Causes of energy density inhomogenisation with $f\mathcal {(G)}$ formalism

Abstract

Here, we analyse the distribution of self-gravitating collapsing fluid to identify the factors accountable for the energy–density inhomogeneity with the systematic construction in modified Gauss–Bonnet (GB) gravity, by taking the space–time which is spherically symmetric. The modified Einstein’s equations help us to observe the variation in the mass function due to different quantities. The dynamical equations and two differential equations for Weyl curvature are formulated, and used to explore the quantities responsible for the inhomogeneity. Irregularity in the fluid is analysed by taking various cases of fluid, under the effects of $f\mathcal {(G)}$ theory, where ${\mathcal {G}}$ is a Gauss–Bonnet term.

Appendix

In eq. (15) we have $\chi _{2}, \chi _3$ and $\chi _{4}$, which are given as

$$\begin{aligned} \chi _{2}&=\bigg (\frac{8\dot{C}'^{2}}{Y^{2}C^{2}} +\frac{16\dot{C'}C'\dot{Y}}{C^{2}Y^{3}} +\frac{16\dot{C'}\dot{C}X'}{Y^{2}C^{2}X}\\&\quad +\frac{16C'\dot{C}X'\dot{Y}}{C^{2}XY^{3}} +\frac{8\dot{C^{2}}X'^{2}}{C^{2}X^{2}Y^{2}} +\frac{8\dot{Y^{2}}C'^{2}}{C^{2}Y^{4}}\\&\quad +\frac{8C''C'X'X}{Y^{4}C^{2}} +\frac{8C''\dot{C}\dot{X}}{XY^{2}C^{2}} -\frac{4R'^{2}{\ddot{Y}}}{C^{2}Y^{3}}\\&\quad -\frac{12C'^{2}Y'X'X}{C^{2}Y^{5}} +\frac{8Y'C'{\ddot{C}}}{Y^{3}C^{2}} -\frac{8C'Y'\dot{C}\dot{X}}{C^{2}Y^{3}X} \\&\quad -\frac{12\dot{C^{2}}\dot{Y}\dot{X}}{YC^{2}X^{3}} +\frac{8{\ddot{C}}\dot{C}\dot{Y}}{YC^{2}X^{2}} -\frac{8C'X'\dot{C}\dot{X}}{C^{2}Y^{2}X^{2}}\\&\quad -\frac{8C'X'\dot{C}\dot{Y}}{C^{2}Y^{3}X} -\frac{8C''{\ddot{C}}}{C^{2}Y^{2}} -\frac{4\dot{Y^{2}}\dot{X}\dot{C}}{Y^{2}X^{3}C} +\frac{4{\ddot{Y}}}{C^{2}Y}\\&\quad - \frac{4X''X}{C^{2}Y^{2}} -\frac{4\dot{Y}\dot{X}}{YC^{2}X} +\frac{4Y'X'X}{C^{2}Y^{3}} +\frac{4C'^{2}X''X}{C^{2}Y^{4}}\\&\quad +\frac{4C'^{2}\dot{Y}\dot{X}}{C^{2}Y^{3}X} +\frac{4\dot{C^{2}}{\ddot{Y}}}{C^{2}X^{2}Y} -\frac{4\dot{C^{2}}X''}{C^{2}Y^{2}X}\\&\quad + \frac{4\dot{C^{2}}Y'X'}{C^{2}Y^{3}X} +\frac{8X''X'Y'}{Y^{5}} +\frac{4\dot{Y^{2}}\dot{X^{2}}}{X^{4}Y^{2}}\bigg )f_{{\mathcal {G}}},\\ \chi _{3}&=\bigg (\frac{4X^{2}\mathcal {G''}}{C^{2}Y^{2}}-\frac{4\dot{Y}\dot{{\mathcal {G}}}}{YC^{2}} -\frac{4Y'X^{2}\mathcal {G'}}{C^{2}Y^{3}}\\&\quad +\frac{8C''\dot{C}\dot{{\mathcal {G}}}}{Y^{2}C^{2}} -\frac{8C''X^{2}C'\mathcal {G'}}{C^{2}Y^{4}} -\frac{4C'^{2}X^{2}\mathcal {G''}}{Y^{4}C^{2}} \\&\quad +\frac{12C'^{2}X^{2}Y'\mathcal {G'}}{Y^{5}C^{2}} +\frac{8C'Y'\dot{C}\mathcal {\dot{G}}}{Y^{3}C^{2}} +\frac{4\dot{C^{2}}\mathcal {G''}}{Y^{2}C^{2}}\\&\quad -\frac{4\dot{C^{2}}Y'\mathcal {G'}}{C^{2}Y^{3}} +\frac{8\dot{C^{2}}C'\mathcal {G'}}{Y^{2}C^{3}} -\frac{12\dot{C^{2}}\dot{Y}\dot{{\mathcal {G}}}}{YC^{2}X^{2}} \\&\quad +\frac{4\dot{C^{2}}{\mathcal {G}}'^{2}}{Y^{2}C^{2}} +\frac{8\dot{C}\dot{Y}C'\mathcal {G'}}{C^{2}Y^{3}}\bigg )f_{{\mathcal {G}}{\mathcal {G}}},\\ \chi _{4}&=\bigg (\frac{4X^{2}}{Y^{2}C^{2}} -\frac{4X^{2}C'^{2}}{C^{2}Y^{4}}\bigg )f_{{\mathcal {G}}{\mathcal {G}}{\mathcal {G}}}{\mathcal {G}}'^{2}. \end{aligned}$$

