A GL(4,R) Yang-Mills Framework for Gravity with a Non-Dynamical Background Metric

Abstract

We formulate a classical GL(4,R) Yang--Mills framework for gravity in the presence of a non-dynamical background metric. The local GL(4,R) symmetry is taken to characterize the admissible local geometric setting, and sixteen Yang--Mills gauge fields are introduced accordingly. The corresponding gauge-invariant action and field equations are constructed, with the background metric entering algebraically rather than dynamically. A central feature of the construction is the explicit dictionary between the GL(4,R) Yang--Mills gauge potential and connection-like geometric variables; under this dictionary, the Yang--Mills field strength is rewritten in curvature form, and the classical equations become closely related to quadratic-curvature theories with independent metric and connection variables. The restricted torsionless, metric-compatible sector formally coincides with the historical Stephenson--Kilmister--Yang/Fairchild-type system, whose non-unique solution branches and phenomenological difficulties are well known. We therefore do not claim the restricted vacuum equations themselves as new. Rather, the present work reinterprets this structure as a connection-primary GL(4,R) Yang--Mills framework with a non-dynamical metric and a gauge-theoretic balance law. In the restricted classical sector, Ricci-flat geometries including the Schwarzschild metric remain admissible exact solutions, while alternative solution branches are retained at the level of their formal mathematical status. The present paper is intended as the classical foundation of a broader GL(4,R) gauge program, whose phenomenological and quantum developments are currently in progress.