Non-Stationary Wormhole Structure

• Non-stationary wormhole structure is a dynamic spacetime configuration defined by time-dependent metric components that govern evolving throat geometry.
• This topic examines mechanisms such as cosmological embedding and thin-shell dynamics that influence gravitational wave emission and black hole evaporation.
• Its study bridges classical and quantum gravity by addressing energy condition dynamics, stability under perturbations, and the role of modified gravity.

A non-stationary wormhole structure is a spacetime configuration exhibiting a wormhole geometry whose metric components, throat properties, or matter content evolve explicitly with respect to time, as opposed to static or stationary wormholes which are invariant under time translations. Non-stationary wormholes are of central interest in gravitational theory, cosmology, quantum gravity, and high-energy physics due to their connections with black hole evaporation, dynamical topology change, gravitational wave emission, and the possible realization of traversable signals or entanglement transfer in quantum spacetime.

1. Definitional Framework and Geometric Structure

Non-stationary wormhole spacetimes generalize the static Morris–Thorne class by allowing explicit time dependence in the metric or stress–energy. The general line element supporting such structures is

$$ds^2 = -e^{2\phi(r,t)} dt^2 + a^2(t) \left[ \frac{dr^2}{1 - \frac{b(r)}{r}} + r^2 d\Omega^2 \right],$$

where $\phi(r,t)$ is the redshift function (possibly time-dependent), $a(t)$ is a cosmological scaling factor or conformal prefactor, and $b(r)$ is the shape function governing the wormhole throat. The flare-out and regularity conditions remain as in the stationary case:

• Throat at $r_0$ defined by $b(r_0) = r_0$.
• Flare-out condition $b'(r_0) < 1$. Time dependence enters either through $a(t)$ (embedding in a cosmological background), $\phi(r,t)$ (time-varying redshift), $b(r,t)$ (dynamical shape), or their combinations. In more exotic settings, additional degrees of freedom such as evolving scalar fields or higher-curvature terms play a central structural role (Bhattacharya et al., 2021, Zangeneh et al., 19 Jul 2025, Alencar et al., 7 Aug 2025).

2. Energy Conditions and Exotic Matter Requirements

A central obstacle in wormhole physics is the need for matter violating the null energy condition (NEC) at or near the throat. For non-stationary solutions, explicit expressions for energy density and pressure components acquire time-dependent corrections: $$\begin{aligned} \rho(r,t) &= \frac{1}{8\pi a^2(t)} \left[3H^2 e^{-2\phi(t)} + \frac{b'(r)}{r^2} \right], \ p_r(r,t) &= \frac{1}{8\pi a^2(t)} \left[ -2\dot H - 3H^2 + 2H\dot\phi(t) \right] e^{-2\phi(t)} - \frac{b(r)}{a^2(t)r^3}, \ p_t(r,t) &= ... \end{aligned}$$ Here, $H = \dot a / a$, and the rightmost terms involve $b(r)$ and its derivatives; see (Bhattacharya et al., 2021, Zangeneh et al., 19 Jul 2025). The key point is that certain choices of $a(t)$ and $\phi(t)$—especially during rapid cosmic expansion or near a cosmological bounce—can render $\rho + p_r \geq 0$ or at least minimize NEC violation over a specific domain and cosmic epoch (Zangeneh et al., 19 Jul 2025, Chakraborty et al., 2 Sep 2024). In settings with anisotropic matter, noncommutative geometry, or modified gravity (e.g., $f(R)$ or hybrid metric–Palatini gravity), further freedom sometimes allows full satisfaction of the NEC and WEC (Rahman et al., 2022, Kavya et al., 2023, Ziaie et al., 17 Feb 2025).

In many scenarios, the supporting matter content is dynamically exotic: canonical scalar fields with a wrong-sign kinetic term (phantom or ghost), cuscuton fields, non-minimally coupled scalar/tensor sectors, or “effective” stress–energy from geometry itself (as in $f(R)$ theories). Constraints and auxiliary conditions can, in principle, eliminate ghost pathologies while retaining the desired spacetime regularity (Alencar et al., 7 Aug 2025).

3. Dynamical Evolution Mechanisms

Time dependence in non-stationary wormholes can arise from several physically distinct mechanisms:

• Cosmological embedding: The wormhole is situated in a dynamic FLRW or de Sitter background with $a(t)$ scaling the wormhole throat and distances, leading to “breathing” wormholes whose sizes evolve with the universe (Zangeneh et al., 19 Jul 2025, Chakraborty et al., 2 Sep 2024).
• Evaporation and black hole–wormhole transition: In the “Black Wormhole” model (Kim et al., 2015), quantum gravity with anisotropic scaling (Hořava gravity) yields a transition from a classical black hole interior (hidden non-traversable wormhole) to an exposed wormhole state as the black hole mass radiatively decreases below a critical value. The Hawking temperature smoothly vanishes at the merge point ($M^2=1/2$), marking a thermodynamically stable/merging configuration.
• Thin-shell dynamics: In time-dependent thin-shell constructions, the wormhole's throat position $r = f(u)$ evolves according to a nonlinear equation with retarded time $u$. The solution can asymptotically settle to a static configuration—such as the Schwarzschild throat—after emitting gravitational radiation, with the approach governed by nonlinear stability under perturbations (Svitek et al., 2016, Akai et al., 2017, Forghani et al., 2018).
• Regularization procedures: The Simpson–Visser–like technique applied to wormhole and cosmological metrics replaces $t^2 \to t^2 + b^2$ and $r^2 \to r^2 + b^2$, yielding non-singular bouncing cosmological evolutions combined with temporal thickness in the throat region (Alencar et al., 7 Aug 2025).

