Special Relativity as Emergence of Information Rate Conservation: Orthogonal Decomposition of Internal Time Flow and External Displacement

Abstract Classical special relativity takes the constancy of the speed of light and the principle of relativity as axioms, adopting Minkowski spacetime as an a priori geometric stage. In this traditional picture, metric structure and Lorentz factors are viewed as geometric facts, with their relationship to information ow and computational capacity not made explicit. On the other hand, quantum information and condensed matter theory reveal that physical systems are universally constrained by nite information propagation speed and nite quantum evolution rate: local quantum lattice systems satisfy the LiebRobinson nite group velocity bound, dening an eective "signal speed of light"; quantum speed limit theorems provide the maximum evolution rate of pure states in projective Hilbert space, determined by energy uncertainty; information-physical limit analysis further indicates that every physical device has a maximum computational rate determined jointly by energy and number of degrees of freedom. This paper proposes an emergence scheme for special relativity centered on information rate within the framework of discrete quantum cellular automaton (QCA) ontology. For any local excitation, we dene two classes of information update rates: one is the external group velocity vext of the envelope center on the lattice, characterizing changes in spatial coordinates; the other is the FubiniStudy velocity in internal Hilbert space projective space, appropriately scaled and denoted as internal velocity vint , characterizing the rate of internal quantum state evolution. We propose the information rate conservation axiom : there exists a universal constant c such that at any moment v2 ext +v2 int =c2, and dene the proper time ow rate by internal velocity as dτ/dt=vint/c . On this basis, we prove the following main results: (1) From rate conservation and the proper time denition, one directly derives dτ= dt/γ , where γ= (1 −v2 ext/c2)−1/2 is the standard Lorentz time dilation factor. (2) Rewriting the information rate circle in terms of dt , dx , dτ yields the invariant line element ds2=−c2dτ2=−c2dt2+ dx2 , thus recovering Minkowski metric structure. (3) From proper time and velocity relations, constructing four-velocity and four-momentum yields standard E=γmc2 , p=γmvext , and the invariant relation E2=p2c2+m2c4 . (4) In the DiracQCA one-dimensional model, the internal Zitterbewegung-type oscillation frequency ω0 and eective mass m satisfy mc2=ℏω0 , so mass can be interpreted as the minimum information rate required to maintain the particle's internal frequencya kind of informationtheoretic impedance. In this perspective, proper time is no longer a presupposed external parameter but the distance traveled by internal quantum state in projective Hilbert space; the impossibility of accelerating massive particles to light speed stems from the fact that as vext →c , internal rate vint →0 , and the energy required to maintain the same internal structure tends to innity. We discuss the relationship of this framework with QCArelativistic emergence literature, quantum speed limit theory, and LiebRobinson nite group velocity, analyze limitations regarding global update time parameters and many-body entanglement, and propose several feasible engineering tests based on QCA quantum simulation and extreme-environment information systems. Keywords: Information rate conservation; Quantum cellular automaton; Proper time; Lorentz factor; Quantum speed limit; FubiniStudy metric; Four-momentum 1 Introduction and Historical Context Einstein proposed special relativity in 1905, starting from two axioms: (1) Principle of relativity: Physical laws take the same form in all inertial frames. (2) Constancy of light speed: The propagation speed c of light in vacuum is the same in all inertial frames. 