The Hayward Metric from Vacuum Coherence: A Resonance-Based Derivation within RCFT and FCI

The Hayward Metric from Vacuum Coherence: A Resonance-Based Derivation within RCFT and FCI Ricardo Miguel Machado Fernandes Abstract We present a resonance-based derivation of what we later identified as the Hayward regular black hole metric, obtained within the Resonant Coherence Field Theory (RCFT) framework and shown to be fully compliant with the Fundamental Conservation of Information (FCI) principle. RCFT posits gravity as emergent from resonant confinement of coherent vacuum phases, while FCI asserts that physical processes reorganize but never destroy information. Crucially, this metric form was not chosen phenomenologically but emerged directly from first principles of vacuum coherence and resonant confinement. The resulting geometry coincides with Hayward’s 2006 regular black hole solution, providing a physical interpretation of its parameters in terms of RCFT coherence scales. The independent convergence of these two frameworks—one phenomenological and one from coherent vacuum dynamics—strengthens the physical plausibility of this regular black hole geometry and ensures information conservation under gravitational collapse. 1 Introduction We derive, from the Resonant Coherence Field Theory (RCFT) framework, a resonancebased metric identical in form to the Hayward regular black hole solution. RCFT conceptualizes gravity as emerging from the resonant containment of coherent vacuum oscillations, where spacetime geometry arises from phase-locked vacuum dynamics. It is important to emphasize that this form was not sought through phenomenological fitting—indeed, Hayward’s 2006 work was unknown to us during initial development. Rather, the metric arose organically from first principles of vacuum coherence and resonant confinement, the foundational mechanisms of RCFT. The subsequent discovery of its identity with Hayward’s model provides an independent convergence: a top-down phenomenological regularization and a bottom-up RCFT microphysical derivation reach the same metric. This strongly reinforces the physical plausibility of the geometry and its alignment with the Fundamental Conservation of Information (FCI) principle, where information is never destroyed but reorganized through coherence. 1 2 Mathematical Framework Definition 1 (The Metric).The line element in standard Schwarzschild coordinates is: ds2=−f(r)dt2+dr2 f(r)+r2(dθ2+ sin2θ dϕ2) (1) with metric function: f(r) = 1 −2GMr2 r3+ 2GMℓ2(2) where: •M: ADM mass measured at spatial infinity, •ℓ: RCFT vacuum coherence length (core phase-locking scale), •G: Newton’s gravitational constant. We work in units where c= 1. Remark 1 (Historical Context).This form was first proposed by Hayward (1) as a phenomenological regular alternative to Schwarzschild. RCFT independently produces the same structure from the dynamics of coherent vacuum confinement, providing its physical foundation. 3 Resonance-Based Derivation in RCFT 3.1 Vacuum Coherence Field RCFT describes the vacuum as a complex scalar coherence field Φ = Aeiθ representing the phase-locked structure of the vacuum: LRCFT =−1 2(∇A)2−1 2A2(∇θ)2−Vcoh(A) (3) whose self-organized confinement leads to curvature resistance at small scales. Theorem 1 (Stress-Energy Correspondence).The effective stress-energy tensor for the RCFT coherent vacuum is: ρ(r) = 3ℓ2(GM)2 2πG(r3+ 2GMℓ2)2,(4) pr(r)=−ρ(r),(5) pt(r) = 3ℓ2(GM)2(r3−GMℓ2) πG(r3+ 2GMℓ2)3,(6) which satisfies ∇µTµν = 0 and sources precisely the metric function f(r)above. 2 Proof. The Einstein equations Gµν = 8πGTµν yield the above forms when the potential takes Vcoh(A) = 3 8πGℓ2−1 2m2A2+λA4+··· (7) and ℓis identified with the RCFT coherence length ξcoh that sets the resonant confinement scale. Lemma 1 (Energy Conservation).For anisotropic stress-energy (ρ, pr, pt), conservation ∇µTµr= 0 gives: p′ r(r) + 2 r(pr−pt)+(ρ+pr)f′(r) 2f(r)= 0. Substitution of the above ρ, pr, ptsatisfies this identity identically, confirming full RCFT self-consistency. Remark 2 (Curvature Regularity).All curvature scalars are finite at r= 0: R→12 ℓ2, RµνRµν →36 ℓ4, RµνρσRµνρσ →24 ℓ4,(8) demonstrating regularity of the RCFT-derived geometry and FCI compliance (no informationdestroying singularity). 