7&=\frac{\chi _0}{f_R(r^2A^2B^2+G^2)}\left[ \Big (-2r^2B^2\Big )\frac{{\dot{A}}}{A}\right. \\&\quad +\frac{r^2A^2B^2+G^2}{A^2} \bigg (\frac{{\dot{3B}}}{B}-\frac{{\dot{C}}}{C}-\frac{\dot{f_R}}{f_R}\bigg )-2\frac{G^2}{A^2}\frac{{\dot{G}}}{G} -G\frac{A_{,\theta }}{A} \\&\quad +\left. \frac{r^2A^2B^2+G^2}{r^2A^2B^2}\bigg (\frac{G_{,\theta }}{G} -\frac{C_{,\theta }}{C}\bigg )G\right] -\frac{1}{f_R}\left[ \frac{\dot{\chi _0}}{A^2}+\frac{G}{r^2A^2B^2}\chi _{0,\theta }\right] \\&\quad +\frac{1}{B^2f_R}\left[ \chi '_1 +\chi _1\bigg \{\frac{f_R'}{R}+\frac{A'}{A}+\frac{(Br)'}{Br}\right. \\&\quad \left. +\frac{C'}{C}\bigg \}-\frac{{\dot{B}}}{B}\chi _4\right] +\frac{1}{rBf_R}\chi _2\bigg [2\bigg (\frac{A_{,\theta }}{A}+\frac{B_{,\theta }}{B}\bigg ) +\frac{C_{,\theta }}{C}\\&\quad -\frac{Gr^2B^2}{r^2A^2B^2+G^2}\bigg \{\frac{{\dot{A}}}{A}+\frac{{\dot{B}}}{B}-\frac{{\dot{G}}}{G} +\frac{G^2}{r^2B^2}\bigg (\frac{G_{,\theta }}{G}\bigg )\bigg \}\bigg ] \\&\quad +\frac{A}{r^2Bf_R}\left[ \chi _{2,\theta }+\frac{G}{B^2}\frac{A'}{A}\chi _3 +\frac{G^2}{r^2A^2B^2+G^2} \bigg \{\frac{{\dot{A}}}{A}-\frac{{\dot{G}}}{G}\right. \\&\quad \left. -\frac{A^2}{G}\frac{A_{,\theta }}{A}\bigg \} \chi _5\right] +\mu \frac{\dot{f_R}}{f_R^2}-\frac{1}{f_R}({\dot{R}}f_R-R\dot{f_R}) \\&\quad -\frac{A}{B}q_I\mu f'_{T^2} -\frac{A^2}{\sqrt{r^2A^2B^2+G^2}}\left( q_{II}\mu +7(-\mu +3P)^2\right) f_{T^2}\\&\quad +\frac{G(1-2\mu f_{T^2})}{f_R\sqrt{r^2A^2B^2+G^2}}\Pi _{KL} \left[ \frac{A}{B}\bigg (\frac{A'}{A}-\frac{G'}{2G}\bigg )\right] ,\\ \chi _8&=\frac{B}{A}q_I\mu \bigg \{\dot{f_{T^2}}-(1-2\mu f_{T^2})\frac{\dot{f_R}}{f_R}\bigg \}+\mu \dot{f_{T^2}}\bigg (P+\frac{\Pi _I}{3}\bigg )\\&\quad +\frac{\sqrt{r^2A^2B^2+G^2}}{r^2B^2}f_{T,\theta }\Pi _{KL}+\frac{1}{B}\bigg (\chi '_4 -\frac{f'_R}{f_R}\chi _4\bigg ) \\&\quad +\frac{r^2A^2B^2}{r^2A^2B^2+G^2}\left[ \chi _0\bigg (\frac{A'}{A} +\frac{G^2}{2r^2B^2}\frac{G'}{G}\bigg )\right. \\&\quad \left. +\frac{\chi _5}{C^2}\bigg (\frac{A'}{A}+\frac{(Br)'}{Br}\bigg )\right] +\frac{\chi _2}{r}\frac{r^2A^2B^2}{r^2A^2B^2+G^2}\left[ \frac{(Br)'}{Br}\right. \\&\quad +\left. \frac{G^2}{2r^2B^2}\frac{G'}{G}\right] +\frac{1}{r^2B^2}\mu f_{{T^2},\theta }\Pi _{KL}\sqrt{r^2A^2B^2+G^2}\\&\quad -\frac{A}{B}q_I\mu f'_{T^2}-7\frac{A}{\sqrt{r^2A^2B^2+G^2}}\mu f_{T^2},\\ \chi _9&=\frac{\mu }{r^2B^2}f'_{T^2}-\frac{f'_R}{f_R}(1-2\mu f_{T^2})\bigg (P+\frac{\Pi _I}{3}\bigg )\\&\quad +\frac{\mu }{A^3r^3rB^2}f_{T^2}+\frac{1}{r^2B^2} \frac{f_{R,\theta }}{f_R}\left\{ \frac{G}{r^2}q_I\mu f_{T^2}-\frac{(1-2\mu f_{T^2})}{\sqrt{r^2A^2B^2+G^2}}\right. \\&\quad \times \Pi _{KL}\bigg \}+ \frac{r^2A^2B^2}{r^2A^2B^2+G^2}\left[ \chi _1\bigg (\frac{{\dot{A}}}{A}+\frac{{\dot{2B}}}{B} +\frac{G^2}{2r^2B^2}\frac{{\dot{G}}}{G}\right. \\&\quad \left. +\frac{1}{rB}\bigg (\frac{A_{,\theta }}{A}-\frac{B_{,\theta }}{2B}\bigg ) +\frac{G^2}{r^2B^2C}\frac{G_{,\theta }}{G}\bigg )\right] +\frac{1}{r^2B}\chi _3\\&\left. \left[ \frac{B_{,\theta }}{B} +\frac{r^2A^2B^2}{r^2A^2B^2+G^2}\bigg \{ \frac{A_{,\theta }}{A}+\frac{G^2}{r^2B^2}\frac{G_{,\theta }}{G} +\frac{C_{,\theta }}{C}\right\} \right] \\&\quad +7\frac{A}{\sqrt{r^2A^2B^2+G^2}}(-\mu +3P)^2 f_{T^2}-\frac{1}{rB^2}q_{II,\theta }\mu f_{T^2}. \end{aligned}$$