1 Minkowski subsequently geometrized special relativity, constructing four-dimensional pseudoEuclidean spacetime with signature (−+ ++) and viewing free particle motion as geodesics in spacetime. Lorentz transformations are equivalent in this spacetime to the group of linear transformations preserving the line element ds2=−c2dt2+ dx2 . This geometry is formal and successful, but the ontological question of why is it so has long remained unresolved: Why does a limiting speed exist and equal the constant c ? Why must the metric necessarily be Minkowski-type rather than some other form? On the other hand, developments in quantum information and quantum statistical mechanics have revealed a series of universal constraints related to velocity and time. Lieb and Robinson proved in studying quantum spin systems that information propagation has a nite group velocity bound; the inuence of local operators spreads spatially in a nite light cone manner, dening the so-called LiebRobinson velocity. This result provides a light-speed-like upper bound in nonrelativistic lattice systems, embodying the profound connection between locality and nite signal speed. In quantum dynamics, MandelstamTamm and MargolusLevitin theorems provide quantum speed limits: the minimum time required for evolving from a pure state to one orthogonal to it is bounded by energy uncertainty and average energy lower bounds. Aharonov and Anandan introduced the FubiniStudy metric on projective Hilbert space, restating the timeenergy inequality as the relationship between quantum state curve length and energy uctuation. In research on information-physical limits, Lloyd analyzed the limiting computational rate determined by c , ℏ , and G , pointing out that the number of computational steps and information storage capacity achievable by any physical system are both limited by energy and degrees of freedom. These works collectively indicate that time and velocity are not merely geometric quantities, but reections of information propagation and processing capacity. Quantum cellular automata provide a discrete and strictly causal dynamical framework for the conception of universe as quantum computation. Local unitary updates acting on lattice sites naturally have nite propagation speed; under appropriate continuum limits, a series of works have proven that Dirac, Weyl, and even Maxwell equations can emerge from the long-wavelength limit of QCA. These results suggest that relativistic-type eld theory can be viewed as an eective description of underlying discrete information processing. Against this background, this paper attempts to answer a more fundamental question: if we interpret c as the total information update rate available to a single local excitation per unit coordinate time, can the entire dynamics of special relativity emerge from the constraint of information rate conservation? Specically, we distinguish two types of information updates: (1) External displacement: propagation of the excitation on the lattice, characterized by group velocity vext . (2) Internal evolution: motion of the internal Hilbert space state in projective space, yielding vint by scaling FubiniStudy velocity. The core axiom proposed in this paper is: for any excitation, the total information rate vector (vext, vint) has modulus constrained at Planck scale by constant c , viewed as the information rate budget available to that excitation. External and internal evolution are orthogonal allocations of the same budget. We will then prove that time dilation, Minkowski metric, and standard energymomentum relations of special relativity can all be viewed as geometric rewrites of this circular budget constraint. 2 2 Model and Assumptions This section provides the QCA background, rigorous denitions of external and internal velocities, and formalizes the information rate conservation axiom and proper time denition. 