Optical Metric from Local Information Volume Conservation and the Entropic Derivation of Einstein Equations
Abstract
From the perspective of Quantum Cellular Automata (QCA) and information ontology, spacetime geometry should be understood as an emergent representation of the information processing capacity and connectivity structure of an underlying discrete quantum network. In this framework, we propose and formalize the principle of "Local Information Volume Conservation": in the coarse-graining process from discrete QCA to a continuous effective manifold, the "total amount of distinguishable quantum degrees
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Optical Metric from Local Information Volume Conservation and the Entropic Derivation of Einstein Equations Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 24, 2025 Abstract From the perspective of Quantum Cellular Automata (QCA) and information ontology, spacetime geometry should be understood as an emergent representation of the information processing capacity and connectivity structure of an underlying discrete quantum network. In this framework, we propose and formalize the principle of "Local Information Volume Conservation": in the coarse-graining process from discrete QCA to a continuous eective manifold, the "total amount of distinguishable quantum degrees of freedom capable of being hosted per unit coordinate volume and per unit coordinate time" within any local volume element must remain invariant. In the static, isotropic case, this principle uniquely selects a class of dual-factor scaled optical metrics ds2=−n−2(x)c2dt2+n2(x)δijdxidxj, where n(x)≥1 can be interpreted as an eective refractive index eld induced by the local information processing load. In the weak eld limit, taking n(x) = 1 −Φ(x)/c2+O(Φ2/c4) , where Φ is the Newtonian potential, the resulting line element expands as ds2=−(1 + 2Φ/c2)c2dt2+ (1 −2Φ/c2)δijdxidxj+O(Φ2/c4), which corresponds exactly to the general relativistic weak eld metric with γ= 1 in the Parametrized Post-Newtonian (PPN) formalism. This yields observables such as light deection and Shapiro delay, with a rst-order deection angle coecient of 4GM/(bc2) , thereby resolving the historical diculty of Einstein's 1911 scalar gravity theory (which yielded only half the deection angle) within a scalar-elddriven "refractive index gravity" framework. Furthermore, this paper proposes the InformationGravity Variational Principle (IGVP): regarding the geometric action as part of a generalized entropy functional and the local information processing load as a source term for entanglement entropy density, we perform variation on the total "information entropy" functional in the sense of local equilibrium. Drawing on Jacobson's idea of "Einstein equation as equation of state," we prove that in generalized scattering regions where the optical metric is viable, the equilibrium condition of the IGVP is equivalent to the Einstein eld equations with an information stress-energy tensor. 1
Thus, this paper establishes bridges on three levels: (1) deriving an experimentally testable optical metric starting from QCA unitary evolution and local information volume conservation; (2) aligning perfectly with the PPN structure of standard General Relativity in the weak eld limit; (3) reinterpreting the Einstein equations as a "geometricinformation equilibrium condition" within the framework of entropic forces and information geometry, providing several engineering proposals testable in numerical QCA and gravity-analog optical experiments. Keywords: Quantum Cellular Automaton; Optical Metric; Local Information Volume Conservation; Gravitational Lensing; Shapiro Delay; Entropic Force; Einstein Equations; Information Geometry 1 Introduction & Historical Context General Relativity (GR) takes a four-dimensional manifold (M, gµν) with Lorentz signature as the fundamental stage for gravitational theory, interpreting gravity as the curvature of spacetime geometry rather than a long-range force acting on a at background. Observational tests in the weak eld limit (such as perihelion precession of