Emergent Time from a Scalar Field: A Canonical Construction of an Internal Clock

Abstract The problem of time is one of the persistent conceptual challenges in gravitational physics. In general relativity, time is considered part of the geometry of spacetime, not an external parameter. In canonical formulations this feature appears through the Hamiltonian constraint. The resulting vanishing of the total Hamiltonian obscures the notion of physical evolution. This problem has been addressed through various strategies. Most of these approaches have defined time either as an internal variable or as a quantity emerging from specific statistical descriptions. In this paper, we do not necessarily claim to be seeking a complete solution to the problem of time, but perhaps this is a step in the right direction, as we focus on a specific, albeit simple, question: under what conditions can a dynamical field within a covariant gravitational framework provide a physically meaningful internal clock? In fact, this wasn't our first attempt to address the problem of time. The idea of emergent time was one of the hypotheses we previously proposed to find a comprehensive solution to the problem of the cosmological constant. Therefore, we chose to address it individually first, and then, in a later step, continue the work we had begun earlier. In attempting to answer the previous question, we explored a simple extension of general relativity using a scalar field 𝑆, described by a standard covariant action with a canonical kinetic term and potential. Our central idea is that any dynamic deviation (deformation) of this field from its equilibrium state defines a natural notion of structural relaxation. The energy of this deformation does not directly determine time but rather controls the rate of evolution of an internal clock variable along the field's relaxation path.