Chapter 2 — The Temporal–Optics Map \(n=e^{\Phi}\)
The key idea: the local rate of time sets an optical index. Light moves as if it is in a medium with \(n=e^{\Phi}\).
Goal of this chapter
- State the lapse-first metric and extract a travel-time functional.
- Show how the eikonal limit gives an effective refractive index.
- Record validity conditions for the static map. Save drift for Chapter 5.
Set-up and assumptions
We write the metric in lapse-first form. The lapse controls clock rate. The spatial metric carries shape.
\[
ds^2 = -e^{2\Phi(t,\mathbf{x})}\,dt^2 \;+\; e^{-2\Phi(t,\mathbf{x})}\,\gamma_{ij}(t,\mathbf{x})\,dx^i dx^j.
\]
Null rays satisfy \(ds^2=0\). We work in the eikonal limit. The geometry varies slowly across a wavelength.
Convention note. This guide uses \(n=e^{\Phi}\). Parts of the paper use \(n_{\gamma}=e^{-2\Phi}\). They are equivalent in the optical limit with \(n_{\gamma}=n^{-2}\). We flag the choice when needed.
Derivation: from metric to index
Step 1 — Turn null condition into a time functional
Set \(ds^2=0\) and solve for \(dt\) along a spatial path with element \(d\ell\).
\[
e^{2\Phi}\,dt^2 \;=\; e^{-2\Phi}\,\gamma_{ij}\,dx^i dx^j
\quad\Rightarrow\quad
dt \;=\; e^{-2\Phi}\,\frac{\sqrt{\gamma_{ij}\,dx^i dx^j}}{1}
\;=\; e^{-2\Phi}\,d\ell_{\gamma}.
\]
Here \(d\ell_{\gamma}=\sqrt{\gamma_{ij}\,dx^i dx^j}\) is the spatial line element for \(\gamma_{ij}\).
Step 2 — Factor out a baseline
The coordinate travel time is a path integral. Subtract the vacuum baseline to define the excess time.
\[
T \;=\; \int dt \;=\; \int e^{-2\Phi}\,d\ell_{\gamma},
\qquad
\Delta T \;\equiv\; T - \frac{1}{c}\int d\ell.
\]
Step 3 — Identify the effective index
Define the physical path element \(d\ell\) so that the travel-time functional reads like Fermat’s principle.
\[
\Delta T \;\simeq\; \frac{1}{c}\int \big(n-1\big)\,d\ell,
\qquad
\boxed{\,n(\mathbf{x},t)=e^{\Phi(\mathbf{x},t)}\,}.
\]
This is the temporal–optics map. The same physics appears if you use \(n_{\gamma}=e^{-2\Phi}\). Then \(n_{\gamma}=n^{-2}\).
Step 4 — Ray equation (optional peek)
Stationary travel time yields Fermat’s equation. Rays curve where the index changes across the path.
\[
\boldsymbol{\alpha}(\hat{\mathbf n}) \;\simeq\; \int \nabla_{\!\perp}\Phi\,d\ell,
\]
We use this for bending in Chapter 4.
Step 5 — Static limit vs. evolving fields
The map above is static. If \(\Phi\) evolves during propagation, the arrival time drifts. We handle that in Chapter 5.
\[
\frac{d}{dt_{\mathrm{obs}}}\Delta T \;\simeq\; -\,\frac{2}{c}\int \partial_t\Phi\,d\ell.
\]
Validity and small parameters
- Quasi-static along the path: \(\bigl|\partial_t\Phi/\Phi\bigr|\,T_{\mathrm{path}} \ll 1\).
- Slow geometry change: \(\epsilon_{\mathrm{lens}} \equiv L_{\mathrm{lens}}/(c\,\tau_{\mathrm{var}}) \ll 1\).
- Eikonal limit: background varies slowly across a wavelength.
Sanity checks
- \(\Phi=0 \Rightarrow n=1\). Travel time reduces to \(\ell/c\).
- Weak field: \(\Phi\ll 1 \Rightarrow n \approx 1+\Phi\). You recover standard linearized optics.
- Isotropic static lens: Chapter 3 reproduces Shapiro timing from \(\Delta T\).
Common pitfalls
- Do not mix \(n=e^{\Phi}\) and \(n_{\gamma}=e^{-2\Phi}\) in the same calculation without converting. Use \(n_{\gamma}=n^{-2}\).
- Use \(d\ell\) consistently for the path element. Keep \(dz\) for the unperturbed ray if you need a longitudinal coordinate.
- Do not include drift terms in static forecasts. Use Chapter 5 for \(\partial_t\Phi\neq 0\).
What you have now
The map \(n=e^{\Phi}\). A travel-time functional \(\Delta T \simeq \frac{1}{c}\int (n-1)\,d\ell\). A clear split between static predictions and time-variable drift.