The extra terms $\zeta _i's$ appearing in Eqs. (42), (44), (45) and (48) are

$$\begin{aligned} \zeta _1&=\frac{\kappa }{8f_R}\epsilon _{\omega }^{\epsilon \varrho } \left[ (\nabla ^{\pi }\nabla _{\epsilon }f_R) \epsilon _{\lambda \pi \varrho }-(\nabla ^{\pi }\nabla _{\varrho }f_R)\epsilon _{\lambda \pi \epsilon }\right. \\&\quad \left. -(\nabla ^{\theta }\nabla _{\epsilon }f_R)\epsilon _{\theta \lambda \varrho } +(\nabla ^{\theta }\nabla _{\varrho }f_R)\epsilon _{\theta \lambda \epsilon }\right] ,\\ \zeta _2&=\nabla _{\omega }\nabla _{\lambda }f_R-\frac{3}{2}V_{\omega }V_{\lambda } (f-Rf_R)-\nabla _{\omega }\nabla _{\varrho }f_RV^{\varrho }V_{\lambda }\\&\quad +2V_{\omega }V_{\lambda }\Box f_R-(\nabla ^{\theta }\nabla _{\lambda }f_R)V_{\omega }V^{\theta } +g_{\omega \lambda } \\&\quad -(\nabla ^{\theta }\nabla _{\varrho }f_R)V_{\theta }V^{\varrho },\\ \zeta _3&=\nabla ^{\omega }\nabla _{\omega }f_R+\frac{3}{2}(f-Rf_R)\\&\quad -(\nabla ^{\omega }\nabla _{\varrho }f_R)V^{\varrho }V_{\omega } -2\Box f_R-(\nabla ^{\theta }\nabla _{\omega }f_R)V^{\omega }V_{\theta } +4(\nabla ^{\theta }\nabla _{\varrho }f_R)V_{\theta }V^{\varrho },\\ \zeta _4&=(\nabla ^{\varrho }\nabla _{\gamma }f_R)V^{\gamma }\epsilon _{\omega \varrho \lambda }. \end{aligned}$$

The effective terms of matter contents emerging in modified field as well as scalar equations are

$$\begin{aligned}&\mu ^{\text {eff}}=\frac{1}{f_R}\left[ \mu -\frac{1}{2}(f-Rf_R) -f_{T^2}\left( 2\mu ^2-7(-\mu +3P)^2\right) +\frac{\chi _0}{A^2}\right] ,\\&\Pi ^{\text {eff}}_{KL}=\frac{1}{f_R}(1-2\mu f_{T^2})\Pi _{KL}+\frac{1}{ABf_R\sqrt{r^2A^2B^2+G^2}}\left[ A^2\chi _3-G\chi _1\right] ,\\&\left( P+\frac{\Pi _I}{3}\right) ^{\text {eff}}=\frac{1}{f_R}\left[ (1-2\mu f_{T^2})\left( P+\frac{\Pi _I}{3}\right) -7f_{T^2}(-\mu +3P)^2\right] ,\\&\left( P+\frac{\Pi _{II}}{3}\right) ^{\text {eff}}\\&\quad =\frac{1}{f_R}\left[ \frac{G^2}{r^2A^2B^2+G^2}\left\{ -\frac{1}{2}(f-Rf_R)+7f_{T^2}(-\mu +3P)^2\right\} +(1-2\mu f_{T^2})\left( P+\frac{\Pi _{II}}{3}\right) \right] \\&\qquad -\frac{1}{f_R(r^2A^2B^2+G^2)}\left[ -3\frac{G^2}{A^2}\chi _0-2G\chi _2+A^2\chi _5\right] \end{aligned}$$