4. Stability and Dynamical Response

The question of nonlinear and linear stability under time-dependent perturbations is central to non-stationary wormhole physics.

• Analytical and numerical stability analyses show that evolving (non-spherically symmetric) thin-shell wormholes in radiative spacetimes (e.g., Robinson–Trautman) ultimately “settle” to spherically symmetric, stationary wormholes, with exponential decay of aspherical perturbations governed by generalized heat-like evolution equations for the shape function or auxiliary metric degrees of freedom (Svitek et al., 2016).
• Nonlinear stability under finite perturbations: For brane-supported wormholes, explicit criteria for post-perturbation survival are derived by matching extrinsic curvatures and analyzing the effective potential for throat motion. Domains exist in the parameter space (throat radius, brane/dust-shell mass, energy) for which the wormhole persists and does not collapse (Akai et al., 2017).
• Role of symmetry and variable equations of state: Asymmetry in the construction (as in asymmetric thin-shell wormholes) typically reduces the parameter range for stability, while additional dynamical variables (such as a pressure–radius derivative term $\gamma$) may enhance or shrink stability regions depending on their sign and functional behavior (Forghani et al., 2018).

5. Field-Theoretic and Quantum Aspects

Several field-theoretic and quantum gravity perspectives emerge in the paper of non-stationary wormholes:

• Black hole as a wormhole factory: Quantum black holes, especially in gravity models with scale asymmetry, may produce microscopic wormholes via infalling negative energy quanta associated with Hawking radiation. In this scenario, microscopic wormholes induced by entangled negative-energy quanta can coalesce into a macroscopic throat, making the “ER=EPR” proposal concrete in this setting (Kim et al., 2015). This links the topology change to quantum entanglement and spacetime foam models.
• Double field theory, string traversability: In string theory, wormhole solutions may be traversable by strings but not by point-particles due to non-Riemannian structure and H-flux. The massless string sector, with negative kinetic dilaton and chirality constraints, supports traversability across non-Riemannian “transition” spheres—manifesting a fundamental non-stationarity at the level of string worldsheet dynamics (Jang et al., 5 Dec 2024).
• Non-local reconstruction and entangled black holes: Non-locally reconstructed vacua near entangled black hole horizons lead to scenarios where wormholes, though present, are non-traversable due to “frozen” vacuum states that prevent information transfer and superluminal signaling. This realization—motivated by the ER=EPR correspondence—demonstrates the subtle interplay between quantum nonlocality, causal structure, and traversability in dynamically evolving geometries (Hadi, 2023).

6. Non-Stationary Wormholes in Modified Gravity

Modified gravitational theories play a crucial role in realizing dynamically regular, non-singular, or energy-condition-satisfying non-stationary wormhole models:

• Hybrid metric–Palatini gravity introduces additional scalar degrees of freedom through an independent Palatini connection. By tuning the field evolution and potential, traversable, evolving wormholes can be constructed without explicit exotic matter and can satisfy NEC and WEC for all times and over all spatial slices (Zangeneh et al., 19 Jul 2025).
• $f(R)$ gravity and elimination of ghosts: In the context of $f(R)$ gravity, certain regular wormhole solutions are constructed without requiring phantom or ghost scalar fields, in contrast to the standard general relativistic setup. This points toward the utility of curvature terms as effective “exotic” sources, especially in time-dependent, nonsingular geometries (Alencar et al., 7 Aug 2025).
• Non-commutative geometry and higher-dimensional extensions: Smeared matter distributions arising from non-commutative geometry and higher-order curvature corrections (e.g., Gauss–Bonnet terms) yield thin-shell wormholes whose throat motion is inherently dynamical. Stability analyses in these settings often reduce to linearized equations for radial perturbations, with the time-dependent throat radius serving as a natural dynamical variable (Rahman et al., 2022, Kavya et al., 2023).

7. Physical Implications, Open Problems, and Future Directions

The paper of non-stationary wormholes has led to a number of new insights and open research directions:

• The possibility of transient satisfaction of energy conditions (NEC/WEC) due to time-dependent background evolution, suggesting that exotic matter requirements can be “dynamically confined” or mitigated.
• The relationship between black hole evaporation, information loss, and wormhole topology change, with non-stationary wormhole interiors providing a potential resolution to cosmic censorship and information paradoxes (Kim et al., 2015).
• The prediction of observable signatures (e.g., ringdown echoes, gravitational radiation patterns, or lensing features) unique to dynamically evolving throats or to complex multi-throat configurations (Svitek et al., 2016, Crispim et al., 6 Dec 2024).
• The use of constraints, auxiliary fields, or higher-curvature terms to model regular, non-stationary traversable wormholes without pathologies (e.g., via constrained ghost/cuscuton fields or $f(R)$ modifications) (Alencar et al., 7 Aug 2025).
• Open problems include the full nonperturbative stability analysis of non-stationary throats, the realization of traversability or signal transfer in quantum-dynamical spacetimes, the embedding of such solutions into realistic astrophysical or cosmological models, and the classification of throat/bounce types in higher dimensions.

Overall, the rapidly developing field of non-stationary wormhole structures continues to bridge classical and quantum gravity, highlighting new connections between dynamic topology, energy conditions, black hole physics, and the fundamental architecture of spacetime.