Axially symmetric relativistic structures and the Riemann curvature tensor

Abstract

Despite the existence of structure scalars in different geometries and modified theories, their importance of static and axially symmetric systems for f(T) gravity (where T is responsible for torsional effects) is still questionable. The novel approach to comprehending the role of structure scalars on static axial symmetric systems in the presence of f(T) gravity is performed in this manuscript. An extensive structure for static and axially symmetric geometry is presented. The line element with compatible anisotropic fluid (which serves as the origin for exterior Weyl spacetime) is contemplated. We initiate by exploring f(T) field equations with the support of non-diagonal tetrads for static axial symmetry. The structure scalars are determined in our scenario. We attain eight distinct sorts of scalars, which are trace and trace-free parts. The three distinct scalars $\mathbb {Y}_{TF_1}$, $\mathbb {Y}_{TF_2}$ and $\mathbb {Y}_{TF_3}$ with f(T) corrections are responsible for the complexity of the system; whereas, the inhomogeneity of the system is controlled by the three other scalars $\mathbb {X}_{TF_1}$, $\mathbb {X}_{TF_2}$ and $\mathbb {X}_{TF_3}$ with f(T) corrections. We assemble the hydrostatic equilibrium equations in terms of f(T) gravity and construct two conformal equations with the support of these equations. In the end, we computed a few analytical solutions in the frame of f(T) gravity. One of them is about the dense spheroid comprising isotropic pressure, and the other is about anisotropic fluid content dealing with inhomogeneity. In the first case, our findings indicate that joining with the Weyl exterior geometry is not possible. On the other hand, the solution associated with anisotropic fluids has smooth joining with the Weyl exterior geometry.

