Basic idea: we pick one scalar field, \(\Phi\), to be our handle on time. We then define the lapse \(N\) (how fast proper time runs compared to coordinate time) as the exponential of that field: \(N=e^{\Phi}\). This guarantees \(N>0\) automatically (so time doesn’t flip sign), and it lets us talk about “time curvature” by just studying \(\Phi\). Setting \(\Phi=0\) at some reference clock makes that clock our baseline; everywhere else, \(e^{\Phi}\) tells you whether local clocks tick faster (\(\Phi>0\)) or slower (\(\Phi<0\)) than the baseline. This single choice is what makes the framework “time-first”: we commit to a clean, universal time variable before solving anything about space.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(\Phi(t,\mathbf{x})\) | time potential | scalar field controlling clock rate | dimensionless function (free up to a constant offset) | “Altitude of time” (higher = faster clocks) |
| \(N(t,\mathbf{x})\) | lapse | conversion from coordinate time to proper time | \(N = e^{\Phi} > 0\) (dimensionless) | “Gear ratio” between your watch and coordinate time |
| \(t\) | coordinate time | label we use to parametrize slices | arbitrary choice; not directly measured | “Page number” of the universe’s flipbook |
| \(\tau\) | proper time | time read by a local ideal clock | \(d\tau = N\,dt\) | “Clock-on-the-wrist time” |
| (baseline) | normalization | choose where \(\Phi=0\) | convention (e.g., at infinity or “today”) | “Zero mark on a ruler” |