Curvature as Non-uniformity of Time Flow Density: Reconstructing Gravitational Geometry from Unified Time Scale \kappa(\omega)

Curvature as Non-uniformity of Time Flow Density: Reconstructing Gravitational Geometry from Unified Time Scale κ(ω) Anonymous Author November 26, 2025 Abstract General relativity attributes gravitational phenomena to spacetime curvature, while in special relativity and quantum theory time often appears as an external parameter or local “proper time”, creating a long-standing conceptual divide. In this paper, within the framework of unified time scale and information rate conservation, we propose and systematically demonstrate the following perspective: geometric curvature can in essence be rewritten as non-uniformity of “time flow density” field κ(x). More specifically, underlying scattering/spectral theory gives the unified time identity κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where φ(ω) is scattering phase shift, ρrel(ω) is relative state density, Q(ω) is Wigner– Smith time delay operator. Localizing and coarse-graining this frequency-domain scale in space, we obtain a scalar field κ(x) describing “internal evolution density per unit external time”. This paper proves that under natural axiomatization: 1. There exists a class of metric families in one-to-one correspondence with κ(x), whose time component satisfies gtt(x) = −c2 η2(x), η(x)≡κ(x) κ∞ , and in weak-field limit gives gravitational potential Φgrav(x)≃c2ln η(x)∝c2ln κ(x). 2. If further requiring “local information volume (state number ×spatial volume) conservation”, then η(x)’s spatial scaling must enter spatial metric in η−1(x) manner, thus selecting a class of double-conformal-scaling optical metrics: ds2=−η2(x)c2dt2+η−2(x)γijdxidxj, automatically reproducing standard gravitational redshift and light deflection formulas in weak-field limit. 3. In quantum cellular automaton (QCA) framework with “information rate conservation” axiom v2 ext +v2 int =c2 κ(x) can be specifically realized as: near a given spatial point, average density of “internal state evolution steps per unit external time”. We demonstrate a concrete construction in one-dimensional Dirac-type QCA, where κ(x) associates with local energy gap/internal frequency, with mass and gravitational effects controlled by same κ-field. In summary, this paper provides a geometric–information-theoretic picture reinterpreting curvature as “time flow density texture”: at each point, time is no longer a uniformly 1 flowing parameter, but a “time density field” κ(x)jointly defined by scattering spectra, state density and local information processing structure. Traditional Einstein geometry is viewed as effective description of κ-field in continuum limit, whose observations can be tested through atomic clock networks, gravitational redshift and gravitational lensing experiments. Appendices provide scattering-theoretical derivation of unified time identity, weak-field curvature calculation of κ-metric, and explicit examples in QCA models. Keywords: Unified time scale; Time flow density; Curvature; Gravitational geometry; Scattering theory; Quantum cellular automaton; Information rate conservation; Wigner–Smith time delay 1 Introduction In standard physics picture, “time” plays dramatically different roles at different theoretical levels:  In special and general relativity, time is part of four-dimensional spacetime coordinates, as one component of metric gµν, whose geometric structure is constrained by energy– momentum tensor.  In non-relativistic quantum mechanics, time is often viewed as external parameter t, wavefunction ψ(t) evolves unitarily with respect to this parameter, not represented by operator.  