Mercury, light deection by the Sun, Shapiro delay, etc.) have provided highly precise support for this geometric gravitational narrative. On the other hand, the information ontology perspective represented by "It from Qubit" posits that the physical world is, at its deepest level, constituted by quantum information and its processing rules, with continuous spacetime, matter elds, and measurement results being emergent properties of some underlying discrete information structure. In this vein, the Quantum Cellular Automaton (QCA) is viewed as a natural model unifying quantum elds and discrete causal structures: dening nite-dimensional local Hilbert spaces on discrete lattices, realizing global time evolution through nite-neighborhood, spatially homogeneous unitary evolution rules, and subsequently emerging standard relativistic eld equations in the continuum limit. As early as 1911, when discussing the inuence of gravity on light, Einstein proposed a scalar gravity model where the gravitational potential Φ acted only as a scalar eld aecting the local speed of light, deriving a light deection angle of 2GM/(bc2) for light passing the Sun, which is half of the correct result later given by General Relativity. This discrepancy can be traced to the model modifying only the temporal component g00 of the metric while ignoring the curvature contribution of the spatial components gij . The complete General Relativity, in the static, weak eld limit, gives a line element writable as ds2=−(1 + 2Φ/c2)c2dt2+ (1 −2γΦ/c2)(dx2+dy2+dz2), compatible with experiments only when the PPN parameter γ= 1 . In literature studying gravitational lensing and wave propagation, the concept of "optical metric" has gradually formed: for static spacetimes, the projection of null geodesics can be equivalent to a geodesic problem on a three-dimensional Riemann manifold with metric γij =−g−1 00 gij , thereby introducing an eective refractive index N e (x) to study light deection and wave propagation in weak gravitational elds. However, such optical metrics are usually rewrites starting from a known gµν , rather than constraints derived from a more fundamental information or computational structure. 2
On the other hand, Jacobson's work showed that if one assumes the entropy of any local Rindler horizon is proportional to its area and requires the Clausius relation δQ =TdS to hold for all local horizons, the Einstein eld equations can be interpreted as an "equation of state," linking energy ux to curvature in some thermodynamic limit. Verlinde subsequently proposed that gravity could be viewed as an entropic force of positiondependent information, deriving Newton's law of gravity and its relativistic generalization within a holographic framework. Although these works provided inspiration for the "entropic origin of gravity," their original formulations usually assumed continuous spacetime, local thermal equilibrium, and holographic screen structures, without directly combining with the discrete QCA framework and constraints on local information processing capacity; they also face various criticisms and suggestions for modication. The goal of this paper is to unify these threads from a QCA perspective: 1. Starting from QCA unitarity and local information processing density, propose the principle of "Local Information Volume Conservation"; 2. Prove that in the static, isotropic case, this principle uniquely selects a class of dual-factor scaled optical metrics ds2=−n−2c2dt2+n2δijdxidxj , where n(x) is given by the local information processing load; 3. Prove that in the weak eld limit, this metric is equivalent to the linearized metric of General Relativity in PPN form, thus yielding correct light deection and Shapiro delay; 4. Inspired by Jacobson and Verlinde, interpret the geometric action as an informationgeometric entropy functional, propose the InformationGravity Variational Principle (IGVP), and prove its local equilibrium condition is equivalent to the standard Einstein eld equations. In this sense, gravity is no longer a "continuous geometric background added to QCA," but an eective optical geometry that the QCA network is forced to adopt to maintain global unitarity and information volume conservation under non-uniform local information processing loads. 