Radiating cylindrical gravitational collapse with structure scalars in f(R) gravity

Abstract

In this paper, we discuss dynamical properties of dissipative collapsing cylindrical self-gravitating systems with account of f(R)=R+γR ²+β ₁ R ³ gravity model. In this perspective, we see effects of higher curvature terms in the formulations of structure scalars already obtained from the orthogonal decomposition of Weyl curvature scalar in general relativity. We compute mass function by generalizing Misner-Sharp formalism and discuss the contribution of relaxation time in the radiating collapsing process. The contribution of scalar functions in the modeling of static anisotropic as well as isotropic fluid configurations are explored. We conclude that all static anisotropic cylindrical solutions of f(R) field equations can be written explicitly by means of triplet of these scalar functions.

Appendix

The f(R) corrections in Eqs. (21) and (22) are

$$\begin{aligned} D_0 =&\frac{1}{\kappa} \biggl[ \biggl\{ \biggl( \dot{f_R}\frac{A'}{A} +f'_R \frac{\dot{A}}{A}-\dot{f'_R} \biggr)\frac{1}{A^4} \biggr\} _{,1} \\ &{} +\frac{1}{A^2} \biggl\{ \biggl(\frac{Rf_R-f}{2} \biggr) +\frac{f'_R}{A^2} \biggl(\frac{B'}{B}-\frac{A'}{A}+ \frac{C'}{C} \biggr) \\ &{} - \biggl(\frac{\dot{C}}{C}+\frac{\dot{A}}{A}+ \frac{\dot{B}}{B} \biggr) \frac{\dot{f_R}}{A^2}+\frac{f''_R}{A^2} \biggr\} _{,0} \\ &{} +\frac{\dot{A}}{A} \biggl\{ \frac{\ddot{f_R}}{A^2} +\frac{f''_R}{A^2}-2\frac{A'f'_R}{A^3}-2 \frac{\dot{A} \dot{f_R}}{A^3} \biggr\} \frac{1}{A^2} \\ &{} +\frac{\dot{B}}{B} \biggl\{ \frac{ \ddot{f_R}}{A^2}- \biggl(\frac{\dot{B}}{B}+\frac{\dot{A}}{A} \biggr) \frac{\dot{f_R}}{A^2}+\frac{f'_R}{B^2} \biggl(\frac{B'}{B}-\frac{A'}{A} \biggr) \biggr\} \frac{1}{A^2} \\ &{}+ \biggl(\frac{\dot{A}}{A}f'_R- \dot{f'_R} +\frac{A'}{A}\dot{f_R} \biggr) \biggl(\frac{4A'}{A}+\frac{C'}{C} +\frac{B'}{B} \biggr) \frac{1}{A^4} \\ &{}+\frac{2}{A^2} \biggl\{ - \biggl(\frac{A'}{A} -\frac{C'}{C} \biggr)\frac{f'_R}{B^2}+ \frac{\ddot{f_R}}{A^2} \\ &{} -\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{C}}{C}+ \frac{\dot{A}}{A} \biggr) \biggr\} \frac{\dot{C}}{C} \biggr], \end{aligned}$$

(A.1)