In eq. (16) we have $Z_2, Z_3$ and $Z_4$ whch are given as

$$\begin{aligned} Z_{2}&=\bigg (-\frac{8\dot{C'^{2}}}{C^{2}X^{2}} +\frac{16\dot{C'}C'\dot{Y}}{X^{2}C^{2}Y} +\frac{24\dot{C'}\dot{C}X'}{X^{3}C^{2}}\\&\quad -\frac{8C'^{2}\dot{Y^{2}}}{X^{2}Y^{2}C^{2}} -\frac{8C'\dot{C}X'\dot{Y}}{C^{2}X^{3}Y} -\frac{8\dot{C^{2}}X'^{2}}{C^{2}X^{4}} \\&\quad -\frac{4\dot{C^{2}}Y'X'}{C^{2}X^{3}Y} -\frac{4{\ddot{Y}}Y}{X^{2}C^{2}} +\frac{4X''}{XC^{2}} +\frac{4\dot{Y}\dot{X}Y}{C^{2}X^{3}} \\&\quad -\frac{4Y'X'}{XYC^{2}} +\frac{4C'^{2}{\ddot{Y}}}{C^{2}X^{2}Y} -\frac{4C'^{2}X''}{XC^{2}Y^{2}} +\frac{12C'^{2}X'Y'}{XC^{2}Y^{3}} \\&\quad - \frac{4C'^{2}\dot{Y}\dot{X}}{C^{2}YX^{3}} -\frac{8C''C'X'}{Y^{2}C^{2}X} +\frac{8{\ddot{C}}C''}{X^{2}C^{2}}\\&\quad -\frac{8{\ddot{C}}C'Y'}{X^{2}C^{2}Y} -\frac{8{\ddot{C}}\dot{C}\dot{Y}Y}{X^{4}C^{2}} -\frac{4\dot{C^{2}}{\ddot{Y}}Y}{X^{4}C^{2}} +\frac{12\dot{C^{2}}\dot{Y}\dot{X}Y}{C^{2}X^{5}} \\&\quad +\frac{4\dot{C^{2}}X''}{C^{2}X^{3}} -\frac{8C''\dot{C}\dot{X}}{X^{3}C^{2}} -\frac{4\dot{Y^{2}}\dot{X^{2}}}{X^{6}}\bigg )f_{{\mathcal {G}}},\\ Z_{3}&=\bigg (\frac{4Y^{2}\ddot{{\mathcal {G}}}}{C^{2}X^{2}} -\frac{4Y^{2}\dot{X}\dot{{\mathcal {G}}}}{C^{2}X^{3}} -\frac{4X'\mathcal {G'}}{C^{2}X}\\&\quad -\frac{4C'^{2}\ddot{{\mathcal {G}}}}{X^{2}C^{2}} +\frac{4C'^{2}\dot{X}\dot{{\mathcal {G}}}}{C^{2}X^{3}} -\frac{12C'^{2}X'\mathcal {G'}}{C^{2}Y^{2}X} -\frac{8C'X'\dot{C}\dot{{\mathcal {G}}}}{C^{2}X^{3}}\\&\quad - \frac{8{\ddot{C}}C'{\mathcal {G}}}{C^{2}X^{2}} +\frac{8{\ddot{C}}Y^{2}\dot{C}\dot{{\mathcal {G}}}}{C^{2}X^{4}} -\frac{12\dot{C^{2}}\dot{X}Y^{2}\dot{{\mathcal {G}}}}{X^{5}C^{2}}\\&\quad +\frac{4\dot{C^{2}}Y^{2}\ddot{{\mathcal {G}}}}{C^{2}X^{4}} -\frac{4\dot{C^{2}}X'\mathcal {G'}}{C^{2}X^{3}} +\frac{8\dot{C}\dot{X}C'\mathcal {G'}}{C^{2}X^{3}}\bigg )f_{{\mathcal {G}}{\mathcal {G}}},\\ Z_{4}&=\bigg (\frac{4Y^{2}}{C^{2}X^{2}} -\frac{4C'^{2}}{X^{2}C^{2}} +\frac{4\dot{C^{2}}Y^{2}}{C^{2}X^{4}}\bigg )f_{{\mathcal {G}}{\mathcal {G}}{\mathcal {G}}}\dot{{\mathcal {G}}^{2}}. \end{aligned}$$

