We now say exactly how the math’s “page number” \(t\) turns into the time a real clock measures, \(\tau\). The conversion factor is the lapse \(N\). If \(N\!>\!1\) at your location, your wristwatch accrues more proper time than the coordinate label \(t\); if \(N\!<\!1\), it accrues less. Because \(N=e^{\Phi}\), variations of the single scalar \(\Phi\) directly are gravitational time dilation. Reparametrize \(t\) however you like\(d\tau\) stays invariant, so physics doesn’t care about your labeling scheme, only about \(N\) (hence \(\Phi\)) along the clock’s path.
Quick intuition: imagine two identical clocks, A where \(N=1.00\) and B where \(N=0.99\). Over the same coordinate interval \(dt=1\ \text{hour}\), A experiences \(d\tau=1.00\ \text{h}\) while B experiences \(0.99\ \text{h}\). B “runs slow” purely because its local \(N\) is smaller. No extra assumptions needed.
| Symbol | Name | Meaning | Value / Units | Metaphor |
|---|---|---|---|---|
| \(d\tau\) | proper-time increment | physical time a perfect local clock accumulates | \(d\tau = N\,dt\); time units (e.g., s) | “Wristwatch tick” |
| \(dt\) | coordinate-time increment | change in the chart label \(t\) | arbitrary; not directly measured | “Flipbook page step” |
| \(N(t,\mathbf{x})\) | lapse | conversion factor from \(dt\) to \(d\tau\) | \(N=e^{\Phi}>0\) (dimensionless) | “Gear ratio” of time |
| \(\Phi(t,\mathbf{x})\) | time potential | log-lapse controlling clock rate | dimensionless; offset is a convention | “Altitude of time” |
| worldline | clock path | trajectory of a clock through spacetime | path; \(d\tau\) integrated along it | “Route a hiker walks, counting steps” |