Spontaneous Emergence of Lorentzian Signature from Curvature-Minimizing Geometry

TL;DR

This work presents a covariant model in which spacetime signature is a dynamical outcome of a curvature-minimizing geometry, encoded by an internal symmetric field $H_{ab}$ that acts as an order parameter for causality. A quartic potential $V_{ ext{sig}}$ yields degenerate minima corresponding to Euclidean, Lorentzian, and mixed signatures, while a curvature sector with quadratic invariants drives the geometry toward flatness; only the Lorentzian phase supports hyperbolic, causally propagating dynamics. Transitions between signatures are exponentially suppressed, and perturbations around the Lorentzian vacuum reproduce general relativity with controlled corrections, linking emergent causality with flatness in a unified framework. The model connects emergent-spacetime ideas with quadratic gravity, offering a dynamical origin for why our universe appears Lorentzian and nearly flat, and outlining testable phenomenological avenues and future research directions.

Abstract

A simple covariant model is presented where the signature of the metric is a dynamical field. Degenerate minima of a curvature-minimizing potential correspond to Euclidean, Lorentzian or mixed phases of geometry. The Lorentzian phase emerges as the only stable configuration supporting causal propagation.