Unified Time Scale and Time Geometry:\\ Equivalence, Domains, and Solvable Models of Spectral--Scattering--Causal--Entropy

Unied Time Scale and Time Geometry: Equivalence, Domains, and Solvable Models of SpectralScatteringCausalEntropy Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract We propose and rigorously characterize a unied time scale framework, aligning phase gradient readings, relative state density, and trace of WignerSmith group delay within strict scattering theory domain, thus dening time scale as monotonic reparametrization of a class of spectralscattering invariants. The identity φ′(ω) π=ρrel(ω) = 1 2πTr Q(ω), Q(ω) = −i S(ω)†∂ωS(ω), φ =1 2arg det S holds within energy windows satisfying elasticunitary scattering and Birman Kren assumptions; for absorptive/non-unitary and long-range potential cases, we propose veriable generalizations: introducing complex time delay , dwell time , and phase renormalization , using Poissonconvolution to give existence and ane uniqueness of windowed clocks . Paper further constructs model-based proof of eikonal phase derivative = geometric Shapiro delay in general relativity end (Schwarzschild exterior scalar wave, high-frequency/high-angularmomentum limit), expresses redshift as phase rhythm ratio in cosmological end, and states entropy extremality → geometric equations as conditional proposition in informationholography end with relative entropy monotonicity and QNEC as core assumptions. Entire text emphasizes domain of equivalence relations and solvable examples , giving engineering-realizable multi-frequency group delay metrology and lensing delay inversion schemes. Keywords: WignerSmith group delay; spectral shift function; BirmanKren formula; eikonal phase; Shapiro delay; BondiSachs time; TolmanEhrenfest redshift; QNEC; generalized entropy MSC 2020: 81U40, 47A40, 83C57, 83C45  1 Introduction and Historical Context Group delay introduced by Wigner and Smith in elastic scattering, dened as derivative of group phase with respect to frequency; trace of its matrix form Q=−iS†∂ωS 1 equals derivative of total scattering phase Φ = arg det S , thus xing experimental reading of time delay = phase gradient as invariant. On other hand, BirmanKren formula connects scattering determinant with spectral shift function ξ via det S(ω) = e−2πiξ(ω) , giving 1 2π∂ωΦ = −ξ′=ρrel . This bridge establishes unication of phase sloperelative state densitygroup delay trace. In gravity end, eikonal amplitude method and geometric optics show: eikonal phase derivative with respect to energy/frequency gives deection angle and time delay (Shapiro delay). In cosmology, FRW redshift relation 1 + z=a(t0)/a(te) can be written as phase rhythm ratio (dϕ/dt)e/(dϕ/dt)0 . Far-eld null innity BondiSachs framework uses retarded time u to regularize outgoing null surface, providing natural boundary time for gravitational scattering and phase readings. In informationholography end, relative entropy monotonicity and QNEC have been proven in general QFT, QFC as conjecture veried in wide range of cases; these inequalities connect second-order deformation of generalized entropy with energy conditions, forming conditional route from entropy extremality to geometric equations. Goal of this paper is: within strict domains organize above bridges, give unied clock scale covering elasticnon-unitary, short-rangelong-range cases, and conrm phase gradient = geometric time delay alignment with solvable models .  