Dynamical analysis of self-gravitating stars in f(R,T) gravity

Abstract

We study the factors affecting the stability of a locally isotropic spherical self-gravitating systems in f(R,T) gravity, where R is the Ricci curvature invariant and T is the trace of stress-energy tensor. Specifically, the collapse equation is obtained from conservation laws with non-null expansion scalar at Newtonian and post-Newtonian approximations. Initially, we consider the hydrostatic phase of the system which upon radial perturbation provides linearized field equations. This approach gives rise to specific instability constraints to ensure the collapsing behavior of the spherical isotropic fluid distribution. Finally, we discuss the role played by matter variables in this perspective.

Appendix

The terms D₀ and D₁ mentioned in Eqs. (16) and (17) are given by

$$\begin{aligned} D_0 =&\frac{1}{f_R(R,T)} \biggl(\frac{2r}{H^2}-\frac{A'}{A}-\frac{H'}{H} -\frac{2AA'}{H^2} \biggr) \\ &{}\times\biggl[f_{RRR}\dot{R}R'+2f_{RRT}\bigl(R'\dot{T}+T'\dot{R}\bigr)+f_{RTT}T'\dot{T} \\ &{}+f_{RR} \biggl(\dot{R}'-\frac{A'\dot{R}}{A} -\frac{\dot{H}R'}{H} \biggr) \\ &{}+f_{RT} \biggl(\dot{T'}-\frac{A'\dot{T}}{A}-\frac{\dot{H}T'}{H} \biggr) \biggr] \\ &{}+\biggl[f_{RRR}\dot{R}R' +2f_{RRT}\bigl(R'\dot{T}+T'\dot{R}\bigr) \\ &{}+f_{RTT}T'\dot{T}+f_{RR} \biggl(\dot{R}'-\frac{A'\dot{R}}{A}-\frac{\dot{H}R'}{H} \biggr) \\ &{}+f_{RT} \biggl(\dot{T'}-\frac{A'\dot{T}}{A}-\frac{\dot{H}T'}{H} \biggr) \biggr]_{,1}+ \biggl[\frac{1}{f_R(R,T)} \\ &{}\times \biggl\{ \biggl(\frac{f(R,T)-Rf_R(R,T)}{2}\biggr)+f_{RRR} \frac{R'^2}{H^2} \\ &{}+2f_{RRT}\frac{R'H'}{H^2}+f_{RTT} \\ &{}\times\frac{T'^2}{H^2}+\frac{f_{RR}}{H^2} \biggl(\frac{2R'}{r} -\frac{R'H'}{H}-\frac{H\dot{R}\dot{H}}{A^2}+R''\biggr) \\ &{}+\frac{f_{RT}}{H^2} \biggl(\frac{2T'}{r}-\frac{T'H'}{H}-\frac{H\dot{T}\dot{H}}{A^2}+T''\biggr) \biggr\} \biggr]_{,0}A^2 \\ &{}-\frac{H\dot{H}}{f_R(R,T)}\biggl[f_{RRR} \biggl(\frac{R'^2}{H^2}+\frac{\dot{R}^2}{A^2} \biggr)+2\biggl(\frac{R'T'}{H^2} \\ &{}+\frac{\dot{R}\dot{T}}{A^2} \biggr)f_{RRT}+f_{RTT}\biggl( \frac{T'^2}{H^2}+\frac{\dot{T}^2}{A^2} \biggr) \\ &{}+f_{RR} \biggl(\frac{R''}{H^2} -\frac{R'H'}{H^3}-\frac{\dot{R}\dot{H}}{A^2H} \\ &{}-\frac{R'A'}{AH^2}+\frac{\ddot{R}}{A^2}-\frac{\dot{R}\dot{A}}{A^3} \biggr) \\ &{}+f_{RT} \biggl(\frac{T''}{H^2}+\frac{\ddot{T}}{A^2}-\frac{T'H'}{H^3} -\frac{\dot{T}\dot{H}}{A^2H}-\frac{\dot{T}\dot{A}}{A^3} \\ &{}-\frac{T'A'}{AH^2} \biggr) \biggr], \end{aligned}$$

