Charged anisotropic strange stars in $f(\mathcal{G},\mathcal{T})$ gravity

Abstract

In this work, we explore charged anisotropic strange stars in the framework of $f(\mathcal{G},\mathcal{T})$ gravity ($\mathcal{G}$ and $\mathcal{T}$ are the Gauss-Bonnet invariant and trace of the energy-momentum tensor, respectively) by considering the minimal model $f(\mathcal{G},\mathcal{T})=\omega \mathcal{G}^{k}+\chi \mathcal{T}$, $\omega \in \mathbb{R}$, $k\in \mathbb{R}^{+}$ and $\chi$ is the coupling parameter. For this purpose, we employ the condition of embedding class-I to obtain the ansatz describing the geometry inside the stellar configuration. Further, to determine the complete solution of modified Einstein-Maxwell equations, we use the MIT bag model equation of state, which relates the density and pressure by involving the bag constant (which balances the inward directed bag pressure) and describes the essential features of strange quark matter distribution inside the stellar system. We match the interior and exterior geometries to evaluate the set of unknown constants by choosing Reissner-Nordström metric as an exterior geometry. The graphical analysis of the developed solution is done by using observed mass and radius of the star candidate 4U 1608-52 corresponding to different values of the bag constant and coupling parameter. To examine the physical viability and stability of the model, we investigate the behavior of energy constraints, causality condition, Herrera cracking approach, compactness factor, and redshift function. We conclude that the charged anisotropic strange stars model is physically viable and stable under all considerations.

Appendix

$$\begin{aligned} \rho &=\frac{1}{2r^{2}(8\pi +\chi )(1+c_{3}r^{2})^{2}}\Big[r\big\{ r \big(\frac{6\sqrt{c_{3}}c_{2}}{\sqrt{c_{3}}c_{2}r^{2}+2c_{1}} \\ &+c_{3}(- \frac{12c_{1}}{\sqrt{c_{3}}c_{2}r^{2}+2c_{1}} +4Br^{2} (8\pi +\chi )+9) \\ &+2c_{3}^{2}Br^{4}(8\pi +\chi )+12c_{3} \mathcal{G}''+2B(8\pi +\chi )\big) \\ &+ \frac{12\sqrt{c_{3}}\mathcal{G}'(2-c_{3}r^{2})(2\sqrt{c_{3}}c_{1}+3c_{3}c_{2}r^{2}+2c_{2})}{(1+c_{3}r^{2})(\sqrt{c_{3}}c_{2}r^{2}+2c_{1})}\big\} \Big], \end{aligned}$$

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$$\begin{aligned} p_{r} =&\frac{1}{2r^{2}(8\pi +\chi )(1+c_{3}r^{2})^{2}}\Big[r\big\{ r \big(\frac{2\sqrt{c_{3}}c_{2}}{\sqrt{c_{3}}c_{2}r^{2}+2c_{1}} \\ +&c_{3}(- \frac{4c_{1}}{\sqrt{c_{3}}c_{2}r^{2}+2c_{1}} -4Br^{2} (8\pi +\chi )+3) \\ -&2c_{3}^{2}Br^{4}(8\pi +\chi )+4c_{3} \mathcal{G}''-2B(8\pi +\chi )\big) \\ +& \frac{4\sqrt{c_{3}}\mathcal{G}'(2-c_{3}r^{2})(2\sqrt{c_{3}}c_{1}+3c_{2}c_{3}r^{2}+2c_{2})}{(1+c_{3}r^{2}) (\sqrt{c_{3}}c_{2}r^{2}+2c_{1})}\big\} \Big], \end{aligned}$$

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$$\begin{aligned} p_{t} =& \frac{1}{2r(8\pi +\chi )(24\pi +5\chi )(1+c_{3}r^{2})^{3}(\sqrt{c_{3}}c_{2}r^{2}+2c_{1})} \\ \times& \Big[r\Big\{ 2 c_{3}^{7/2}c_{2}r^{6}(8\pi +\chi )(3\mathcal{G}^{2}r^{2}+7Br^{2}\chi \\ +&24\pi Br^{2}-3)+3c_{3}^{5/2}c_{2}r^{4} \big(\chi (6\mathcal{G}^{2}r^{2}+14Br^{2}\chi+11) \\ +&8\pi (6\mathcal{G}^{2}r^{2}+20Br^{2}\chi +3)+384\pi ^{2}Br^{2} \big) \\ +&c_{3}^{3/2}c_{2}\Big(18\mathcal{G}^{2}r^{4} (8\pi +\chi )+576\mathcal{G}(8\pi +\chi ) \\ +& r^{2}\big(24\pi (20Br^{2}\chi +19) +\chi (42Br^{2}\chi +97) \\ +&1152\pi ^{2}Br^{2}\big)\Big)+4c_{1}c_{3}^{3}r^{4}(8\pi +\chi )(3 \mathcal{G}^{2}r^{2} \\ +&7Br^{2}\chi +24\pi Br^{2}-3) +2c_{1}c_{3}^{2}r^{2}\big(24\pi (6\mathcal{G}^{2}r^{2} \\ +&20Br^{2} \chi -7)+\chi (18\mathcal{G}^{2}r^{2}+42Br^{2} \chi -13) \\ +&1152\pi ^{2}Br^{2}\big)-4\sqrt{c_{3}}\mathcal{G}''(1\,{+}\,c_{3}r^{2}) \big(2\sqrt{c_{3}}c_{1}(24\pi \,{-}\,5\chi \!) \\ +&c_{2}c_{3}r^{2}(24\pi -5\chi )-24c_{2}(8\pi +\chi )\big) \\ +&2c_{1}c_{3}\big(24\pi (6 \mathcal{G}^{2}r^{2}+20Br^{2}\chi -5)+\chi (18\mathcal{G}^{2} r^{2} \\ +&42Br^{2}\chi -7)+1152\pi ^{2}Br^{2}\big) \\ +&2\sqrt{c_{3}}c_{2} \big(\chi (3\mathcal{G}^{2}r^{2}+7Br^{2}\chi +29) \\ +&8\pi (3\mathcal{G}^{2}r^{2}+10Br^{2}\chi +21)+192\pi ^{2}Br^{2} \big) \\ +&4c_{1}(8\pi +\chi )(7B\chi +24\pi B+3\mathcal{G}^{2})\Big\} \\ -&4\sqrt{c_{3}}\mathcal{G}'\big(2c_{3}^{3/2}c_{1}r^{2}(5 \chi -24\pi ) +3c_{3}^{2}c_{2}r^{4}(40\pi +13\chi ) \\ +&4\sqrt{c_{3}}c_{1}(24\pi -5\chi )+4c_{2}c_{3}r^{2}(72\pi +\chi ) \\ -&4c_{2}(23 \chi +120\pi )\big)\Big]. \end{aligned}$$

