Traversable Wormhole Solutions in f (Q, Lm) Gravity

Abstract: We investigate traversable wormhole solutions within the framework of $f(\mathscr{Q},\mathscr{L}_m)$ gravity, a symmetric teleparallel theory featuring non-minimal coupling between geometry and matter. Adopting a linear functional form $f(\mathscr{Q},\mathscr{L}_m) = -α\mathscr{Q} + 2\mathscr{L}_m + β$, we derive the field equations for a static, spherically symmetric Morris-Thorne wormhole geometry with vanishing redshift function. Four distinct shape functions are considered: $b(r)=\sqrt{r_0 r}$, $b(r)=r_0\left(\dfrac{r}{r_0}\right)^γ$ (with $0<γ<1$), and $b(r)=\dfrac{r_0 \ln (r+1)}{\ln (r_0+1)}$. The geometric viability of each configuration is verified through standard traversability conditions, including the flaring-out requirement and asymptotic flatness. We analyze the energy conditions and demonstrate that, consistent with known results in wormhole physics, the null energy condition is violated in the vicinity of the throat, indicating the presence of exotic matter. In addition, we employ embedding diagrams to visualize the spatial geometry of the wormhole solutions, providing a clear geometric interpretation of the flaring-out condition at the throat. Our results suggest that $f(\mathscr{Q},\mathscr{L}_m)$ gravity provides a viable framework for constructing traversable wormholes, with the non-minimal matter-geometry coupling influencing both the geometry and the matter sector.