$D_{3}$ in eq. (17) is given by

$$\begin{aligned} D_{3}&=\bigg (\frac{8\dot{C'}\dot{C}\dot{{\mathcal {G}}}}{C^{2}X^{2}} -\frac{8\dot{C'}C'\mathcal {G'}}{C^{2}Y^{2}} -\frac{8C'\dot{Y}\dot{C}\dot{{\mathcal {G}}}}{YX^{2}C^{2}}\\&\quad +\frac{8C'^{2}\dot{Y}\mathcal {G'}}{C^{2}Y^{3}} -\frac{8\dot{C^{2}}X'\dot{{\mathcal {G}}}}{C^{2}X^{3}} +\frac{8\dot{C}X'C'\mathcal {G'}}{C^{2}XY^{2}}\\&\quad +\frac{4\dot{{\mathcal {G}}}\mathcal {G'}}{C^{2}} -\frac{4C'^{2}\dot{{\mathcal {G}}}{\mathcal {G'}}}{C^{2}Y^{2}} +\frac{4\dot{C^{2}}\dot{{\mathcal {G}}}\mathcal {G'}}{C^{2}X^{2}} +\frac{4\dot{\mathcal {G'}}}{C^{2}}\\&\quad -\frac{4C'^{2}\dot{\mathcal {G'}}}{Y^{2}C^{2}} +\frac{4\dot{C^{2}}\dot{\mathcal {G'}}}{C^{2}X^{2}} -\frac{4X'\dot{{\mathcal {G}}}}{C^{2}X} +\frac{4C'^{2}X'\dot{{\mathcal {G}}}}{C^{2}Y^{2}X} \\&\quad -\frac{4\dot{C^{2}}X'\dot{{\mathcal {G}}}}{C^{2}X^{3}} +\frac{4C'^{2}\dot{Y}\mathcal {G'}}{C^{2}Y^{3}} -\frac{4\dot{C^{2}}\dot{Y}\mathcal {G'}}{C^{2}X^{2}Y}\bigg )f_{{\mathcal {G}}{\mathcal {G}}}. \end{aligned}$$

Values of $F_{2}, F_{3}$ and  $F_{4}$ in (18) are given as follows:

