Traversable Wormholes in Loop Quantum Gravity

• Traversable wormholes from Loop Quantum Gravity are Lorentzian geometries where quantum spacetime discreteness modifies Einstein’s equations, reducing the need for exotic matter.
• Key LQG parameters, such as the polymerization parameter and self-dual radius, regulate the wormhole throat to ensure regularity and improved stability.
• This framework minimizes energy condition violations while providing equilibrium solutions with macroscopic traversability and finite curvature invariants.

Traversable wormholes constructed from the quantum gravity program of Loop Quantum Gravity (LQG) represent a class of Lorentzian geometries in which classical energy conditions are effectively violated by modifications to the Einstein field equations rooted in quantum discreteness of spacetime. These solutions describe macroscopic, static, spherically symmetric wormhole throats regularized by LQG corrections, exhibiting reduced requirements for exotic matter and improved stability properties when compared to classical (General Relativistic) counterparts. The archetypal construction connects with the effective anisotropic stress‐energy of self‐dual black holes discovered by Modesto and incorporates key quantum parameters linked to LQG's underlying structure (Cruz et al., 2024).

1. Metric Structure and LQG Parameters

The canonical ansatz for a static traversable wormhole is the Morris–Thorne metric: $$ds^2 = -e^{2\Phi(r)} dt^2 + \frac{dr^2}{1 - b(r)/r} + r^2 (d\theta^2 + \sin^2\theta\,d\varphi^2)$$ where $\Phi(r)$ is the redshift function, required to be finite for horizon avoidance, and $b(r)$ is the shape function, which must satisfy $b(r_0) = r_0$ at the throat and $b'(r_0) < 1$ for the flaring-out condition.

Within LQG, two main quantum parameters modify the solution:

• The polymerization parameter $\epsilon = \gamma \delta_b$, with $\gamma$ the Barbero–Immirzi parameter and $\delta_b$ the fundamental polymer scale, yielding the polymeric function $$\mathcal{P} = [\sqrt{1+\epsilon^2}-1]/[\sqrt{1+\epsilon^2}+1] \approx \epsilon^2/4$$ for small $\epsilon$.
• The self-dual radius $a_0 = A_{\min}/(8\pi)$, encoding the minimal area gap predicted by the quantum geometry of LQG.

These parameters alter the source terms of the Einstein equations, leading to effective matter contents and modified geometry that interpolate between fully classical and strongly quantum-regulated regimes (Cruz et al., 2024). For macroscopic (astrophysically large) throats, these parameters remain small, yet exert significant influence near the throat.

2. Self-Dual Black Hole Source and Stress–Energy Components

LQG modifications are encapsulated by the effective matter stress–energy tensor: $$T^\mu{}_\nu = \mathrm{diag}(-\rho(r),\,p_r(r),\,p_t(r),\,p_t(r))$$ with the energy density $\rho(r)$ for the self-dual source: $$\rho(r) = \frac{4 r^4 \, \left[a_0^4\,\tfrac{r_0}{2}(1+\mathcal P)^2 + \tfrac{r_0^2}{2} \mathcal P (1+\mathcal P)^2\, r^7 - a_0^2 r^2 (r_0 \mathcal P + r)\, \mathcal N(r) \right]}{(r_0 \mathcal P + r)^3\,(a_0^2 + r^4)^3}$$ where $\mathcal{N}(r)$ is a polynomial in $r,r_0,\mathcal{P}$. The radial and tangential pressures, $p_r(r)$ and $p_t(r)$, are fixed by the modified Einstein equations and depend algebraically on $(r, r_0, a_0, \mathcal{P})$.

The 'self-dual' property ensures local regularity: all curvature invariants, including the Ricci scalar $R$ and Kretschmann scalar $K = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}$, remain finite everywhere due to the presence of $a_0$ and $\mathcal{P}$ (Cruz et al., 2024).

3. Classes of Traversable Wormhole Solutions

Solutions fall into several families distinguished by the form of the redshift function $\Phi(r)$:

• Zero-tidal wormholes (Case A): $\Phi'(r) = 0$. Shape function approximated as

$$b(r) \approx r_0 (1 + 2\mathcal{P}) + \frac{a_0^2}{r_0^3}(\tfrac{\mathcal{P}}{5}-\tfrac12) - \frac{2\mathcal{P} r_0^2}{r} + \frac{4a_0^2}{r^3} - \dots$$

Asymptotically flat, satisfies all standard wormhole geometric constraints.

