Matter–curvature gravity modification and the formation of cylindrical isotropic systems

Abstract

In this paper, we constructed the three-layered gravastar model in cylindrical space–time. We considered one of the modified gravity theories to investigate the structural progression of the celestial object. The matter we considered in this model is effective, which further constituted the perfect fluid and extra degrees of freedom due to the modification of Einstein gravity. For the modelling of the three regions of gravastar, we used a specific barotropic equation of state. We then evaluated the subsequent field equations, hydrostatic equilibrium condition and gravitational mass. Furthermore, the metric coefficients for the three regions of the system were determined. Eventually, we discussed the important features of the gravastar and deduced its physical significance along with its graphical representations.

Appendix

The values of $\delta _{i}$’s where $i=$1–10 and that of $D_{j}$’s where $j=$6,7,8 appeared in eqs (9), (10) and (11) are discussed as follows:

$$\begin{aligned} \delta _{1}&=1+\frac{Rf_{Q}}{2}-\frac{9f_{Q}H'^{2}}{8KH^{2}}-\frac{2f_{Q}}{r^{2}K}\\&\quad -\frac{K'H'f_{Q}}{4HK^{2}}+\frac{f_{Q}'H'}{4H^{2}} +\frac{3f_{Q}H''}{4KH}-\frac{f_{Q}''}{2K}\\ {}&\quad -\frac{3f_{Q}'H'}{4KH} +\frac{K'f_{Q}'}{4K^{2}}+\frac{5f_{Q}H'}{4rKH},\\ \delta _{2}&=\frac{K'f_{Q}}{4K^{2}}-\frac{f_{Q}'}{K}-\frac{f_{Q}H'}{2H^{2}},\\ \delta _{3}&=-\frac{5H'^{2}f_{Q}}{8KH^{2}}-\frac{f_{Q}''}{K} -\frac{f_{Q}'K'}{4K^{2}}\\&\quad -\frac{f_{Q}H''}{4KH}+\frac{f_{Q}'}{2rK}-\frac{3f_{Q}H'}{4rKH}\\&\quad -\frac{f_{Q}'}{rKH}\frac{2f_{Q}'}{rH}-\frac{f_{Q}}{r^{2}KH},\\ \delta _{4}&=\frac{f_{Q}'}{K}-\frac{f_{Q}K'}{4K^{2}}+\frac{f_{Q}}{2rK},\\ D_{6}&=-\frac{f}{2}+\frac{f_{R}R}{2}.