$$\begin{aligned} D_1 =&\frac{1}{\kappa} \biggl[ \biggl\{ \frac{1}{A^4} \biggl(\frac{A'}{A} \dot{f_R}-\dot{f'_R}+ \frac{\dot{A}}{A}f'_R \biggr) \biggr\} _{,0} \\ &{} + \frac{1}{A^2} \biggl\{ \frac{\ddot{f_R}}{A^2} +\frac{f-Rf_R}{2} - \frac{f'_R}{B^2} \biggl(\frac{B'}{B}+\frac{A'}{A}+ \frac{C'}{C} \biggr) \\ &{}-\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{B}}{B}- \frac{\dot{A}}{A} +\frac{\dot{C}}{C} \biggr) \biggr\} _{,1}+ \frac{1}{A^2} \biggl\{ \frac{ f''_R}{A^2}- \biggl(\frac{A'}{A} +\frac{B'}{B} \biggr) \\ &{} \times \frac{f'_R}{B^2} + \biggl( \frac{\dot{B}}{B}-\frac{\dot{A}}{A} \biggr)\frac{\dot{f_R}}{A^2} \biggr\} \frac{B'}{B}+ \biggl\{ \frac{f''_R}{A^2}+\frac{\ddot{f_R}}{ A^2}-2 \frac{A'f'_R}{A^3} \\ &{} -2\frac{\dot{A}\dot{f_R}}{A^3} \biggr\} \frac{A'}{A^3} +\frac{1}{A^2} \biggl\{ \frac{f''_R}{A^2} -\frac{f'_R}{A^2} \biggl(\frac{A'}{A}+ \frac{C'}{C} \biggr) \\ &{} -\frac{\dot{f_R}}{A^2} \biggl(\frac{\dot{C}}{C}- \frac{\dot{A}}{A} \biggr) \biggr\} \frac{C'}{C}-\frac{1}{A^4} \biggl(\dot{f'_R}- \frac{\dot{A}}{A}f'_R-\frac{A'}{A} \dot{f_R} \biggr) \\ &{}\times \biggl(\frac{\dot{B}}{B}+\frac{\dot{C}}{C} + \frac{4\dot{A}}{A} \biggr) \biggr]. \end{aligned}$$

(A.2)

The quantities ϕ _γi and ϕ _βi mentioned in Eqs. (28) and (30) are

$$\begin{aligned} \phi_{\gamma0} =&\frac{R^2}{4}+\frac{R''}{A^2}- \frac{R'}{A^2} \biggl(\frac{B'}{B}-\frac{A'}{A}+ \frac{C'}{C} \biggr) \\ &{} -\frac{ \dot{R}}{A^2} \biggl(\frac{\dot{B}}{B}+ \frac{\dot{C}}{C}+\frac{ \dot{A}}{A} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\gamma1} =& -\frac{R^2}{4}+\frac{\ddot{R}}{A^2}+ \frac{ \dot{R}}{A^2} \biggl(\frac{\dot{B}}{B}-\frac{\dot{A}}{A}+ \frac{ \dot{C}}{C} \biggr) \\ &{} -\frac{R'}{A^2} \biggl(\frac{B'}{B}+ \frac{C'}{C} -\frac{A'}{A} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\gamma2}=-\frac{R^2}{4}+\frac{1}{A^2} \biggl( \ddot{R}-R'' +\frac{\dot{C}\dot{R}}{C}-\frac{C'R'}{C} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\gamma3}=-\frac{R^2}{4}+\frac{1}{A^2} \biggl( \ddot{R}-R'' +\frac{\dot{C}\dot{R}}{C}-\frac{B'R'}{B} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\beta0} =& \frac{R^3}{6}+\frac{RR''+R'^2}{A^2}- \frac{RR'}{ A^2} \biggl(\frac{B'}{B}-\frac{A'}{A}+ \frac{C'}{C} \biggr) \\ &{} -\frac{R \dot{R}}{A^2} \biggl(\frac{\dot{B}}{B}+ \frac{\dot{C}}{C}+\frac{ \dot{A}}{A} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\beta1} =& -\frac{R^3}{6}+\frac{R\ddot{R}+\dot{R}^2}{A^2} + \frac{R\dot{R}}{A^2} \biggl(\frac{\dot{B}}{B}-\frac{\dot{A}}{A} + \frac{\dot{C}}{C} \biggr) \\ &{} -\frac{RR'}{A^2} \biggl(\frac{B'}{B} + \frac{C'}{C}-\frac{A'}{A} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\beta2} =& -\frac{R^3}{6}+\frac{1}{A^2} \biggl( \dot{R}^2 +R\ddot{R}-RR''-R'^2 \\ &{} + \frac{R\dot{C}\dot{R}}{C}-\frac{RC'R'}{C} \biggr), \end{aligned}$$

$$\begin{aligned} \phi_{\beta3} =& -\frac{R^3}{6}+\frac{1}{A^2} \biggl( \dot{R}^2 +R\ddot{R}-RR''-R'^2 \\ &{} + \frac{R\dot{C}\dot{R}}{C}-\frac{RB'R'}{B} \biggr). \end{aligned}$$