Derivation 3.5A — Particle Response and Internal Cycling
theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology
Purpose
A stationary deformation can preserve a particle’s reference core while changing its local momentum and motion. This derivation obtains that result from the free-carrier action, constructs §3.5’s directional shell from reference-expressed four-momentum, and derives the cycle rate under its explicit clock hypothesis.
Starting assumptions
Use the free-carrier action of Derivation 3.4A, with fixed free-space reference \(p_f\), positive local \(M\), and signature \((-+++)\). For the reference-root calculation take a zero-shift map,
\[ \phi^0{}_i=\phi^i{}_0=0, \qquad \phi^0{}_0>0, \qquad g_{ij}=\delta_{kl}\phi^k{}_i\phi^l{}_j. \]
The inverse spatial block is \(g^{ij}\). Momentum indices refer to the stated frame; when local and coordinate spatial momenta occur together, the local ones are labelled explicitly. Work in the test-particle limit with a prescribed background.
For an ideal massive clock, assume that a physical phase follows the action according to \(d\varphi=-dS/\hbar\), with one recurrence per \(2\pi\) phase increment. This cycle coupling is an additional model premise.
Derivation
Carrier variation and the reference root
Variation of the constrained action with respect to the multiplier, momentum and position gives
\[ \begin{aligned} g^{\mu\nu}p_\mu p_\nu+p_f^2&=0,\\ \dot x^\mu&=e\,g^{\mu\nu}p_\nu,\\ \dot p_\gamma&=-\frac e2(\partial_\gamma g^{\mu\nu})p_\mu p_\nu. \end{aligned} \tag{1}\]
The dot refers to the trajectory parameter \(\lambda\). In the zero-shift branch the positive root is
\[ -p_0=\phi^0{}_0 M, \qquad M=\sqrt{p_f^2+g^{ij}p_ip_j}, \qquad p_i=p_j^{\mathrm{local}}\phi^j{}_i. \tag{2}\]
Eliminating the constraint and using \(x^0\) as parameter gives the reduced action
\[ S=\int\left[p_i\frac{dx^i}{dx^0}-(-p_0)\right]dx^0. \]
Hamilton’s equations therefore differentiate the same positive root:
\[ \begin{aligned} \frac{dx^i}{dx^0} &=\frac{\partial(-p_0)}{\partial p_i} =\frac{\phi^0{}_0}{M}g^{ij}p_j,\\ \frac{dp_i}{dx^0} &=-\frac{\partial(-p_0)}{\partial x^i} =-M\partial_i\phi^0{}_0 -\frac{\phi^0{}_0}{2M}(\partial_i g^{jk})p_jp_k. \end{aligned} \tag{3}\]
At an instant of local rest, differentiation of the first equation gives
\[ \frac{d^2x^i}{(dx^0)^2} =-\phi^0{}_0g^{ij}\partial_j\phi^0{}_0. \tag{4}\]
The term differentiating the velocity coefficient vanishes at that instant because \(p_j=0\). Acceleration per reference time \(t=x^0/c\) is \(c^2\) times this expression.
