Chapter 7 — Lapse vs Shift: What Lives Where
The lapse \(\Phi\) controls how fast clocks tick. The shift \(\beta_i\) encodes motion of frames. Clocks read \(\Phi\). Sagnac and gyros read \(\beta\).
Goal of this chapter
- Write the metric with lapse and shift.
- Show which observables read each sector.
- Give null tests that separate the two cleanly.
Metric with lapse and shift
We keep the lapse-first scaling and allow a nonzero shift. The spatial metric is \(\gamma_{ij}\).
\[
ds^2
\;=\;
-e^{2\Phi}\,dt^2
\;+\;
e^{-2\Phi}\,\gamma_{ij}\,\big(dx^i+\beta^i dt\big)\big(dx^j+\beta^j dt\big).
\]
Set \(\beta^i=0\) to recover the earlier chapters. Keep \(\beta^i\neq 0\) to model rotation and frame transport.
Convention note. For optics we sometimes use \(n=e^{\Phi}\) or \(n_{\gamma}=e^{-2\Phi}\). Both are compatible with the shift form above when you work in the optical limit.
What lives where
- Lapse \(\Phi\): Achromatic dilation and drift. Read out with synchronized clocks and phase links.
- Shift \(\beta_i\): Rotation and near-zone gravitomagnetism. Read out with Sagnac loops and gyros.
Sagnac timing from the shift
A closed optical loop measures the circulation of the shift along the loop \(C\).
\[
\Delta T_{\mathrm{Sagnac}}
\;\simeq\;
\frac{2}{c^2}\oint_{C} \boldsymbol{\beta}\cdot d\boldsymbol{\ell}.
\]
Reverse the loop orientation. The sign flips. The lapse contribution does not. This is a null test.
For a rigid loop of area \(A\) with effective rotation rate \(\Omega_{\mathrm{eff}}\), the Sagnac frequency split is
\[
\Delta f_{\mathrm{Sagnac}}
\;=\;
\frac{4A}{\lambda P}\,\Omega_{\mathrm{eff}}.
\]
Here \(\lambda\) is the wavelength and \(P\) is the loop perimeter. \(\Omega_{\mathrm{eff}}\) includes mechanical rotation and any gravitomagnetic term.
Near-zone frame dragging (Lense–Thirring)
Near a rotating mass with angular momentum \(\mathbf{J}\), gyroscopes precess. The leading magnitude at distance \(r\) is
\[
\Omega_{\mathrm{LT}}
\;\approx\;
\frac{2GJ}{c^2 r^3}.
\]
The full vector form depends on geometry. The effect belongs to the shift sector. It does not mimic the achromatic timing drift from \(\Phi\).
Operational split: how to read each sector
- Read \(\Phi\) (lapse): Compare clocks across a baseline. Track a common, color-independent drift \(\big(d\Delta T/dt_{\mathrm{obs}}\big)\). Use Chapter 5.
- Read \(\beta\) (shift): Measure loop time asymmetry or ring-laser frequency split. Flip loop orientation as a check.
- Combine: Run both at once. The clock link provides the lapse. The loop provides the shift. Cross-compare for self-consistency.
Null tests that separate signals
- Color swap. Change \(\nu\). The lapse signal stays fixed. Plasma terms move with \(\nu^{-2}\).
- Loop reversal. Reverse \(C\). The Sagnac sign flips. Any lapse-driven time offset does not.
- Closed triangle. Compare A→B, B→C, C→A phases. The sum isolates the shift circulation.
Sanity checks
- \(\beta=0 \Rightarrow \Delta T_{\mathrm{Sagnac}}=0\).
- \(\Phi=\text{const} \Rightarrow\) no drift, no bending.
- Increase loop area \(A\). The Sagnac split grows linearly with \(A\).
Common pitfalls
- Mixing lapse and shift responses in one observable. Use separate channels.
- Dropping the orientation flip test for Sagnac. Always check the sign.
- Ignoring platform rotation and Earth tides. Calibrate with local sensors.
What you have now
A clear separation of sectors. Clocks probe \(\Phi\). Sagnac and gyros probe \(\beta\). Simple null tests keep the channels independent.