Local--Relational Constants: A Two-Field Framework for Geometry and Causality

Abstract
I propose a covariant framework in which the circumference–to–diameter ratio pi and the speed of light c are promoted to smoothly varying scalar fields, pi(x) and c(x), constrained by a local relational invariance. Operationally, pi(x) is defined as the geodesic ratio of circumference to diameter in the physical (optical) metric within an infinitesimal Riemann normal neighborhood; c(x) encodes the light-cone slope as measured by local standard clocks and rods. The central hypothesis is an invariant current constraint, nabla_mu(pi(x)c(x)) = 0, or, in an equivalent gauge form, a Weyl weight assignment that renders the product pi*c locally conserved under smooth evolution. This realizes the idea that “constants” are locally constant for any freely falling observer but need not be globally universal across curved space–time; what is invariant is a relation between them. We derive field equations from a minimal generally covariant action with a Lagrange multiplier enforcing the relational constraint, analyze linearized solutions around Minkowski and FLRW backgrounds, and discuss black-hole spacetimes where hair is generically suppressed. The framework unifies two invariances---geometric and causal---into a single conserved relation and yields testable consequences distinct from varying-alpha or varying-c models in the literature. We outline observational bounds and falsifiability pathways.

I. Motivation and Principle The status of “constants” has been debated from Dirac's large-numbers hypothesis to modern scalar–tensor and varying-alpha/c scenarios [1, 2, 3, 4, 5, 6]. Empirically, local experiments uphold Lorentz symmetry and local position invariance at exquisite precision, yet global curvature implies that “Euclidean” constructs (like the circumference-to-diameter ratio) pick up curvature corrections when probed over extended regions.

We posit: Local Relational Invariance (LRI): For any freely falling observer, the combination pi(x)c(x) is conserved along physical evolution, while pi(x) and c(x) may each vary across space–time. Thus, each observer inhabits a locally constant “patch” (equivalence principle), but different patches can be related by smooth fields obeying a conservation law. The object of invariance is not an absolute constant but a relation.

II. Operational Definitions

Geometric field pi(x). In Riemann normal coordinates about a point p, define a geodesic circle of radius r << R^(-1/2) (curvature radius) in the optical metric g_opt_munu seen by null rays. Let C(r) be its geodesic circumference and D(r)=2r the geodesic diameter. We define pi(x) = lim_{r->0} C(r)/D(r). In flat Euclidean geometry, pi(x)=pi_E; curvature induces finite-r corrections C(r) = 2pi_Er * [1 - (K/6)*r^2 + O(r^4)] with Gaussian curvature K, so the operational extrapolation r->0 isolates the local value.

Causal field c(x). Define c(x) via the null-cone condition g_munu(x) dx^mu dx^nu = 0 measured against local standards (atomic clocks, interferometric rods). In units where c is dimensionless, c(x) encodes the local light-cone slope in the observer's orthonormal frame.

III. Action and Field Equations

We consider the following generally covariant action for gravity plus the two fields and matter: S = integral d^4x sqrt(-g) * [ (M_^2)/2 * f(pi,c) * R - 1/2(nabla pi)^2 - Z(pi,c)/2*(nabla c)^2 - V(pi,c) ] + integral d^4x sqrt(-g) * Lambda(x) * C(pi,c) + S_m[g_munu, psi; pi,c]. (2) Here f(pi,c) is an effective Planck coupling, Z(pi,c) a kinetic function for c, V a potential, and S_m allows for minimal or non-minimal couplings to matter fields psi. The LRI constraint is enforced either as a holonomic constraint C = pic - Pi_0C_0 or as a current conservation constraint C = nabla_mu(pic) * nabla^mu(pic), (minimum at nabla_mu(pi*c)=0). (3)

Varying S w.r.t. g_munu, pi, c, and Lambda yields (4a) M_^2 * f * G_munu = T^(pi,c)munu + M^2 * (nabla_munabla_nu - g_munuBox)f + T^(m)munu, (4b) Box pi - 1/2partial_piZ*(nabla c)^2 - partial_pi*V + (M^2)/2partial_pifR + Lambdapartial_piC = 0, (4c) nabla_mu(Znabla^mu c) - partial_cV + (M_^2)/2partial_cfR + Lambdapartial_cC = 0, (4d) C(pi,c) = 0 or nabla_mu(pi*c)=0.