2 Model and Assumptions 2.1 Scattering Pair and Spectral Shift Framework Let (H, H0) be pair of self-adjoint operators satisfying trace-class/quasi-trace-class perturbation assumption (e.g., H−H0∈S1 or (H−i)−1−(H0−i)−1∈S1 ). Then there exists spectral shift function ξ(λ) such that for suciently smooth f Trf(H)−f(H0)=ZR f′(λ)ξ(λ) dλ. If absolutely continuous spectral energy window I⊂R has wave operators existing and scattering matrix S(ω) dierentiable and unitary , then BirmanKren formula det S(ω) = e−2πi ξ(ω) holds and continuous branch of Φ(ω) = arg det S(ω) can be chosen. Denition 2.1 (Relative State Density (Denition 2.1)) . Denote ρrel(ω) := −ξ′(ω) . At Lebesgue-a.e. points in I have 1 2π∂ωΦ(ω) = ρrel(ω). Domain remark: Above equality may hold only in distributional or bounded variation (BV) sense at thresholds, bound states, and resonance points ; branch of Φ xed jointly by analytic continuation of S(ω) and far-eld normalization (Appendix A). 2 2.2 WignerSmith Group Delay For unitary S(ω) dene Q(ω) = −i S(ω)†∂ωS(ω), then Q self-adjoint, and trace identity ∂ωΦ(ω) = Tr Q(ω) holds in I , thus φ′(ω) π=ρrel(ω) = 1 2πTr Q(ω), φ =1 2Φ. This is scale identity in elasticunitary domain. Counterexample and lower bound: Group delay can take negative values near anti-resonances (anomalous delay); but Wigner causality gives lower bound on energy derivative and overall sum constraint. This paper obtains weak monotonicity and ane uniqueness under windowed clocks (4.2, Appendix B). 2.3 Non-Unitary/Absorptive and Generalized Time Delay When external visible channels incomplete or absorption exists (black hole horizon, lossy media, open cavities), S non-unitary. Take Qgen(ω) := −i S(ω)−1∂ωS(ω), whose trace generally complex; can dene real part as generalized Wigner delay, imaginary part related to absorption/gain; can also introduce dwell time and transmission reection decomposition. This paper in 4.3 gives relationship with ∂ωarg det S and metrological meaning. 2.4 Long-Range Potential and Phase Renormalization For Coulomb/gravity 1/r long-range potentials, need use modied wave operators and phase renormalization (Dollard/IsozakiKitada type), removing logarithmic terms in asymptotic phase. This paper for Schwarzschild exterior scalar wave under tortoise coordinates and ReggeWheeler equation constructs renormalized phase Φren(ω) , proving ∂ωΦren(ω)=∆TShapiro(ω) + o(1) holds in high-frequency/high-angular-momentum limit (5, Appendix D). 2.5 Geometry and Boundary Time Local clock rate/redshift in static spacetime controlled by gtt or TolmanEhrenfest law; lapse N in ADM decomposition gives ratio of coordinate time to proper time; remote boundary BondiSachs retarded time u provides natural scattering time at null innity. 3 2.6 InformationHolography Assumption Domain Relative entropy monotonicity and QNEC hold in general QFT; QFC as conjecture provides stronger structure. This paper states entropy extremality → eld equations as conditional proposition , asserting only under small causal diamonds, Hadamard states, weak curvature, and appropriate deformation classes (6, Appendix F).  3 Main Results (Theorems and Alignments) 3.1 Domain Theorem of Scale Identity Theorem 3.1 (ElasticUnitary Domain (Theorem 3.1)) . Let (H, H0) be self-adjoint scattering pair satisfying 2.1 trace-class assumption. Let I⊂R be absolutely continuous spectral energy window, S(ω)∈C1(I;U(N(ω))) with isolated set Σ⊂I of thresholds and resonances absent. Then in I\Σ have φ′(ω) π=ρrel(ω) = 1 2πTr Q(ω) (Lebesgue-a.e.) . On Σ this equality holds in BV/distributional sense, jumps of Φ with bound state resonance contributions given by Levinson/Friedel integral (Appendix A). Proof: See Appendix A (BirmanKren + trace identity + dierentiability and branch choice). Note (long-rangerenormalization): If potential long-range, then there exists renormalized phase Φren such that identity holds after renormalization; proof in Appendix D.1 (Dollard/IsozakiKitada framework). 