Chapter 3 — Travel Time & Shapiro Delay
We turn the temporal–optics map into a travel-time integral. We recover the classic Shapiro delay in the static, weak-field limit.
Goal of this chapter
- Write the travel-time functional in terms of \(n=e^{\Phi}\).
- Define the excess time \(\Delta T\) relative to vacuum.
- Recover the standard Shapiro form for a static point mass.
Travel time from the index
Fermat’s principle gives travel time as index times path length over \(c\).
\[
T \;=\; \frac{1}{c}\int n(\mathbf{x})\,\mathrm{d}\ell,
\qquad
\Delta T \;\equiv\; T \;-\; \frac{1}{c}\int \mathrm{d}\ell.
\]
With the temporal–optics map, the excess time is
\[
\Delta T \;\simeq\; \frac{1}{c}\int \big(e^{\Phi}-1\big)\,\mathrm{d}\ell.
\]
In weak fields, expand \(e^{\Phi}\approx 1+\Phi\).
\[
\Delta T \;\approx\; \frac{1}{c}\int \Phi\,\mathrm{d}\ell.
\]
Static, spherically symmetric example
Take a point mass \(M\). Use the weak-field potential \(\Phi(\mathbf{x}) \approx -\,\frac{GM}{c^2 r}\). Assume a straight path with impact parameter \(b\). Integrate along the unperturbed line-of-sight coordinate \(z\).
\[
r(z) \;=\; \sqrt{b^2+z^2},
\qquad
\Delta T \;\approx\; \frac{1}{c}\int_{z_S}^{z_R} \Phi\big(r(z)\big)\,\mathrm{d}\ell
\;\approx\; -\,\frac{GM}{c^3}\int_{z_S}^{z_R} \frac{\mathrm{d}z}{\sqrt{b^2+z^2}}.
\]
Evaluate the line integral between the finite endpoints \(z_S\) and \(z_R\).
\[
\int \frac{\mathrm{d}z}{\sqrt{b^2+z^2}}
\;=\; \ln\!\Big|\,z+\sqrt{b^2+z^2}\,\Big|\;+\;\text{const}.
\]
Collect terms in the usual geometric form. You recover the logarithmic Shapiro delay.
\[
\Delta T_{\mathrm{Shapiro}}
\;=\; \frac{2GM}{c^3}\,
\ln\!\left(\frac{r_S+r_R+D}{r_S+r_R-D}\right),
\]
Here \(r_S\) and \(r_R\) are source and receiver distances from the mass. \(D\) is their separation along the path. This matches the standard result in the static limit.
Consistency with the paper
The paper also presents the equivalent form with \(n_{\gamma}=e^{-2\Phi}\). In that convention \(\Delta T\simeq -\,\frac{2}{c}\int \Phi\,\mathrm{d}\ell\) at leading order. Both routes give the same Shapiro law once you keep track of factors in the optical limit.
Assumptions and limits
- Weak field: \(|\Phi|\ll 1\).
- Static potential along the path.
- Straight-line (Born) approximation for the path integral.
- No plasma dispersion in this chapter.
Sanity checks
- \(\Phi\to 0 \Rightarrow \Delta T \to 0\).
- Increase \(M\) and \(\Delta T\) grows logarithmically through the path geometry.
- Move the path farther from the mass. The delay weakens as \(b\) grows.
Common pitfalls
- Mixing \(n=e^{\Phi}\) and \(n_{\gamma}=e^{-2\Phi}\) without converting.
- Forgetting that the log form comes from finite endpoints. Infinite limits hide geometry in a divergent constant.
- Using a curved ray while also assuming the straight-line integral. Be consistent with the Born approximation.
What you have now
A clean travel-time functional. The weak-field expansion. The standard Shapiro delay as the static limit of temporal optics.