On dynamical instability of spherical star in f(R,T) gravity

Abstract

This work is based on stability analysis of spherically symmetric collapsing star surrounding in locally anisotropic environment in f(R,T) gravity, where R is Ricci scalar and T corresponds to the trace of energy momentum tensor. Field equations and dynamical equations are presented in the context of f(R,T) gravity. Perturbation scheme is employed on dynamical equations to find the collapse equation. Furthermore, condition on adiabatic index Γ is constructed for Newtonian and post-Newtonian eras to address instability problem. Some constraints on physical quantities are imposed to maintain stable stellar configuration. The results in this work are in accordance with f(R) gravity for specific case.

Appendix

$$\begin{aligned} Z_1(r,t) =&f_R A^2 \biggl[ \biggl\{ \frac{1}{f_R A^2} \biggl(\frac{f-Rf_R}{2} - \frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{B}}{B}+\frac{2\dot{C}}{C} \biggr) \\ &{} -\frac{f'_R}{B^2} \biggl(\frac{B'}{B} -\frac{2C'}{C} \biggr)+\frac{f''_R}{B^2} \biggr) \biggr\} _{,0}+ \biggl\{ \frac{1}{f_RA^2B^2} \\ &{} \times \biggl(\dot{f'_R}- \frac{A'}{A}\dot{f_R} -\frac{\dot{B}}{B}f_R' \biggr) \biggr\} _{,1} \biggr] -\frac{\dot{f_R}}{A^2} \biggl\{ \biggl( \frac{\dot{B}}{B} \biggr)^2 \\ &{}+2 \biggl(\frac{\dot{C}}{C} \biggr)^2 + \frac{3\dot{A}}{A} \biggl(\frac{\dot{B}}{B}+\frac{2\dot{C}}{C} \biggr) \biggr\} + \frac{\ddot{f_R}}{A^2} \biggl(\frac{\dot{B}}{B}+\frac{2\dot{C}}{C} \biggr) \\ &{}+ \frac{\dot{A}}{A}(f-Rf_R)- \frac{2f'_R}{B^2} \biggl\{ \frac{\dot{A}}{A} \biggl( \frac{B'}{B}-\frac{C'}{C} \biggr) \\ &{} +\frac{\dot{B}}{B} \biggl(\frac{2A'}{A}+\frac{B'}{B}+\frac{C'}{C} \biggr)+ \frac{\dot{C}}{C} \biggl(\frac{A'}{A}-\frac{3C'}{C} \biggr) \biggr\} \\ &{}+\frac{f_R''}{B^2} \biggl(\frac{2\dot{A}}{A}+ \frac{\dot{B}}{B} \biggr) +\frac{1}{B^2} \biggl(\dot{f'_R}- \frac{A'}{A}\dot{f_R} \biggr) \\ &{} \times \biggl(\frac{3A'}{A} + \frac{B'}{B}+\frac{2C'}{C} \biggr), \end{aligned}$$

(A.1)

$$\begin{aligned} Z_2(r,t) =&f_RB^2 \biggl[ \biggl\{ \frac{1}{f_RA^2B^2} \biggl(\dot{f}_R'-\frac{A'}{A} \dot{f_R} -\frac{\dot{B}}{B}f_R' \biggr)\biggr\} _{,0} \\ &{}+ \biggl\{ \frac{1}{f_RB^2} \biggl(\frac{Rf_R-f}{2} -\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{A}}{A}- \frac{2\dot{C}}{C} \biggr) \\ &{}-\frac{f'_R}{B^2} \biggl(\frac{A'}{A} + \frac{2C'}{C} \biggr)+\frac{\ddot{f_R}}{A^2} \biggr) \biggr\} _{,1} \biggr]+(Rf_R-f)\frac{B'}{B} \\ &{}-\frac{\dot{f_R}}{A^2} \biggl\{ \frac{A'}{A} \biggl( \frac{\dot{A}}{A}+\frac{\dot{B}}{B} \biggr) +\frac{B'}{B} \biggl( \frac{\dot{A}}{A}-\frac{2\dot{C}}{C} \biggr) \\ &{}+\frac{2C'}{C} \biggl( \frac{\dot{B}}{B}-\frac{\dot{C}}{C} \biggr) \biggr\} + \biggl(\frac{\dot{A}}{A} +\frac{3\dot{B}}{B} +\frac{2\dot{C}}{C} \biggr) \\ &{}\times \biggl( \dot{f}_R'-\frac{A'}{A}\dot{f_R} - \frac{\dot{B}}{B}f_R' \biggr)\frac{1}{A^2} \\ &{}- \frac{f'_R}{B^2} \biggl\{ \frac{A'}{A} \biggl(\frac{A'}{A}+ \frac{3B'}{B} \biggr) +\frac{2C'}{C} \biggl(\frac{3B'}{B} + \frac{C'}{C} \biggr) \biggr\} \\ &{}+\frac{\ddot{f_R}}{A^2} \biggl(\frac{A'}{A}+ \frac{2B'}{B} \biggr)+\frac{f''_R}{B^2} \biggl(\frac{A'}{A} + \frac{2C'}{C} \biggr). \end{aligned}$$