Four-momentum and the directional shell
For a massive carrier, define \(u^\mu=dx^\mu/d\tau\) with \(g_{\mu\nu}u^\mu u^\nu=-c^2\). The shell and trajectory relations in Equation 1 give
\[ g_{\mu\nu}\dot x^\mu\dot x^\nu=-e^2p_f^2, \qquad c\,\frac{d\tau}{d\lambda}=e p_f \]
on the future-directed branch. Consequently the local momentum is
\[ (p^\mu)_{\mathrm{local}} =\phi^\mu{}_\nu g^{\nu\gamma}p_\gamma =\frac{p_f}{c}\phi^\mu{}_\nu u^\nu. \]
Express every local component with the same distant normalization:
\[ p_r^\mu=\phi^0{}_0(p^\mu)_{\mathrm{local}} =\frac{p_f\phi^0{}_0}{c}\phi^\mu{}_\nu u^\nu. \tag{5}\]
The output axes remain local, while \(u^\nu\) has coordinate components. In the zero-shift map, \(M_r=p_r^0=\phi^0{}_0M=-p_0\). Contracting the reference vector gives
\[ (p_r^0)^2-\delta_{ij}p_r^ip_r^j =-\left(\frac{p_f\phi^0{}_0}{c}\right)^2 g_{\mu\nu}u^\mu u^\nu =(\phi^0{}_0p_f)^2. \tag{6}\]
Thus \(p_{f,r}=\phi^0{}_0p_f\) and \(p_r=\sqrt{\delta_{ij}p_r^ip_r^j}\) obey
\[ M_r^2=p_{f,r}^{\,2}+p_r^2. \tag{7}\]
For any unit spatial direction \(k^i\) in the same output axes, \(p_{k,r}=\delta_{ij}k^ip_r^j\). The opposed values follow by projection:
\[ p_{k,r}^{\pm} =M_r\pm\frac12p_{k,r} =\frac{p_f\phi^0{}_0}{c} \left(\phi^0{}_\nu\pm\frac12\delta_{ij}k^i\phi^j{}_\nu\right)u^\nu. \tag{8}\]
In a plane containing the bosic direction, \(p_{k,r}=p_r\cos\alpha\). Plotting the plus value along each ray therefore gives \(R(\alpha)=M_r+(p_r/2)\cos\alpha\); the opposite ray gives the minus value. The core and fermic reference circles have radii \(M_r\) and \(p_{f,r}\), respectively.
For light, use \(p_r^\mu=\phi^0{}_0(p^\mu)_{\mathrm{local}}\) with nonzero null momentum. Its norm vanishes and \(p_r=M_r>0\); the proper-time formula above requires \(p_f>0\).
Stationary core and contextual components
Along a trajectory, the total derivative of the positive root equals its explicit reference-time derivative:
\[ \frac{dM_r}{dx^0}=\frac{d(-p_0)}{dx^0} =M\partial_0\phi^0{}_0 +\frac{\phi^0{}_0}{2M}(\partial_0g^{ij})p_ip_j. \tag{9}\]
It vanishes for a stationary map. At fixed \(M_r\), the squared fermic and bosic changes in Equation 7 cancel. Local \(M=M_r/\phi^0{}_0\) need not be constant along the trajectory. The local shell retains \(p_f\) as its coefficient, while the reference shell uses \(p_{f,r}\).
The momenta conjugate to the coordinates \(x^i\) are \(p_i\). Their canonical one-form uses the spatial map,
\[ p_i\,dx^i=p_j^{\mathrm{local}}\phi^j{}_i\,dx^i, \]
The reference triangle instead uses \(\phi^0{}_0\mathbf p_{\mathrm{local}}\). Substituting those components into the action without transforming the coordinates and symplectic structure would change the dynamics.
Reference normalization and physical internal length
The common factor is a declared comparison. If the reference observer has temporal coefficient \(N_o\), replace \(N=\phi^0{}_0\) in the normalized readings by \(N/N_o\); the chapter chooses \(N_o=1\). A constant change \(\widetilde x^0=a x^0\) sends both \(N\) and \(N_o\) to their values divided by \(a\). The normalized readings are unchanged, whereas the coordinate generator \(-p_0\) scales as \(1/a\). Its conservation statement always refers to the chosen stationary time.
The reference triangle does not uniquely determine a physical internal radius. For a selected cyclic pair \((\vartheta,I)\) with \(I>0\), \(R_0>0\) and \(p_f=I/R_0\), compare the two extended line elements
\[ d\sigma_1^2=-(dx^0)^2+N^{-2}g_{ij}dx^idx^j +(R_0/N)^2d\vartheta^2, \qquad d\sigma_2^2=N^2d\sigma_1^2. \]
Their null constraints differ by the positive factor \(N^{-2}\) and give the same root \(-p_0=N\sqrt{I^2/R_0^2+g^{ij}p_ip_j}\) after rescaling the constraint multiplier. They have the same reduced trajectories and selected cycle phase, although the displayed circle radii are \(R_0/N\) and \(R_0\). These comparison observables alone therefore do not select which internal length is physical. A separate internal constitutive law must make that assignment.
