Inertial Geometry from Information Rate Conservation: Microscopic Unification of Mass, Time Dilation, and Energy--Frequency Relations

Abstract In traditional formulations, time dilation and relativistic mass increase in special relativity are usually viewed as two independent kinematic eects, while quantum mechanics directly relates energy to frequency through the Planck relation E=ℏω . For a massive particle, when its velocity approaches the speed of light, total energy grows rapidly, corresponding to growth of plane wave phase frequency ω ; yet proper time ow of the same particle exhibits Lorentz dilation, with internal clocks appearing to slow down. This image of slower clock corresponding to greater energy creates tension at the intuitive level. This paper introduces a two-dimensional information rate circle within the framework of quantum cellular automata (QCA) and information rate conservation (optical path conservation): viewing total information update rate as constant c and making an orthogonal decomposition between external spatial displacement velocity vext and internal state evolution velocity vint , satisfying v2 ext +v2 int =c2 . Under this structure, rest mass m0 is characterized as the Compton frequency ω0=m0c2/ℏ of internal quantum state in the rest frame, while the moving state corresponds to resource reallocation between the two frequency components of internal evolution and external translation in Hilbert space. We prove that: total frequency of plane matter wave satises ωtot =γω0 ; internal clock frequency undergoes redshift with velocity as ωclock =ω0/γ ; and these two frequency types are both uniformly embedded in the frequency identity ω2 tot =ω2 0+ω2 space induced by energy momentum relation E2=m2 0c4+p2c2 . On the information rate circle, this corresponds to a simple rule of inertial geometry: when external velocity approaches c , internal evolution velocity vint →0 , and corresponding eective inertial impedance meff ∝v−3 int diverges. Inertia is thus interpreted as energy cost required to maintain topological structure from collapsing in the internal-time freezing limit. This framework gives geometric and information-theoretic uni- cation between E=mc2 and E=ℏω without modifying any veried relativistic and quantum mechanical predictions. Keywords: Information rate conservation; Quantum cellular automaton; Inertial geometry; Compton clock; Relativistic time dilation; Energyfrequency correspondence; FubiniStudy metric 1 Introduction and Historical Context 1.1 Apparent Contradiction of Mass, Time, and Frequency Special relativity centers on the energymomentum relation E2=m2 0c4+p2c2, dening m0 as a Lorentz invariant, while relativistic mass γm0 mainly served as a pedagogical convenience in early teaching; its physical content has gradually been weakened or abandoned in modern literature, shifting to emphasizing invariant mass and covariant description of fourmomentum. On the other hand, quantum theory directly relates energy to frequency through the Planck relation E=ℏω. For a plane wave solution of a free particle ψ∼exp[i(kx −ωt)] , total energy is determined by temporal phase frequency ω . De Broglie further proposed that every particle with rest mass m0 carries an intrinsic oscillation frequency in its rest frame: ω0=m0c2 ℏ, now commonly called the Compton frequency or internal clock. This gives rise to an apparently contradictory picture: 1  Relativity predicts: when a particle moves with velocity v , its proper time satises dτ= dt/γ , i.e., the internal clock runs slower relative to laboratory time.  Under the quantum wave picture, total energy E=γm0c2 corresponds to plane wave phase frequency ωtot =E/ℏ=γω0 which increases with γ , i.e., frequency becomes faster. If frequency is naively understood as rhythm of internal oscillations of the particle, then time slowing and total frequency increasing seem mutually contradictory. Indeed, there has been extensive discussion and experimental proposals regarding the relationship between de Broglie internal clock, Compton frequency, and matter wave frequency, including rocks are clocks and detection of Compton clocks in atomic interferometers. 