Unified Time Scale and Time Geometry:\\ Causal Ordering, Unitary Evolution, and Generalized Entropy

Unied Time Scale and Time Geometry: Causal Ordering, Unitary Evolution, and Generalized Entropy Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 20, 2025 Abstract We propose a unied time scale equivalence class rigorously gluing three ends of relativity, quantum scattering, and informationholography. Core scale identity unies derivative of total scattering phase, relative state density, and trace of WignerSmith group delay as dierent projections of same object: φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), Q(ω) = −iS(ω)†∂ωS(ω), φ =1 2arg det S. In geometry end, Killing time, ADM lapse, null geodesic ane parameter, and FRW conformal time proven mutually rescalable within unied equivalence class; in informationholography end, taking TomitaTakesaki modular ow as intrinsic time, controlling generalized entropy extremality on small causal diamonds with QFC/QNEC and relative entropy monotonicity, thus deriving Einstein equations in semiclassicalholographic window. Framework obtains three alignments: (i) phase proper time equivalence ϕ= (mc2/ℏ)Rdτ ; (ii) gravitational time delay = group delay trace ∆T=∂ωΦ = Tr Q ; (iii) FRW redshift = phase rhythm ratio 1 + z=a(t0)/a(te) = [(dϕ/dt)e]/[(dϕ/dt)0] . This paper under three axioms of causal orderingunitary evolutionentropy monotonicity/extremality , establishes existence and ane uniqueness of unied time scale, giving realization schemes for experimental and engineering metrology. Keywords: Time Geometry; Unied Time Scale; WignerSmith Group Delay; Spectral Shift Function; Killing/ADM/Null/Conformal/Modular Time; Generalized Entropy; QFC/QNEC MSC 2020: 83C45, 81U40, 81T20, 83C57  1 Introduction and Historical Context Role of time splits in dierent theories: general relativity scales causal structure with proper time, quantum theory generates unitary evolution with external parameter, information holography views modular ow as intrinsic thermal time. Yet when three end readings 1 synchronize, still lacking rigorous and metrologically measurable common scale. Wigner and Smith introduced derivative of phase with respect to energy in scattering theory dening time delay, trace of WignerSmith group delay matrix Q=−iS†∂ωS equals derivative of total phase Φ = arg det S , making idea of time = phase gradient rst land on experimentally readable ruler. BirmanKren formula characterizing state density change caused by interaction with spectral shift function tightly connects phase of scattering determinant with spectral geometry, thus deriving phase derivative = relative state density. In relativity side, redshift/clock rate in static spacetime given by gtt or Tolman Ehrenfest law; lapse N in ADM (3+1) decomposition characterizes ratio of coordinate time to proper time; tortoise coordinates and (u, v) in asymptotically at exterior provide natural null time at innity; in FRW cosmology 1+z=a(t0)/a(te) linearizes null geodesics with conformal time. In informationholography side, TomitaTakesaki modular theory endows any (state, algebra) pair with family of intrinsic one-parameter automorphisms (modular ow); ConnesRovelli thermal time hypothesis views this modular ow as physical time candidate; relative entropy monotonicity, QFC, and QNEC bind changes of generalized entropy together with stress tensor constraints, connecting to eld equations and reconstruction in small causal diamond limit. Above historical threads suggest: unifying phasegroup delayspectral shift with clock rateredshiftane/conformal time and modular timegeneralized entropy into single scale , hopeful to obtain cross-scale timegeometry framework.  