In scattering theory and spectral theory, time appears as “delay”, “dwell time”, “phase derivative”, typically characterized through frequency derivative of scattering phase shift φ(ω): τ(ω) = ℏdφ dE=dφ dω. Above perspectives are mathematically compatible, but ontologically lack unified definition of “what time is”. In particular, gravitational curvature seems to require time to “flow at different rates” at different positions, while in quantum theory time is treated as uniform external parameter, creating various conceptual tensions. The core question of this paper can be briefly stated as: Can we start from a unified “time flow density” scale κ(x, ω), rewrite geometric curvature as spatial texture of this density field, thus understanding time dilation, gravitational redshift and quantum scattering time delay in same framework? Our answer is affirmative. Based on Eisenbud–Wigner–Smith time delay, spectral shift function and information rate conservation, we propose: 1. Unified time identity: In appropriate scattering/spectral cases, there exists gaugeindependent unified scale κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), where κsimultaneously characterizes phase derivative, relative state density and Wigner– Smith delay trace, thus can be interpreted as “time flow density”. 2 2. Localization and coarse-graining: When considering scattering/propagation problems with spatial structure, we can introduce local κ(x, ω), and do appropriate weighting and coarse-graining over frequency (or energy) windows, obtaining macroscopic scalar field κ(x), whose physical meaning is “total internal evolution steps per unit external time, per unit spatial volume”. 3. Geometric reconstruction: Under requirements:  gµν has Lorentz signature;  Local QCA/quantum field local structure in local free-fall maintains covariant form;  Total “information volume” conserved (state number ×spatial volume); can prove metric must inherit specific double-conformal-scaling structure from κ(x), with gravitational potential in weak-field limit given by ln κ(x), curvature tensor directly related to second derivatives of ln κ(x). 4. Embedding with QCA: If underlying ontology is quantum cellular automaton (QCA), each cell executes finite local unitary updates in each discrete time step. For effective field mode in long-wavelength limit, can directly associate local internal frequency ωint(x) with κ(x), unifying mass, time dilation and curvature through “information rate conservation” axiom v2 ext +v2 int =c2 as different allocation methods of same κ-budget. This paper is organized as follows: Section 2 gives notation, preliminaries and brief review of unified time identity; Section 3 defines local time flow density field κ(x, ω) and macroscopic κ(x); Section 4 constructs effective metric from κ(x) and derives gravitational potential and curvature in weak-field limit; Section 5 discusses consistency conditions with Einstein field equations; Section 6 analyzes observable effects; Section 7 gives concrete realization in QCA framework; appendices provide detailed scattering-theoretical derivation and curvature calculations. 2 Notation, Preliminaries and Unified Time Scale 2.1 Notation Conventions  Greek indices µ, ν, ρ, σ = 0,1,2,3 represent spacetime coordinate components, where x0= t, spatial indices i, j, k = 1,2,3.  Metric signature chosen as (−+ ++), i.e., ds2=gµνdxµdxν.  cis vacuum light speed; in some derivations set c= 1 to simplify notation, restore when necessary.  ωis angular frequency, E=ℏωis corresponding energy.  κ(ω) represents frequency-domain unified time scale; κ(x, ω) represents its spatial localization; κ(x) is macroscopic time density field after appropriate coarse-graining over frequency.  ρ(E) is state density, ρrel(E) is “relative state density” relative to some reference background. 3 2.2 Time Dilation and Curvature in Relativity In special relativity, flat Minkowski metric ds2=−c2dt2+δijdxidxj gives proper time τfor static particle: dτ=r−ds2 c2=r1−v2 c2dt. In general relativity, for observer static in coordinate system (dxi= 0), proper time satisfies dτ=p−gtt(x)dt c. For static, weak-field and isotropic case, metric can be written as ds2=−1 + 2Φ(x) c2c2dt2+1−2Φ(x) c2δijdxidxj, where Φ(x) is Newtonian potential. Then dτ≃1 + Φ(x) c2dt, manifesting gravitational time dilation. Here, relationship between Φ(x) and gtt(x) is geometrically defined, while Φ(x) connects to matter distribution through Einstein field equations. In this paper, we propose reverse construction: directly define Φ(x)and gtt(x)using time flow density κ(x). 