2 Model & Assumptions This section constructs the modeling framework from discrete QCA to continuous optical metrics and lists explicit assumptions. 2.1 1. QCA Universe and Local Information Processing Density Assume the underlying universe is described by a QCA object U QCA = (Λ,H cell , U, ω0), where: Λ⊂Z3 is the set of three-dimensional lattice sites with lattice spacing a ; Each cell carries a nite-dimensional Hilbert space H cell ∼ =Cd ; The global Hilbert space is the quasi-local tensor product H=Nx∈ΛH cell ; Time evolution is realized by a unitary operator U with nite propagation radius Ra : ρn+1 =UρnU†, 3
where ρn is the global state at the n -th discrete time step; ω0 is the given initial state or family of initial states. Given a cuto scale L≫a , T≫∆t , we can coarse-grain the QCA, dening continuous coordinates xµ= (ct, x) , and introducing an eective eld theory description. The unitarity of the QCA ensures the conservation of the total system Hilbert space dimension and von Neumann entropy, but locally, quantum entanglement and information processing loads can be highly non-uniform. Dene the dimensionless local information processing density eld ρ info (x)∈[0,1), representing the proportion of the computational budget consumed by "intrinsic evolution" such as internal spins, local entanglement, and feedback loops, relative to the maximum vacuum budget, per unit coordinate volume and per unit coordinate time. Correspondingly, introduce the local information congestion factor n(x) = 1 1−αρ info (x)≥1, where α∈(0,1) is a coupling constant for scale normalization. n(x) = 1 corresponds to vacuum (no extra load), and n(x)>1 corresponds to regions of high information load. In the continuum limit of the QCA, the maximum group velocity of the free light cone is given by the lattice spacing and time step c=a/∆t. When local information load increases, the budget available for external signal propagation decreases, manifesting as a reduction in eective propagation speed, i.e., v ext (x)< c . 2.2 2. Local Information Volume Conservation and Scaling Factors Consider a small four-dimensional coordinate volume element d4x= dtd3x. In QCA coarse-graining, this corresponds to the evolution of a certain nite cluster of cells Ω⊂Λ over a nite number of time steps. Due to QCA unitarity, the number of distinguishable quantum states within Ω is constrained by the local Hilbert space dimension and entanglement structure, but should not change under coordinate reparametrization. To characterize metric deformation, introduce local scaling factors for time and space: The relation between physical proper time and coordinate time is dτ=ηt(x) dt; The relation between physical length element and coordinate length element is dℓ=ηx(x)|dx|. 4
Then the physical four-volume element is dV phys 4= dτd3ℓ=ηt(x)η3 x(x) dtd3x. Dene the "information capacity" per unit coordinate volume per unit coordinate time as the maximum logarithmic number of distinguishable quantum states C(x) . Under information ontology, assume the C(x) given by the QCA local structure in coarse-graining is proportional to the physical four-volume element but suppressed by the local information load ρ info (x) : C(x)∝ηt(x)η3 x(x) n(x). Here n(x) characterizes the compression of available evolution steps per unit physical time. To ensure local information capacity remains invariant under pure coordinate reparametrization, we introduce: Axiom 2.1 (Local Information Volume Conservation) . In the mapping from QCA to continuous manifold, for any suciently small manifold segment, the local information capacity satises ηt(x)η3 x(x)∝n(x), and degenerates to Minkowski values ηt=ηx=n= 1 in the perturbative weak eld limit. In the static, isotropic, and weak eld cases focused on in this paper, actual observational constraints mainly come from light deection and time delay, which are sensitive primarily to the metric structure in the t r subspace. To this end, we further