About this guide: This step-by-step walkthrough develops the complete theory of light propagation in temporal geometry. Each chapter states the goal, shows the math, and explains the meaning.
Chapter 1: Foundations & Intuition
Why time first. What the lapse \(\Phi\) represents. The core picture: light behaves as if it moves in a medium set by the local rate of time.
Chapter 2: The Temporal–Optics Map \(n=e^{\Phi}\)
The organizing equation. How \(\Phi\) defines a refractive index. Why this unifies lensing, Shapiro timing, and drift at leading order.
Chapter 3: Travel Time & Shapiro Delay
From Fermat’s principle to travel time: \(\Delta T \simeq \frac{1}{c}\int (e^{\Phi}-1)\,\mathrm{d}\ell\).
Recover the classic Shapiro timing in the static limit.
Chapter 4: Deflection from Transverse Time-Gradients
Bending ties to sky-plane gradients of clock rate:
\(\mathbf{\alpha}(\hat{\mathbf{n}}) \simeq \int \nabla_{\!\perp}\Phi\,\mathrm{d}\ell\).
Chapter 5: Time-Variable Potentials & Achromatic Drift
When \(\partial_t\Phi \neq 0\) the arrival time drifts:
\(\frac{\mathrm{d}}{\mathrm{d}t_{\mathrm{obs}}}\Delta T \simeq -\frac{2}{c}\int \partial_t\Phi\,\mathrm{d}\ell\).
Why the signal is frequency-independent.
Chapter 6: Discriminant — Plasma vs Temporal
Model the data as \(\epsilon_{\mathrm{dis}}(\nu)=A\nu^{-2}+B\).
Plasma is chromatic. Temporal geometry is achromatic.
Chapter 7: Lapse vs Shift — What Lives Where
\(\Phi\) controls achromatic dilation and drift. The shift carries rotation and waves.
How clocks, Sagnac links, and gyros separate these sectors.
Chapter 8: Observables & Forecasts
Clock networks, pulsar timing, and lensed quasars. What scales to expect.
Practical tactics and near-term benchmarks.
Chapter 9: Systematics, Checks, & Summary
Control path delays and instrument drift. Use null tests and multi-band checks.
Summarize the program and next steps.