Derivation 5.4A — Lightlike propagation and matched endpoint standards
theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology
Purpose
This appendix supports Chapter 5.4. It derives the lightlike propagation kernel, proves the neighboring-feature endpoint relation, and states the rod and clock assumptions needed to read that relation as wavelength redshift and duration stretching.
The derivation keeps three objects distinct: the conserved external spatial wavenumber, the drifting generator frequency, and the wavelength reported relative to local rods.
Role, regime, premises, and status
- Role: lightlike kinematics plus a conditional matched-endpoint measurement dictionary.
- Regime: homogeneous yield history, geometric propagation, neighboring signal features, negligible peculiar motion and endpoint evolution during one feature separation.
- Premises: bosic-slot dressing, spatial homogeneity, the clean bound-frame branch, co-scaling rods, and matched endpoint clocks.
Notation ledger
- \(\chi(t)>0\) — normalized yield entering the generator; \(\chi_o=1\) in the compact formulas.
- \(\chi_e,\chi_o\) — endpoint yield values when normalization is restored.
- \(p_f\) — fermic momentum scale; \(p_f=0\) for a lightlike carrier.
- \(\mathbf k\) — external spatial wavevector on the homogeneous true-frame chart.
- \(\mathbf p=\hbar\mathbf k\) — conserved spatial momentum content.
- \(\omega_\chi\) — instantaneous generator angular frequency.
- \(x_T,t\) — true-frame propagation coordinate and common history parameter.
- \(\delta t_e,\delta t_o\) — neighboring-feature endpoint separations in \(t\).
- \(\Delta t_e,\Delta t_o\) — durations reported by local endpoint clocks.
- \(\ell_T\) — local true-frame rod standard.
- \(a_B=\chi_o/\chi\) — clean-branch bound-frame scale factor.
- \(\eta\) — conformal propagation time.
Whenever the absolute normalization is not \(\chi_o=1\), replace the generator’s \(\chi\) by \(X=\chi/\chi_o\). Endpoint observables depend only on \(\chi_e/\chi_o\).
Lightlike dispersion and group velocity
The bosic-slot generator magnitude is
\[ M_\chi = \sqrt{p_f^2+\chi^2p^2}. \]
For a lightlike carrier, \(p_f=0\), so
\[ \boxed{ M_\chi=\chi|p| }. \]
The energy-unit generator and angular frequency are
\[ H_\chi=cM_\chi=c\chi|p|, \qquad \omega_\chi=\frac{H_\chi}{\hbar} = \chi c|\mathbf k|. \]
The group velocity is
\[ \dot x_k = \frac{\partial H_\chi}{\partial p_k} = \chi c\,\frac{p_k}{|p|}. \]
For forward propagation,
\[ \boxed{ \dot x_T=\chi c }. \]
Thus the \(\chi^2\) massive theorem reduces to the original \(\chi\) lightlike law because the dressed magnitude itself contributes one factor of \(\chi\) in the denominator.
Conserved external wavenumber and drifting frequency
The generator is homogeneous in space:
\[ \frac{\partial H_\chi}{\partial x_k}=0. \]
Hamilton’s equation therefore gives
\[ \dot p_k=0, \]
and, with fixed \(\hbar\),
\[ \boxed{ \dot k_k=0 }. \]
The external spatial phase gradient and true-frame coordinate wavelength
\[ \lambda_T=\frac{2\pi}{|\mathbf k|} \]
remain fixed in flight. The generator frequency does not:
\[ \omega_\chi(t)=\chi(t)c|\mathbf k|, \qquad \frac{\dot\omega_\chi}{\omega_\chi} = \frac{\dot\chi}{\chi}. \]
The signal therefore undergoes generator drift at fixed external spatial content. This is not an in-flight stretching of the external wavelength. A changing wavelength is reported only after the fixed coordinate pattern is compared with endpoint rods whose true-frame size depends on \(\chi\).
Propagation kernel
Integrating the forward lightlike velocity from emission to observation gives the true-frame path interval
\[ \boxed{ \mathcal D_\chi(t_e,t_o) = c\int_{t_e}^{t_o}\chi(t)\,dt }. \]
The integral is a propagation kernel. It is not by itself a luminosity distance, angular-diameter distance, ruler distance, or flux law.
On the clean branch,
\[ a_B=\frac{1}{\chi} \]
in the normalized convention. Define conformal time by
\[ d\eta\equiv\frac{dt}{a_B}. \]
Then
\[ \boxed{ d\eta=\chi\,dt } \]
and
\[ \mathcal D_\chi=c\int_{\eta_e}^{\eta_o}d\eta. \]
With an unnormalized yield, the exact identity is
\[ d\eta=\frac{\chi}{\chi_o}\,dt; \]
the conventional \(\chi_o=1\) choice gives the boxed form. The lightlike kernel is therefore the standard conformal propagation kernel under the clean correspondence. It does not by itself change the usual horizon accounting.
Neighboring-feature equal-path derivation
Consider two neighboring signal features emitted by the same source and received by the same observer. Assume that, over the small endpoint separations, source and observer keep the same true-frame coordinate separation and that changes in peculiar motion, lensing, and inhomogeneity are negligible.