adopt the following simplifying assumptions. Hypothesis 2.2 (Isotropy Simplication) . At the considered scale, the local information load is spatially isotropic, i.e., ηx(x) = ηs(x), where ηs scales all three spatial directions uniformly, and n(x) depends only on the potential function Φ(x) . Hypothesis 2.3 (2D Light Cone Conformality) . Require the area element of any radial light cone in the (t, r) two-dimensional subspace to remain conformally invariant, thereby preserving the "causal budget" conservation of null geodesics in the QCA. This is equivalent to requiring the determinant of the 2D sub-metric to be aected only by an overall scale factor under coordinate transformation. In the case of a static, spherically symmetric metric ds2=−A(r)c2dt2+B(r)(dr2+r2dΩ2), the determinant of the metric in the (t, r) subspace is det g(t,r)=−A(r)B(r). 2D Light Cone Conformality combined with Local Information Volume Conservation yields: Proposition 2.4. Under the above settings, A(r)B(r) = 1 . 5
Intuitively, this is because: if time is stretched by a factor ηt due to information load, then to keep the light cone area and the "discrete step shell" of null geodesics in the QCA invariant, the radial direction must scale by ηr=η−1 t , thus A∝η2 t , B∝η2 r giving AB =η2 tη2 r= 1 . This conclusion will be derived more systematically via Liouville-type arguments in Appendix A. Thus we can choose the parameterization A(r) = n−2(r), B(r) = n2(r), obtaining a family of optical metrics ds2=−n−2(r)c2dt2+n2(r)dr2+r2dΩ2. This is what we term the "InformationOptical Metric." 2.3 3. Matching Information Refractive Index with Newtonian Potential To recover the Newtonian gravitational potential at macroscopic scales, consider the geodesic equation of a slow test particle in this metric. Let the general form of the metric be ds2=−(1 + 2Φ/c2)c2dt2+ (1 −2γΦ/c2)δijdxidxj, where γ= 1 in PPN form corresponds to General Relativity. In the weak eld, slow speed limit, the dispersion relation determined by the time component gives the eective potential Φ , and particle acceleration satises d2x/dt2≈ −∇Φ. Expand the InformationOptical Metric to rst order: n(x) = 1 + ϵ(x),|ϵ| ≪ 1, then n−2= (1 + ϵ)−2≈1−2ϵ, n2≈1+2ϵ. Thus the line element becomes ds2≈ −(1 −2ϵ)c2dt2+ (1 + 2ϵ)δijdxidxj. Comparing with the PPN form, taking ϵ(x) = −Φ(x)/c2 yields g00 =−(1 + 2Φ/c2), gij = (1 −2Φ/c2)δij, which is the weak eld metric of General Relativity with γ= 1 . We can thus dene: Denition 2.5 (Information Refractive Index) . Let the Newtonian potential Φ(x) satisfy ∇2Φ = 4πGρ . The information congestion factor is dened as n(x)=1−Φ(x)/c2. In gravitational potential wells where Φ<0 , we have n(x)>1 , corresponding to regions with higher local information load and reduced eective light speed. 6
3 Main Results (Theorems and Alignments) Under the above model and assumptions, the main results of this paper can be summarized in the following theorems and corollaries. Theorem 3.1 (Local Information Volume Conservation ⇒ Optical Metric) . In a static, spherically symmetric, weak eld region obtainable by QCA coarse-graining, if the following are satised: 1. The underlying evolution is a local unitary QCA; 2. The Local Information Volume Conservation axiom holds; 3. Isotropy Simplication and 2D Light Cone Conformality hypotheses hold; then the metric allowed in the continuum limit must belong to the InformationOptical Metric family ds2=−n−2(r)c2dt2+n2(r)(dr2+r2dΩ2), where n(r)≥1 is a scalar eld determined by the local information processing density. Furthermore, in the weak eld limit, requiring the existence of a Newtonian potential Φ(r) such that slow particle motion satises d2x/dt2≈ −∇Φ uniquely determines n(r) = 1 −Φ(r)/c2+O(Φ2/c4). Theorem 3.2 (PPN Alignment of InformationOptical Metric) . Under the setting of Theorem 1, the line element expands in the limit |Φ|/c2≪1 as ds2=−(1 + 2Φ/c2)c2dt2+ (1 −2Φ/c2)(dr2+r2dΩ2) + O(Φ2/c4). Thus, in PPN notation, γ= 1, automatically yielding: 1. The weak eld deection angle for light passing a point mass M with impact parameter b : ∆θ=4GM bc2+OG2M2 b2c4; 2. Shapiro delay: ∆t Shapiro ∝(1 + γ)GM c3ln 4rErR b2, where 1 + γ= 2 , fully consistent with standard General Relativity. Thus, within the framework of a scalar refractive index eld, considering the coordinated scaling of temporal and spatial components satisfying A(r)B(r)=1 naturally avoids the historical diculty of traditional scalar gravity theories yielding only half the deection angle. Theorem 3.3 (InformationGravity Variational Principle and Einstein Equations) . Let the total "Information Entropy" functional be dened as S tot [g, Ψ] = S geom [g] + S info [g, Ψ], where: 7
