Paper XIII - Gravity as Inhomogeneous Influence Propagation

DOI: 10.5281/zenodo.18009130 Gravity as Inhomogeneous Influence Propagation Paper XIII of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Abstract. We reconstruct classical gravity as an emergent consequence of spatially inhomogeneous influence propagation within an information-theoretic framework where space and locality are derived rather than assumed. Building on Paper XII, where distance is defined operationally via minimal signaling delay and space emerges as a sparse interaction graph, we introduce a local influence-delay field encoding resistance to information transport. Spatial variation in this delay produces path-dependent propagation times, which we identify as curvature. Extremal-delay trajectories reproduce free fall, gravitational time dilation, redshift, and light bending without assuming spacetime geometry or Einstein’s equations. In the continuum coarse-grained limit, general relativity emerges as the unique effective second-order description compatible with locality, reversibility, and finite influence speed. Observable deviations are expected only beyond the classical regime, providing a principled foundation for quantum gravity extensions. Contents 1 Introduction 1 2 Recap of Emergent Space 2 3 Influence Delay Field 2 3.1 Definition 2 3.2 Path dependence and curvature 2 3.3 Mass as information saturation 2 3.4 Universality of coupling 3 4 Gravitational Phenomena 3 4.1 Time dilation and redshift 3 4.2 Free fall 3 4.3 Light bending 3 5 Continuum Limit and Effective Geometry 3 5.1 Effective metric 3 5.2 Newtonian limit 3 6 Effective Field Theory and Einstein Gravity 4 6.1 Why Einstein’s equations emerge 4 7 Schwarzschild Solution 4 8 Predictions Beyond Classical Gravity 5 9 Discussion and Outlook 5 10 Conclusion 5 1 Introduction General relativity encodes gravity as spacetime curvature, yet leaves open why geometry should be dynamical and why all forms of matter couple to it universally. In Paper XII we addressed a logically prior question: why space, distance, and locality exist at all. We showed that bounded influence propagation forces a sparse interaction graph with an operationally defined distance and a dynamically selected spatial dimension. In this paper we ask the next question: what follows when influence propagation is not uniform across this emergent space? We show that spatial variation in operational signaling delay is sufficient to reproduce all classical gravitational phenomena. Gravity arises not as a fundamental interaction but as a manifestation of inhomogeneous information transport in a system with finite influence capacity. No spacetime manifold, metric tensor, or field equations are assumed. These structures emerge only in the continuum limit as effective descriptions. – 1 – 2 Recap of Emergent Space Paper XII established the following results: •Distance d(A, B) is defined by minimal signaling delay, •Finite influence capacity forces a bounded-degree interaction graph, •Polynomial growth defines an emergent spatial dimension, •Stability of macroscopic records selects d= 3, •The graph is generically disordered and statistically isotropic. These results are taken as fixed in what follows. 3 Influence Delay Field 3.1 Definition We introduce a coarse-grained influence-delay factor Z(x) defined on regions xof the interaction graph. Influence traversing region xaccumulates operational time d˜ t=Z(x)dt, (3.1) where tis the underlying global evolution parameter used to parametrize dynamics, and ˜ t denotes operational time reconstructed from records (see Paper XI for the general distinction between ordering time and operational clock time). Z(x) is introduced operationally as a delay factor and carries no fundamental geometric meaning. It quantifies resistance to influence propagation due to local saturation of information-processing capacity. A metric description is introduced only later as an effective bookkeeping representation in the continuum limit. 3.2 Path dependence and curvature For a protocol following path γ, τ[γ] = X i Z(xi) ∆ti.(3.2) When Z(x) is inhomogeneous, different paths between the same endpoints accumulate different delays. This path dependence is the operational definition of curvature. 3.3 Mass as information saturation Regions with large Z(x) are those with high record density. Incoming influence competes for finite processing capacity, producing delay. Mass is therefore identified with localized resistance to influence propagation. This identification follows directly from finite influence capacity and does not require additional postulates. – 2 – 3.4 Universality of coupling All forms of influence are subject to the same capacity constraints. The delay field Z(x) therefore couples universally to all degrees of freedom, yielding the equivalence principle without assumption. 