From Phase Potential to Covariant Gravity: Effective Metric and Action Principle

From Phase Potential to Covariant Gravity: Effective Metric and Action Principle This paper provides the covariant completion of the weak-field phase-potential framework introduced in Paper 1. The same dimensionless phase potential ψ(r) = Φ(r)/c² is embedded into a minimal effective spacetime metric, unifying clock slowing, free fall, and light propagation within a single covariant structure. An explicit action principle for ψ is introduced, promoting it from a chosen parametrization to a bona fide field with dynamics. In the static weak-field limit, the resulting field equation reduces to Poisson’s equation, ensuring continuity with Newtonian gravity and the successful predictions of Paper 1. The aim of this paper is structural rather than revisionist: it does not claim to replace General Relativity, but to show that the phase-potential skeleton can be expressed in a coordinate-independent way and endowed with a consistent variational foundation. The formulation is deliberately conservative, matching GR in the verified weak-field regime. Series context: • Builds directly on Paper 1 (weak-field unification). • Provides the formal framework required for Paper 3 (controlled deviations and falsifiability). • Supplies the covariant “landing strip” used in Paper 4 to connect ψ to quantum interference.