Ψ represents matter and information degrees of freedom; The geometric part takes the form of JacobsonIyerWald type geometric entropy, i.e., under appropriate normalization δS geom [g] = 1 4GZM √−g(Rµν −1 2Rgµν )δgµν d4x; The variation of the information part denes the information stress-energy tensor T info µν =−2 √−g δS info δgµν . If we require local equilibrium to exist on Rindler small neighborhoods seen by all local observers, such that for any compactly supported metric variation δS tot = 0, then this condition is equivalent to Rµν −1 2Rgµν = 8πG T info µν . In the macroscopic limit, identifying T info µν with the standard stress-energy tensor Tµν recovers the usual Einstein eld equations. 4 Proofs This section provides proof outlines for the main theorems above; detailed calculations are placed in the Appendices. 4.1 Proof of Theorem 1 (Outline) 1. **2D Light Cone and Liouville Invariance** In the geometric optics limit of the QCA, consider radial light rays in the (t, r) subspace. Each discrete light ray can be viewed as a Hamiltonian ow on the phase space (t, r;pt, pr) , where Liouville's theorem guarantees the invariance of the phase space volume element dtdrdptdpr . Coarse-graining to a continuous metric, requiring the area element of the null geodesic shell in (t, r) subspace p−det g(t,r)dtdr to be proportional to the corresponding QCA step shell implies that det g(t,r) scales only by an overall factor. In the static, spherically symmetric metric ds2=−A(r)c2dt2+B(r)dr2+. . . , we have det g(t,r)=−A(r)B(r) . Matching the QCA step shell with the continuous light cone area element proves A(r)B(r) = const. Requiring the metric to degenerate to Minkowski ( A→1, B →1 ) at innity r→ ∞ , the constant must be 1, yielding A(r)B(r) = 1 . 2. **Isotropy and Scaled Parameterization** Under the assumption of 3D spatial isotropy, assigning B(r) uniformly to radial and angular parts yields ds2=−A(r)c2dt2+B(r)(dr2+r2dΩ2). 8
From A(r)B(r)=1 , the two functions can be expressed by a single scalar n(r) : A(r) = n−2(r), B(r) = n2(r). 3. **Uniqueness of Matching with Newtonian Potential** Expand the line element to rst order: n(r) = 1 + ϵ(r)⇒A(r)≈1−2ϵ(r), B(r)≈1+2ϵ(r). The geodesic equation in the slow limit can be written as d2x/dt2=−c2 2∇g00 +O(v2/c2). Substituting g00 =−A(r)≈ −(1 −2ϵ) , we get d2x/dt2≈ −∇(c2ϵ). To match Newton's law d2x/dt2=−∇Φ , the unique possibility is c2ϵ(r) = Φ(r), i.e., ϵ(r) = Φ(r)/c2 . By the convention Φ<0 , to keep n≥1 , redene ϵ(r) = −Φ(r)/c2 , determining n(r)=1−Φ(r)/c2. This completes the outline of Theorem 1. Full phase space derivation is in Appendix A. 4.2 Proof of Theorem 2 (Outline) 1. **PPN Form Expansion** Substituting n(r) = 1 −Φ(r)/c2 into the InformationOptical Metric and expanding to rst order gives g00 =−n−2≈ −(1 + 2Φ/c2), gij =n2δij ≈(1 −2Φ/c2)δij. Comparing with standard PPN notation, we read γ= 1 . 2. **Optical Metric and Eective Refractive Index** For any static spacetime, the 3D optical metric is dened as γij =−g−1 00 gij. Null geodesic projections on 4D spacetime are equivalent to geodesics on this optical 3D manifold. In the InformationOptical Metric: γij =n4(r)δij. Thus the eective refractive index for light in 3D space is N e (r) = n2(r) = 1−Φ(r)/c22≈1−2Φ(r)/c2. 9