Appendices

Appendix A

The values for $\mathbb {W}_{a}$ where $a=1,2,3,...,15$ are stated as below

$$\begin{aligned} \mathbb {W}_{1}&=H(\theta ,\phi )J(\theta ,\phi )(\frac{K'}{K}-1)+H^2(\theta ,\phi ) (\frac{K'}{K}+\frac{1}{r}+1)\\&+\frac{J(\theta ,\phi )\textrm{sin}\theta }{Kr}(L-L')+\frac{K'}{K}\\&\times H(\theta ,\phi ) \textrm{sin}\theta -\frac{J(\theta ,\phi ) \textrm{sin}\theta }{L}\times (K'r+K),\\ \mathbb {W}_{2}&=H(\theta ,\phi )J(\theta ,\phi ) cot\phi (\frac{K'}{K}-1) -H(\theta ,\phi )J(\theta ,\phi ) cot\theta (\frac{K'r}{K}+1-r)\\&-\frac{J^2(\theta ,\phi ) tan\phi }{K cos\phi }\\&(L-L')-\frac{K H^2(\theta ,\phi )}{L \textrm{sin}\phi }-\frac{Kr J(\theta ,\phi ) \textrm{cos}\theta }{L},\\ \mathbb {W}_{3}&=\frac{H(\theta ,\phi )J(\theta ,\phi ) cot\phi }{2}\times (\frac{K'}{K}-1) \\&+\frac{H^2(\theta ,\phi ) cot\phi }{2}\times (\frac{K'}{K}+1-r)+\frac{J^2(\theta ,\phi ) tan\phi }{2\textrm{cos}\phi }\\&(L'-L)+\frac{H^2(\theta ,\phi )K}{Lr^2\textrm{sin}\phi }+\frac{J(\theta ,\phi ) r K \textrm{cos}\theta }{L},\\ \mathbb {W}_{4}&=\frac{H(\theta ,\phi )J(\theta ,\phi )}{r}(\frac{K'}{K}-1)+H^2(\theta ,\phi ) (\frac{K'}{K}+\frac{1}{r}-1)\\&+\frac{J(\theta ,\phi )\textrm{sin}\theta }{Kr}(L-L')-\frac{K'}{L}\\&\times H(\theta ,\phi ) \textrm{sin}\theta +\frac{J(\theta ,\phi ) \textrm{sin} \theta }{L}\times (K'r+K),\\ \mathbb {W}_{5}&=\frac{K'}{2L}H(\theta ,\phi ) \textrm{sin}\theta -\frac{J(\theta ,\phi ) \textrm{sin}\theta }{2L}(K'r+K)\\&-\frac{H(\theta ,\phi )J(\theta ,\phi )}{r}(\frac{K'}{K}-1) -H^2(\theta ,\phi )\\&(\frac{K'}{K}+\frac{1}{r}-1)-\frac{J(\theta ,\phi ) \textrm{sin}\theta }{Kr} \times (L-L'),\\ \mathbb {W}_{6}&=(\frac{K}{L}+\frac{Kr}{L})\times H(\theta ,\phi )\textrm{cos}\theta +(cot\phi (\frac{K'}{K}-1)\\&+ cot\theta (\frac{K'r}{K}+1-r))H(\theta ,\phi )J(\theta ,\phi )\\&+\frac{H(\theta ,\phi ) tan\theta \textrm{sin}\theta }{K}\times (L'-L),\\ \mathbb {W}_{7}&=\frac{H(\theta ,\phi )J(\theta ,\phi )}{r}(\frac{K'}{K}-1) +H^2(\theta ,\phi )(\frac{K'}{K}+\frac{1}{r}-1)\\&+\frac{J(\theta ,\phi )\textrm{sin}\theta }{Kr}(L-L') -\frac{K'}{L}\\&\times H(\theta ,\phi )\textrm{sin}\theta +\frac{J(\theta ,\phi )\textrm{sin}\theta }{L} \times (K'r+k),\\ \mathbb {W}_{8}&=2\bigg \{(Tf_{T}-f)-f_{TT}S_{\upsilon }^{~\omega \rho } \delta ^{\upsilon }_{\omega }\delta ^{\rho }_{\rho }\nabla _{\rho } T\bigg \},\\ \mathbb {W}_{9}&=\bigg [\frac{Tf_{T}-f}{2}g_{\omega \upsilon }-f_{TT}S_{\upsilon }^{~\omega \rho } g_{\omega \omega }\delta ^{\rho }_{\rho }\nabla _{\rho } T\bigg ],\\ \mathbb {W}_{10}&=\frac{1}{3}\bigg [\frac{Tf_{T}-f}{2}-f_{TT}S_{\upsilon }^{~\omega \rho } \delta ^{\upsilon }_{\omega }\delta ^{\rho }_{\rho }\nabla _{\rho } T\bigg ],\\ \mathbb {W}_{11}&=-\frac{2}{3}\bigg [(Tf_{T}-f)-f_{TT}S_{\upsilon }^{~\omega \rho } \delta ^{\upsilon }_{\omega }\delta ^{\rho }_{\rho }\nabla _{\rho } T\bigg ],\\ \mathbb {W}_{12}&=\frac{}{}\bigg \{(Tf_{T}-f)-f_{TT}S_{\upsilon }^{~\omega \rho } \delta ^{\upsilon }_{\omega }\delta ^{\rho }_{\rho }\nabla _{\rho } T+2f_{TT}S^{\rho }\nabla _{\rho } T\\&+2f_{TT}u^{\sigma }S_{\sigma }^{~\rho }\nabla _{\rho } T\bigg \}+\bigg \{\frac{Tf_{T}-f}{2}\\&-f_{TT}S_{\upsilon }^{~\omega \rho }\delta ^{\upsilon }_{\omega } \delta ^{\rho }_{\rho }\nabla _{\rho } T\bigg \},\\ \mathbb {W}_{13}&=\frac{1}{2}\bigg \{\frac{Tf_{T}-f}{2}g_{\omega \upsilon } -f_{TT}S_{\upsilon }^{~\omega \rho }g_{\omega \omega }\delta ^{\rho }_{\rho }\nabla _{\rho } T\\&+f_{TT}u_{\upsilon }S_{\omega }^{~\rho }\nabla _{\rho }T+f_{TT}u_{\omega }S_{\upsilon }^{~\rho } \nabla _{\rho }T+f_{TT}\\&g_{\omega \upsilon }u^{\sigma }S_{\upsilon }^{~\rho }\nabla _{\rho }T\bigg \},\\ \mathbb {W}_{14}&=\frac{1}{3}\bigg \{\frac{Tf_{T}-f}{2}-f_{TT}S_{\upsilon }^{~\omega \rho } \delta ^{\upsilon }_{\omega }\delta ^{\rho }_{\rho }\nabla _{\rho }T\bigg \}h_{\omega \upsilon },\\ \mathbb {W}_{15}&=\bigg [\frac{1}{2}\bigg \{(Tf_{T}-f)-f_{TT}S_{\upsilon }^{~\omega \rho } \delta ^{\upsilon }_{\omega }\delta ^{\rho }_{\rho }\nabla _{\rho }T+2f_{TT}S^{\rho }\nabla _{\rho }T +2f_{TT}u^{\sigma }S_{\sigma }^{\rho }\nabla _{\rho }T\bigg \}+\bigg (\frac{Tf_{T}-f}{2}\\&-f_{TT}S_{\upsilon }^{~\omega \rho }\delta ^{\upsilon }_{\omega } \delta ^{\rho }_{\rho }\nabla _{\rho }T\bigg )\bigg ]h_{\omega \upsilon }. \end{aligned}$$

Appendix B

Here, the values for corrections $\mathbb {Z}_{1}^{(D)}$, $\mathbb {Z}_{2}^{(D)}$ are given below