$$\begin{aligned} F_{2}&=\bigg (-\frac{8{\ddot{C}}C'Y'}{Y^{3}X^{2}} -\frac{4X''C'^{2}}{Y^{4}X} +\frac{4X''\dot{C^{2}}}{Y^{2}X^{3}}\\&\quad +\frac{4X''}{XY^{2}} -\frac{8C''C'X'}{Y^{4}X} +\frac{12C'^{2}X'Y'}{Y^{5}X} \\&\quad -\frac{4X'Y'\dot{C^{2}}}{Y^{3}X^{3}} -\frac{4X'Y'}{XY^{3}} +\frac{8{\ddot{C}}C''}{X^{2}Y^{2}} +\frac{4{\ddot{Y}}C'^{2}}{Y^{3}X^{2}}\\&\quad -\frac{4{\ddot{Y}}\dot{C^{2}}}{YX^{4}} -\frac{4{\ddot{Y}}}{YX^{2}} -\frac{8{\ddot{C}}\dot{C}\dot{Y}}{YX^{4}} +\frac{8\dot{C}X'C'\dot{Y}}{X^{3}Y^{3}} \\&\quad +\frac{8\dot{C^{2}}\dot{Y}\dot{X}}{YX^{5}} -\frac{8C''\dot{C}\dot{X}}{X^{3}Y^{2}} +\frac{8\dot{C}\dot{X}C'Y'}{X^{3}Y^{3}}\\&\quad -\frac{4C'^{2}\dot{X}\dot{Y}}{X^{3}Y^{3}} +\frac{4\dot{X}\dot{Y}}{X^{3}Y} -\frac{8\dot{C'^{2}}}{X^{2}Y^{2}} -\frac{8C'^{2}\dot{Y^{2}}}{X^{2}Y^{4}}\\&\quad - \frac{8X'^{2}\dot{C^{2}}}{X^{4}Y^{2}} +\frac{16C'\dot{C'}\dot{Y}}{X^{2}Y^{3}} +\frac{16X'\dot{C}\dot{C'}}{X^{3}Y^{2}}\bigg )f_{\mathcal {G}},\\ F_{3}&=\bigg (-\frac{4C''C\ddot{{\mathcal {G}}}}{X^{2}Y^{2}} +\frac{4C''C\dot{X}\dot{{\mathcal {G}}}}{X^{3}Y^{2}} +\frac{4C''CX'\mathcal {G'}}{X^{Y^{4}}}\\&\quad -\frac{4{\ddot{C}}C\mathcal {G''}}{X^{2}Y^{2}} +\frac{4{\ddot{C}}C\dot{Y}\dot{{\mathcal {G}}}}{YX^{4}} +\frac{4{\ddot{C}}CY'\mathcal {G'}}{Y^{3}X^{2}}\\&\quad + \frac{16{\ddot{C}}\dot{C}\dot{{\mathcal {G}}}}{X^{4}} -\frac{16{\ddot{C}}C'\mathcal {G'}}{X^{2}Y^{2}} +\frac{4C'Y'C\ddot{{\mathcal {G}}}}{Y^{3}X^{2}}\\&\quad -\frac{4C'Y'C}{Y^{3}X^{3}} -\frac{4C'Y'CX'\mathcal {G'}}{XY^{5}} +\frac{4C'X'C\mathcal {G''}}{Y^{4}X}\\&\quad -\frac{4C'X'C\dot{Y}\dot{{\mathcal {G}}}}{Y^{3}X^{3}} -\frac{4C'X'CY'\mathcal {G'}}{Y^{5}X} +\frac{4\dot{C}\dot{Y}C\ddot{{\mathcal {G}}}}{YX^{4}}\\&\quad -\frac{4\dot{C}\dot{Y}C\dot{X}\dot{{\mathcal {G}}}}{YX^{5}} -\frac{4\dot{Y}\dot{C}CX'\mathcal {G'}}{X^{3}Y^{3}} +\frac{4\dot{C}\dot{X}C\mathcal {G''}}{X^{3}Y^{2}} \\&\quad - \frac{4\dot{C}\dot{X}C\dot{Y}\dot{{\mathcal {G}}}}{YX^{5}} -\frac{4\dot{C}\dot{X}CY'\mathcal {G'}}{Y^{3}X^{3}} +\frac{8\dot{C^{2}}\dot{X}\dot{{\mathcal {G}}}}{X^{5}}\\&\quad -\frac{8\dot{C}\dot{X}C'\mathcal {G'}}{X^{3}Y^{2}} -\frac{4X''\dot{C}C\dot{{\mathcal {G}}}}{Y^{2}X^{3}} +\frac{4X''CC'\mathcal {G'}}{Y^{4}} \\&\quad + \frac{4X'Y'C\dot{C}\dot{{\mathcal {G}}}}{X^{3}Y^{3}} -\frac{4X'Y'C'C\mathcal {G'}}{XY^{5}} +\frac{4{\ddot{Y}}C\dot{C}\dot{{\mathcal {G}}}}{YX^{4}}\\&\quad -\frac{4{\ddot{Y}}C'C\mathcal {G'}}{Y^{3}X^{2}} -\frac{4\dot{X}\dot{Y}\dot{C}\dot{{\mathcal {G}}}}{X^{5}Y} +\frac{4\dot{X}\dot{Y}C'\mathcal {G'}}{X^{3}Y^{3}}\\&\quad - \frac{8C'^{2}\dot{C}\dot{{\mathcal {G}}}}{CY^{2}X^{2}} +\frac{8C'^{3}\mathcal {G'}}{CY^{4}} -\frac{8\dot{R^{2}}\dot{X}\dot{{\mathcal {G}}}}{X^{5}} +\frac{8\dot{C}\dot{X}C'\mathcal {G'}}{X^{3}Y^{2}}\\&\quad +\frac{8C'^{2}\dot{C}\dot{{\mathcal {G}}}}{CX^{4}} -\frac{8C'^{3}\mathcal {G'}}{CX^{2}Y^{2}} +\frac{8\dot{C'}C\dot{{\mathcal {G}}}\mathcal {G'}}{X^{2}Y^{2}} \\&\quad + \frac{8\dot{C'}C\dot{\mathcal {G'}}}{X^{2}Y^{2}} -\frac{8X'C\dot{C'}\dot{{\mathcal {G}}}}{X^{3}Y^{2}} -\frac{8\dot{C'}C\dot{Y}\mathcal {G'}}{X^{2}Y^{3}}\\&\quad -\frac{8C'\dot{Y}\mathcal {G'}\dot{{\mathcal {G}}}}{X^{2}Y^{3}} -\frac{8C'\dot{Y}C\dot{\mathcal {G'}}}{X^{2}Y^{3}} +\frac{8C'\dot{Y}CX'\dot{{\mathcal {G}}}}{X^{3}Y^{3}} \\&\quad +\frac{8C'C\dot{Y^{2}}\mathcal {G'}}{X^{2}Y^{4}} -\frac{8\dot{C}CX'\dot{{\mathcal {G}}}\mathcal {G'}}{Y^{2}X^{3}} -\frac{8C\dot{C}X'\dot{\mathcal {G'}}}{X^{3}Y^{2}}\\&\quad +\frac{8C\dot{C}X'^{2}\dot{{\mathcal {G}}}}{Y^{2}X^{4}} +\frac{8\dot{C}CX'\dot{Y}\mathcal {G'}}{Y^{3}X^{3}}\bigg )f_{{\mathcal {G}}{\mathcal {G}}},\\ F_{4}&=\bigg (-\frac{4C''C\dot{{\mathcal {G}}^{2}}}{Y^{2}X^{2}} +\frac{4C'Y'C\dot{{\mathcal {G}}^{2}}}{X^{2}Y^{3}} +\frac{4\dot{C}\dot{Y}C\dot{{\mathcal {G}}^{2}}}{YX^{4}}\\&\quad -\frac{4{\ddot{C}}C\mathcal {G'}^{2}}{X^{2}Y^{2}} +\frac{4C'X'C\mathcal {G'}^{2}}{Y^{4}X} \\&\quad + \frac{4\dot{C}\dot{X}C\mathcal {G'}^{2}}{X^{3}Y^{2}}\bigg )f_{{\mathcal {G}}{\mathcal {G}}{\mathcal {G}}}. \end{aligned}$$