(A.1)

$$\begin{aligned} D_1 =& \biggl[\frac{1}{f_R(R,T)} \biggl\{ f_{RRR}\dot{R}R'+2f_{RRT}\bigl(R'\dot{T}+T'\dot{R}\bigr) \\ &{}+f_{RTT}T'\dot{T}+f_{RR}\biggl(\dot{R}'-\frac{A'\dot{R}}{A}-\frac{\dot{H}R'}{H} \biggr) \\ &{}+f_{RT} \biggl(\dot{T'}-\frac{A'\dot{T}}{A}-\frac{\dot{H}T'}{H} \biggr) \biggr\} \biggr]_{,0}-\frac{1}{f_R(R,T)} \\ &{}\times \biggl(\frac{\dot{A}}{A}+\frac{\dot{H}}{H}+\frac{2H\dot{H}}{A^2} \biggr) \biggl[f_{RRR}\dot{R}R' \\ &{}+2f_{RRT}\bigl(R'\dot{T} +T'\dot{R}\bigr)+f_{RTT}T'\dot{T} \\ &{}+f_{RR} \biggl(\dot{R}'-\frac{A'\dot{R}}{A}-\frac{\dot{H}R'}{H} \biggr) \\ &{}+f_{RT} \biggl(\dot{T'}-\frac{A'\dot{T}}{A}-\frac{\dot{H}T'}{H} \biggr) \biggr]-\frac{AA'}{f_R(R,T)} \\ &{}\times \biggl[f_{RRR} \biggl(\frac{R'^2}{H^2}+\frac{\dot{R}^2}{A^2} \biggr) +2 \biggl(\frac{R'T'}{H^2}+\frac{\dot{R}\dot{T}}{A^2}\biggr)f_{RRT} \\ &{}+f_{RTT} \biggl( \frac{T'^2}{H^2}+\frac{\dot{T}^2}{A^2} \biggr) \\ &{}+f_{RR} \biggl(\frac{R''}{H^2}-\frac{R'H'}{H^3}-\frac{\dot{R}\dot{H}}{A^2H}+\frac{\ddot{R}}{A^2}-\frac{\dot{R}\dot{A}}{A^3}\biggr) \\ &{}+f_{RT} \biggl(\frac{T''}{H^2}+\frac{\ddot{T}}{A^2}-\frac{T'H'}{H^3} \\ &{}-\frac{\dot{T}\dot{H}}{A^2H}-\frac{\dot{T}\dot{A}}{A^3} -\frac{T'A'}{AH^2} \biggr) \biggr] \\ &{}+\frac{2r}{H^2} \biggl[f_{RRR}\frac{R'^2}{H^2} +2f_{RRT}\frac{R'{T'}}{H^2}+f_{RTT}\frac{T'^2}{H^2} \\ &{}+f_{RR} \biggl(\frac{R''}{H^2}-\frac{R'}{rH^2}-\frac{\dot{R}\dot{H}}{A^2H}-\frac{R'H'}{rH^2} \biggr) \\ &{}+f_{RT}\biggl(-\frac{\dot{T}\dot{H}}{A^2H} -\frac{H'{T'}}{H^3}+\frac{T''}{H^2}-\frac{T'}{rH^2} \biggr) \biggr] \\ &{}+H^2\biggl[\frac{1}{f_R(R,T)} \biggl\{ - \biggl(\frac{f(R,T)-Rf_R(R,T)}{2}\biggr) \\ &{}+f_{RRR}\frac{\dot{R}^2}{A^2}+2f_{RRT}\frac{\dot{R}\dot{T}}{A^2}+f_{RTT}\frac{\dot{T}^2}{A^2} \\ &{}+\frac{f_{RR}}{A^2} \biggl(\ddot{R} -\frac{\dot{R}\dot{A}}{A}-\frac{A{R'A'}}{H^2} -\frac{2A^2R'}{rH^2} \biggr) \\ &{}+\frac{f_{RT}}{A^2} \biggl(\ddot{T}-\frac{\dot{A}\dot{T}}{A}-\frac{AA'T'}{H^2}-\frac{2A^2{T'}}{rH^2} \biggr) \biggr\} \biggr]_{,1}. \end{aligned}$$

(A.2)

