A Strain Based Effective Field Theory of Gravity: Formulation, Equations, and Horizon Scale Tests

We construct a classical, Wilsonian Effective Field Theory (EFT) of gravity formulated in terms of a symmetric, dimensionless strain field ϵµν that perturbs physical intervals via gphys ≈ g + 2ϵ in the weak-field regime. In this theory, gravity arises from a two-step mechanism where mass sources a strain field, which in turn modifies the spacetime geometry. The unique massless spin-2 Fierz–Pauli (FP) kinetic operator supplies the principal (hyperbolic) part of the dynamics, while controlled nonlinearities enter through an algebraic quartic self-interaction and a compact set of higher-derivative operators suppressed by an EFT cutoff. We derive the Euler–Lagrange equations, explain their structure and invariances, and provide explicit expressions for the static O((rg/r)3ln(r/rg)) tail with the coefficient clog = 3/2 derived from the Fierz–Pauli Green’s function normalization. In the strong-field regime, we compute the photon-sphere radius rph, the critical impact parameter bc, and the ring diameter 2θc using a deformation of gtt, and extract strong-deflection coefficients with the Bozza formalism, rigorously calibrated to reproduce the Schwarzschild limits Aˆ = 1.000000 and Bˆ = −0.400230. Using Event Horizon Telescope (EHT) ring diameters—Sgr A∗ 51.8 ± 2.3 µas and M87∗ 43.0 ± 1.5 µas—we perform a diameter-only joint fit and find c4 = 0.564 ± 0.444 (1σ), a two-sided 95% CI of −0.306, 1.434, and, with a positivity prior c4 ≥ 0, a one-sided 95% upper limit of c4 < 1.280. All results are statistically consistent with General Relativity at the 1.3σ level. The EFT validity thresholds (|ΦEFT|ΦGR| ≈ 0.103 at r = 3rg) are comfortably satisfied. We include theoretical health checks (hyperbolicity, positivity causality priors, constraint propagation), quantify validity thresholds, and report numerical stability/robustness tests, providing a self-contained formulation with forward-predictions for higher-precision EHT/ngEHT tests.