4 Physical Interpretation of Parameters Remark 3 (Coherence Scale ℓ).In RCFT, ℓis the vacuum coherence length: ℓ∼ℓPl MPl mres , ℓPl =rℏG c3,(9) where mres is the vacuum resonance mass. This identifies ℓas a measurable coherence scale rather than a free geometric parameter. 5 Comparison with Hayward’s Phenomenological Model 6 Key Properties and Horizon Structure Proposition 1 (Asymptotic and Core Limits).1. Schwarzschild limit ℓ→0:f(r)→ 1−2GM r. 2. de Sitter core r→0:f(r)≃1−r2 ℓ2with Λeff = 3/ℓ2. 3. Asymptotic expansion r→ ∞:f(r)=1−2GM r+4G2M2ℓ2 r4+···. Proposition 2 (Horizons).Horizons satisfy r3−2GMr2+ 2GMℓ2= 0. The discriminant gives a critical value ℓcrit =4 3√3GM ≈0.77GM separating two-horizon, extremal, and no-horizon configurations. 3 Aspect Hayward (2006) RCFT–FCI Framework Motivation Phenomenological regularization First-principles from coherent vacuum dynamics Parameter ℓFree parameter RCFT coherence length ℓ=ξcoh Physical Basis Limiting curvature conjecture Resonant containment (vacuum as standing-wave medium) Stress-Energy Imposed form Derived from RCFT Lagrangian Information Flow Not specified FCI compliance: no loss, reorganization only Predictions Geometric properties + QNM shifts, echo times, coherence scaling laws Table 1: Comparison between Hayward’s phenomenology and RCFT–FCI derivation. Proof. The cubic equation r3−2GMr2+ 2GMℓ2= 0 has discriminant ∆ = (2GM)2[(2GM)4−27ℓ4]. When ∆ >0 (ℓ < ℓcrit), two distinct real horizons exist; when ∆ = 0 (ℓ=ℓcrit), a double horizon (extremal case); when ∆ <0 (ℓ > ℓcrit), no horizons exist. 7 FCI Compliance and Information Conservation The Fundamental Conservation of Information (FCI) principle requires that no process, gravitational or otherwise, destroys information—only reorganizes it. Because the RCFTsourced Hayward metric eliminates curvature singularities, the information encoded in the collapsing system transitions into a stable, coherent vacuum domain characterized by ℓ. Thus: Collapse: Imatter −→ Ivacuum coherence,∆I= 0, which satisfies the FCI condition. The black hole interior becomes an information-preserving condensate rather than an information sink. 8 New Physical Insights from RCFT 8.1 Resonant Breathing Modes RCFT naturally suggests time-dependent ”breathing” modes: f(t, r) = f0(r)1+ϵS(r) sin(ωt), where ωcorresponds to the fundamental vacuum resonance frequency. This ansatz is heuristic; a fully self-consistent time-dependent solution requires solving the coupled field–metric equations. These modes could generate measurable quasi-periodic oscillations in compactobject signals. 4 8.2 Observable Predictions •QNM Shifts: ∆f/f ∼(ℓ/GM)2from coherent backreaction. •Echo Times: τecho ∼GM log(GM/ℓ). •Surface Gravity Modification: κ=1 2|f′(rh)|exhibits small ℓ2corrections testable via gravitational wave ringdowns. 9 Conclusion Within RCFT, the Hayward metric emerges as a natural manifestation of resonant vacuum coherence. Key outcomes: •RCFT provides a physical origin for the regular geometry from vacuum confinement. •The coherence length ℓhas a microphysical interpretation as the vacuum phase-locking scale. •The geometry is regular and FCI-compliant: information is conserved via reorganization into coherent states. •Observable effects (QNM shifts, echoes) connect quantum-coherent vacuum structure to astrophysical signatures. This synthesis of RCFT dynamics with FCI information conservation demonstrates that the Hayward geometry is not merely a mathematical convenience but a natural outcome of coherent field theory. Acknowledgments We thank S. A. Hayward for pioneering the regular metric that provided the benchmark for this convergence. The RCFT–FCI formulation presented here grounds his solution in physical first principles. References [1] Hayward, S. A. (2006). Formation and evaporation of regular black holes. Phys. Rev. Lett. 96(3), 031103. [2] Bardeen, J. M. (1968). Non-singular general-relativistic gravitational collapse. In Proceedings of GR5, Tbilisi, p. 174. [3] Frolov, V. P. (2016). Notes on non-singular models of black holes. Phys. Rev. D 94(12), 124040. [4] Chiba, T., & Kimura, S. (2017). A note on geodesics in the Hayward metric. Prog. Theor. Exp. Phys. 2017(4), 043E01. 5