2.1 QCA Background and Causal Structure Consider a quantum cellular automaton dened on a d -dimensional regular lattice Λ⊂Zd . For each lattice site x∈Λ , associate a nite-dimensional local Hilbert space Hx ; the global Hilbert space is H=O x∈ΛHx. Time is indexed by integer steps n∈Z , and each step is realized by a global unitary operator U decomposable into a nite-depth array of local unitary gates. Locality means: there exists a nite neighborhood N(r) such that in each evolution step, the operator at any lattice site couples only with operators on its nite neighborhood. Under this setup, one can dene the support region of a local operator AX after n evolution steps, whose growth rate is controlled by the LiebRobinson bound: there exist constants vLR >0 and decay function F such that the commutator norm of operators on region Y suciently far from X with AX(n) decays exponentially or super-polynomially. Therefore, information propagation in QCA has an upper bound, which can be identied in the continuum limit as the eective speed of light c . In this paper, we assume QCA evolution converging to Weyl/Dirac equations in the longwavelength limit has been obtained through known constructions. Thus, external group velocity vext in the eective continuum limit equals the group velocity of relativistic wavepackets, but the underlying structure of discrete lattice and global update steps is retained. 2.2 Local Excitations and External Velocity Consider the single-excitation subspace H1p of QCA, spanned by position and internal degrees of freedom of a single particle on the lattice. Let |ψp⟩ be a single-excitation eigenmode with quasi-momentum p and eective dispersion relation ω(p) . On this basis, construct a narrowband wavepacket |Ψ(t)⟩=Zdp f(p−p0)e−iω(p)t|ψp⟩, where f is a weight function sharply concentrated around p0 . The time derivative of the envelope center position x(t) gives the group velocity vext(p0) =  dx dt = ∂ω(p) ∂p p=p0 . By the nite signal speed property of QCA, |vext(p)| ≤ c holds for all p . In what follows, we abbreviate vext as v , whose physical meaning is: the average displacement rate of the particle envelope center in space per unit coordinate time. 2.3 Internal Hilbert Space and FubiniStudy Velocity Each excitation carries a quantum state in a nite-dimensional internal Hilbert space Hint representing spin, avor, internal oscillation modes, etc. Let |ψ(t)⟩∈Hint be the internal state at coordinate 3 time t , evolving under eective Hamiltonian H : iℏd dt|ψ(t)⟩=H|ψ(t)⟩. Physical equivalence classes of internal states are represented by projective Hilbert space P(Hint) , i.e., pure state orbits with overall phase removed. Projective space is naturally equipped with the FubiniStudy metric, whose innitesimal line element is ds2 FS = 4 ⟨˙ ψ(t)|˙ ψ(t)⟩−|⟨ψ(t)|˙ ψ(t)⟩|2dt2, and the corresponding instantaneous geometric velocity is vFS(t) = dsFS dt=2∆H(t) ℏ, where ∆H(t) = p⟨H2⟩t−⟨H⟩2 t is the energy uncertainty. This result shows that the internal state's evolution speed in projective space is controlled by energy uctuation, and quantum speed limit theorems give the minimum time to evolve from the initial state to one orthogonal to it at this speed. To place this geometric velocity in the same dimension as external group velocity vext , introduce an internal length scale ℓint >0 and dene internal velocity vint(t) = ℓintvFS(t) = ℓint 2∆H(t) ℏ. ℓint can be viewed as a proportionality factor converting FubiniStudy distance to spatial equivalent length, its value chosen by calibration: in the rest frame, take some maximally active internal state such that its internal velocity saturates at c . Quantitatively, on a two-level system with xed spectral support, choose a superposition state simultaneously saturating MandelstamTamm and MargolusLevitin quantum speed limits and require it to satisfy vint =c in the rest frame, thus determining ℓint . Under this normalization, for any internal state we have 0≤vint ≤c . 2.4 Information Rate Conservation Axiom This paper proposes the following axiom as the starting point for special relativity emergence: Axiom 1 (Information Rate Conservation) . For any local excitation in QCA and any coordinate time t , its external group velocity vext(t) and internal velocity vint(t) obtained by scaling internal FubiniStudy velocity satisfy v2 ext(t) + v2 int(t) = c2, where 0≤vext(t)≤c , 0≤vint(t)≤c . This axiom can be physically understood as: the total information update rate available to each excitation per unit coordinate time is bounded by upper limit c ; external displacement and internal evolution can only make orthogonal allocations within this budget. The squared sum form is adopted to obtain geometric structure corresponding to a Euclidean circle, thereby introducing the invariant form of Minkowski metric. 