1.2 Quantum Cellular Automata and Discrete Relativistic Dynamics Quantum cellular automata (QCA) model the universe as discrete, local, and strictly unitary update rules acting on lattice sites Λ⊂Zd , whose continuum limits can give rise to eld equations such as Dirac, Weyl, and Maxwell. In particular, one-dimensional Dirac-type QCA has been constructed and proven to precisely reproduce the evolution operator and dispersion relation of the free Dirac equation in the long-wavelength limit. In Feynman's checkerboard model, propagation of spin1/2 fermions is represented as path sums on spacetime lattice points advancing at light speed and weighted by mass parameters at turning points; this model likewise yields the Dirac equation in the continuum limit. These works indicate: (1) Free relativistic particle dynamics can emerge in discrete, nite-information frameworks. (2) Mass parameters are naturally connected to local structures such as turning rate and internal ip rate in discrete models. This provides a natural entry point for understanding inertia and mass from information and computation perspectives. 1.3 Quantum Evolution Geometry and Internal Evolution Speed Another important clue comes from quantum state space geometry. Anandan and Aharonov pointed out that on projective Hilbert space P(H) , quantum state evolution can be described by Fubini Study metric geodesic length, with evolution velocity satisfying ds dt=2 ℏ∆H, where ∆H is energy uncertainty. These results, together with quantum speed limits (Mandelstam Tamm and MargolusLevitin bounds), indicate that under xed energy resources, the evolution rate of quantum states has an upper bound. If we understand internal time ow as rate at which states evolve in Hilbert space, then for a given free particle, its available evolution bandwidth can be viewed as nite resource. When this resource is used more to change spatial position (external motion), the share for internal phase/spin/topological structure evolution must necessarily decrease. 1.4 Goals and Contributions of This Paper Against the above background, this paper introduces the concept of information rate conservation: viewing global evolution of a massive particle as information rate allocation between external 2 position degrees of freedom and internal state degrees of freedom, and assuming there exists a universal upper bound c such that v2 ext +v2 int =c2, where vext is external group velocity and vint characterizes the rate of internal quantum state evolution under FubiniStudy metric, both jointly forming a two-dimensional information rate circle. On this basis, this paper achieves the following: (1) Provides an inertial geometry model starting from QCA and quantum evolution geometry, viewing special-relativistic time dilation as reallocation of information rate between internal and external components. (2) Proves that for Dirac-type free particles, plane wave total frequency, spatial frequency, and rest Compton frequency satisfy ω2 tot =ω2 0+ω2 space , obtaining concise parametric relations ωtot =γω0 and ωclock =ω0/γ on the information rate circle. (3) Starting from classical relativistic mechanics, rewrites longitudinal eective inertial mass meff =γ3m0 as meff ∝v−3 int , thus interpreting inertia as topological impedance in the internal time freezing limit. (4) Combined with existing Compton clock experiments and matter wave interference experiments, discusses how to test this inertial geometryinformation rate picture and its possible correction terms at the experimental level. This paper's position is: without modifying veried energymomentum relations and quantum measurement rules, but imposing a unied geometricinformation-theoretic interpretation on these