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Here, the expressions of $\mathcal{G}$ and its derivatives are given as

$$\begin{aligned} \mathcal{G} =&- \frac{48c_{3}^{3/2}c_{2}}{(1+c_{3}r^{2})^{3}(\sqrt{c_{3}}c_{2}r^{2}+2c_{1})}, \\ \mathcal{G}' =& \frac{96c_{3}^{2}c_{2}r(6\sqrt{c_{3}}c_{1}+4c_{2}c_{3}r^{2}+c_{2})}{(1+c_{3}r^{2})^{4} (\sqrt{c_{3}}c_{2}r^{2}+2c_{1})^{2}}, \\ \mathcal{G}'' =&- \frac{1}{(1+c_{3}r^{2})^{5}(\sqrt{c_{3}}c_{2}r^{2}+2c_{1})^{3}} \\ \times&\Big[96c_{3}^{2}c_{2} \big\{ c_{3}^{3/2} c_{2}r^{4}(106\sqrt{c_{3}}c_{1}+15c_{2}) +36c_{3}^{5/2}c_{2}^{2}r^{6} \\ +&\sqrt{c_{3}}r^{2}(84c_{1}^{2}c_{3}+8 \sqrt{c_{3}} c_{1}c_{2}+3c_{2}^{2}) \\ -&2c_{1}(6\sqrt{c_{3}}c_{1}+c_{2}) \big\} \Big]. \end{aligned}$$

$$\begin{aligned} \Delta =& \frac{3}{r(8\pi +\chi )(24\pi +5\chi )(1+c_{3}r^{2})^{3} (\sqrt{c_{3}}c_{2}r^{2}+2c_{1})} \\ \times&\Big[r\Big\{ c_{3}^{7/2}c_{2}r^{6}(8\pi +\chi ) (\mathcal{G}^{2}r^{2}+4Br^{2}\chi +16\pi Br^{2}-1) \\ +&3c_{3}^{5/2}c_{2}r^{4} \big(\mathcal{G}^{2}r^{2}(8\pi +\chi )+4Br^{2} \chi ^{2} +48\pi Br^{2}\chi \\ +&128\pi ^{2}Br^{2}+\chi \big) +c_{3}^{3/2}c_{2} \big(3\mathcal{G}^{2}r^{4}(8\pi +\chi ) \\ +&96\mathcal{G}(8\pi {+}\chi ) {+}4r^{2}(3\chi (1+Br^{2}\chi )\,{+}\,2\pi (18Br^{2}\chi {+}7) \\ +&96\pi ^{2}Br^{2}) \big)+2c_{1}c_{3}^{3}r^{4} (8\pi +\chi )(\mathcal{G}^{2}r^{2}+4Br^{2}\chi \\ +&16\pi Br^{2}-1)+2c_{3}^{2}c_{1}r^{2} \big(3\chi (\mathcal{G}^{2}r^{2}+4Br^{2}\chi -1) \\ +&8\pi (3\mathcal{G}^{2}r^{2}+18Br^{2}\chi -4)+384\pi ^{2}Br^{2} \big) \\ -&16\sqrt{c_{3}} \mathcal{G}''(1+c_{3}r^{2})\big(4\pi \sqrt{c_{3}}c_{1} +2\pi c_{2}c_{3}r^{2} \\ -&c_{2}(8\pi +\chi )\big) +2c_{1}c_{3}\big(24 \pi (\mathcal{G}^{2}r^{2}+6Br^{2}\chi -1) \\ +&\chi (3\mathcal{G}^{2}r^{2} +12Br^{2}\chi -2)+384\pi ^{2}Br^{2}\big) \\ +& \sqrt{c_{3}}c_{2}\big( \chi (\mathcal{G}^{2}r^{2}+4Br^{2}\chi +8)+8\pi (\mathcal{G}^{2}r^{2} +6Br^{2}\chi \\ +&6)+128\pi ^{2}Br^{2}\big)+2c_{1}(8\pi +\chi )\big(4B(4 \pi +\chi ) +\mathcal{G}^{2}\big)\Big\} \\ -&16\sqrt{c_{3}}\mathcal{G}'\big(-4\pi c_{3}^{3/2}c_{1}r^{2}+c_{3}^{2}c_{2}r^{4}(2 \pi +\chi ) \\ +&8\pi \sqrt{c_{3}}c_{1} +c_{2}c_{3}r^{2} (16\pi +\chi ) -c_{2}(16\pi +3\chi )\big)\Big]. \end{aligned}$$

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