Weyl-gauge formulation. An equivalent formulation introduces a local Weyl symmetry under g_munu -> Omega^2 * g_munu, pi -> Omega^(-1)pi, c -> Omegac, so that pic is Weyl-invariant. Taking f(pi,c) = xi(pic)^2 / M_^2 reproduces an induced-gravity coupling reminiscent of scalar–tensor theories while keeping the invariant relation explicit.

IV. Linearized Analysis and Cosmology

Perturbations around Minkowski Set g_munu = eta_munu + h_munu, pi = pi_bar + delta_pi, c = c_bar + delta_c, with constant backgrounds satisfying partial_mu(pi_bar * c_bar)=0. To first order, the LRI implies c_barpartial_mudelta_pi + pi_barpartial_mudelta_c = 0 ==> delta_c = -(c_bar / pi_bar) * delta_pi, (5) so only one physical scalar fluctuation remains. The scalar couples to curvature via partialf/partialpi and partialf/partialc; choosing f(pi,c) = xi*(pic)^2 / M_^2 suppresses long-range fifth forces in the decoupling limit, consistent with tight local tests [6].

FLRW background For spatially flat FLRW, ds^2 = -dt^2 + a(t)^2 d(x)^2, the LRI yields d/dt(pi(t)c(t)) = 0 ==> pi(t)c(t) = const. If c(t) drifts (VSL-like), pi(t) compensates. Unlike pure VSL models [4, 5], redshift–distance relations are controlled by the pair (pi,c), offering a clean null-test: any monotonic drift in one must be countered by the other to keep pi*c constant.

V. Black Holes and No-Hair Expectations Stationary, asymptotically flat black holes in broad scalar–tensor classes typically exhibit no scalar hair unless symmetry breaking or nonminimal structures are introduced. In the LRI framework, the conserved product suppresses independent profiles for pi and c outside the horizon: a nontrivial radial profile in one field forces a compensating one in the other, and with a single physical scalar combination the standard no-hair intuition reappears. This coheres with known constraints in Brans–Dicke–like limits [2] and with general lessons summarized in [6].

VI. Observational Windows and Falsifiability

• Local clocks and rulers. Atomic clock comparisons and cavity experiments constrain drifts of dimensionless combinations. In LRI, predictions map to drifts orthogonal to the pi*c = const direction, tightening consistency conditions beyond single-field varying-alpha scenarios [3, 6].

• Cosmology. Distance duality, CMB sound horizon, and BAO encode causal and geometric scales. Joint fits can isolate LRI patterns distinct from VSL-only models [4, 5].

• Strong gravity. Lensing rings and black-hole shadows probe local geometry (sensitive to pi(x) through geodesic curvature corrections) while ringdown encodes causal structure (sensitive to c(x)). Correlated deviations that preserve pi*c would be a smoking gun for LRI.

VII. Discussion We have replaced the notion of absolute constants by a conserved relation between two locally measurable structures: geometry (pi) and causality (c). The proposal satisfies the equivalence principle (locally constant for any freely falling observer), respects covariance, and reduces to GR with universal constants when the fields settle to global fixed points. It generalizes prior varying-constant approaches by (i) promoting a geometrically meaningful pi(x) tied to the optical metric, (ii) pairing it with c(x) under a conserved relation, and (iii) making falsifiable joint predictions.

Metaphysical note. This is a concrete instance of “coherence over absolutes”: what physics preserves is not a numeric idol but an invariant relation—an echo of the broader internal coherence that governs successful physical theories.

Acknowledgments The author thanks the broader community for foundational insights on varying constants and scalar–tensor gravity.

References

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