The first feature obeys
\[ D_T = c\int_{t_e}^{t_o}\chi(t)\,dt. \]
The second feature follows the same path:
\[ D_T = c\int_{t_e+\delta t_e}^{t_o+\delta t_o}\chi(t)\,dt. \]
Subtracting gives
\[ 0 = c\int_{t_o}^{t_o+\delta t_o}\chi(t)\,dt - c\int_{t_e}^{t_e+\delta t_e}\chi(t)\,dt. \]
For neighboring features, \(\chi\) varies negligibly across each endpoint interval. To first order,
\[ 0 = c\chi_o\delta t_o - c\chi_e\delta t_e. \]
Hence
\[ \boxed{ \chi_o\delta t_o = \chi_e\delta t_e } \]
and
\[ \boxed{ \frac{\delta t_o}{\delta t_e} = \frac{\chi_e}{\chi_o} }. \]
This equation follows from the common path and the lightlike propagation law. Turning it into an observed duration relation requires the clock assumption below.
Matched endpoint rods
On the clean branch, local true-frame rods scale as
\[ \ell_T(t)=\ell_o\frac{\chi(t)}{\chi_o}. \]
The conserved external wavelength \(\lambda_T=2\pi/|\mathbf k|\) is therefore reported in local rod units as
\[ \lambda_B(t) \propto \frac{\lambda_T}{\ell_T(t)} \propto \frac{\chi_o}{\chi(t)}. \]
For the same transported phase pattern compared at emission and observation,
\[ \frac{\lambda_{B,o}}{\lambda_{B,e}} = \frac{\chi_e}{\chi_o}. \]
Thus the measured wavelength relation is
\[ \boxed{ 1+z \equiv \frac{\lambda_o}{\lambda_e} = \frac{\chi_e}{\chi_o} }. \]
The same ratio appears in frequency language. An emitted feature with local angular frequency \(\omega_e\) fixes an external wavevector through
\[ \omega_e = \frac{\chi_e}{\chi_o}c|\mathbf k|. \]
Spatial homogeneity preserves \(\mathbf k\) in flight, so the observation-epoch generator frequency is
\[ \omega_o = c|\mathbf k| = \frac{\chi_o}{\chi_e}\omega_e. \]
Therefore
\[ \frac{\omega_e}{\omega_o} = \frac{\chi_e}{\chi_o} = 1+z. \]
This frequency statement assumes that emitter and observer use matched local spectral standards. Derivation 5.6A supplies a hydrogenic proxy in which those standards remain fixed while rods co-scale.
Matched endpoint clocks and duration stretching
The clean branch takes local tick frequency to be fixed relative to the common history parameter:
\[ \nu_T\propto\chi^0. \]
Reported endpoint durations are therefore proportional to the corresponding background intervals with the same conversion factor at emission and observation:
\[ \frac{\Delta t_o}{\Delta t_e} = \frac{\delta t_o}{\delta t_e}. \]
Using the equal-path result,
\[ \boxed{ \frac{\Delta t_o}{\Delta t_e} = \frac{\chi_e}{\chi_o} = 1+z }. \]
Equivalently,
\[ \boxed{ \Delta t_o=(1+z)\Delta t_e }. \]
One endpoint ratio therefore governs wavelength redshift and neighboring-feature duration stretching, but only after the co-scaling rod and matched-clock assumptions are applied.
Limiting checks
Inertial history
If \(\chi\) is constant, then
\[ \omega_\chi=\text{constant}, \qquad \delta t_o=\delta t_e, \qquad z=0. \]
Normalization invariance
Under a constant rescaling \(\chi\mapsto C\chi\), the normalized generator uses \(X=\chi/\chi_o\), while endpoint observables use
\[ \frac{C\chi_e}{C\chi_o} = \frac{\chi_e}{\chi_o}. \]
No measured relation depends on the arbitrary normalization.
Direction reversal
Reversing the propagation orientation changes the sign of \(\dot x_T\) but not \(M_\chi\), \(\omega_\chi\), or the endpoint ratios.
Result and boundaries
This appendix establishes:
- \(p_f=0\) implies \(M_\chi=\chi|p|\) and \(\dot x_T=\chi c\);
- spatial homogeneity conserves external \(\mathbf k\) while \(\omega_\chi=\chi c|\mathbf k|\) drifts;
- the path kernel is \(c\int\chi\,dt\);
- neighboring equal paths give \(\chi_o\delta t_o=\chi_e\delta t_e\);
- co-scaling rods give \(1+z=\chi_e/\chi_o\);
- matched clocks give \(\Delta t_o/\Delta t_e=1+z\);
- \(d\eta=\chi\,dt\) in the normalized clean convention.
It does not derive luminosity distance, angular-diameter distance, flux or photon-number transport, lensing, full radiative transfer, source evolution, blackbody evolution, recombination, thermal history, or Friedmann dynamics. It also does not claim that the external wavelength stretches during flight.
Chapter 5.4 may cite this appendix for the lightlike path, generator drift, endpoint comparison, and conditional matched-standard signal dictionary.