2.3 Scattering Delay and Unified Time Identity Consider a pair of self-adjoint operators (H, H0), where H0is “free” Hamiltonian, His “full” Hamiltonian containing potential field or scattering center. Scattering matrix S(ω) can be written as unitary operator family of energy parameter ω, whose determinant phase det S(ω) = e2iφ(ω) defines total phase shift φ(ω). Eisenbud–Wigner–Smith theory shows time delay operator Q(ω) = −iS†(ω)dS(ω) dω has trace satisfying tr Q(ω)=2dφ(ω) dω. On the other hand, Lifshits–Kre˘ın spectral shift function ξ(ω) satisfies dξ(ω) dω=ρrel(ω), where ρrel(ω) is relative state density. Birman–Kre˘ın formula gives det S(ω) = e−2πiξ(ω). Combining above equations yields dφ(ω) dω=−πdξ(ω) dω=−π ρrel(ω), 4 thus obtaining unified time scale (absorbing constant differences in sign convention): κ(ω)≡φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω). This identity shows: there exists an essentially unique scale κ(ω)among time delay (phase derivative), spectral density change and scattering dynamics. We interpret it as time flow density: on mode at frequency ω, “effective time resource” carried per unit energy (or frequency) interval. Appendix A provides more detailed scattering-theoretical derivation. 3 Construction of Local Time Flow Density Field κ(x, ω) 3.1 Local State Density and Spatial Resolution In systems with spatial structure, state density can be localized as ρ(x, E), satisfying ρ(E) = Zρ(x, E) d3x. In scattering theory and Green function formalism, local state density can be given by ρ(x, E) = −1 πIm tr G+(x, x;E) where G+is retarded Green function. Similarly, local version of relative state density can be written as ρrel(x, E) = ρ(x, E;H)−ρ(x, E;H0). In spirit of unified time identity, we can spatially decompose κ(ω) as κ(ω) = Zκ(x, ω) d3x, where κ(x, ω) can be intuitively viewed as “relative time density localized near xat frequency ω”. More specifically, we define: Definition 3.1 (Local time flow density) Under given Hamiltonian pair (H, H0), local time flow density κ(x, ω) is defined as κ(x, ω)≡ρrel(x, ω), where ρrel(x, ω) = −1 πIm tr G+(x, x;ω;H)−G+(x, x;ω;H0). Then Zκ(x, ω) d3x=ρrel(ω) = κ(ω). Intuitively, κ(x, ω) represents at frequency ω, local state density increased or decreased by “structured universe” relative to some simple reference background; in our subsequent interpretation, it will be proportional to “time density of local internal information evolution”. 5 3.2 Frequency Window and Macroscopic κ(x) Macroscopic experiments (such as atomic clocks, gravitational redshift observations) often only sensitive to modes within certain frequency window ∆ω. We therefore define weighted coarsegraining: κ(x)≡ZW(ω)κ(x, ω) dω, where W(ω) is gauge-selected weight function satisfying W(ω)≥0,ZW(ω) dω= 1. Typically, W(ω) can be taken as window function matching transition frequency of used atomic clock; in QCA continuum limit, W(ω) naturally concentrates in low-energy effective band. We call κ(x) the local time flow density field, taking it as basic scalar field for subsequent geometric construction. 4 Curvature as Geometric Texture of κ(x) This section constructs metric from κ(x) and directly relates it to gravitational potential and curvature in weak-field limit. 4.1 Time Factor and κ(x): Metric Ansatz We first propose following axiom: Axiom 4.1 (Time scale axiom) At each point x, ratio of proper time τto external coordinate time tfor local static observer is determined by κ(x), i.e., there exists constant κ∞such that dτ dt(x) = κ∞ κ(x). where κ∞is time density value in “reference gravity-free region” (e.g., spatial infinity). In other words: larger κ(x), more internal evolution steps can occur per unit external time, thus local proper time is “slower” relative to coordinate time (because internal changes are more densely “filled”). This is consistent with gravitational time dilation intuition: deeper gravitational potential, slower time, corresponding to larger κ(x). For static observer (dxi= 0), have ds2=gtt(x) dt2=−c2dτ2, thus gtt(x) = −c2dτ dt2 =−c2κ2 ∞ κ2(x). Define dimensionless time factor η(x)≡κ(x) κ∞ , then gtt(x) = −c2 η2(x). 