4 2.5 Internal Denition of Proper Time Based on the above internal velocity denition, proper time τ is dened as the cumulative amount of internal evolution in the information length sense: dτ dt=vint(t) c. When the excitation is in the rest frame, if the internal state is chosen as an extremal state saturating the quantum speed limit, then vint =c , thus τ=t+ const . In a general moving state, increasing external velocity will compress internal velocity vint , thus slowing proper time ow. This denition will be proven in later sections to exactly reproduce the Lorentz relation between proper time and coordinate time in special relativity. 3 Main Results: Theorems and Alignments Based on the above model and axioms, we obtain the following main results. Theorem 2 (Emergence of Lorentz Time Dilation) . Let a local excitation in some inertial frame have external velocity vext(t) and internal velocity vint(t) satisfying information rate conservation v2 ext(t) + v2 int(t) = c2, and proper time τ dened as dτ/dt=vint(t)/c . Then dτ dt=r1−v2 ext(t) c2, i.e., dτ=dt γ(t), γ(t) = 1 p1−v2 ext(t)/c2. This is the standard special-relativistic Lorentz time dilation formula between proper time and coordinate time. Theorem 3 (Minkowski Line Element as Information Rate Identity) . In the same inertial frame, let the excitation's spatial coordinate be x(t) satisfying  dx dt =vext(t). Dene the spacetime line element ds2=−c2dt2+ dx2. Then under information rate conservation and proper time denition, ds2=−c2dτ2. Thus ds2 depends only on proper time increment, is an invariant independent of specic inertial frame, and has a form consistent with Minkowski metric. 5 Theorem 4 (Four-Velocity and Four-Momentum Structure) . Dene four-coordinates xµ= (ct, x) , proper time τ as before, four-velocity Uµ=dxµ dτ. Let m > 0 be the excitation's rest mass; four-momentum is dened as Pµ=mUµ. Then: (1) Four-velocity components are U0=cγ, U=γvext, γ =1 p1−v2 ext/c2, and satisfy the invariant normalization condition UµUµ=−c2. (2) Four-momentum components are P0=E c=mcγ, P=p=mγvext, and satisfy the invariant mass shell condition PµPµ=−m2c2, thus E2=p2c2+m2c4. Theorem 5 (Mass as Internal Frequency and Information Impedance) . Suppose a stationary excitation in its rest frame has internal oscillation angular frequency ω0 , with internal state evolving in proper time as e−iω0τ|ψ0⟩ . Dene rest mass as mc2=ℏω0. In a general inertial frame, total energy E is given by internal phase evolving with coordinate time e−iEt/ℏ . If information rate conservation and proper time denition hold, then: (1) Total energy satises E=γmc2, where γ is the Lorentz factor determined by vext . (2) Internal velocity satises vint =c γ, thus energy can also be expressed as E=mc2c vint . When vext →c , vint →0 , E→ ∞ . Therefore, mass can be interpreted as: under information rate conservation constraint, the minimum information impedance required to maintain the particle's internal oscillation frequency ω0 ; the higher the external velocity, the greater the total energy needed to preserve that internal structure. 6 4 Proofs This section provides proofs of the above theorems, with technical details supplemented in appendices. 4.1 Proof of Theorem ?? From information rate conservation v2 ext(t) + v2 int(t) = c2 holding for any t , we obtain vint(t) = qc2−v2 ext(t). Proper time is dened as dτ dt=vint(t) c, substituting the above gives dτ dt=1 cqc2−v2 ext(t) = r1−v2 ext(t) c2. Dene γ(t) = 1 p1−v2 ext(t)/c2, which can be written as dτ=dt γ(t), completely consistent with the time dilation relation in special relativity. 4.2 Proof of Theorem ?? In the same inertial frame, the spatial line element satises dx2=v2 ext(t) dt2. By rate conservation, we can write v2 ext(t) = c2−v2 int(t), thus the spacetime line element ds2=−c2dt2+ dx2=−c2dt2+c2−v2 int(t)dt2=−v2 int(t) dt2. On the other hand, the proper time denition gives dτ2=vint(t) c2 dt2⇒v2 int(t) dt2=c2dτ2. Substituting back yields ds2=−c2dτ2. Thus the line element depends only on proper time, is an invariant under Lorentz transformations, and has a form consistent with Minkowski metric. 