Derivation 5.3A — Free-carrier correspondence under a yield history
theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology
Purpose
This appendix supports Chapter 5.3. It converts the bosic-slot group velocity into the clean bound frame and proves that a free massive M1 carrier follows the exact special-relativistic FLRW peculiar-velocity law when its conserved spatial content is identified with comoving momentum.
The theorem is conditional on a prescribed homogeneous yield history. It checks the kinematics produced by that history; it does not derive \(H(t)\) or a Friedmann equation.
Role, regime, premises, and status
- Role: exact kinematic correspondence theorem and placement diagnostic.
- Regime: free carrier, homogeneous yield history, no local gravitational field, clean bound-frame branch.
- Premises: Derivation 5.2A, normalized yield, co-scaling rods, matched clocks, and subtraction of the common background recession term when peculiar velocity is defined.
Notation ledger
- \(\chi(t)>0\) — normalized yield entering the dressed generator.
- \(\chi_e,\chi_o\) — unnormalized endpoint values when a ratio must be displayed.
- \(X\equiv\chi/\chi_o\) — normalized yield ratio if \(\chi_o\ne1\).
- \(a_B\equiv\chi_o/\chi=1/X\) — clean-branch bound-frame scale factor.
- \(H\equiv\dot a_B/a_B=-\dot X/X\) — prescribed fractional history rate.
- \(p_f=m_0c\) — fixed fermic momentum scale.
- \(\mathbf p\) — conserved M1 spatial momentum content; \(p=|\mathbf p|\).
- \(M_X\equiv\sqrt{p_f^2+X^2p^2}\) — normalized dressed generator magnitude.
- \(v_B\) — peculiar velocity measured in the bound frame.
- \(q\) — standard FLRW comoving momentum used only in the correspondence comparison.
For compactness, the derivation first keeps \(X=\chi/\chi_o\) explicit. Setting \(\chi_o=1\) gives \(X=\chi\) and recovers the notation of Derivation 5.2A.
Clean bound-frame conversion
On the clean branch, local true-frame rods scale with the normalized yield:
\[ \ell_T=\ell_oX. \]
A true-frame coordinate interval is therefore reported in bound-frame length units as
\[ x_B=\frac{x_T}{X}=a_Bx_T, \qquad a_B=\frac{1}{X}=\frac{\chi_o}{\chi}. \]
Differentiating \(x_B=a_Bx_T\) gives a background term \(\dot a_Bx_T\) and a local-motion term \(a_B\dot x_T\). Peculiar velocity is the local-motion term after the common recession of the bound-frame coordinate grid is subtracted:
\[ \boxed{ v_B=a_B\dot x_T }. \]
The clean branch uses matched clocks, so no additional time conversion appears.
Bosic-slot free-carrier velocity
With the normalized yield \(X\), Derivation 5.2A gives
\[ M_X=\sqrt{p_f^2+X^2p^2}, \]
and, along the direction of motion,
\[ \dot x_T = X^2c\,\frac{p}{M_X}. \]
The bound-frame peculiar velocity is therefore
\[ \begin{aligned} v_B &= a_BX^2c\,\frac{p}{M_X} \\ &= Xc\,\frac{p}{\sqrt{p_f^2+X^2p^2}} \\ &= c\,\frac{p/a_B} {\sqrt{p_f^2+(p/a_B)^2}}. \end{aligned} \]
Thus
\[ \boxed{ v_B = c\,\frac{p/a_B} {\sqrt{p_f^2+(p/a_B)^2}} }. \]
The numerator contains the bound-frame momentum expression \(p/a_B\), while the M1 content label \(p\) itself remains conserved.
Exact FLRW correspondence
For a freely moving particle in a spatially flat FLRW description, spatial homogeneity conserves comoving momentum \(q\). The physical momentum measured by a comoving observer is \(q/a\), and special-relativistic velocity is
\[ v_{\mathrm{FLRW}} = c\,\frac{q/a} {\sqrt{m_0^2c^2+(q/a)^2}}. \]
Using
\[ p_f=m_0c, \qquad a=a_B, \qquad \boxed{q\equiv p}, \]
gives
\[ v_{\mathrm{FLRW}}=v_B \]
at every speed. M1’s conserved spatial content is therefore the variable that standard FLRW kinematics calls comoving momentum in this homogeneous correspondence regime.