relations, making massfrequencytime dilation integrate into one. 2 Model and Assumptions 2.1 DiracQCA Eective Model and Free Particle Sector Consider a Dirac-type quantum cellular automaton on one-dimensional space, with lattice set Λ=∆xZ ; each lattice site carries internal Hilbert space Hcell ∼ =C2 , corresponding to left-right propagation modes or spin up/down two internal degrees of freedom. Overall Hilbert space is H=O n∈Λ H(n) cell. Evolution is given by local unitary operator U , viewable as combination of internal rotation and conditional translation. Existing research has shown that under conditions of homogeneity, locality, and discrete causality, one can construct such a QCA class whose eective Hamiltonian on the single-particle sector in the long-wavelength limit is H=cαˆp+βm0c2, where ˆp is one-dimensional momentum operator, α, β are Pauli matrices satisfying α2=β2=I and anticommutation relation {α, β}= 0 . This Hamiltonian's eigenvalues on plane wave basis ψp(x)∼exp(ipx/ℏ) are E(p) = ±qm2 0c4+p2c2, the standard energymomentum dispersion relation. Therefore, in subsequent derivations, we need only use this energymomentum relation and QCA's discrete ontology as conceptual background, without depending on specic cell rule details. 3 2.2 Information Rate Circle and InternalExternal Decomposition We introduce the following basic axiom. Axiom 1 (Information Rate Conservation) . For any free particle, on one of its worldlines parametrized by some external reference time t , there exist two non-negative functions vext(t) and vint(t) , denoted respectively as external displacement rate and internal state evolution rate, satisfying v2 ext(t) + v2 int(t) = c2. where:  vext(t) in the continuum limit equals the group velocity of particle center position v(t) = ∂E/∂p .  vint(t) characterizes the quantum state evolution rate in the internal degrees of freedom direction of the same particle, with dimensions of velocity, obtained linearly from Fubini Study evolution velocity ds/dt via a xed scale factor. This relation geometrically constrains (vext, vint) to a two-dimensional circle of radius c , hence called the information rate circle. To remain consistent with special relativity, we identify external velocity as vext =v, and accordingly dene vint(v) := pc2−v2=cp1−β2, β := v c. This denition gives vint =c in the rest frame (v= 0) , i.e., all information rate is used for internal evolution; while in the limit v→c , vint →0 , corresponding to the extreme case of internal time freezing. 2.3 Time Dilation and Internal Clock Frequency In special relativity, proper time τ and laboratory time t satisfy dτ dt=p1−β2=1 γ, γ := 1 p1−β2. Comparing with the above, we can naturally view vint c=dτ dt as the normalized rate of internal time ow per unit laboratory time. That is, the magnitude of internal velocity vint is equivalent to measuring proper time ow rate. If in the rest frame the particle carries Compton frequency ω0=m0c2 ℏ, then along the worldline parametrized by proper time, its internal phase can be written as φ(τ) = ω0τ. 4 From the laboratory time t perspective, the observable internal clock frequency is ωclock := dφ dt=ω0 dτ dt=ω0 vint c=ω0 γ. This is precisely the frequency form of time dilation: the internal clock of a moving particle undergoes redshift relative to laboratory time. 2.4 EnergyFrequency Relation and Spatial Frequency Component On the other hand, in momentum eigenstates, the plane wave solution of Dirac particles has phase ψ(x, t)∼exp i ℏ(px −Et)= exp [i (kx −ωtott)] , where k:= p ℏ, ωtot := E ℏ. From the energymomentum relation: ω2 tot =E2 ℏ2=m2 0c4+p2c2 ℏ2=m0c2 ℏ2 +pc ℏ2=ω2 0+ω2 space, where ωspace := c|k|=pc ℏ can be interpreted as spatial direction frequency component, corresponding to spatial oscillations brought by the wave vector. This frequency identity has close parallelism with the information rate circle: ω0 plays the role of rest internal frequency, corresponding to vint at rest; ωspace corresponds to external momentum and group velocity. In what follows, we will show how to unify ωtot , ω0 , ωspace and vext , vint under QCA's internal external degrees of freedom decomposition into a framework of inertial geometry. 