Chapter 1 — Foundations & Intuition
Time sets the pace. Light keeps the beat. When the pace changes, paths and arrival times change too.
Goal of this chapter
- State the lapse-first picture and the role of \(\Phi\).
- Explain why light behaves like a ray in a medium.
- Preview the key relations we prove later.
The set-up
We write the metric in lapse-first form. The lapse controls clock rate. The spatial part carries geometry.
\[ds^2 = -e^{2\Phi(t,\mathbf{x})}\,dt^2 + e^{-2\Phi(t,\mathbf{x})}\,\gamma_{ij}(t,\mathbf{x})\,dx^i dx^j.\]
Think of \(\Phi\) as the local tempo of time. Bigger \(\Phi\) means faster ticks in coordinate time \(t\). Proper time is \(d\tau = e^{\Phi} dt\).
Convention note. In this guide we present the refractive map as \(n=e^{\Phi}\). Some parts of the paper use \(n_\gamma=e^{-2\Phi}\). Both describe the same physics once you track factors in the optical limit. We flag the choice at the start of each chapter.
Why a refractive index?
Light follows paths that make travel time stationary. This is Fermat’s principle. If time flows unevenly, light acts as if the medium is uneven.
Picture a runner on moving walkways. The ground speeds up and slows down. The quickest route is not a straight line anymore.
Optional math peek
In the eikonal limit we write the line element and factor out the lapse. The optical metric produces the same ray equations as a medium with index \(n\).
\[n(\mathbf{x},t) = e^{\Phi(\mathbf{x},t)}.\]
Chapter 2 derives this carefully and sets the limits of validity.
What lives where
- Lapse \(\Phi\): Governs achromatic time dilation and drift. This is the temporal signal.
- Shift \(\beta_i\): Carries rotation and waves. Sagnac links and gyros respond to it.
Three pictures to hold
- Glass picture. Space behaves like glass with thickness set by time’s pace. Thicker glass slows light. Variations bend rays.
- Clock picture. Clocks define the medium. Where clocks tick faster, the optical index is higher or lower depending on convention. Paths respond to that field.
- Network picture. A network of clocks and links can read the field. Achromatic signals point to \(\Phi\). Chromatic signals point to plasma.
Quick checks
- Flat space. \(\Phi=0\) gives \(n=1\). Rays are straight. Travel time is \(\ell/c\).
- Weak field near a mass. \(\Phi\ll1\). Then \(n \approx 1+\Phi\). Paths bend toward stronger \(\Phi\).
- Slow evolution. If \(\partial_t\Phi\) is tiny, there is little drift. We quantify this in Chapter 5.
What you need to proceed
- Comfort with basic calculus and line integrals.
- Familiarity with Fermat’s principle helps. We explain the needed parts.
- No tensor gymnastics here. The heavy lifting stays in the paper.
Where we are heading
Chapter 2 writes the temporal–optics map and its limits. Chapter 3 turns it into travel time and the Shapiro result. Chapter 4 ties bending to transverse gradients. Chapter 5 introduces time-variable drift. Later chapters separate temporal signals from plasma and identify tests.