This equality is an overlap theorem. It demonstrates that the M1 stage-dressing variables reproduce known free-particle kinematics under a prescribed history. It is not an independent observational prediction and does not explain why the history has any particular form.
Exact drag law
Define the bound-frame expressed momentum
\[ y\equiv\frac{p}{a_B}=Xp. \]
Since \(p\) is conserved,
\[ \dot y = -\frac{\dot a_B}{a_B}y = -Hy. \]
Write the velocity as
\[ v_B(y) = c\,\frac{y}{\sqrt{p_f^2+y^2}}. \]
Its derivative is
\[ \frac{dv_B}{dy} = c\,\frac{p_f^2}{(p_f^2+y^2)^{3/2}}. \]
Therefore
\[ \dot v_B = -Hc\,\frac{yp_f^2}{(p_f^2+y^2)^{3/2}}. \]
The velocity itself satisfies
\[ 1-\frac{v_B^2}{c^2} = \frac{p_f^2}{p_f^2+y^2}, \]
so the previous expression becomes
\[ \boxed{ \dot v_B = -Hv_B \left( 1-\frac{v_B^2}{c^2} \right) }. \]
This is the exact relativistic FLRW geodesic drag law in velocity form. The history enters only through the supplied function \(H(t)=-\dot X/X\).
Limiting checks
Inertial history
If \(X=1\) is constant, then \(a_B=1\), \(H=0\), and
\[ v_B=c\,\frac{p}{\sqrt{p_f^2+p^2}}, \qquad \dot v_B=0. \]
The inertial free-particle result is recovered.
Nonrelativistic massive carrier
For \(p/a_B\ll p_f\),
\[ v_B \simeq c\,\frac{p}{a_Bp_f} = \frac{p}{m_0a_B}. \]
Hence
\[ v_B\propto a_B^{-1}, \qquad \dot v_B\simeq-Hv_B. \]
The standard nonrelativistic peculiar-momentum decay is recovered.
Lightlike carrier
For \(p_f=0\),
\[ v_B=c \]
for forward propagation, independent of \(a_B\). The true-frame speed is still \(\dot x_T=Xc\), while the bound-frame rod conversion supplies the compensating factor \(a_B=1/X\).
Diagnostic for the old placement
The old Chapter 5 map used
\[ \dot x_T^{(A)} = Xc\,\frac{p}{M}, \qquad M=\sqrt{p_f^2+p^2}. \]
Its bound-frame peculiar velocity is
\[ \begin{aligned} v_B^{(A)} &= a_B\dot x_T^{(A)} \\ &= \frac{1}{X} Xc\,\frac{p}{M} \\ &= c\,\frac{p}{\sqrt{p_f^2+p^2}}. \end{aligned} \]
Therefore
\[ \boxed{ \dot v_B^{(A)}=0 } \]
for every freely moving massive carrier, regardless of \(H(t)\). Placement A agrees with the photon limit because \(v_B=c\) there, but it fails the massive FLRW overlap already in the nonrelativistic limit.
As shown in Derivation 5.2A, this old velocity can be generated by the whole-generator Hamiltonian \(H_A=cXM\). Its failure here is empirical and structural: whole-generator dressing plus the clean bound-frame conversion gives constant massive peculiar velocity.
Result and boundaries
Given
\[ a_B=\frac{\chi_o}{\chi}, \qquad \hat M_X = \beta p_f+X\boldsymbol\alpha\!\cdot\!\hat{\mathbf p}, \]
the bosic-slot group velocity yields
\[ v_B = c\,\frac{p/a_B} {\sqrt{p_f^2+(p/a_B)^2}} \]
and
\[ \dot v_B = -Hv_B \left(1-\frac{v_B^2}{c^2}\right). \]
These are exactly the relativistic FLRW free-carrier relations under \(p\equiv q\). The result establishes the homogeneous kinematic overlap and excludes the old placement on the same clean branch.
It does not derive \(a_B(t)\), \(H(t)\), a Friedmann constraint, a gravitational source law, structure growth, or any observational fit. Chapter 5.3 may cite it only as a free-carrier correspondence under a prescribed yield history.