In eqs (28) and (29) we have $Z_0$ and $Z_{1}$, which are written as

$$\begin{aligned} Z_0&=\bigg (T^{{11}^{(\textrm{eff})}},1 +T^{{22}^{(\textrm{eff})}},2\bigg )\\&\quad +\bigg (\frac{\dot{X}}{X} +\frac{3X'}{2{X}} +\frac{\dot{Y}}{2Y} +\frac{Y'}{2Y} +\frac{Y\dot{Y}}{2X^{2}} +\frac{\dot{C}}{C} +\frac{C'}{C} \\&\quad +(1+\sin ^{2}\theta )\frac{C\dot{C}}{2X^{2}}\bigg )f +\bigg (\frac{2\dot{X}}{X} +\frac{\dot{Y}}{Y} +\frac{2\dot{C}}{C}\bigg )T^{{11}^{(\textrm{eff})}}\\&\quad +\bigg (\frac{3X'}{X}+\frac{Y'}{Y} +\frac{2C'}{C}\bigg )T^{{12}^{(\textrm{eff})}} \\&\quad + \frac{Y\dot{Y}}{X^{2}}T^{{22}^{(\textrm{eff})}} +\frac{C\dot{C}}{X^{2}}T^{{33}^{(\textrm{eff})}} +\frac{C\dot{C}}{X^{2}}\sin ^{2}\theta T^{{44}^{(\textrm{eff})}},\\ Z_1&=\bigg (T^{{22}^{(\textrm{eff})}},2 +T^{{21}^{(\textrm{eff})}},1\bigg )\\&\quad +\bigg (\frac{XX'}{2Y^{2}} +\frac{3\dot{Y}}{2Y} +\frac{Y'}{Y} +\frac{\dot{C}}{C} +\frac{C'}{C} + \frac{X'}{2X} +\frac{\dot{X}}{2X} \\&\quad - \cos ^{2}\frac{CC'}{2Y^{2}}\bigg )f +\bigg (\frac{3\dot{Y}}{Y} +\frac{2\dot{C}}{C} +\frac{\dot{X}}{X}\bigg )T^{{12}^{(\textrm{eff})}} \\&\quad +\bigg (\frac{2Y'}{Y} +\frac{2C'}{C} +\frac{X'}{X}\bigg )T^{{22}^{(\textrm{eff})}} +\\&\quad +\frac{XX'}{Y^{2}}T^{{11}^{(\textrm{eff})}} -\frac{CC'}{Y^{2}}T^{{33}^{(\textrm{eff})}} +\frac{CC'}{Y^{2}}\sin ^{2}\theta T^{{44}^{(\textrm{eff})}}. \end{aligned}$$