$$\begin{aligned} \mathbb {Z}_{1}^{(D)}&=\frac{I'}{I^3K^2}T_{00}^{(T)}+\bigg \{\bigg (\frac{I'}{I}+\frac{1}{r} +\frac{L'}{L}\bigg )\frac{1}{K^4}+\bigg (\frac{2}{K^2}+1\bigg )\frac{K'}{K^3}\bigg \}T_{11}^{(T)}\\&-\bigg (K'-\frac{1}{Kr}\bigg )\frac{T_{22}^{(T)}}{K^3r^2}\\&-\frac{L'}{L^3K^2}T_{33}^{(T)} +T_{,1}^{11(T)}++T_{,2}^{12(T)},\\ \mathbb {Z}_{2}^{(D)}&=\frac{I_{\theta }}{I^3K^2r^2}T_{00}^{(T)}- \frac{K_{\theta }}{K^5r^2}T_{11}^{(T)}\bigg \{\bigg (\frac{I'}{I}+\frac{3K'}{K} \bigg )\frac{1}{K^4r^2}+\bigg (\frac{2}{r}+\frac{K'}{K}\bigg )\frac{1}{K^4r^2}\bigg \}T_{12}^{(T)}\\&+ \bigg \{\bigg (\frac{I_{\theta }}{I}+\frac{3K_{\theta }}{K} \bigg )\frac{1}{K^4r^4}+\frac{L_{\theta }}{K^4Lr^4}\bigg \}T_{22}^{(T)}\\&-\frac{L_{\theta }}{K^2r^2L^3}T_{33}^{(T)}+T_{,2}^{22(T)}++T_{,1}^{21(T)}. \end{aligned}$$

The values for conformal scalars $\psi _{a}$ where $a=1,2,3$ are as

$$\begin{aligned} \psi _{1}&=\frac{1}{2K^2}\bigg [\frac{1}{r}\bigg \{\frac{I'_{\theta }}{I}-\frac{L'_{\theta }}{L} -\frac{K_{\theta }I'}{KI}+\frac{L' K_{\theta }}{LK}-\frac{K' I_{\theta }}{KI}+ \frac{L_{\theta }K'}{LK}\bigg \}\\&+\frac{1}{r^2}\bigg (\frac{L_{\theta }}{L} -\frac{I_{\theta }}{I}\bigg ) \bigg ],\\ \psi _{2}&=-\frac{1}{2K^2}\bigg [-\frac{I''}{I}+\frac{K''}{K}+\frac{I'K'}{IK}+\frac{I'L'}{IL} -\bigg (\frac{K'}{K}\bigg )^2-\frac{K'L'}{KL}\\&+\frac{1}{r}\bigg (\frac{K'}{K}-\frac{L'}{L} \bigg )\bigg ]\\&-\frac{1}{2K^2r^2}\bigg [\frac{K_{\theta \theta }}{K}-\frac{L_{\theta \theta }}{L} -\frac{I_{\theta }K_{\theta }}{IK}+\frac{I_{\theta }L_{\theta }}{IL} -\bigg (\frac{K_{\theta }}{K}\bigg )^2 +\frac{K_{\theta }L_{\theta }}{KL}\bigg ],\\ \psi _{3}&=-\frac{1}{2K^2}\bigg [\frac{K''}{K}-\frac{L''}{L}-\frac{I'K'}{IK}+\frac{I'L'}{IL} -\bigg (\frac{K'}{K}\bigg )^2+\frac{K'L'}{KL}\\&+\frac{1}{r}\bigg (\frac{K'}{K}-\frac{I'}{I} \bigg )\bigg ]\\&-\frac{1}{2K^2r^2}\bigg [\frac{K_{\theta \theta }}{K}-\frac{I_{\theta \theta }}{I} +\frac{I_{\theta }K_{\theta }}{IK}+\frac{I_{\theta }L_{\theta }}{IL} -\bigg (\frac{K_{\theta }}{K}\bigg )^2 -\frac{K_{\theta }L_{\theta }}{KL}\bigg ]. \end{aligned}$$

The values for f(T) correction terms $T_{aa}^{(D)}$ where $a=1,2,3$ are given below

$$\begin{aligned} T_{00}^{(D)}&=-\frac{I^2}{8\pi }\bigg \{\frac{Tf_{T}-f}{2}+\frac{f_{TT}}{2K^2}\bigg (\frac{I'}{2I} +\mathbb {W}_{1}\bigg )T'+\frac{1}{r^2}\bigg (\frac{I_{\theta }}{2I}-\mathbb {W}_{2}\bigg )T_{\theta } \bigg \},\\ T_{11}^{(D)}&=\frac{K^2}{8\pi }\bigg \{\frac{Tf_{T}-f}{2}+\frac{f_{TT}}{2K^2r^2}\bigg ( \frac{I_{\theta }}{I}+\mathbb {W}_{3}\bigg )T'\bigg \},\\ T_{22}^{(D)}&=-\frac{K^2r^2}{8\pi }\bigg \{\frac{Tf_{T}-f}{2} -\frac{f_{TT}}{2K^2}\bigg (\mathbb {W}_{4}-\frac{I'}{I}\bigg )T'\bigg \},\\ T_{33}^{(D)}&=\frac{L^2}{8\pi }\bigg \{\frac{Tf_{T}-f}{2}-\frac{f_{TT}}{2K^2} \bigg (\mathbb {W}_{5} -\frac{I'}{I}\bigg ) T' +\frac{1}{r^2}\bigg (\mathbb {W}_{6}-\frac{I_{\theta }}{I}\bigg )T_{\theta }\bigg \},\\ T_{12}^{(D)}&=\frac{f_{TT}}{16\pi }\bigg \{\mathbb {W}_{7}-\frac{I'}{I}\bigg \}T_{\theta }. \end{aligned}$$