• Nonconstant redshift wormholes:
  • Case B: $\Phi(r) = r_0/r$ yields a monotonic redshift, with no added horizons.
  • Case C: $\Phi(r)$ logarithmically depends on quantum parameters, further amplifying curvature near the throat.
  • Case D: Barotropic equation of state $p_r = \omega \rho$. Imposing regularity at $r = r_0$ fixes $\omega$; for $r_0\gg\sqrt{a_0}$, one obtains $\omega<-1$ (phantom), with regular, traversable throats.

Increasing $\mathcal{P}$ steepens the throat profile (via embedding diagrams) and enhances localized curvature, but does not induce singular behavior (Cruz et al., 2024).

4. Embedding, Curvature, and Energy Condition Analysis

Embedding diagrams, constructed on equatorial slices using

$\frac{dz}{dr} = \pm \sqrt{\frac{b(r)}{r-b(r)}}$

display standard ‘flaring out’ for the wormhole throat, with increasing $\mathcal{P}$ leading to sharper curvatures.

Curvature invariants behave as follows:

• For zero-tidal solutions, $R$ is finite everywhere and peaked just outside the throat.
• For nonconstant redshift cases, both $R$ and $K$ increase close to the throat as LQG corrections intensify, denoting strong—but finite—quantum gravity effects.

The Null Energy Condition (NEC), $\rho + p_r \geq 0$, is violated in a finite region surrounding the throat for all solution classes, signifying unavoidable exoticity. However, as $\mathcal{P}$ increases, the spatial extent and magnitude of NEC violation decrease linearly in $1-2\mathcal{P}$, approaching null exoticity for $\mathcal{P} \to 1/2$. Volume Integrals Quantifying (VIQ) the total exotic matter yield

$$\mathcal{I}_q \longrightarrow -8\pi \sqrt{a_0} (1-2\mathcal{P})$$

in the macroscopic limit, which vanishes as quantum corrections approach their maximal allowed value (Cruz et al., 2024).

Weak and strong energy conditions are also violated near the throat, but in all cases, violations are minimized for larger quantum corrections.

5. Equilibrium, Stability, and Traversability

The generalized Tolman–Oppenheimer–Volkoff (TOV) equilibrium is governed by anisotropy and quantum corrections: $$-\,\frac{dp_r}{dr} - \Phi'(r)(\rho+p_r) + \frac{2}{r}(p_t-p_r) = 0$$ Decomposing into hydrostatic, gravitational, and anisotropy forces: $$F_h = -\frac{dp_r}{dr},\quad F_g = -\Phi'(r)(\rho+p_r),\quad F_a = \frac{2}{r}(p_t-p_r)$$ In zero-tidal solutions, equilibrium reduces to $F_h + F_a = 0$. For $\Phi' \neq 0$, stronger LQG effects (“softening” of forces) render equilibrium easier to maintain and enhance the parameter space for stable throats.

Tidal forces remain finite at the throat for all $\mathcal{P}$, ensuring the traversability criteria are satisfied without introducing divergent accelerations (Cruz et al., 2024).

6. Comparative Context and Physical Implications

LQG-induced traversable wormholes interpolate between purely classical (General Relativity) geometries (recovered for $\mathcal{P} = 0$) and quantum-dominated regimes with strong lattice-scale spacetime effects. The quantum origin of exoticity is manifest: the effective matter source is purely geometric (arising from LQG corrections), not requiring ad hoc introduction of phantom or anisotropic physical fields. Increasing LQG corrections exponentially reduces the requirement or total volume of exotic matter—indicating that quantum spacetime effects may generically 'naturalize' traversable wormhole geometries under otherwise realistic, non-pathological stress–energy (Cruz et al., 2024).

In summary, traversable wormholes from Loop Quantum Gravity provide a technically consistent and quantum-geometrically motivated mechanism for microscopic or macroscopic throats supported by greatly diminished exotic matter, regular everywhere, and possessing equilibrium and stability domains beyond those accessible to classical General Relativity. The quantum parameters inherent in LQG directly dictate the throat properties, curvature maxima, and the admissibility of solutions, making spacetime discreteness a central actor in the viability of such wormholes.