6 This gives unique form for time component: higher time density, smaller |gtt|, corresponding to proper time slower relative to coordinate time. 4.2 Spatial Factor and Information Volume Conservation Time factor alone cannot determine entire metric. We introduce second axiom: Axiom 4.2 (Local information volume conservation) Consider physical system containing many degrees of freedom, whose information volume is defined as Vinfo ≡Zκ(x)√γd3x, where γis determinant of spatial three-metric γij. We require in regions without macroscopic in/outflow, Vinfo remains invariant during slow evolution. Intuitive interpretation: in QCA or quantum field ontology, total information processing capacity (total internal evolution steps across all space per unit external time) should be conserved; if local time flow density κ(x) rises, corresponding available spatial volume should contract to maintain total information volume constant. Assume metric has static, isotropic form: ds2=−c2 η2(x)dt2+a2(x)δijdxidxj, then spatial three-metric is γij =a2(x)δij, giving √γ=a3(x). Information volume is Vinfo =Zκ(x)a3(x) d3x=κ∞Zη(x)a3(x) d3x. If we require Vinfo extremal under arbitrary local deformations and recovering flat metric in weak-field limit, i.e., η→1, a→1, natural choice is η(x)a3(x)=1, or a(x) = η−1/3(x). However, this choice has subtle differences from observations like light deflection. Another choice more consistent with gravitational lensing and optical metric experiments is letting spatial scaling factor inversely proportional to time factor, i.e., a(x) = η−1(x), corresponding to √γ=η−3(x), then information volume is Vinfo =κ∞Zη(x)η−3(x) d3x=κ∞Zη−2(x) d3x. 7 To make Vinfo equivalent to flat case in weak-field limit, we require spatial coordinates also do appropriate rescaling with η; under natural isotropic exchange, a class of stable solutions corresponds to double-conformal-scaling optical metric: ds2=−η2(x)c2dt2+η−2(x)γ(0) ij dxidxj, where γ(0) ij is background flat or slowly varying spatial metric. Through appropriate redefinition of η, can unify aforementioned gtt =−c2/η2with this form; in weak-field limit both are equivalent rescaling choice issues. For brevity, hereafter adopt ds2=−η2(x)c2dt2+η−2(x)δijdxidxj, and identify η(x)≃1 + Φ(x) c2 as time scale factor of gravitational potential. 4.3 Weak-Field Limit and Gravitational Potential In weak-field limit, let η(x) = 1 + ϵ(x),|ϵ(x)| ≪ 1, then gtt =−η2(x)c2≃ −(1 + 2ϵ(x)) c2, comparing with standard form gtt =−1 + 2Φ(x) c2c2 yields ϵ(x) = Φ(x) c2. On the other hand, from η(x) = κ(x)/κ∞get Φ(x) = c2ln η(x)≃c2η(x)−1=c2κ(x) κ∞−1, more precisely, Φ(x) = c2ln κ(x) κ∞. This shows: Newtonian gravitational potential in weak-field approximation can be viewed as logarithm of time flow density field κ(x). 8 4.4 Curvature Tensor and Second Derivatives of ln κ(x) Under above static, isotropic metric, Christoffel symbols and curvature tensor can be explicitly written. In particular, R00 component in weak-field approximation satisfies R00 ≃ −∇2Φ(x)/c2=−∇2ln η(x) = −∇2ln κ(x) + constant term, where ∇2is Laplace operator of background Euclidean space. This means: if writing Einstein equation as R00 −1 2g00R=8πG c4T00, in weak-field static case degenerates to familiar Poisson equation ∇2Φ(x) = 4πGρ(x), then in our time density perspective, this is equivalent to ∇2ln κ(x)∝ −ρ(x). Appendix B provides detailed calculation of above relation. 5 Consistency with Einstein Field Equations 5.1 Time Density and Energy–Momentum Tensor In standard general relativity, gravity source is energy–momentum tensor Tµν. In our framework, κ(x) directly characterizes “internal information evolution rate per unit external time, per unit spatial volume”, thus more naturally related to “information energy density”. We propose following correspondence: Hypothesis 5.1 (κ–source correspondence) At appropriate coarse-graining scale, there exists function Fsuch that ln κ(x) = FTµν(x), gµν(x), and in weak-field low-velocity limit degenerates to ln κ(x)≃αT00(x) ρ0c2+β, where α, β, ρ0are constants or slowly varying parameters. Combining previous section’s Φ(x) = c2ln κ(x) κ∞, yields Φ(x)≃αc2 ρ0c2T00(x) + constant = α ρ0 T00(x) + constant. For matter as static dust, T00 ≃ρc2, this degenerates to Φ(x)≃αρ(x) ρ0 c2+ constant, consistent with Newtonian potential equation form. More generally, there should exist some “information–energy correspondence relation” viewing κ(x) as function containing all components of Tµν; in this work we focus on static, lowvelocity situations dominated by T00. 