3.2 Existence and Ane Uniqueness of Windowed Clocks Denition 3.1 (PoissonWindowed Clock (Denition 3.2)) . Take Poisson kernel of width ∆>0 P∆(x) = 1 π ∆ x2+ ∆2,ZR P∆(x) dx= 1. Dene windowed scale density Θ∆(ω) := ρrel ∗P∆(ω) = 1 2πTr Q∗P∆(ω) and clock t∆(ω)−t∆(ω0) = Zω ω0 Θ∆(˜ω) d˜ω. Theorem 3.2 (Weak Monotonicity and Ane Uniqueness (Theorem 3.3)) . If S analytic in upper half-plane with no upper half-plane poles, and ∆ of constant order larger than minimum resonance width/spacing within given energy window, then Θ∆(ω)>0 holds in measure sense, thus t∆ strictly increasing; if ˜ t∆ is clock given by another window family satisfying same window condition, then there exist a > 0, b ∈R such that ˜ t∆=a t∆+b. 4 Proof key points: log det S is NevanlinnaHerglotz type function, whose boundary imaginary part is distribution −2πξ′ ; Poisson smoothing gives harmonic continuation and suppresses oscillation terms of local negative delay; window width condition ensures positive margin covers anti-resonance negative lobes (Appendix B; counterexamples and numerics in 5.3). Comment: This theorem responds to fact that group delay can be locally negative: clock driven by windowed state density , satisfying weak monotonicity and ane uniqueness, not pointwise monotonicity. 3.3 Generalized Identity for Non-Unitary/Absorptive Proposition 3.3 (Generalized Time Delay and Phase (Proposition 3.4)) . For nonunitary S dene Qgen =−iS−1∂ωS . Then ∂ωlog det S(ω) = iTr Qgen(ω), ∂ωarg det S=ℜTr Qgen, can dene real delay τRe := (1/2π)ℜTr Qgen and absorption rate α:= (1/2π)ℑTr Qgen . In small absorption limit |S†S−1| ≪ 1 have τRe = (2π)−1Tr Q+O(|S†S−1|) . 3.4 Eikonal Phase and Geometric Shapiro Delay Theorem 3.4 (High-Frequency/Highl Limit (Theorem 3.5)) . Renormalized phase Φren(ω) of Schwarzschild exterior scalar wave (frequency ω ) satises in eikonal limit ∂ωΦren(ω) = ∆TShapiro(ω) + O(ω−1), where ∆TShapiro is Shapiro delay of geometric ray path. Proof: See 5 (WKB phase dierence = action dierence, using tortoise coordinates and high-frequency decomposition of ReggeWheeler potential; phase branch normalized with eld-free reference). 3.5 Redshift = Phase Rhythm Ratio and Boundary Time Under FRW metric, time derivative of photon phase ϕ proportional to observed frequency, obtaining 1 + z=νe ν0 =(dϕ/dt)e (dϕ/dt)0 =a(t0) a(te), this formula unies cosmological redshift as boundary phase rhythm ratio . 3.6 Entropy Extremality → Geometric Equations: Conditional Proposition Proposition 3.5 (Conditional (Proposition 3.6)) . Under small causal diamond limit, Hadamard state, weak curvature, and appropriate deformation class, if assuming relative entropy monotonicity and QNEC , then second-order deformation of generalized entropy combined with Raychaudhuri equation yields Rµν −1 2Rgµν + Λgµν = 8πG ⟨Tµν⟩. 5 Explanation: QFC not universal theorem, this paper does not use it as sucient condition; proposition only holds under above assumptions and local window, technically supported by Jacobson equation of state and subsequent JLMS/deformation modular Hamiltonian (Appendix F).  4 Proofs (Summary; Details in Appendices) 4.1 Theorem 3.1 BirmanKren gives det S=e−2πiξ ; dierentiating with respect to ω gives Φ′=−2πξ′= 2πρrel . On other hand Tr Q=∂ωTr log S=∂ωΦ . Combining gives identity; understood as BV/distribution at thresholds and resonances (Appendix A). 