3 Main Results: Theorems and Alignments This section presents main theorems and structural conclusions of this paper. 3.1 Inertial Geometry Theorem: Geometric Rewrite of Time Dilation Theorem 2 (Inertial Geometry and Time Dilation) . Under Axiom ?? (information rate conservation), dening external velocity as vext =v and internal velocity as vint =pc2−v2, the following statements are equivalent: (1) Special-relativistic time dilation formula dτ= dtr1−v2 c2, i.e., dτ/dt=vint/c . 5 (2) The particle's velocity vector in the two-dimensional information rate plane u= (vext, vint) has xed modulus |u|=c . In other words, time dilation can be understood as: when an object's motion speed in external space increases, internal evolution speed is forced to decrease on a circle of radius c . 3.2 Frequency Geometry Theorem: Coordination of Internal Clock and Total Frequency Theorem 3 (Frequency Geometry and EnergyMomentum Relation) . For a plane wave state of a free Dirac particle, dene rest Compton frequency ω0=m0c2 ℏ, total frequency ωtot =E ℏ, and spatial frequency ωspace =pc ℏ. Then there holds frequency identity ω2 tot =ω2 0+ω2 space. Furthermore, dening internal clock frequency ωclock := ω0 γ=ω0 vint c, then total frequency and internal clock frequency satisfy ωtot =γω0=ω2 0 ωclock , i.e., ωclock ·ωtot =ω2 0. This shows:  For given invariant ω0 , when internal clock slows down in laboratory time ( ωclock decreases), total frequency ωtot necessarily increases.  Frequency slowing and speeding actually point to two dierent projections: internal clock and external plane wave phase. 6 3.3 Information Geometric Expression of Inertial Mass Amplication In relativistic mechanics, acceleration a∥ along velocity direction and applied force F∥ satisfy F∥=m0γ3a∥, from which one can dene longitudinal eective inertia m∥ eff := γ3m0. Using vint =c/γ , this can be rewritten as m∥ eff =m0c vint 3 . Theorem 4 (Inertia as Inverse Cube of Internal Time Rate) . Under the information rate circle framework, longitudinal eective inertial mass varies as the inverse cube of internal evolution velocity vint : m∥ eff ∝v−3 int . Therefore, when v→c , vint →0 , and m∥ eff → ∞ . From an information-theoretic perspective, this corresponds to: in the limit of nearly frozen internal time, for a system to maintain stability of its own topology and quantum correlation structure, its response rigidity to external forces tends to innity. 3.4 EnergyInternal Rate Identity and Unication of E=ℏω Starting from time dilation relation dτ/dt=vint/c and invariant m0 , one can dene an energy quantity directly related to internal rate: Etot := m0c2dt dτ=m0c2c vint =m0c3 vint . On the other hand, from energymomentum relation, Etot =γm0c2=ℏωtot . Theorem 5 (EnergyInternal Rate Identity) . Under the information rate circle framework, total energy of a free particle can be written both as Etot =γm0c2, and as Etot =m0c3 vint , compatible with the Planck relation Etot =ℏωtot. This establishes ωtot =Etot ℏ=m0c3 ℏvint =γω0. This shows:  E=mc2 and E=ℏω are essentially two ways of writing the same identity in dierent variables.  The information rate circle gives a triple unication among energyfrequencyinternal time rate. Proofs of these theorems will be given in subsequent Proofs section and appendices. 7 4 Proofs This section provides main derivations of above theorems, with more technical operator and geometric arguments moved to appendices. 