Hyperbolically symmetric sources, a comprehensive study in f(T) gravity

Abstract

The theme of this work is to carry a comprehensive analysis on static fluid distribution enclosed with hyperbolically symmetric source in the framework of f(T) gravity as an extension of the work of Herrera et al. (Phys Rev D 103:024037, 2021). We study their physical characteristics in depth. We characterize the stress-energy tensor via thoroughly examining the constituents of the tetrad field in the Minkowski co-ordinate structure and set forth modified field equations. The emergence of negative density implies quantum impacts, as well as extreme limitations, are required to explain any physical implementation for this sort of fluid. Along with this, it is important to note that the region near the center of symmetry is null. This kind of region is designated as a vacuum cavity about the center. We apply the definition of mass and Tolman mass to calculate its value in the f(T) scenario. The value of the mass function is positive, whereas the Tolman mass discloses its negative nature, exhibiting that gravitational interaction is repulsive. Also, we calculate the structure scalars in the f(T) scenario by implying Herrera’s strategy. A standard strategy to attain an accurate solution is given and some accurate analytical solutions are determined in structure of f(T) gravity.

Appendices

Appendix A

The dark source term is given as

$$\begin{aligned} l_{1}&=-\frac{\nu 'e^{-2\lambda }}{32\pi }\bigg (\frac{\nu '}{4}-\frac{2}{r}\bigg )T'+\frac{\lambda 'e^{-\lambda }}{16\pi } \left[ \left\{ Tf_{T}-f\right\} _{,1}-(Tf_{T}-f)\right] \\&\quad +\frac{e^{-2\lambda }}{r}\left[ \frac{f_{TT}}{16\pi }\bigg (\frac{3}{r}-\nu '\bigg )\right] . \end{aligned}$$

The value for $\Theta ^{(T)}_{00}$,$\Theta ^{(T)}_{11}$ $\Theta ^{(T)}_{22}$ and $k_1(r)$ is as

$$\begin{aligned} \Theta ^{(T)}_{00}&=\frac{e^{\nu }}{16\pi }\left\{ (Tf_{T}-f)-f_{TT}e^{-\lambda }\bigg (\frac{\nu '}{4}-\frac{2}{r}\bigg )T'\right\} ; \\ \Theta ^{(T)}_{11}&=-\frac{e^{\lambda }}{8\pi }\bigg (\frac{Tf_{T}-f}{2}\bigg ), \\ \Theta ^{(T)}_{22}&=-\frac{r^{2}}{16\pi }\left\{ (Tf_{T}-f)+\frac{f_{TT}}{2}\bigg (\frac{3e^{-\lambda }}{r}-e^{-\lambda }\nu '\bigg )T'\right\} ; \\ K_1(r)&=e^\frac{{(\nu +\lambda })}{2}\bigg (\Theta ^{(T)}_{00}+\Theta ^{(T)}_{11}+\Theta ^{(T)}_{22}\bigg ){\mathrm {d}}\tilde{r}^{3}. \end{aligned}$$

The value of $\chi _{1}$,$\chi _{2}$ and $\chi _{3}$ are as below

$$\begin{aligned} \chi _{1}&=-\frac{4\pi }{f_{T}}\int ^{r}_{0}\bigg (2\Theta ^{(T)}_{00}-\Theta ^{(T)}_{11}+\Theta ^{(T)}_{22}\bigg ){\mathrm {d}}r -4\pi \Theta ^{(T)}_{00}-4\pi \Theta ^{(T)}_{22}\\&\qquad -\frac{1}{4\pi r^{4}}\left\{ k_{1}(r)+\int ^{r}_{0} \left\{ k_1(r){\mathrm {d}}\tilde{r}\right\} _{,1}\right\} , \\ \chi _{2}&=\frac{1}{2}\left\{ \bigg (\frac{Tf_{T}-f}{2}\bigg )\delta _{\delta \gamma }+f_{TT} S_{\delta \gamma \rho }\delta ^{\delta }_{\delta }\delta ^{\rho }_{\rho }\nabla _{\rho }T\right\} \\&\quad +\frac{1}{3}\left\{ \bigg (\frac{Tf_{T}-f}{2}\bigg )-f_{TT}\delta ^{\rho }_{\rho }S^{\delta \rho }\nabla _{\rho }T\right\} \mathfrak {h}_{\delta \gamma }, \\ \chi _{3}&=\frac{1}{2}\left\{ \bigg (\frac{Tf_{T}-f}{2}\bigg )g_{\delta \gamma }-f_{TT}S^{\delta \rho }_{\gamma }g_{\delta \delta } \delta ^{\rho }_{\rho }\nabla _{\rho }T+\frac{f_{TT}}{2}(V_{\gamma }+V_{\delta }-g_{\delta \gamma }V^{\mu })S^{\rho }_{\delta } \nabla _{\rho }T\right\} \\ {}&\quad -\frac{\mathfrak {h}_{\delta \gamma }}{2}\left\{ \bigg (\frac{Tf_{T}-f}{2}\bigg )-f_{TT}S^{\delta \rho }\nabla _{\rho }T\right\} . \end{aligned}$$