In particular, replacing the local shell coefficient by \(Np_f\) while retaining the same temporal conversion gives the different root \(N\sqrt{(Np_f)^2+|\mathbf p_{\mathrm{local}}|^2}\). Its held value is \(N^2p_f\), not \(Np_f\). The intended no-exchange redistribution requires a consistently derived internal response and source, not a second use of one reference factor.
Axial motion and the shell figures
For motion along \(+x\) with a diagonal spatial map, \(p_1=\phi^1{}_1p^1_{\mathrm{local}}\) and \(g^{11}=1/(\phi^1{}_1)^2\). Substitution into Equation 3 gives
\[ \frac{v}{c} =\frac{dx^1}{dx^0} =\frac{\phi^0{}_0}{\phi^1{}_1}\frac{p^1_{\mathrm{local}}}{M} =\frac{\phi^0{}_0}{\phi^1{}_1}\frac{p_r}{M_r}. \tag{10}\]
At the same reference velocity, the bosic-to-core ratio depends on the local map. The shell comparison in §3.5 holds the reference velocity fixed and calculates the two bosic momenta separately.
For release from rest in the distant limit, \(\phi^0{}_0\to1\) fixes \(M_r=p_f\). The reference triangle then gives
\[ p_r^2=p_f^2-p_{f,r}^{\,2} =p_f^2\left(1-[\phi^0{}_0]^2\right). \tag{11}\]
This gives the free-fall sequence in §3.5. A later interaction that stops the carrier at a fixed depth leaves \(p_r=0\) and \(M_r=p_{f,r}\), transferring \(p_f-p_{f,r}\) out of the carrier. The endpoint balance fixes this amount; the stopping interaction must be supplied separately.
The specified moving and held cycle
For the same massive carrier, the constrained part of the action vanishes on the shell. Using \(c\,d\tau/d\lambda=e p_f\), the action along the trajectory becomes
\[ dS=p_\mu dx^\mu=-cp_f\,d\tau. \]
The chosen action phase consequently satisfies
\[ d\varphi=\frac{cp_f}{\hbar}d\tau. \tag{12}\]
In reference coordinates, \(\dot x^0=eM/\phi^0{}_0\). Hence
\[ \frac{d\tau}{dt}=\phi^0{}_0\frac{p_f}{M}, \qquad \frac{d\varphi}{dt} =\frac{cp_f}{\hbar}\phi^0{}_0\frac{p_f}{M}. \tag{13}\]
Equivalently, \(d\varphi/dt=(c/\hbar)p_{f,r}^{\,2}/(-p_0)=(c/\hbar)p_{f,r}^{\,2}/M_r\). The same result follows directly from the reduced action: \(p_i(dx^i/dx^0)-(-p_0)=-\phi^0{}_0p_f^2/M\). In flat space, \(\phi^0{}_0=1\), and the cycle-rate factor reduces to the Foundations ratio \(p_f/M\).
For a held ideal carrier, \(\mathbf p_{\mathrm{local}}=0\) and \(M=p_f\) in this free kinetic description. If the support does not perturb the chosen cycle,
\[ \left(\frac{d\varphi}{dt}\right)_{\mathrm{held}} =\frac{cp_f}{\hbar}\phi^0{}_0=\frac{cp_{f,r}}{\hbar}. \tag{14}\]
Applying this ideal-cycle rate to a material clock requires its internal motion, stresses and response to the support to justify the approximation.
Result
The free-carrier action supplies §3.5’s motion equations and its four-momentum-to-shell construction. In a prescribed zero-shift stationary map it conserves \(M_r=-p_0=\phi^0{}_0M\), while the reference triangle exhibits fermic-to-bosic redistribution at fixed reference core. The additional action-phase hypothesis supplies the stated moving and held ideal-clock rates.
Notes
The shell represents the particle’s momentum state; a microscopic density or spatial boundary needs further structure. In the free-carrier calculation the bosic momentum describes translation. Internal bosic compensation, source stresses and physical recurrence in a structured particle require its constrained internal dynamics. Derivation 3.10A develops the operational comparison for the cycle hypothesis used here.