2 Model and Assumptions (A) Causal and Global Structure Let (M, g) be stably causal Lorentzian manifold; under global hyperbolicity condition there exist smooth time function and smooth decomposition M∼ =R×Σ . (B) Scattering and Spectral Shift On absolutely continuous spectral energy window I⊂R , scattering matrix S(ω) unitary and smooth; dene total phase Φ = arg det S , group delay Q=−iS†∂ωS . There exists spectral shift function ξ(ω) and relative state density ρrel =−ξ′ ; BirmanKren formula det S(ω) = exp[−2πi ξ(ω)] holds. (C) Unied Scale Identity (Core Assumption) φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), φ =1 2Φ. (D) Boundary Entropy and Modular Flow Take small causal diamond Dp,r through point p , null generator ane parameter λ ; generalized entropy Sgen(λ) = Area(Σλ) 4Gℏ+Sout(λ) satises QFC/QNEC type inequalities and relative entropy monotonicity; modular Hamiltonian K=−ln ρ generates modular ow σs . (E) Unitary Evolution On state space H there exists strongly continuous unitary group U(t) = e−iHt ; semiclassical worldline limit can relate t and τ through phase density (see Section 4.1).  2 3 Unied Time Scale: Denition and Three Axioms 3.1 Unied Time Scale Equivalence Class Denition 3.1 (Unied Time Scale (Denition 3.1)) . There exists equivalence class [T]∼ {τ, t, tK,(N, Ni), λnull, u, v, η, ω−1, z, smod}, whose members mutually convertible through monotonic rescaling and geometric/entropy structure, making dynamics local, causally ordered, and entropy structure simplest. 3.2 Three Axioms Axiom 3.1 (Causal Ordering (Axiom I)) . In local hyperbolic domain there exists strictly increasing time function, making fundamental equations local (hyperbolic/rst-order) form. Axiom 3.2 (Unitary Evolution (Axiom II)) . There exists strongly continuous unitary group U(t) ; in semiclassical limit phasetime relation determined by Lagrangian stationary phase (Section 4.1). Axiom 3.3 (Entropy Monotonicity/Extremality (Axiom III)) . Along null cut family {Σλ} , Sgen satises relative entropy monotonicity and QFC/QNEC monotonicity/convexity and takes extremum under physical evolution; modular ow parameter s makes organization law of Sgen simplest. Theorem 3.1 (Mutual Implication in SemiclassicalHolographic Window (Theorem 3.2)) . Under small causal diamond limit and relative entropy monotonicity/QNEC holding: Axiom I + Axiom II ⇐⇒ Scale Identity =⇒ Axiom III =⇒ Einstein Equations . Proof in Section 5 and Appendices D/E.  4 Main Results (Theorems and Alignments) 4.1 PhaseProper Time Equivalence Theorem 4.1 (Worldline Principal Phase (Theorem 4.1)) . For narrow wave packet of mass m , in semiclassical limit ϕ=−1 ℏS[γcl] = mc2 ℏZγcl dτ, dϕ dτ=mc2 ℏ. (Proof: worldline path integral stationary phase; see Appendix B.) 4.2 Gravitational Time Delay = Group Delay Trace Theorem 4.2 (EikonalScattering Alignment (Theorem 4.2)) . In geometric optics limit of static or asymptotically at background, ∆T(ω) = ∂ωΦ(ω) = Tr Q(ω). Weak eld limit returns to Shapiro delay. 3 4.3 Redshift = Phase Rhythm Ratio Proposition 4.3 (FRW Phase Expression (Proposition 4.3)) . Under at FRW metric comoving observers measure 1 + z=νe ν0 = dϕ dte dϕ dt0 =a(t0) a(te). (See Appendix C.) 4.4 Four Bridges of GR Time Structure Bridge B (Killing TimeClock RateRedshift) In static metric ds2=−V(x)c2dt2+ ··· stationary observers have dτ=√Vdt , √V is local redshift/clock rate factor (Tolman Ehrenfest). Bridge C (ADM LapseLocal Clock Rate) ADM decomposition ds2=−N2dt2+ hij(dxi+Nidt)(dxj+Njdt) ; Euler family orthogonal to slicing satises dτ=Ndt . Bridge D (Null Ane ParameterRetarded/Advanced/Conformal Time) Asymptotically at exterior denes tortoise r∗ and u=t−r∗, v =t+r∗ , monotonically equivalent to null geodesic ane parameter; in FRW dη= dt/a(t) linearizes null geodesics. Bridge E (Modular TimeEntropy GradientGeometric Equations) Parameter s of modular ow σs provides information-theoretic time; relative entropy monotonicity and QNEC/QFC bind extremality/monotonicity of ∂sSgen with ⟨Tkk⟩ . 