Under hydrostatic phase, the component of Eq. (29) is given by

$$\begin{aligned} D_{1S} =&-\frac{2{\epsilon}A_0A'_0}{(1+2{\epsilon}R_0)} \biggl(\frac{R''_0}{H_0^2}-\frac{R'_0H'_0}{H_0^3} \biggr) \\ &{} +\frac{4r{\epsilon}}{(1+2{\epsilon}R_0)} \biggl(\frac{R''_0}{H_0^2}-\frac{R'_0}{rH_0^2}-\frac{R'_0H'_0}{rH_0^2} \biggr) \\ &{}+H_0^2 \biggl[\frac{1}{(1+2{\epsilon}R_0)} \biggl\{ \frac{{\lambda}T_0-{\epsilon}R_0^2}{2}-\frac{2{\epsilon}}{A_0^2} \biggl(\frac{A_0R'_0A'_0}{H_0^2} \\ &{}-\frac{2A_0^2R'_0}{rH_0^2} \biggr) \biggr\} \biggr]_{,1}. \end{aligned}$$

(A.3)

The radial dependent portion $\mathcal{P}_{0}$ is given by

$$\begin{aligned} \mathcal{P}_0 =&\frac{2{\epsilon}}{H_0^2} \biggl(\frac{A'_0}{ A_0}-\frac{1}{r^2} \biggr)+\frac{2(1+2{\epsilon}R_0)}{rH_0^2} \biggl\{ \biggl(\frac{a}{A_0} \biggr)' \\ &{}-\frac{2hA'_0}{H_0A_0} + \frac{2h}{rH_0} \biggr\} -\frac{2{\epsilon}}{r^2}+e{\epsilon}R_0 \\ &{}+\frac{2{\epsilon}}{H_0^2} \biggl[\frac{e'A'_0}{A_0}+R'_0 \biggl\{ \biggl(\frac{a}{A_0} \biggr)'-\frac{2hA'_0}{H_0A_0}\biggr\} \biggr] \\ &{}+\frac{4{\epsilon}R'_0}{rH_0^2} \biggl( \frac{e'}{R'_0}- \frac{2h}{H_0} \biggr)-\frac{2{\epsilon}e\omega^2}{A_0^2}. \end{aligned}$$

(A.4)

The component of Eq. (31) is

$$\begin{aligned} D_2 =&\frac{2\epsilon}{(1+2{\epsilon}R_0)} \biggl(\frac{2r}{H_0^2} -\frac{A'_0}{A_0}-\frac{H'_0}{H_0}-\frac{2A_0A'_0}{H_0^2} \biggr) \\ &{}\times\biggl(e'-\frac{eA'_0}{A_0}-\frac{hR'_0}{H_0} \biggr) \\ &{}+ \biggl[\biggl(e'-\frac{eA'_0}{A_0}-\frac{hR'_0}{H_0} \biggr)\frac{2\epsilon}{(1+2{\epsilon}R_0)} \biggr]' \\ &{}+\frac{A_0^2}{(1+2{\epsilon}R_0)} \biggl[\frac{{\lambda}T_0-{\epsilon}R_0^2}{2} \\ &{}+\frac{2{\epsilon}}{H_0^2} \biggl(\frac{2R'_0}{r}-\frac{R'_0H'_0}{H_0}+R''_0\biggr)-\frac{2{\epsilon}h}{H_0(1+2{\epsilon}R_0)} \\ &{}\times\biggl(R''_0-\frac{R'_0H'_0}{H_0}-\frac{R'_0A'_0}{A_0} \biggr) \biggr]. \end{aligned}$$

(A.5)