(A.2)

$$\begin{aligned} Z_{1p} =& 2\alpha A_0^2 \biggl[ \frac{1}{A_0^2B_0^2Y} \biggl\{ e'-e\frac{A'_0}{A_0}- \frac{b}{B_0}R'_0 \biggr\} \biggr]_{,1}+ \frac{1}{Y} \biggl[e \\ &{} -\bigl[\lambda T_0-\alpha R_0^2\bigr] \biggl( \frac{a}{A_0}+\frac{e}{Y} \biggr) +\frac{2\alpha}{B_0^2} \biggl\{ \biggl( \frac{B'_0}{B_0}-\frac{2}{r} \biggr) \\ &{} \times \biggl(e'-2R'_0 \biggl(\frac{a}{A_0} +\frac{b}{B_0} \biggr) +\frac{2 \alpha e}{Y}R'_0 \biggr) \\ &{} + R''_0 \biggl(\frac{2a}{A_0}+ \frac{b}{B_0} \biggr)-2R'_0 \biggl( \frac{b}{B_0} \biggl(\frac{2A'_0}{A_0}+ \frac{B'_0}{B_0}+ \frac{1}{r} \biggr) \\ &{} +\frac{\bar{c}}{r} \biggl( \frac{A'_0}{A_0}-\frac{3}{r} \biggr) \biggr) + \biggl(e'-e \frac{A'_0}{A_0} \biggr) \\ &{}\times \biggl(\frac{3A'_0}{A_0} + \frac{B'_0}{B_0}+ \frac{2}{r} \biggr) \biggr\} \biggr] \end{aligned}$$

(A.3)

$$\begin{aligned} Z_{2p} =&B_0^2Y \biggl[ \frac{1}{B_0^2Y} \biggl\{ e-\frac{2\alpha}{B_0^2} \biggl\{ \biggl( \frac{A'_0}{A_0}+\frac{2}{r} \biggr) \\ & {}\times \biggl(e-\biggl[\frac{2 \alpha e}{Y}+ \frac{4b}{B_0}\biggr]R'_0 \biggr) +R'_0\biggl[ \biggl(\frac{a}{A_0}\biggr)' + \biggl(\frac{\bar{c}}{r} \biggr)'\biggr] \biggr\} \\ & {}-\bigl[\lambda T_0 -\alpha R_0^2\bigr] \biggl(\frac{b}{B_0}+\frac{e}{Y} \biggr) \biggr\} \biggr]_{,1} +bB_0Y \biggl[ \frac{1}{B_0^2Y} \\ & {}\times \biggl\{ \lambda T_0 -\alpha R_0^2 -\frac{4\alpha}{B_0^2} \biggl(\frac{A'_0}{A_0}+ \frac{2}{r} \biggr)R'_0 \biggr\} \biggr]_{,1} \\ & {}+\frac{2\alpha}{B_0^2} \biggl[R''_0 \biggl\{ \biggl(\frac{a}{A_0} \biggr)'-2 \biggl( \frac{A'_0}{A_0} +\frac{2}{r} \biggr) \biggl(\frac{b}{B_0}+ \frac{e}{Y} \biggr) \\ & {}+ \biggl(\frac{\bar{c}}{r} \biggr)' \biggr\} -R'_0 \biggl\{ \frac{A'_0}{A_0} \biggl[2 \biggl(\frac{a}{A_0} \biggr)'+3 \biggl(\frac{b}{B_0} \biggr)' \biggr] \\ & {}+3\frac{B'_0}{B_0} \biggl[ \biggl( \frac{a}{A_0} \biggr)'+2 \biggl(\frac{\bar{c}}{r} \biggr)' \biggr] + \frac{2}{r} \biggl[ \biggl(3 \frac{b}{B_0} \biggr)' \\ &{} +2 \biggl(\frac{\bar{c}}{r} \biggr)' \biggr] \biggr\} + \biggl(\frac{2b}{B_0}R'_0 -e \biggr) \biggl\{ 3\frac{B'_0}{B_0} \biggl(\frac{A'_0}{A_0}+ \frac{2}{r} \biggr) \\ & {}+ \biggl(\frac{A'_0}{A_0} \biggr)^2+\frac{2}{r^2} \biggr\} \biggr]+e \frac{B'_0}{B_0} \\ &{} -\bigl[\lambda T_0 -\alpha R_0^2\bigr] \biggl(\frac{b}{B_0}+\frac{2e}{Y}\frac{B'_0}{B_0} \biggr) \end{aligned}$$