4.1 Proof of Theorem ??: Minkowski Geometry and Circular Reparametrization Four-velocity is dened as uµ=dxµ dτ= (γc, γv). Its Minkowski norm satises uµuµ=−c2γ2+v2γ2=−c2. Let vext := v, vint := cr1−v2 c2. Then v2 ext +v2 int =v2+c21−v2 c2=c2. On the other hand, from time dilation dτ dt=r1−v2 c2=vint c, we see the relation between dτ/dt and vint is precisely the radial projection of the information rate circle. Thus, Minkowski four-velocity identity and information rate circle are just dierent parametrizations of the same constraint. This completes Theorem ?? . 4.2 Proof of Theorem ??: Minkowski Geometry of Frequency Energymomentum relation E2=m2 0c4+p2c2 divided on both sides by ℏ2 gives E ℏ2 =m0c2 ℏ2 +pc ℏ2. Let ωtot := E ℏ, ω0:= m0c2 ℏ, ωspace := pc ℏ, we obtain ω2 tot =ω2 0+ω2 space. On the other hand, velocity can be written as v=∂E ∂p =pc2 E=ωspacec2/c ωtot =cωspace ωtot . Thus β=v/c =ωspace/ωtot . From this: γ=1 p1−β2=ωtot ω0 , 8 hence ωtot =γω0. Writing time dilation as dτ/dt= 1/γ , internal clock frequency ωclock =ω0 dτ dt=ω0 γ obviously satises ωclock ·ωtot =ω0 γ·(γω0) = ω2 0. Theorem ?? follows. 4.3 Proof of Theorem ??: Relativistic Dynamics and Internal Rate Rewrite Relativistic mechanics along velocity direction gives F∥=d dt(γm0v) = m0γ3a∥, a standard textbook result. Introducing m∥ eff := γ3m0, we have F∥=m∥ effa∥. Using vint =cr1−v2 c2=c γ, we obtain γ=c vint , γ3=c vint 3 , thus m∥ eff =γ3m0=m0c vint 3 . When v→c , vint →0 ; from the above equation we directly see m∥ eff diverges, i.e., Theorem ?? . 4.4 Proof of Theorem ??: Triple Identity of EnergyInternal RateFrequency From time dilation dt dτ=γ and Etot =γm0c2, this can be viewed as Etot =m0c2dt dτ. On the other hand, from vint =c/γ we have dt dτ=c vint , 9 [10] P. M. Brown, On the concept of mass in relativity, arXiv:0709.0687 (2007). [11] H. Ma, Universal Conservation of Information Celerity, preprint (2025). A DiracQCA Hamiltonian and Operator Proof of Frequency Orthogonality This appendix provides operator-form proof of frequency orthogonal relation in Theorem ?? and demonstrates its connection with DiracQCA eective Hamiltonian. A.1 Dirac Hamiltonian Square and EnergyMomentum Relation Consider one-dimensional Dirac-type Hamiltonian H=cαˆp+βm0c2, where α, β are 2×2 Pauli matrices satisfying α2=β2=I,{α, β}=αβ +βα = 0. Computing H2 : H2=c2αˆp·αˆp+cαˆp·βm0c2+βm0c2·cαˆp+β2m2 0c4 =c2α2ˆp2+cm0c2ˆp(αβ +βα) + β2m2 0c4. Using α2=β2=I and {α, β}= 0 , middle cross term completely cancels, yielding H2=c2ˆp2+m2 0c4. On momentum eigenstate ˆpψp=pψp , H2ψp=E2ψp , from which we obtain E2=c2p2+m2 0c4, the standard energymomentum relation. This derivation remains valid in DiracQCA's continuum limit, as QCA's one-step evolution operator U has eective representation U= exp(−iH∆t/ℏ) . A.2 Frequency Orthogonal Relation Dividing both sides of above by ℏ2 gives E ℏ2 =m0c2 ℏ2 +pc ℏ2. Let ωtot := E ℏ, ω0:= m0c2 ℏ, ωspace := pc ℏ, we obtain ω2 tot =ω2 0+ω2 space, precisely the Pythagorean relation in frequency space. It embodies orthogonality of mass term and momentum term in the Hamiltonian at operator level: anticommutation relation ensures no cross term appears in H2 , thus preserving right triangle algebraic structure. 16 A.3 Correspondence with Information Rate Geometry In information rate circle, vext and vint satisfy v2 ext +v2 int =c2. Via energymomentum relation, velocity and frequency can be connected: v=cωspace ωtot , γ =ωtot ω0 , vint =cr1−v2 c2=c γ=cω0 ωtot . Therefore vext c2=ωspace ωtot 2 ,vint c2=ω0 ωtot 2 . This shows that geometric structure of information rate circle in velocity space completely parallels orthogonal relation of ω0 and ωspace in frequency space, further supporting rationality of inertial geometry as unied description. B Detailed Derivation of Longitudinal Eective Inertia This appendix provides detailed expansion of derivation of F∥=m0γ3a∥ and gives its rewrite in information rate variables. B.1 Relativistic Momentum and Acceleration Decomposition Relativistic momentum is dened as p=γm0v. Acceleration along velocity direction a∥:= dv dt, applied force F∥:= dp dt. Computing F∥=d dt(γm0v) =m0γdv dt+vdγ dt. From γ=1 p1−β2, β =v c, we get dγ dt=dγ dβ dβ dt=βγ3 c dv dt=γ3v c2a∥. Substituting F∥=m0γa∥+vγ3v c2a∥=m0γ+γ3v2 c2a∥. 