4.2 Theorem 3.3 log det S is NevanlinnaHerglotz function; its boundary imaginary part is distribution −2πξ′ . Poisson smoothing equals boundary value of harmonic continuation to upper half-plane; choosing ∆ larger than minimum resonance width, local uctuations of negative delay covered by positive envelope, thus Θ∆>0 a.e.; ane uniqueness from unit normalization and additive constant freedom (Appendix B). Counterexamples (negative delay) and window threshold quantitatively shown in one-dimensional solvable potentials (5.3). 4.3 Proposition 3.4 For invertible S use Jacobi identity ∂ωlog det S= Tr(S−1∂ωS) = iTr Qgen . Taking real and imaginary parts gives statement; small absorption expansion in Appendix C. 4.4 Theorem 3.5 In Schwarzschild exterior, express transmission/reection phase using WKB solution of ReggeWheeler equation; at high frequency/high l phase dierence equals geometric action dierence, ∂ω gives Shapiro delay; long-range phase treated with tortoise coordinates and reference phase renormalization (Appendix D). 4.5 Proposition 3.6 Relative entropy monotonicity gives linear relationship between modular Hamiltonian and energy-momentum tensor; QNEC relates lower bound of second-order deformation of generalized entropy with Tkk , combined with Raychaudhuri equation and extremality condition yields tensor form in each null direction; Λ as integration constant (Appendix F). 6 5 Model Applications 5.1 Schwarzschild Exterior: ∂ωΦ and Shapiro Delay Starting from ReggeWheeler equation, construct eikonal solution and phase renormalization Φren , numerical/asymptotic comparison shows ∂ωΦren(ω) consistent with geometric ∆TShapiro (deviation O(ω−1) ). Provides end-to-end chain from wave equation → Smatrix → phase derivative → geometric time delay . 5.2 Lensing: ∂ω(Φi−Φj) = ∆tij Derivative of phase of Kirchho integral amplication factor F(ω) with respect to ω gives Fermat arrival time delay; in thin lens limit with point mass/SIS model obtains unied frequency-domaintime-domain tting of multi-image time delays. 5.3 One-Dimensional Solvable Potential and Negative Delay Choose solvable potential containing anti-resonance, showing local negative values and sum rule of Tr Q(ω) ; verify weak monotonicity critical width of windowed clock with ∆ as variable. Reference Winful's review on Hartman/anomalous delay and electromagnetic/acoustic extensions.  6 Engineering Proposals 1. Multi-frequency Shapirogroup delay parallel inversion: Measure phase Φ(ω) in planetary occultation geometry, compute ∂ωΦ parallel deconvolution with coronal plasma dispersion, combined with hydrogen clock and stable link gives absolute phase reference . 2. On-chip WignerSmith tomography metrology: Construct Q=−iS†∂ωS in multi-port S-parameter metrology, use trace invariance for device tolerance inversion and group delay imaging. 3. Wave lensing broadband time delay spectrum: Fit multi-image arrival time delays and dispersion using ∂ωΦ , reducing time delay cosmology systematic errors.  7 Discussion (Risks, Boundaries, Past Work)  Domain and regularity: Scale identity clearest under elasticunitary and short-range classes; needs BV/distributional understanding at thresholds/resonances; long-range potentials need renormalization.  Negative delay and windowing: Group delay can be locally negative; Poisson windowing provides weakly monotonic clock. Sucient conditions and minimum window width of this construction depend on resonance spectrum. 7  Non-unitary generalization: In absorptive/open systems, ℜTr Qgen gives measurable real delay, ℑTr Qgen measures absorption; quantitative relationship with dwell time/energy storage exists.  Geometry end: Eikonalgeometric optics connection most direct in static/weak elds; strong elds and rotation need more rened coherent transport and numerical ray tracing.  