9 B.1 B.1 Christoffel Symbols Metric components: g00 =−η2c2, gij =η−2δij, g0i= 0. Inverse: g00 =−1 η2c2, gij =η2δij. Christoffel symbol Γρ µν =1 2gρσ∂µgσν +∂νgσµ −∂σgµν. Due to metric being static, ∂0gµν = 0. Main nonzero components: 1. Γ0 0i: Γ0 0i=1 2g00∂ig00 =1 2−1 η2c2∂i(−η2c2) = 1 η∂iη. 2. Γi 00: Γi 00 =1 2gij (−∂jg00) = −1 2η2δij∂j(−η2c2) = η2δijηc2∂jη=η3c2∂iη. 3. Γi jk: Γi jk =1 2giℓ (∂jgℓk +∂kgℓj −∂ℓgjk) = 1 2η2δiℓ∂jη−2δℓk+··· Expanding using ∂jη−2=−2η−3∂jηyields Γi jk =−1 ηδi j∂kη+δi k∂jη−δjk∂iη. B.2 B.2 Weak-Field Approximation of Ricci Tensor R00 Ricci tensor defined as Rµν =∂λΓλ µν −∂νΓλ µλ −Γλ µνΓσ λσ + Γσ µλΓλ νσ. For R00: R00 =∂λΓλ 00 −∂0Γλ 0λ−Γλ 00Γσ λσ + Γσ 0λΓλ 0σ. Staticity gives ∂0Γλ 0λ= 0. In weak-field limit, let η= 1 + ϵ,|ϵ| ≪ 1, keeping linear terms. Then: ∂iη=∂iϵ, η3≃1,1 η≃1. 1. First term: ∂λΓλ 00 =∂iΓi 00 ≃∂i(c2∂iϵ) = c2∇2ϵ. 2. Second term vanishes. 3. Third and fourth terms are second-order small (ϵ ∂ϵ), can be ignored in weak-field linear approximation. Therefore 16 R00 ≃c2∇2ϵ. On the other hand, from η= 1 + ϵget ln η≃ϵ, thus R00 ≃c2∇2ln η(x). Relating ηwith κ: η(x) = κ(x) κ∞⇒ln η(x) = ln κ(x)−constant, thus R00 ≃c2∇2ln κ(x). Comparing with general relativity weak-field result R00 ≃ −∇2Φ(x)/c2 and using Φ(x) = c2ln η(x), can verify both are consistent, only differing in constant and sign conventions. C Appendix C: Construction Sketch of κ(x)in One-Dimensional Dirac–QCA This appendix provides simplified one-dimensional Dirac–QCA model, demonstrating how to construct discrete version of κ(x) from discrete update rules, corresponding to time density field in continuum geometric limit. C.1 C.1 Model Definition On one-dimensional integer lattice n∈Z, each site carries two-component internal degree of freedom (like spin), total Hilbert space is H=O n∈Z C2. Define one-step evolution operator U=S+⊗P++S−⊗P−, where  S+shifts state right one site, S−shifts left;  P+, P−are complementary projections on internal space satisfying P++P−=I. Through appropriate choice of P±parameters (e.g., including mass angle), can prove Ugives Dirac equation in long-wavelength limit. 17 C.2 C.2 Local Internal Frequency and Discrete Time Density Consider large-scale approximately stationary background, energy spectrum of local plane wave solution can be written as ψk(n, t)∼ei(kn−ω(k)t)u(k), where ω(k) is discrete energy eigenvalue. Internal “Zitterbewegung” oscillation frequency can be extracted by ωint(k), directly related to mass parameter. If we change local parameters of QCA in different lattice site regions (e.g., mass angle θ(n)), then ωint(n) becomes function of position, corresponding to local time density κ(n)∝ωint(n). In scattering construction, there exist two types of Hamiltonians: H0is QCA with uniform parameters, His QCA with local mass perturbations. Calculating scattering phase shift and spectral shift function between these two constructs discrete version of unified time scale κ(ω) and κ(n, ω). C.3 C.3 Continuum Limit and Geometric Correspondence In limit of lattice spacing ℓ→0, time step ∆t→0, define continuous coordinate x=nℓ, t=m∆t, and let η(x)∝κ(x) κ∞ . Through appropriate rescaling and coarse-graining, can obtain continuous time density field κ(x), substituting into metric constructed in Section 4, thus realizing mapping from discrete QCA to continuous geometry. In this mapping:  Spatial non-uniformity of local update rules in QCA manifests as spatial texture of κ(x);  Curvature in continuous geometry relates to second derivatives of κ(x);  Phenomena like gravitational redshift and light deflection manifest in QCA through signal propagation time and path differences. Thus “curvature as non-uniformity of time flow density” is not mere formal reinterpretation, but structural proposition constructible and verifiable in concrete discrete models through operators and spectra. 18