4.5 Entropic Geometric Form of Einstein Equations Theorem 4.4 (EntropyGeometry (Theorem 4.4)) . Under Axiom III and Raychaudhuri equation, on small causal diamond have Rµν −1 2Rgµν + Λgµν = 8πG ⟨Tµν⟩. (Proof: combining second-order area variation with QNEC/relative entropy, see Appendix D; cf. Jacobson and subsequent holographic arguments.) 4.6 Unied Scale Identity (SpectralScatteringGeometry) Corollary 4.5 (Corollary 4.5) . φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), obtained by combining BirmanKren and Tr Q=∂ωΦ (Appendix A).  4 5 Proofs (Key Points) 5.1 Theorem 4.1: Worldline action stationary phase along timelike geodesic, uctuations only change quantum prefactor, principal phase gives ϕ= (mc2/ℏ)Rdτ (Appendix B). 5.2 Theorem 4.2: Eikonal phase dierence ∆S≃ −ℏω∆T aligns with scattering ∂ωΦ = Tr Q , yielding ∆T= Tr Q ; weak eld test returns to Shapiro delay. 5.3 Proposition 4.3: Null geodesic and scale factor give ν∝1/a(t) , redshift as phase rhythm ratio (Appendix C). 5.4 Bridges BE: Static clock rate, ADM lapse, null coordinates, and modular ow each consistent with standard conclusions and literature (Appendix E). 5.5 Theorem 4.4: Raychaudhuri's second-order area variation ∝ −RRkk combines with QNEC S′′ out ≥(2π/ℏ)R⟨Tkk⟩ , extremality condition S′ gen(0) = 0 yields tensor equation (Appendix D); QFC provides stronger monotonicity background.  6 Model Applications A. Frequency-Domain Reconstruction of Solar System Geometric Delay Differentiating multi-frequency radar echo phase Φ(ω) gives ∆T(ω) = ∂ωΦ = Tr Q , compare with Shapiro delay, can parallelly strip plasma dispersion term. B. PhaseGroup Delay Unication of Gravitational Lensing Image pair (i, j) Fermat potential dierence ∆tij equals ∂ω[Φi−Φj] ; broadband electromagnetic/gravitational wave joint use for Hubble constant and mass model systematic error suppression. C. Cosmological Phase Ruler Directly estimate a(t) using phase rhythm ratio of pulsar/FRB, avoiding specic spectral line systematics; phaseredshift synchronization guaranteed by Proposition 4.3. D. Eective Time Refractive Index Tomography Invert nt= (−gtt)−1/2 from spatial distribution of ρrel(ω) or tr Q(ω) , cooperating with optical metricFermat principle for weak eld time delay imaging.  7 Engineering Proposals 1. On-chip group delay tomography metrology: Measure S(ω) in integrated photonics and real-time compute Tr Q(ω) , generating equivalent gravitational time delay map for device inversion and robust design. 2. Double-height matter wave standard: COW geometric arrangement compare ∆ϕ and ∆τ , test linear regime of ∆ϕ= (mc2/ℏ)∆τ . 3. Broadband lens group delay spectrum: Synchronously t each image arrival time delay and dispersion using ∂ωΦ , reducing time delay cosmology systematic errors. 4. Entropy light cone platform: Measure second-order deformation and energy ow of Sout on controllable quantum system, test QNEC/QFC coecients and saturation conditions. 5  8 Discussion (Risks, Boundaries, Past Work) (i) Spectral endpoints and regularity: Scale identity requires S(ω) dierentiable and belonging to appropriate determinant class; near resonances and thresholds need contour displacement and trace-class regularization. (ii) Geometric optics and strong elds: Strong spin/non-static backgrounds need generalized optical metric and coherent transport; near-horizon regions better use null coordinates and numerical ray tracing. (iii) Entropygeometry assumption domain: QNEC has general QFT proofs and holographic proofs continuously strengthening (including latest new proof approaches), but still advancing in high curvature, strong quantum gravity regions. (iv) GR time structure unication: Killing/ADM/null/conformal/modular times are coordinatizations of unied scale in dierent projections; BernalSánchez's global time functions and ADM foliation provide rigorous foundation.  