The value of $\Psi _{1}$,$\Psi _{2}$,$\Psi _{3}$,$\Psi _{4}$ $\Psi _{4i}$ and $\Psi _{5}$ are given as

$$\begin{aligned} \Psi _{1}&=\frac{1}{9}\left\{ \frac{Tf_{T}-f}{2}-f_{TT}\delta ^{\rho }_{\rho } S^{\delta \rho }\nabla _{\rho }T\right\} ; \\ \Psi _{2}&=\frac{1}{2}\left\{ \frac{Tf_{T}-f}{2}\delta _{\delta \gamma }+f_{TT} S_{\delta \gamma \rho }S^{\rho }_{\delta }\delta ^{\rho }_{\rho }\nabla _{\rho }T\right\} , \\ \Psi _{3}&=-\frac{1}{6}\left\{ \frac{Tf_{T}-f}{2}-f_{TT}S^{\delta \rho }\nabla _{\rho }T\right\} ; \\ \Psi _{4}&=\frac{1}{2}\left\{ (\frac{Tf_T-f}{2})g_{\delta \gamma }-f_{TT}S^{\delta \rho }_{\gamma } g_{\delta \delta }\delta ^{\rho }_{\rho }\nabla _{\rho }T+\frac{f_{TT}}{2}(V_{\gamma }+V_{\delta }-g_{\delta \gamma }V^{\mu })S^{\rho }_{\delta } \nabla _{\rho }T\right\} , \\ \Psi _{4i}&=-4\pi \Theta ^{(T)}_{00}-4\pi \Theta ^{(T)}_{22}-\frac{1}{4\pi r^{4}}\left\{ k_{1}(r)+\int ^{r}_{0}k_{1}(r){\mathrm {d}}\tilde{r}\right\} ; \\ \Psi _{6}&=\left\{ (Tf_{T}-f)+\frac{f_{TT}}{2}\bigg (\frac{2m}{r}-1\bigg )\bigg (3-2(2z-1\bigg )\bigg (\frac{2m}{r}-1)\bigg )T'\right\} . \end{aligned}$$

Appendix B

The value of $\tau _{1}$, $\tau _{2}$, $\tau _{3}$, $\tau _{4}$, $\tau _{5}$ and $\tau _{6}$ are as

$$\begin{aligned} \tau _{1}&=\frac{24}{r^{4}(9g-4)^{2}}\left[ 2\frac{Tf_{T}-f}{f_{T}}+\bigg (\frac{Tf_{T}-f}{f_{T}}\bigg )'\right] \\ {}&\quad +\frac{6}{r^{3}(2g-1)(9g-4)^{2}f_{T}}\left[ \frac{-2gg'(Tf_{T}-f)}{r} +\frac{(Tf_{T}-f)^{2}}{f_{T}}\right] , \\ \tau _{2}&=\frac{r^{2}(Tf_{T}-f)}{2f_{T}}\left\{ \frac{(Tf_{T}-f)}{8(2g-1)f_{T}}+3g-1\right\} ; \\ \tau _{3}&=-\frac{1}{16\pi }\left\{ (Tf_{T}-f)-f_{TT}(2g-1)\bigg (\frac{\tau _{2}}{4\beta g^{2}(2g-1)}-\frac{2}{r}\bigg )T'\right\} \\ \tau _{4}&=-\bigg (\frac{3r\lambda '}{2}\frac{-2g'}{(2g-1)}-1 +r\bigg )r^{2}\frac{Tf_{T}-f}{f_{T}}-r^{3}\bigg (\frac{Tf_{T}-f}{f_{T}}\bigg )' -\frac{r^{4}}{2f_{T}^{2}}\bigg (\frac{Tf_T-f}{2g-1}\bigg )^{2}, \\ \tau _{5}&=\bigg (\frac{gr}{(2g-1)^{2}}+g(g-1)-\frac{g'r}{4(2g-1)^{2}}\bigg )+\frac{Tf_{T}-f}{f_{T}}-\frac{r}{4(2g-1)} \bigg (\frac{Tf_{T}-f}{f_{T}}\bigg )'\\ {}&\quad +\frac{r^{2}}{8(2g-1)^{3}} \bigg (\frac{Tf_{T}-f}{f_{T}}\bigg )^{2}+\frac{r^{2}}{f_{T}}\left\{ (Tf_{T}-f)\right. \\&\left. \quad +\frac{f_{TT}}{2}(2g-1)\bigg (\frac{3}{r} +\frac{2g}{r(2g-1)}+\frac{r(Tf_{T}-f)}{2f_{T}(2g-1)^{2}}\bigg )T'\right\} , \\ \tau _{6}&=\frac{f_{T}}{8\pi }\left[ \frac{-r}{2(2g-1)}\bigg (\frac{Tf_{T}-f}{f_{T}}\bigg )' +\bigg (\frac{1}{2(2g-1)}+2g'(2g-1)r\right. \\ {}&\quad \left. +\frac{r^{2}}{8f_{T}(2g-1)^{4}}-\frac{1}{(2g-1)^{2}}\bigg )(\frac{Tf_{T}-f}{f_{T}})\right] -\Theta ^{(T)}_{00}. \end{aligned}$$