17 Using γ2=1 1−β2⇒γ2−1 = β2 1−β2=γ2v2 c2, i.e., γ2v2 c2=γ2−1, then γ+γ3v2 c2=γ+γ(γ2−1) = γ3. Thus F∥=m0γ3a∥, the origin of longitudinal eective inertial mass m∥ eff =γ3m0 . B.2 Rewrite Using Internal Rate vint Information rate circle gives vint =cr1−v2 c2=c γ, thus γ=c vint , γ3=c vint 3 . Hence m∥ eff =γ3m0=m0c vint 3 . This shows sensitive dependence of m∥ eff on internal rate vint : when vint decreases due to high external velocity, eective inertia grows rapidly cubically. In information geometric picture, this means:  Large eective inertia does not indicate more matter, but indicates internal time ow compressed to extremely low rate, making any attempt to change its external motion state have to lever a nearly frozen internal structure, thus appearing extremely dicult. C Internal Evolution Speed and FubiniStudy Metric This appendix explains how to relate internal velocity vint to evolution speed in quantum state space and discusses its relationship with quantum speed limits. C.1 FubiniStudy Evolution Velocity In projective Hilbert space P(H) , evolution of state vector |ψ(t)⟩ independent of overall phase can be characterized by FubiniStudy line element ds2= 4 1− |⟨ψ(t)|ψ(t+ dt)⟩|2. AnandanAharonov proved that for evolution driven by Hamiltonian H , evolution velocity satises ds dt=2 ℏ∆H, 18 where ∆H=p⟨H2⟩−⟨H⟩2 is energy uncertainty. This relation shows: under given energy dispersion resources, geometric evolution rate of state in P(H) is limited. When system approaches quantum speed limit, its evolution speed approaches 2∆H/ℏ . C.2 Internal Velocity Scaling and Saturation For single-particle sector of free Dirac particles, overall Hilbert space can be split into external (position) and internal (spin/particleantiparticle) two parts of degrees of freedom. Under high symmetry, energy uncertainty can be expected to be mainly determined by internal structure, while group velocity is related to external momentum. One can then dene internal evolution velocity vint := ℓint ds dt=ℓint 2 ℏ∆Hint, where ℓint is a xed length scale used to convert dimensionless geometric velocity to quantity with velocity dimensions. Choose ℓint such that at rest vint(v= 0) = c, establishing one-to-one correspondence between internal evolution velocity and vint in information rate circle. In continuum limit, DiracQCA naturally provides such length and time units (lattice spacing and step size), so internal evolution velocity can be viewed as ratio of FubiniStudy distance traversed by internal state change per step to step size. When system energy is entirely contributed by rest mass, internal evolution approaches quantum speed limit, vint ≈c ; when most system energy converts to external momentum, internal evolution velocity decreases, vint < c , corresponding to deection on information rate circle. C.3 Quantum Speed Limit and Information Rate Upper Bound Quantum speed limit gives minimum time required to evolve from one state to orthogonal state: t⊥≥max πℏ 2∆E,πℏ 2¯ E, where ∆E and ¯ E are energy uncertainty and average energy respectively. If internal degrees of freedom are viewed as subsystem mainly responsible for state orthogonal change, upper bound of internal evolution velocity vint can be viewed as geometric embodiment of quantum speed limit. Information rate circle assumes vint ≤c as absolute upper bound, saturating this bound at rest. This assumption is formally compatible with quantum speed limit concept:  Stationary particle internal evolution approaches extreme speed, able to complete state distinction in minimum time.  Moving particle allocates part of velocity budget to external displacement, thus internal evolution slows, corresponding to longer state orthogonal time. This picture provides natural quantum information background for information rate circle, also pointing out that future more rened quantum speed limit experiments can further test this framework. 19