9 Conclusion Under three axioms of causal orderingunitary evolutionentropy monotonicity/extremality , obtain unied time scale equivalence class, aligning microscopic (phase/scattering), mesoscopic (group delay/redshift), and macroscopic (entropygeometry) three-end languages. Core conclusions: ϕ=mc2 ℏZdτ, ∆T(ω) = ∂ωΦ(ω) = Tr Q(ω),1+z=(dϕ/dt)e (dϕ/dt)0 , Gµν+Λgµν = 8πG ⟨Tµν⟩, and spectralscatteringgeometry scale identity φ′/π =ρrel = (2π)−1tr Q . Time thus can be characterized as: equivalence class of one-dimensional parameter making dynamics local, causality clear, entropy structure simplest ; its dierent names merely coordinates of same object in dierent projections.  Acknowledgements, Code Availability Thanks to related textbooks and literature. Symbolic derivations and numerical scripts for group delaytime delay reconstruction and FRW phase rhythm demonstration available upon request.  References [1] E. P. Wigner, Lower Limit for the Energy Derivative of the Scattering Phase Shift, Phys. Rev. 98 (1955) 145. 6 [2] F. T. Smith, Lifetime Matrix in Collision Theory, Phys. Rev. 118 (1960) 349. [3] A. Strohmaier and A. Waters, The BirmanKrein formula for dierential forms and electromagnetic scattering, arXiv:2104.13589. [4] A. N. 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[19] Scholarpedia, BondiSachs Formalism (for retarded time u and advanced time v ). 7 A WignerSmith Group Delay and SpectralScattering Geometry Identity A.1 BirmanKren and Spectral Shift For self-adjoint pair (H, H0) with trace-class/quasi-trace-class perturbation, spectral shift function ξ(ω) satises det S(ω) = e−2πiξ(ω)⇒1 2π∂ωΦ(ω) = −ξ′(ω) = ρrel(ω). (See reference 3.) A.2 Trace Identity From Q(ω) = −iS†∂ωS and Tr ln S= ln det S obtain Tr Q(ω) = ∂ωΦ(ω). Combining A.1 gives scale identity: φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). B PhaseProper Time Under Worldline Path Integral Fermi normal coordinate expansion along timelike geodesic γcl S[γ] = −mc2Zdτ+m 2Zdτ δij ˙yi˙yj+··· . Stationary phase gives ϕ=−1 ℏS[γcl] = mc2 ℏZdτ, dϕ dτ=mc2 ℏ. C Phase Expression of Redshift in FRW Cosmology Flat FRW: ds2=−dt2+a(t)2dx2 . Comoving observer uµ= (1,0,0,0) , photon eikonal phase ϕ has kµ=∂µϕ and ν=−1 2πkµuµ=1 2π dϕ dt∝1 a(t)⇒1 + z=(dϕ/dt)e (dϕ/dt)0 =a(t0) a(te). (Reference 7.) 8 D Generalized Entropy Extremality/Monotonicity and Field Equations Let {Σλ} be null cut family through p . Raychaudhuri: ˙ θ=−1 2θ2−σ2−Rkk . Second-order area variation d2A/dλ20∝ −ZRkk dA . QNEC and relative entropy monotonicity: d2Sout/dλ20≥2π ℏZ⟨Tkk⟩dA . Extremum S′ gen(0) = 0 combines to give Rkk = 8πG⟨Tkk⟩ , holding for any kµ , upgrades to tensor equation and gives Λ as integration constant. E Renement of GR Time Bridges E.1 Static Spacetime (Killing): ξµ is timelike Killing vector, stationary observer uµ=ξµ/p−ξ2 , if gtt =−V then dτ=√Vdt . (TolmanEhrenfest temperature redshift law same form.) E.2 ADM Lapse: ds2=−N2dt2+hij(dxi+Nidt)(dxj+Njdt) ; slicing orthogonal family satises dxi+Nidt= 0 ⇒dτ=Ndt . E.3 Null Coordinates: Schwarzschild exterior r∗=r+2Mln |r/2M−1| , u=t−r∗ , v=t+r∗ ; in FRW dη= dt/a(t) . E.4 Modular Time: Given (algebra, state) pair (A, ω) GNS representation, modular ow σs intrinsically denes time; under half-space and small deformations, K localizes to RTkk , isomorphic with ANEC/QNEC, JLMS/relative entropy. F Shapiro Delay and Group Delay Weak eld Schwarzschild exterior ∆t≃4GM c3ln 4rErR b2+··· , consistent with frequency-domain measured ∂ωΦ = Tr Q ; multi-frequency echo tting can separate dispersion and geometric terms. G Existence and Uniqueness of Unied Time Scale Given scattering data (S(ω)) satisfying scale identity. Dene t−t0=Zω ω0 1 2πTr Q(˜ω) d˜ω=Zω ω0 φ′(˜ω) πd˜ω=Zω ω0 ρrel(˜ω) d˜ω. In non-degenerate frequency window derivative positive, t(ω) locally bijective; if another ˜ t satises same condition, then ˜ t=at +b ( a > 0 ), giving ane uniqueness.  9