7 4.3 Proof of Theorem ?? From Theorem ?? , dτ dt=r1−v2 ext c2⇒dt dτ=γ, γ =1 p1−v2 ext/c2. Four-coordinates are dened as xµ= (ct, x) , four-velocity as Uµ=dxµ dτ=d(ct) dτ,dx dτ=cdt dτ,dx dt dt dτ= (cγ, γvext). Its norm UµUµ=−(U0)2+U2=−c2γ2+γ2v2 ext =−c2γ21−v2 ext c2=−c2. This gives the Lorentz-invariant normalization of four-velocity. Four-momentum is dened as Pµ=mUµ , thus P0=mcγ, P=mγvext. Setting E=P0c , p=P , we have E=γmc2,p=γmvext. Computing the invariant PµPµ=−(P0)2+P2=−m2c2γ2+m2v2 extγ2=−m2c2, yields E2=p2c2+m2c4. 4.4 Proof of Theorem ?? In the rest frame ( vext = 0 ), suppose the internal state evolves with proper time as |ψ(τ)⟩= e−iω0τ|ψ0⟩, where ω0 is the internal oscillation frequency. The quantum mechanical energyfrequency relation gives E0=ℏω0. Dene rest mass m satisfying mc2=E0=ℏω0. In a general inertial frame, the coordinate time evolution form is |ψ(t)⟩= e−iEt/ℏ|ψ′ 0⟩, where E is the total energy measured in that frame. From the proper time and coordinate time relation dτ=dt γ⇒d dτ=γd dt, 8 the internal state's derivative with respect to proper time is d dτ|ψ(τ)⟩=−iE ℏγ|ψ(τ)⟩. Comparing with the rest frame case d dτ|ψ(τ)⟩=−iω0|ψ(τ)⟩, we obtain Eγ =ℏω0=mc2, i.e., E=mc2 γ. Note that here γ is dened as the Lorentz factor of the rest frame relative to that inertial frame, so if we adopt the convention of viewing γ as the moving frame relative to rest frame factor, we need to handle the inverse. A more direct approach is to use the already-obtained result from Theorem ?? E=γmc2, and combine with the internal velocity expression. From information rate conservation and Theorem ?? , we have vint =cr1−v2 ext c2=c γ. Thus total energy can be rewritten as E=γmc2=mc2c vint . When vext →c , γ→ ∞ , vint →0 , total energy diverges. This indicates: to maintain the original internal structure under nearly completely frozen internal time ow requires innite energy investment. Mass m here can be viewed as information impedance to internal oscillation frequency ω0 : the higher the frequency, the greater the information rate needed to prevent collapse of that internal structure for xed external velocity component, manifesting as larger m and more dramatic energy growth. 5 Model Applications This section discusses applications and interpretations of the above structure in several typical physical situations. 5.1 Two Limits: Photons and Stationary Particles On the information rate plane (vext, vint) , information rate conservation corresponds to a quartercircle of radius c . Two important limits are: 1. Stationary massive particle : vext = 0 , vint =c . All information budget is used for internal evolution; proper time coincides with coordinate time; the particle's internal clock runs at maximum rate. 9 A.4 Internal Velocity Normalization and Upper Bound This paper denes internal velocity vint =ℓintvFS =ℓint 2∆H ℏ. To x ℓint , we can adopt the following strategy: (1) Select some physically realizable limiting internal state family, e.g., in two-level systems, superposition states saturating MandelstamTamm and MargolusLevitin upper bounds, making them represent maximum internal activity in the rest frame. (2) Require these states to have vint =c in the rest frame. (3) Solve to get ℓint =cℏ 2∆Hmax . Since any state's energy uncertainty does not exceed ∆Hmax under that support, for all states we have vint ≤c. Combined with QCA's nite signal speed |vext| ≤ c , the information rate conservation axiom requires physical states to be restricted within a circle of radius c in the two-dimensional rate plane, naturally introducing subsequent Minkowski geometry structure. B Abstract Derivation from Information Rate Circle to Minkowski Metric This appendix provides derivation from information rate circle to Minkowski metric under more abstract setup, highlighting consistency between geometric structure and algebraic identities. B.1 Abstract Information Rate Plane Consider two-dimensional real vector space R2 with coordinates (u1, u2) representing two types of information rates. Assume physically allowed rate pairs are constrained on a circle of radius constant c > 0 , i.e., u2 1+u2 2=c2. Introduce parameter θ∈[0, π/2] , letting u1=csin θ, u2=ccos θ. Identify u1 as external velocity vext and u2 as internal velocity vint . Assume coordinate time parameter t exists; external displacement satises  dx dt =vext(t) = u1(t) = csin θ(t), and dene proper time dτ dt=vint(t) c=u2(t) c= cos θ(t). 