Gravity as Temporal Geometry IV: Origin of Time via Temporal Vacuum Decay

A calculable how for a beginning of time.
Refactoring GR to a time-first form with one temporal potential Φ (Phi) that sets the lapse and expansion—
dτ = exp(Φ)·dt, a = exp(−Φ), H = −dΦ/dτ (i.e. dτ=eΦdt, a=e−Φ, H=−dΦ/dτdτ=eΦdt, a=e−Φ, H=−dΦ/dτ)—the Big Bang is not a singular point to evolve through. It is boundary + transition: the high-Φ limit (Φ → +∞) is a kinematic boundary where proper time does not extend; a universe begins when a finite-Φ patch nucleates and Φ subsequently rolls, yielding FRW evolution with finite, selected initial data (no ad-hoc “just-so” start).

Paper IV in Gravity as Temporal Geometry. We work in a time-first map where a single lapse field Φ controls proper time and expansion (dτ = e^Φ dt, a = e^−Φ, H = −dΦ/dτ). In this dictionary the “origin of time” is a temporal boundaryat Φ → +∞. Cosmology begins via a quantum nucleation-and-roll transition that selects finite FRW initial data {Φin,Φin′}{Φin,Φin′} by extremizing a Euclidean action in V(Φ). We give thin- and thick-wall rates with a unified, dimensionally consistent prefactor A=αm∗4A=αm∗4 and treat the Hawking–Moss limit; gravitational quenching is noted.

Operationally, we derive a spherical flux law that calibrates the “time dial”:

∂tΦ=− 4πG r Ttr∂tΦ=−4πGrTtr

(with full SI factors), implying no self-restart in spherical vacuum and suggesting clock-network probes (flux-modulated redshift drift; universal dephasing). The FRW handoff follows from symmetry/regularity at the bounce center (isotropic induced 3-metric and extrinsic curvature), so the first proper-time slice is an FRW representative without circular assumptions. We also give the gauge link to Eddington–Finkelstein/Vaidya.

No new graviton DOFs: lapse/shift remain constraints (ADM); only the usual tensor TT modes propagate. Result: a selection principle for initial conditions—finite and calculable—that replaces “singularity crossing” with a boundary-plus-transition mechanism tied to observational and laboratory handles.

Significance: A concrete, testable mechanism for how time can start—boundary + quantum nucleation with selected FRW data—while remaining fully consistent with GR’s propagating content.

Series context

• Paper I: Gravity as Temporal Geometry: A Quantizable Reformulation of General Relativity (Zenodo: 10.5281/zenodo.16878019)

• Paper II: Experimental Signatures of Quantum Temporal Geometry (Zenodo: 10.5281/zenodo.16879129)

• Paper III: FRW Equivalence and Background Constraints (Zenodo: 10.5281/zenodo.16884232)