Chapter 5 — Time-Variable Potentials & Achromatic Drift
When the lapse changes during propagation, the arrival time drifts. The signal is achromatic.
Goal of this chapter
- Write the drift of arrival time when \(\partial_t\Phi\neq 0\).
- Connect the drift to energy flow with the spherical flux law.
- Show simple estimates and conditions for validity.
Drift from an evolving lapse
Start from the travel-time functional and differentiate with respect to the observer’s time.
\[
\frac{\,\mathrm{d}}{\mathrm{d}t_{\mathrm{obs}}}\,\Delta T
\;\simeq\;
-\,\frac{2}{c}\,\int_{\text{path}} \partial_t \Phi(t,\mathbf{x})\,\mathrm{d}\ell.
\]
Sign convention: a growing \(\Phi\) along the path makes arrivals earlier. The result does not depend on radio/optical color.
Why achromatic? \(\Delta T\) depends on the time field along the path. Color enters through plasma dispersion, not through gravity. We use this contrast in Chapter 6.
Link to energy flow (spherical flux law)
Relate \(\partial_t\Phi\) to outward power \(P(t)\) at radius \(R\). Use the SI form.
\[
\partial_t \Phi(t,R)
\;=\;
-\,\frac{G}{c^4}\,\frac{P(t)}{R}.
\]
Insert this into the drift integral.
\[
\frac{\,\mathrm{d}}{\mathrm{d}t_{\mathrm{obs}}}\,\Delta T
\;\simeq\;
\frac{2G}{c^{5}}\,
\int_{\text{path}} \frac{P(t)}{R}\,\mathrm{d}\ell.
\]
Positive outward power produces a positive drift. The scale follows the luminosity and geometry.
From drift to a clock signal
For pulse trains or phase-locked links, the instantaneous fractional frequency shift is the negative time derivative of the delay.
\[
\frac{\dot{\nu}}{\nu}
\;\approx\;
-\,\frac{\mathrm{d}}{\mathrm{d}t_{\mathrm{obs}}}\,\Delta T.
\]
This is the form used in forecasts for clock networks and pulsar timing.
Simple estimates
Assume a segment of length \(L\) near radius \(R\) where \(P(t)\) is approximately constant.
\[
\left.\frac{\,\mathrm{d}}{\mathrm{d}t_{\mathrm{obs}}}\,\Delta T\right|_{\text{seg}}
\;\simeq\;
\frac{2G}{c^{5}}\;\frac{P}{R}\;L.
\]
For multiple segments, add contributions along the path. Geometry enters through \(L/R\) factors.
Numerical benchmarks appear in Chapter 8.
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.
- Use a single index convention per calculation:
\(n=e^{\Phi}\) or \(n_{\gamma}=e^{-2\Phi}=n^{-2}\).
Worked mini-examples
Thin emitting shell
Take a luminous shell of thickness \(L\) at radius \(R\) with power \(P\).
\[
\dot{\Delta T}
\;\simeq\;
\frac{2G}{c^{5}}\;\frac{P}{R}\;L.
\]
Central engine plus quiescent halo
Split the path into a compact zone of size \(L_1\) near \(R_1\) and a quiet zone \(L_2\) near \(R_2\) with \(P_2\!\ll P_1\).
\[
\dot{\Delta T}
\;\simeq\;
\frac{2G}{c^{5}}\left(\frac{P_1}{R_1}L_1 + \frac{P_2}{R_2}L_2\right)
\;\approx\;
\frac{2G}{c^{5}}\;\frac{P_1}{R_1}L_1.
\]
The compact zone dominates.
Sanity checks
- \(\partial_t\Phi\to 0 \Rightarrow \dot{\Delta T}\to 0\).
- Set \(P\to 0\). The flux form gives \(\dot{\Delta T}\to 0\).
- Increase distance \(R\). The drift falls like \(1/R\).
Common pitfalls
- Mixing \(\mathrm{d}\ell\) and \(dz\) without stating the path model.
- Switching between \(n=e^{\Phi}\) and \(n_{\gamma}=e^{-2\Phi}\) mid-derivation.
- Dropping small-parameter checks for \(\bigl|\partial_t\Phi/\Phi\bigr|\,T_{\mathrm{path}}\).
What you have now
A clean achromatic drift law. A direct link to source power via the flux relation. Simple scaling estimates that match the forecasts in Chapter 8.