Metric Signature as an Auxiliary Order Parameter: A Causal Viability Criterion

Abstract

We describe a branchwise formulation of metric gravity in which the spacetime signature is encoded by a real symmetric internal field H through g = e H e^T. An O(4)-invariant algebraic potential has non-degenerate minima satisfying H^2 = 1, with one vacuum orbit for each inertia (p,q). Within a connected region where H is non-degenerate, Sylvester's law permits a change of variables that places all derivatives in the metric sector. The remaining components of H are auxiliary, while the flat directions along its O(4) orbit are gauge. We derive this statement directly from the coupled tetrad and H equations. As a diagnostic example, we study R + alpha R^2 gravity around the five flat signature classes in four dimensions. For fixed universal couplings, only one Lorentzian branch has the standard graviton sign, a non-tachyonic scalaron, and a hyperbolic initial-value problem. Euclidean and split-signature branches do not define ordinary causal phases. Conditional on the existence of persistent observers, this supplies an anthropic interpretation of the causal viability criterion. It is not a dynamical theory of transitions or a probability measure on the branches: any real transition must encounter det H = 0, where the branchwise description fails.