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A Liouville-Type Derivation of Local Information Volume Conservation and A(r)B(r)=1 Consider the restriction of a static, spherically symmetric metric to the (t, r) subspace: ds2 (t,r)=−A(r)c2dt2+B(r)dr2. Let the conjugate momenta be pt=gtt ˙ t=−A(r)c2˙ t, pr=grr ˙r=B(r) ˙r, where the dot denotes dierentiation with respect to an ane parameter λ . The Hamiltonian constraint for null geodesics satises H=1 2gµνpµpν= 0, which on the (t, r) subspace is H=−1 2A−1(r)p2 t/c2+1 2B−1(r)p2 r= 0. Dene the phase space volume element dΓ = dtdrdptdpr. Hamilton's equations preserve dΓ , which is the content of Liouville's theorem. For the null geodesic shell H= 0 , the volume element can be written as dΓ|H=0 = dtdrdprdϕ, where ϕ is an angular variable parameterizing the shell. Since QCA unitarity requires that the "step density" corresponding to the null geodesic shell in the coarse-graining map remains invariant, the scaling factor of p−det g(t,r)dtdr relative to dtdr must remain constant. Specically, q−det g(t,r)=pA(r)B(r). If A(r)B(r) were allowed to vary with r , partial variation could be absorbed by coordinate redenition t→t′(t, r), r →r′(r) , but the "step shell density" on the null geodesic shell would no longer depend solely on local information load, but would mix in pure coordinate degrees of freedom. To ensure "information volume" reects only physical load in the QCA and not coordinate choice, we require A(r)B(r) = const. At innity r→ ∞ , requiring the metric to approach Minkowski implies A(∞) = B(∞) = 1 , so the constant must be 1, yielding A(r)B(r) = 1. Combined with the isotropy assumption, this can be written as A(r) = n−2(r), B(r) = n2(r). 17
B Calculation of Light Deection under Information Optical Metric Consider light propagation in a plane, choosing the plane θ=π/2 , with line element ds2=−n−2(r)c2dt2+n2(r)dr2+r2dφ2. Null geodesics satisfy ds2= 0 , and Killing symmetries give conserved quantities E=−gtt ˙ t=n−2(r)c2˙ t, L =gφφ ˙φ=n2(r)r2˙φ. Here E and L are "energy" and "angular momentum" respectively. The null geodesic condition gives 0 = −n−2c2˙ t2+n2˙r2+n2r2˙φ2. Eliminating ˙ t, ˙φ yields the radial equation ˙r2+L2 n4r2=E2 c2n4. Dene impact parameter b=Lc/E , and let u(φ)=1/r(φ) , transforming the radial equation to du dφ2 +u2=1 b2n−4(r(u)). In the weak eld limit, taking n(r)=1−Φ(r)/c2,Φ(r) = −GM/r, then n−4(r)≈1−4Φ(r)/c2= 1 + 4GM/(rc2). This gives the perturbation equation d2u dφ2+u=2GM c2b2. The solution is u(φ) = sin φ b+GM c2b2(1 + cos φ) + O(G2M2/(b3c4)). When φ varies from −π/2−δ to π/2 + δ , light travels from innity to innity, requiring u(φ)→0 . Solving for the deection angle gives ∆φ= 2δ=4GM bc2. This matches the standard General Relativity calculation exactly. Another more concise derivation uses the paraxial approximation and integral form under the optical metric ∆θ≈Z∞ −∞ ∇⊥N e (r) dl, Here N e =n2≈1+2GM/(rc2) , integrating along the unperturbed line r=√b2+z2 also yields 4GM/(bc2) . 18
C Variational Calculation for InformationGravity Variational Principle Let the geometric entropy functional be S geom [g] = 1 4GZH d2Σκ, where H is the local horizon cross-section and κ is the surface gravity. Jacobson and subsequent work showed that the corresponding volume functional variation can be written as δS geom =1 4GZM (Rµν −1 2Rgµν )δgµν√−gd4x. The information entropy functional takes the form S info [g, Ψ] = ZM s info (g, Ψ)√−gd4x, where s info is the local information entropy density, depending on the metric and matter information degrees of freedom in the QCA. Its variation is δS info =Z∂s info ∂gµν δgµν +∂s info ∂ΨδΨ√−gd4x+Zs info δ√−gd4x. Using δ√−g=−1 2√−ggµν δgµν , this can be rearranged as δS info =−1 2ZT info µν δgµν√−gd4x+Z∂s info ∂ΨδΨ√−gd4x, where we dene T info µν =−2∂s info ∂gµν +s info gµν. The total entropy variation is δS tot =1 4GZ(Rµν −1 2Rgµν )δgµν√−gd4x−1 2ZT info µν δgµν√−gd4x+. . . . Requiring δS tot = 0 for any compactly supported metric variation δgµν and assuming matter elds satisfy their respective EulerLagrange equations, we must have Rµν −1 2Rgµν = 8πG T info µν . In the macroscopic limit, identifying T info µν with the standard stress-energy tensor Tµν recovers the usual Einstein eld equations. This derivation demonstrates the equivalence between IGVP and standard action forms, while endowing geometric entropy and information entropy with explicit informationgeometric meaning. 19