Numerical Indications of Trapped Surface Formation via Resonant Vacuum Polarization without Explicit Exotic Matter Sources

Numerical Indications of Trapped Surface Formation via Resonant Vacuum Polarization without Explicit Exotic Matter Sources Author: [İsmail Başkır]Date: January 2026 Abstract In conventional General Relativity, the formation of traversable wormhole geometries or stable Einstein–Rosen bridges is typically associated with violations of the Null Energy Condition (NEC), often modeled through explicit exotic matter sources with negative energy density. In this work, we present numerical results indicating that NEC-violating effective behavior may arise without prescribing any exotic matter fields. The simulations solve the full, non-linear Einstein equations using the Einstein Toolkit (Cactus framework) with the Method of Lines (MoL). No linearization, perturbative truncation, or reduced-order modeling is employed. The spacetime is initialized as vacuum (Tμν=0T_{\mu\nu}=0Tμν=0), and no matter evolution equations are activated. A fixed gauge choice is imposed, with a static lapse and vanishing shift, while the spatial metric and extrinsic curvature are fully evolved in time. A controlled, time-dependent geometric modulation is introduced through a resonant chirped signal combined with a logarithmically structured spectral envelope inspired by the Riemann zeta function. The numerical evolution reveals the formation of localized marginally trapped surfaces characterized by the vanishing of the outgoing null expansion scalar, Θ+=0\Theta^{+} = 0Θ+=0, accompanied by negative ingoing expansion. These surfaces remain stable over multiple evolution steps and satisfy the geometric criteria of a marginally outer trapped surface (MOTS), consistent with a wormhole throat or apparent horizon. The effective stress–energy contribution inferred from the Einstein tensor exhibits transient NEC violation arising purely from spacetime dynamics rather than from prescribed matter content. While no claim is made regarding physical realizability, the results demonstrate that nontrivial trapped-surface geometries can emerge in vacuum General Relativity through resonant geometric excitation alone. 1. Numerical Framework The simulations are performed using the Einstein Toolkit built on the Cactus framework, with time integration handled by the Method of Lines (MoL). The evolution solves the complete, non-linear 3+1 Einstein equations, without approximation or perturbative expansion. No external matter fields are included: Tμν=0(identically, at all times).T_{\mu\nu} = 0 \quad \text{(identically, at all times)}.Tμν=0(identically, at all times). The observed effective energy behavior is therefore entirely inferred from the geometric sector via: Gμν=8πTμνeff.G_{\mu\nu} = 8\pi T_{\mu\nu}^{\text{eff}} .Gμν=8πTμνeff. 2. Gauge Conditions and Evolution Strategy A deliberately restricted gauge choice is employed to isolate purely geometric effects: α=α0(static lapse),βi=0(vanishing shift).\alpha = \alpha_0 \quad \text{(static lapse)}, \qquad \beta^i = 0 \quad \text{(vanishing shift)}.α=α0(static lapse),βi=0(vanishing shift). No gauge driver or slicing evolution equations are activated. Despite this restriction, the spatial metric γij\gamma_{ij}γij and extrinsic curvature KijK_{ij}Kij are fully evolved in time via MoL. This setup constitutes a gauge-restricted vacuum evolution, not a toy model. The absence of dynamic gauge degrees of freedom is intentional and eliminates gauge-induced horizon artifacts, allowing a clean interpretation of trapped-surface formation driven solely by spacetime dynamics. 3. Resonant Modulation Ansatz The time-dependent geometric excitation applied to the system is defined as: Ψ(t)=A0ei(ω0t+αt2)⏟Chirped carrier × (1+∑n=1Ncos⁡ ⁣(γnln⁡(ln⁡t))t)⏟Logarithmic spectral modulator.\Psi(t) = \underbrace{A_0 e^{i(\omega_0 t + \alpha t^2)}}_{\text{Chirped carrier}} \;\times\; \underbrace{\left( 1 + \sum_{n=1}^{N} \frac{\cos\!\bigl(\gamma_n \ln(\ln t)\bigr)}{\sqrt{t}} \right)}_{\text{Logarithmic spectral modulator}} .Ψ(t)=Chirped carrierA0ei(ω0t+αt2)×Logarithmic spectral modulator(1+n=1∑Ntcos(γnln(lnt))). The accumulated resonant coupling is quantified by: Ires≡∫0t∣ΨChirpΨMod∣2 dt.I_{\text{res}} \equiv \int_{0}^{t} \left| \Psi_{\text{Chirp}} \Psi_{\text{Mod}} \right|^2 \, dt .Ires≡∫0t∣ΨChirpΨMod∣2dt. Numerically, the emergence of localized trapped regions correlates strongly with peaks in IresI_{\text{res}}Ires, indicating a resonance-driven geometric response. 4. Trapped Surface Detection Trapped surfaces are identified using the null expansion scalars. A surface is classified as marginally trapped when: Θ+=0,Θ−<0.\Theta^{+} = 0, \qquad \Theta^{-} < 0 .Θ+=0,Θ−<0. The simulations reveal closed, spatially localized surfaces satisfying these conditions. The detected Θ+=0\Theta^{+} = 0Θ+=0 contours persist over multiple time steps, indicating stability rather than numerical noise. By definition, these surfaces correspond to Marginally Outer Trapped Surfaces (MOTS), consistent with either an apparent horizon or a wormhole throat geometry. 5. Effective Energy Condition Violation Although no matter fields are present, the Einstein tensor evaluated along null vectors kμk^\mukμ satisfies: Tμνeffkμkν<0T_{\mu\nu}^{\text{eff}} k^\mu k^\nu < 0Tμνeffkμkν<0 in localized regions during resonant excitation. This violation of the Null Energy Condition is therefore effective and geometric in origin, arising from the dynamical response of spacetime itself rather than from exotic matter sources. 6. Figure Description Figure 1:Left: Three-dimensional visualization of the outgoing null expansion scalar Θ+\Theta^{+}Θ+, highlighting regions with Θ+<0\Theta^{+} < 0Θ+<0 (blue), indicative of trapped geometry.Right: Two-dimensional cross-section showing the Θ+=0\Theta^{+} = 0Θ+=0 contour (black line), identifying a stable marginally trapped surface consistent with a wormhole throat. 7. Discussion and Scope This work does not claim physical realizability or experimental feasibility. The results demonstrate that, within full non-linear numerical General Relativity, vacuum spacetimes subjected to resonant geometric excitation can develop trapped-surface structures without explicit exotic matter. The findings motivate further investigation into purely geometric mechanisms for effective energy condition violation and their role in horizon and throat formation.