Informationholography: This paper avoids treating QFC as theorem, only giving conditional proposition under QNEC and relative entropy monotonicity.  8 Conclusion Within strict scattering domain, this paper denes time scale as monotonic reparametrization of spectralscattering invariant, core object being φ′(ω) π≡ρrel(ω)≡1 2πTr Q(ω). We specify its domain (elasticunitary, short-range, energy windows away from thresholds/resonances) and generalizations (non-unitary/absorptive, phase renormalization for long-range potentials), propose Poissonwindowed clock proving weak monotonicity and ane uniqueness, give end-to-end model-based proof of eikonal phase Shapiro delay in Schwarzschild exterior, and write cosmological redshift as phase rhythm ratio . In informationholography end, state conditional proposition of entropy extremality → geometric equations based on QNEC/relative entropy monotonicity. Thus forming unied time geometry from spectralscattering to causalentropy.  Acknowledgements, Code Availability Thanks to public textbooks and papers; phase renormalization and Schwarzschild eikonal numerical scripts, windowed clock demonstration, and group delay curve tting code for one-dimensional potentials available upon request.  References [1] E. P. Wigner, Lower Limit for the Energy Derivative of the Scattering Phase Shift, Phys. Rev. 98 , 145 (1955). [2] F. T. Smith, Lifetime Matrix in Collision Theory, Phys. Rev. 118 , 349 (1960). [3] J. Behrndt, M. M. Malamud, H. Neidhardt, Scattering matrices and Weyl functions, Proc. London Math. Soc. 97 , 568598 (2008). [4] D. R. Yafaev, Mathematical Scattering Theory: Analytic Theory , AMS (2010). 8 [5] D. Borthwick, Spectral Theory of Innite-Area Hyperbolic Surfaces , Birkhäuser (2016). [6] A. M. Steinberg, P. G. Kwiat, R. Y. Chiao, Measurement of the single-photon tunneling time, Phys. Rev. Lett. 71 , 708 (1993). [7] H. G. Winful, Tunneling time, the Hartman eect, and superluminality: A proposed resolution of an old paradox, Phys. Rep. 436 , 169 (2006). [8] A. Grabsch, D. P. Karevski, Time delay distributions in chaotic systems, Phys. Rev. E 97 , 052210 (2018). [9] M. Accettulli Huber et al., Eikonal phase matrix, deection angle, and time delay in eective eld theories of gravity, Phys. Rev. D 102 , 046014 (2020). [10] R. Takahashi, Wave eects in the gravitational lensing of electromagnetic radiation by a cosmic string, Astron. Astrophys. 423 , 787 (2004). [11] S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity , Addison Wesley (2004). [12] R. C. Tolman, P. Ehrenfest, Temperature Equilibrium in a Static Gravitational Field, Phys. Rev. 36 , 1791 (1930). [13] D. W. Hogg, Distance measures in cosmology, arXiv:astro-ph/9905116. [14] T. Faulkner et al., Nonlinear Gravity from Entanglement in Conformal Field Theories, JHEP 08 , 057 (2014). [15] S. Balakrishnan et al., A General Proof of the Quantum Null Energy Condition, JHEP 09 , 020 (2019). [16] T. Jacobson, Thermodynamics of Spacetime: The Einstein Equation of State, Phys. Rev. Lett. 75 , 1260 (1995). A Rigorous Domain of Scale Identity (ElasticUnitary, Short-Range) A.1 SSF and BirmanKren Under H−H0∈S1 or resolvent dierence trace-class, spectral shift function ξ exists satisfying trace formula and det S(ω) = e−2πi ξ(ω). Choosing continuous branch satisfying arg det S(ω)→0 ( |ℑω| → ∞ ), obtain Φ(ω) = −2πξ(ω) mod 2π . A.e. derivative with respect to ω gives 1 2πΦ′(ω) = ρrel(ω), ρrel =−ξ′. Understood as BV/distribution at threshold/resonance points Σ ; Levinson/Friedel integral controls Rρrel and bound state counting. 9