The quantity Φ which is common in both second dynamical equation (32) as well as in collapse equation (36) is mentioned as below

$$\begin{aligned} \varPhi =& \biggl[-\frac{2{\epsilon}}{(1+2{\epsilon}R_0)}+\frac{P_0}{(1+2{\epsilon}R_0)^2} \biggl\{ -A_0A'_0(1+2{\epsilon}R_0) \\ &{}\times\biggl(\frac{a}{A_0}+\frac{a'}{A'_0}+\frac{2\epsilon}{(1+2{\epsilon}R_0)} \biggr) +2e'\epsilon \biggr\} \\ &{}-\frac{4\epsilon}{(1+2{\epsilon}R_0)^3} \bigl\{ 2{\epsilon}R'_0-A_0A'_0(1+2{\epsilon}R_0) \bigr\} \\ &{}-\biggl(\frac{a}{A_0}+\frac{a'}{A'_0}-\frac{2{\epsilon}}{(1+2{\epsilon}R_0)} \biggr) \\ &{}+\frac{H_0^2T'_0}{2(1-\lambda)} \biggl(\frac{z'}{T'_0}+\frac{2h}{H_0} \biggr)+\frac{{\epsilon}R_0^2-{\lambda}T_0}{2} \biggr] \\ &{}+ \biggl[\frac{2{\epsilon}A_0A'_0}{H_0^2(1+2{\epsilon}R_0)} \biggl\{ \biggl(R''_0-\frac{R'_0H'_0}{H_0} \biggr) \\ &{}\times \biggl(\frac{a}{A_0}+\frac{a'}{A'_0} -\frac{2{\epsilon}}{(1+2{\epsilon}R_0)} \biggr)+R''_0 \biggl(\frac{e''}{R''_0} -\frac{2h}{H_0}\biggr) \\ &{}-R'_0 \biggl\{ \frac{e'H'_0}{H_0R'_0}+ \biggl(\frac{h}{H_0} \biggr)'-\frac{2hH'_0}{H_0^2} \biggr\} \biggr\} + \frac{4{\epsilon}r}{H_0^2(1+2{\epsilon}R_0)} \\ &{}\times \biggl\{ R''_0\biggl(\frac{e''}{R''_0}-\frac{2h}{H_0} \biggr)-\frac{1}{r}\biggl(e'-\frac{2hR'_0}{H_0} \biggr) \\ &{}+\frac{H_0^2R'_0}{r} \biggl(\frac{h}{H_0} \biggr)'-\frac{hR'_0H'_0}{r} \\ &{}+\frac{e'H'_0}{r}-2\epsilon \biggl(R''_0-\frac{R'_0}{r} -\frac{R'_0H'_0}{r} \biggr) \biggr\} \\ &{}+\frac{2hH_0}{(1+2{\epsilon}R_0)}\biggl\{ \frac{{\epsilon}R_0^2-{\lambda}T_0}{2}-\frac{2\epsilon}{H_0^2}\biggl(\frac{A'_0R'_0}{A_0}+\frac{2R'_0}{r}\biggr) \biggr\} \\ &{}+\frac{2{\epsilon}H_0^2}{(1+2{\epsilon}R_0)} \biggl\{ \frac{eR_0}{2}-\frac{R'_0}{H_0^2} \biggl\{ \biggl(\frac{a}{A_0} \biggr)' -\frac{2hA'_0}{A_0H_0} \biggr\} \\ &{}-\frac{1}{H_0^2} \biggl(\frac{e'A'_0}{A_0} -\frac{4hR'_0}{rH_0}+\frac{2e'}{r} \biggr) \biggr\} \\ &{}+\frac{2{\epsilon}}{H_0^2}\biggl(\frac{R_0A'_0}{A_0} +\frac{2R'_0}{r} \biggr) \biggr]. \end{aligned}$$

(A.6)

The entity ω² described in Eq. (30) is as follows

$$\begin{aligned} \omega^2(r) =&\frac{A_0^2H_0}{h} \biggl[e+\frac{4h}{H_0^3} \biggl\{ \frac{A'_0}{A_0} \biggl(\frac{2}{r}-\frac{H'_0}{H_0} \biggr)-\frac{2H'_0}{H_0r} \\ &{}+\frac{A''_0}{A_0}+\frac{1}{r^2} \biggr\} -\frac{2}{H_0^2} \biggl\{ \biggl(\frac{a}{A_0} \biggr)'\biggl(\frac{2}{r}-\frac{H'_0}{H_0}\biggr)-\frac{a''}{A_0} \\ &{}-\frac{aA''_0}{A_0^2}- \biggl(\frac{h}{H_0}\biggr)' \biggl(\frac{A'_0}{A_0}+\frac{2}{r} \biggr) \biggr\} \biggr]. \end{aligned}$$

(A.7)