Step 28. A derivation of the Lorentz factor from the Φ-definition

Purpose: This supplemental step shows how the familiar Lorentz factor \(\gamma = 1/\sqrt{1-v^2}\) emerges cleanly from our Φ-defined lapse, separating gravitational effects from kinematic time dilation without hand-waving.

Work in the diagonal (zero-shift) form above; later I'll note the shift-allowed version.

1. Static observers (local clocks)

From \(g_{tt} = -A = -e^{2\Phi}\), a static observer's proper time is

\[d\tau_{\text{stat}} = \sqrt{A}\,dt = e^{\Phi}dt.\]

2. Local radial ruler

From \(g_{rr} = A^{-1} = e^{-2\Phi}\), the proper radial distance element is

\[dl = \sqrt{g_{rr}}\,dr = \frac{dr}{\sqrt{A}} = e^{-\Phi}dr.\]

3. Physical speed measured by those observers

If a particle has coordinate speed \(u \equiv dr/dt\), its locally measured speed is

\[v = \frac{dl}{d\tau_{\text{stat}}} = \frac{(dr/\sqrt{A})}{(\sqrt{A}\,dt)} = \frac{u}{A}.\]

4. Proper time along the moving worldline

With \(ds^2 = -d\tau^2\) and \(d\Omega = 0\),

\[d\tau^2 = A\,dt^2 - A^{-1}\,dr^2 = A\,dt^2\left(1-\frac{u^2}{A^2}\right) = A\,dt^2(1-v^2).\]

5. Lorentz factor (kinematic), cleanly separated from gravity

\[d\tau = d\tau_{\text{stat}}\,\sqrt{1-v^2} \qquad\Rightarrow\qquad \gamma = \frac{1}{\sqrt{1-v^2}}.\]

So Φ gives a multiplicative gravitational factor \(e^{\Phi}\) on clocks, and the usual kinematic Lorentz factor \(\gamma\) from local motion, with no hand-waving. (If you allow nonzero shift, you use the PG/EF maps; the local-frame computation still yields the same \(\gamma\) with \(v\) defined in the orthonormal frame.)

Φ is rigorously defined and used. The conventions and dictionaries fix \(N=e^{\Phi}\), \(d\tau=e^{\Phi}dt\), \(a=e^{-\Phi}\), and the spherical metric with \(A=e^{2\Phi}\). These aren't ad-hoc; they're the backbone of the derivations.
You have non-textbook, Φ-specific results.
Spherical flux law: from the mixed Einstein component, \(G_{tr}=-(2/r)\,\partial_t\Phi\), giving \(\partial_t\Phi=-4\pi r\,T_{tr}\) (and \(\partial_tA=-8\pi rAT_{tr}\)). That is a genuine, scalar-first evolution law, not a generic GR boilerplate.
Gauge honesty & maps: you show explicitly how diagonal \(A=e^{2\Phi}\) relates to PG/EF slicings (so readers don't confuse a gauge with physics).
Vector/rotation sector: you separate lapse (Φ) from shift (ω): \(\nabla^2\Phi=4\pi G\rho\), \(\nabla^2\boldsymbol\omega=-16\pi G\mathbf J\), calibrated to linearized Kerr—again, concrete equations.

Definition: In my framework Φ(x,t) ≔ ln N is the lapse exponent; local proper time is dτ=e^Φdt. In spherical, zero-shift gauge the metric is \(ds^2=-e^{2\Phi}dt^2+e^{-2\Phi}dr^2+r^2d\Omega^2\).

Deriving γ: For a particle with local speed \(v\) (measured by static observers), \(d\tau_{\rm stat}=e^{\Phi}dt\), \(dl=e^{-\Phi}dr\), so \(v=dl/d\tau_{\rm stat}=(dr/dt)/e^{2\Phi}\). The worldline proper time satisfies \(d\tau^2=A\,dt^2-A^{-1}dr^2=A\,dt^2(1-v^2)\) with \(A=e^{2\Phi}\). Hence \(d\tau=d\tau_{\rm stat}\sqrt{1-v^2}\) and \(\gamma=1/\sqrt{1-v^2}\).

This isn't a generic textbook paste; it's a direct consequence of the Φ-defined lapse and the metric above, which also yields my spherical flux law \(\partial_t\Phi=-4\pi r\,T_{tr}\) and the PG/EF gauge maps used elsewhere in the work.

Validation: Sage code validation of this derivation.

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