(A.4)

$$\begin{aligned} Z_{3} =&\frac{Y}{B_0^2} \biggl[\frac{a''}{A_0}+ \frac{\bar{c}''}{r}-\frac{A''_0}{A_0} \biggl(\frac{a}{A_0}+ \frac{2b}{B_0} \biggr)+\frac{A'_0}{A_0} \biggr)' \\ & {} \times \biggl\{ \frac{2b}{B_0} \biggl( \frac{B'_0}{B_0} -\frac{1}{r} \biggr)+ \biggl(\frac{\bar{c}}{r}- \biggl(\frac{b}{B_0} \biggr)' \biggr\} +\frac{B'_0}{B_0} \biggl\{ \frac{2bB_0'}{rB_0} \\ & {}- \biggl(\frac{a}{A_0} \biggr)'- \biggl(\frac{\bar{c}}{r} \biggr)' \biggr\} + \frac{1}{r} \biggl\{ \biggl( \frac{a}{A_0} \biggr)'- \biggl(\frac{b}{B_0} \biggr)' \biggr\} \biggr] \\ & {}-\frac{2\alpha e}{Y} \biggl\{ \frac{\lambda T_0 -\alpha R_0^2}{2}-\frac{2\alpha}{B_0^2} \biggl(R'_0 \biggl(\frac{A'_0}{A_0}-\frac{B'_0}{B_0} +\frac{1}{r} \biggr) \\ & {}-R''_0 \biggr)\biggr\} -\frac{2\alpha}{B_0^2} \biggl\{ e''+\frac{2b}{B_0}R''_0 \\ & {} +\biggl(\frac{A'_0}{A_0}-\frac{B'_0}{B_0} +\frac{1}{r} \biggr) \biggl( \frac{2b}{B_0}R'_0-e' \biggr) \biggr\} \end{aligned}$$

(A.5)

$$\begin{aligned} Z_{4} =&-\frac{rA_0^2B_0}{br+2B_0\bar{c}} \biggl[ \frac{e}{2}- \frac{2\bar{c}}{r^3} -\frac{1}{A_0B_0^2} \biggl\{ A_0'' \biggl[\frac{a}{A_0}+\frac{2b}{B_0}\biggr] \\ &{} -\frac{1}{B_0} \biggl(a'B_0'+a''+A_0'b'-A_0'B_0' \biggl[\frac{a}{A_0}+\frac{3b}{B_0}\biggr] \biggr) \\ &{} +\frac{2}{r} \biggl\{ a'+\bar{c}'A_0'-A_0' \biggl[\frac{a}{A_0}+\frac{2b}{B_0}+\frac{\bar{c}}{r}\biggr] \biggr\} +\frac{A_0}{r} \\ &{} \times \biggl\{ \bar{c}''- \frac{b'}{B_0}-\frac{B_0'\bar{c}'}{B_0}+ \frac{3b}{B_0}+\frac{\bar{c}}{r}\biggr\} \\ &{}\times \biggl\{ \bar{c}''- \frac{b'}{B_0}-\frac{B_0'\bar{c}'}{B_0} +\frac{2}{r^2}\biggl[\bar{c}'-\frac{b}{B_0}\frac{\bar{c}}{r}\biggr] \biggr\} \biggr] =0. \end{aligned}$$

(A.6)