\\ \delta _{5}&=f_{T}-\frac{f_{Q}'H'}{4H^{2}}-\frac{f_{Q}K'H'}{4HK^{2}}\\&\quad -\frac{H''f_{Q}}{4KH}-\frac{f_{Q}H'^{2}}{8KH^{2}}+\frac{2f_{Q}}{Kr^{2}}\\&\quad -\frac{f_{Q}H'}{4rHK}+\frac{f_{Q}'H'}{2HK},\\ \delta _{6}&=f_{T}+\frac{Rf_{Q}}{2}+1-\frac{f_{Q}'H'}{4KH}\\&\quad +\frac{2f_{Q}''}{K}-\frac{5H'^{2}f_{Q}}{8KH^{2}}-\frac{f_{Q}H'}{4rKH}-\frac{f_{Q}K''}{2K^{2}} \\&\quad -\frac{3f_{Q}H''}{4HK}-\frac{H'K'f_{Q}}{4HK^{2}}-\frac{3f_{Q}'}{2rK}-\frac{4K'f_{Q}}{rK^{2}},\\ \delta _{7}&=\frac{f_{Q}'}{K}-\frac{3f_{Q}}{2rK}-\frac{f_{Q}H'}{4HK},\\ D_{7}&=\frac{f}{2}-\frac{Rf_{R}}{2}-\frac{H'f_{R}'}{2HK}-\frac{2f_{R}'}{rK},\\ \delta _{8}&=f_{T}-\frac{f_{Q}'H'}{4H^{2}}-\frac{H'^{2}f_{Q}}{8KH^{2}}-\frac{H''f_{Q}}{4HK}\\&\quad +\frac{K'H'f_{Q}}{4HK^{2}}+\frac{2f_{Q}}{Kr^{2}}-\frac{f_{Q}H'}{4HKr},\\ \delta _{9}&=f_{T}+\frac{Rf_{Q}}{2}+1-\frac{3f_{Q}'H'}{4HK}\\&\quad -\frac{f_{Q}H'^{2}}{8KH^{2}}+\frac{f_{Q}''}{2K}+\frac{3f_{Q}'K'}{4K^{2}}-\frac{H''f_{Q}}{4HK}\\&\quad -\frac{f_{Q}H'K'}{4HK^{2}}-\frac{3f_{Q}'}{2Kr}+\frac{3f_{Q}H'}{4HKr}\\&\quad +\frac{f_{Q}'K'}{4K^{2}}-\frac{2f_{Q}}{r^{2}}-\frac{f_{Q}K'}{Kr^{2}}+\frac{f_{Q}'}{rK},\\ \delta _{10}&=-\frac{3H'f_{Q}}{4HK}-\frac{2f_{Q}'}{K}\\&\quad +\frac{f_{Q}K'}{2K^{2}}-\frac{3f_{Q}}{2Kr}-\frac{f_{Q}}{rK},\\ D_{8}&=\frac{f}{2}-\frac{f_{R}R}{2}+\frac{f_{R}'}{rK}-\frac{2f_{R}}{r^{2}K}\\&\quad +\frac{K'f_{R}'}{2K^{2}}-\frac{f_{R}''}{K}-\frac{H'f_{R}'}{2HK}.\\ Z&=\frac{2}{2+Rf_{Q}+2f_{T}}\left[ \frac{f_{Q}'PK'H'}{4K^{2}H}-\frac{f_{Q}'PH''}{2HK}\right. \\&\quad \left. +\frac{f_{Q}'PH'^{2}}{4KH^{2}}+\frac{f_{Q}'PK'}{rK^{2}} +\frac{f_{Q}PK''H'}{4HK^{2}}\right. \\&\quad +\frac{3f_{Q}PK'H''}{4HK^{2}}-\frac{f_{Q}PH'''}{2HK}\\&\quad +\frac{f_{Q}PH''H'}{4H^{2}K}-\frac{f_{Q}PH'^{3}}{2KH^{3}}-\frac{f_{Q}PK''}{rK^{2}}-\frac{f_{Q}PK'}{r^{2}K^{2}}\\&\quad -\frac{f_{Q}PK'^{2}H'}{2HK^{3}}+\frac{f_{Q}P'K'H'}{8K^{2}H}\\&\quad -\frac{f_{Q}P'H''}{4KH}+\frac{f_{Q}P'H'^{2}}{8KH^{2}}-P'f_{T}-Pf_{T}'\\&\quad +\frac{f_{Q}P'}{r^{2}K}+\frac{f_{Q}'PH'}{KHr}+\frac{f_{Q}'P}{Kr^{2}}+\frac{f_{Q}P'H'}{KHr}\\&\quad -\frac{f_{Q}R'P}{2}-\frac{f_{Q}'RP}{2}+\frac{f_{Q}\rho 'K'H'}{8K^{2}H}-\frac{f_{Q}\rho 'H''}{4KH}\\&\quad +\left. \frac{f_{Q}\rho 'H'^{2}}{8KH^{2}}-\rho 'f_{T}-\rho f_{T}'+\frac{\rho 'f_{T}}{2}\right. \\&\quad \left. -\frac{P'f_{T}}{2}-\frac{f_{Q}\rho 'H'}{2KHr} +\frac{f_{Q}P'H'}{2KHr}+\frac{f_{Q}\rho '}{r^{2}K}\right] . \end{aligned}$$