The value of $\zeta _{1}$, $\zeta _{2}$, $\zeta _{3}$, $\zeta _{4}$, $\zeta _{5}$ and $\zeta _{6}$ is as given

$$\begin{aligned} \zeta _{1}&=-\frac{4\pi r^{2}}{f_{T}}\int ^{r}_{0}\bigg (\frac{l_{1}r}{3(8\pi )^{2}}-\frac{r^{2}}{(8\pi )^{2}} \int ^{r}_{0}\frac{r^{3}l_{1}'}{3}dr+\frac{r^{2}\Theta ^{(T)}_{00}}{f_{T}}\bigg ){\mathrm {d}}r; \\ \zeta _{2}&=\frac{r l_{1}}{3(8\pi )^{2}}-\frac{r^{2}}{(8\pi )^{2}}\int ^{r}_{0}\frac{r^{3}l_{1}'}{3}{\mathrm {d}}r, \\ \zeta _{3}&=\int \left[ \frac{8\zeta _{2}}{r}+6\int \bigg (\frac{\zeta _{2}}{r^{2}}+\frac{l_{1}'}{(8\pi )^{2}}\bigg ){\mathrm {d}}r\right] ; \\ \zeta _{4}&=4\pi \int ^{r}_{0}\left( \frac{r^{2}\Theta ^{(T)}_{00}}{f_{T}}-\frac{\zeta _{3}r^{2}}{f_{T}}-\bigg (rd -\frac{r^{3}c}{3}\bigg )\bigg (\frac{1}{f_{T}}\bigg )'\right) {\mathrm {d}}r, \\ \zeta _{5}&=-\frac{4\pi d}{r^{2}_{\Xi ^e}}\int ^{r}_{0}\bigg (r r^{2}_{\Xi ^e}-\frac{r^{3}}{3}\bigg )\bigg (\frac{1}{f_{T}}\bigg )'{\mathrm {d}}r +4\pi d\int ^{r}_{0}\bigg (2\zeta _{3}+\frac{r^{2}\Theta ^{(T)}_{00}}{f_{T}}\bigg ){\mathrm {d}}r; \\ \zeta _{6}&=r(\zeta _{3}+\zeta _{3}')-\frac{d}{r^{2}_{\Xi ^e}}+\frac{rl_{1}}{2(8\pi )^{2}}. \end{aligned}$$

The value of $\nu '$ is as below

$$\begin{aligned} \nu '=\frac{\frac{8\pi }{f_{T}}(rd-\zeta _{3}r^{3})-\frac{16\pi c r^{3}}{3f_{T}}(1+r\Theta ^{(T)}_{11}-\frac{8\pi rd}{f_{T}}(2r^{2}\Theta ^{(T)}_{11}+1)+2\zeta _{4}(1-2r^{2}\Theta ^{(T)}_{11})-2r^{3}\Theta ^{(T)}_{11}}{r(\frac{8\pi rd}{f_{T}}-\frac{8\pi r^{3}c}{3}+2\zeta _{4}-r)^{2}} \end{aligned}$$