16 B.2 Geometric Relation of Lorentz Factor By denition dτ dt= cos θ(t)⇒dt dτ=1 cos θ(t)≡γ(t). On the other hand, vext =csin θ , so sin2θ=v2 ext c2,cos2θ= 1 −v2 ext c2, thus γ=1 cos θ=1 p1−v2 ext/c2, precisely the Lorentz factor. Thus, as long as an information rate circle and proper time denition exist as above, time dilation relation is automatically recovered. B.3 Algebraic Rewrite of Minkowski Line Element Dene spacetime line element ds2=−c2dt2+ dx2. From dx2=v2 extdt2 , we have ds2=−c2dt2+v2 extdt2=−(c2−v2 ext)dt2. Using rate circle c2−v2 ext =v2 int, we obtain ds2=−v2 intdt2. On the other hand, proper time denition gives dτ=vint cdt⇒v2 intdt2=c2dτ2. Thus ds2=−c2dτ2. This relation shows that Minkowski line element structure is equivalent to combined result of information rate circle and proper time denition. In other words, once we admit external velocity and internal velocity form circle of radius c on rate plane, Minkowski metric is no longer an additional assumption but an algebraic rewrite of the circle. C Mass and Internal Frequency in One-Dimensional DiracQCA This appendix uses one-dimensional DiracQCA as example to illustrate relation between internal oscillation frequency and mass parameter, thus providing model-level support for Theorem ?? . 17 C.1 One-Dimensional DiracQCA Construction Consider one-dimensional lattice Λ = aZ with lattice spacing a . Each lattice site carries twodimensional internal Hilbert space Hx∼ =C2 , viewable as spin1 2 degree of freedom. Dene shift operators (S+ψ)x=ψx−1,(S−ψ)x=ψx+1. Construct one-step evolution operator U= exp −iθX x σy x!exp −iπ 2X x σx x!(S+⊗|↑⟩⟨↑|+S−⊗|↓⟩⟨↓|), where σx, σy are Pauli matrices and θ is an adjustable parameter. Fourier transforming this QCA yields diagonalized form in momentum space U(p)=e−iHeff (p)∆t/ℏ, with eective Hamiltonian converging in small p limit to one-dimensional Dirac Hamiltonian HD(p) = cp σz+mc2σx, where c and m are parameters determined by θ , a , ∆t . C.2 Dispersion Relation and Internal Oscillations Diagonalizing the above Dirac Hamiltonian yields energy spectrum E±(p) = ±p(cp)2+m2c4. At rest momentum p= 0 , E±(0) = ±mc2. Consider internal state formed by equal-amplitude superposition of positive and negative energy band eigenstates |+⟩ , |−⟩ : |ψ(0)⟩=1 √2(|+⟩+|−⟩). Its time evolution is |ψ(t)⟩=1 √2e−imc2t/ℏ|+⟩+ eimc2t/ℏ|−⟩. For suitable internal observables (e.g., some Pauli component), expectation values will exhibit oscillations at frequency 2mc2/ℏ this is the discrete version of Zitterbewegung phenomenon. In this paper's framework, the fundamental frequency ω0 of internal oscillations can be taken as mc2/ℏ or a constant multiple of 2mc2/ℏ ; for brevity, we dene mc2=ℏω0, viewing mass as linear function of internal frequency. The above construction shows that this relation has explicit model foundation in DiracQCA. 18 C.3 Correspondence of Internal Velocity and Mass At rest momentum p= 0 , internal state evolves in projective space at frequency ω0 with Fubini Study velocity vFS =2∆H ℏ. For equal-amplitude superposition state of two-level symmetric energy spectrum E±=±mc2 , energy uncertainty is ∆H=mc2, so vFS =2mc2 ℏ= 2ω0. Under this paper's chosen internal length scale ℓint , requiring saturated state in rest frame to satisfy vint =c , i.e., c=ℓintvFS =ℓint2ω0⇒ℓint =c 2ω0 . Thus for general state we have vint =ℓintvFS =c 2ω0 vFS. When considering cases including momentum, energy spectrum expands to E±(p) , internal oscillation frequency and energy uncertainty vary with p , and internal velocity vint will be below c . 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