Traversable Wormhole Solutions in f (Q, Lm) Gravity

TL;DR

The paper tackles the problem of constructing traversable wormholes within a symmetric teleparallel framework that features a non-minimal coupling between geometry and matter, encapsulated by the linear model $f(\mathscr{Q},\mathscr{L}_m) = -\alpha\mathscr{Q} + 2\mathscr{L}_m + \beta$. It derives the corresponding field equations for a Morris–Thorne wormhole with $\Phi(r)=0$ and analyzes three distinct shape functions to assess geometric viability via flaring-out, asymptotic flatness, and horizon-avoidance, as well as energy-condition profiles and embedding diagrams. The results show that all three shape functions satisfy the essential geometric requirements, while the null energy condition is violated near the throat, indicating exotic matter, albeit with the non-minimal coupling redistributing this exoticity into the geometric sector. Embedding diagrams corroborate the throat structure and asymptotic behavior, illustrating smooth, traversable geometries in this theory. Overall, $f(\mathscr{Q},\mathscr{L}_m)$ gravity provides a viable setting for wormholes, with tunable coupling parameters $\alpha$ and $\beta$ and potential to minimize exotic-matter requirements compared to curvature-based alternatives.

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.

Figures (5)

• Figure 1: Geometric constraints for wormhole shape functions. (a) $b(r)<r$ ensures a proper throat. (b) $b'(r)<1$ satisfies the flaring-out condition. (c) Positivity of $\frac{b(r)-rb'(r)}{b(r)^2}$ validates traversability. (d) Asymptotic flatness $\frac{b(r)}{r}\to0$. (e) $b(r)-r<0$ reinforces the throat condition. (f) $1-\frac{b(r)}{r}>0$ ensures horizon absence. Different colors correspond to different shape functions.
• Figure 2: Energy condition profiles for different wormhole shape functions. (a) Energy density $\rho(r)$. (b) NEC term $\rho(r)+p(r)$. (c) DEC term $\rho(r)-p(r)$. (d) SEC term $\rho(r)+3p(r)$. The violation of NEC near the throat is evident, confirming the need for exotic matter.
• Figure 3: Embedding diagram for the shape function $b_1(r)$. The left panel shows the two-dimensional embedding profile $z(r)$, while the right panel depicts the corresponding three-dimensional embedding surface, illustrating a smooth throat and outward flaring geometry.
• Figure 4: Embedding diagram for the shape function $b_2(r)$. The embedding surface exhibits a well-defined throat and satisfies the flaring-out condition, confirming the traversable nature of the wormhole geometry.
• Figure 5: Embedding diagram for the shape function $b_3(r)$. The geometry smoothly opens away from the throat and approaches a nearly flat configuration at large radial distances.