Charged wormhole solutions in Einstein-Cartan gravity

Static solutions representing wormhole configurations in Einstein-Cartan theory ($\mathsf{ECT}$) in the presence of electric charge are obtained. The solutions are described by a constant redshift function with matter content consisting of a Weyssenhoff fluid along with an anisotropic matter and energy momentum tensor ($\mathsf{EMT}$) of the electric field which together generalize the anisotropic energy momentum tensor in Einstein-Maxwell theory in order to include the effects of the intrinsic angular momentum (spin) of the particles. Assuming the equation of state ($\mathsf{EoS}$) $p_r={\mathsf{w}}_1\rho$ and $p_t={\mathsf{w}}_2\rho$, we derive exact wormhole solutions satisfying weak and null energy conditions. Depending on the value of the spin square density at the wormhole throat, these solutions can be asymptotically flat, de Sitter, or anti–de Sitter. Observational aspects of the wormhole solutions are also studied, using gravitational lensing effects. It is found that the throat can act as a photon sphere near which the light deflection angle as arbitrarily large values. Moreover, for a particular class of solutions, when ${\mathsf{w}}_1\to -{\mathsf{w}}_2$, the lensing features of the present model mimic those of the Ellis wormhole in the weak field limit.

Figure 1

The allowed values of $\mathsf{EoS}$ parameters for ${\mathsf{Q}}_0=1.1$, $\mathrm{\Lambda }=0$ and $r_0=1$.

Figure 2

Plot of ${\mathsf{g}}_{\mathrm{rr}}^{-1}(r)$ (left panel) and the ratio $b(r)/r$ (right panel) for ${\mathsf{Q}}_0=1.1$, $\mathrm{\Lambda }=0$ and $r_0=1$, ${\mathsf{w}}_1=-0.33$, ${\mathsf{w}}_2=1$ (solid curve), ${\mathsf{w}}_1=-1$, ${\mathsf{w}}_2=0.33$ (dashed curve), and ${\mathsf{w}}_1=-1$, ${\mathsf{w}}_2=0.63$ (dot-dashed curve).

Figure 3

The behavior of electric field for ${\mathsf{w}}_1=-0.3$, ${\mathsf{w}}_2=0.8$, ${\mathsf{Q}}_0=1.1$ (solid curve) and ${\mathsf{Q}}_0=3.1$ (dashed curve). Dotted and dot-dashed curves are plotted for the same values but negative sign of (25). We have set $\mathrm{\Lambda }=0$ and $r_0=1$.

Figure 4

The behavior of energy density (upper left), $\rho (r)+p_r(r)$ (upper right) and $\rho (r)+p_t(r)$ (lower left) for ${\mathsf{Q}}_0=1.1$, $\mathrm{\Lambda }=0$, $r_0=1$, ${\mathsf{w}}_1=-0.33$, ${\mathsf{w}}_2=1$ (solid curve), ${\mathsf{w}}_1=-1$, ${\mathsf{w}}_2=0.33$ (dashed curve), ${\mathsf{w}}_1=-1$, ${\mathsf{w}}_2=0.63$ (dot-dashed curve), ${\mathsf{w}}_1=-0.2$, ${\mathsf{w}}_2=2$ (dotted curve). (Lower right) The behavior of square of spin density for ${\mathsf{Q}}_0=1.1$, $\mathrm{\Lambda }=0$, $r_0=1$, ${\mathsf{w}}_1=-0.33$, ${\mathsf{w}}_2=1$ (solid curve), ${\mathsf{w}}_1=-1$, ${\mathsf{w}}_2=0.33$ (dashed curve), ${\mathsf{w}}_1=-1$, ${\mathsf{w}}_2=0.63$ (dot-dashed curve).

Figure 5

(Left panel) The embedding diagram of the wormhole (43) obtained from solution (32) for ${\mathsf{w}}_1=-0.33$, ${\mathsf{w}}_2=1$, $\mathrm{\Lambda }=0$, $r_0=1$ and ${\mathsf{Q}}_0=1.1$. The solid and dashed lines correspond to $+$ and $-$ signs in (46), respectively. (Right panel) Plot of the radial proper distance of the wormhole against radial coordinate for the same parameters of the left panel.

Figure 6

The deflection angle as a function of closest distance approach for ${\mathsf{Q}}_0=1.1$, $\mathrm{\Lambda }=0$ and $r_0=1$. The $\mathsf{EoS}$ parameters are picked up from the gray region of Fig. 1.