4 Gravitational Phenomena 4.1 Time dilation and redshift Clocks measure operational time ˜ t. A clock at position xaccumulates dτ =Z(x)dt. (4.1) For two clocks at x1and x2, ν2 ν1 =Z(x1) Z(x2),(4.2) reproducing gravitational redshift. 4.2 Free fall Influence follows extremal-delay paths: δZZ(x)dt = 0.(4.3) Massive particles and light rays therefore follow the same trajectories, yielding universal free fall as a consequence of influence optimization. 4.3 Light bending In inhomogeneous Z(x) profiles, extremal-delay paths curve. This reproduces gravitational lensing as a Fermat-type principle for influence propagation. 5 Continuum Limit and Effective Geometry 5.1 Effective metric When Z(x) varies smoothly and the interaction graph is coarse-grained to a continuum, it is convenient to encode delays via an effective metric g00 =−Z(x)2, gij =δijZ(x)2/(d−1),(5.1) with d= 3. Remark 1.This metric is not fundamental. It is a bookkeeping device encoding influence delays in geometric language. 5.2 Newtonian limit For weak inhomogeneities Z(x) = 1 + Φ(x) with |Φ| ≪ 1, conservation of influence capacity implies ∇2Φ=4πGρ, (5.2) where ρis influence density. This is Poisson’s equation. – 3 – 6 Effective Field Theory and Einstein Gravity 6.1 Why Einstein’s equations emerge At scales where the interaction graph appears smooth, the effective dynamics of Z(x) must satisfy: •locality (finite influence speed), •reversibility, •coordinate independence, •second-order evolution (stability). In this continuum regime, these requirements uniquely select general relativity as the effective field theory governing the metric derived from Z(x). Remark 2.This is an EFT statement. We do not derive Einstein gravity directly from the graph, but from its continuum coarse-grained limit. 7 Schwarzschild Solution For a static, spherically symmetric source, self-consistency yields Z(r) = 1−2GM r−1/2 .(7.1) This reproduces: •gravitational redshift, •light deflection, •perihelion precession, •Shapiro delay. The divergence at r= 2GM corresponds operationally to infinite outward delay: the event horizon. Remark 3.The singularity at r= 0 signals saturation of influence capacity, not a breakdown of spacetime. Quantum corrections must enter beyond this point. Remark 4 (Path delay versus clock processing delay).It is important to distinguish the inhomogeneous delay factor Z(x) = p−g00(x) derived in this paper from the effective clock rescaling αeff introduced later in the program. The quantity Z(x) represents a path-dependent propagation delay: it encodes the accumulated influence delay experienced by signals traversing the interaction graph and therefore affects all probes universally, independent of internal structure. In this sense, Zis geometric and underwrites gravitational redshift and curvature in operational time. By contrast, αeff represents a processing delay associated with finite clocks and recordforming systems. It arises from environment-dependent internal update costs and need not affect simple signals such as photons. Accordingly, αeff couples to the informational complexity of the probe rather than universally to energy. Gravity in the present paper is entirely encoded by path delay Z(x), while clock processing delay is treated separately in Paper XI, where its necessity is derived from chronological consistency of bounded records. – 4 – 8 Predictions Beyond Classical Gravity The framework reproduces general relativity exactly in the classical regime. Deviations arise only when: •influence capacity saturates (black hole interiors), •vacuum fluctuations modify delay (Casimir geometries), •large-scale baseline delays appear (cosmology). These regimes provide falsifiable targets for future experiments and observations. 9 Discussion and Outlook Gravity emerges here not as geometry but as constrained information flow. Spacetime curvature is an effective description of inhomogeneous influence delay. This framework: •explains universality of free fall, •removes singularities as fundamental objects, •clarifies why gravity is weak and geometric, •provides a clean interface to quantum extensions. Subsequent papers derive gauge redundancy, quantum kinematics, and operational time regulation, completing the foundational reconstruction. 10 Conclusion We have shown that classical gravity follows necessarily from spatial variation in influence propagation within an emergent space. General relativity appears as the unique effective description in the continuum limit, not as a fundamental postulate. Gravity is thus reinterpreted as a macroscopic expression of finite information flow. References [1] P. Cooney, Emergent Spatial Locality from Bounded Influence, Paper XII of the Ordered-Dynamics Reconstruction Program (2025). [2] D. Lovelock, J. Math. Phys. 12, 498 (1971). [3] T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995). [4] R. Sorkin, Lectures on Quantum Gravity, Springer (2005). – 5 –