Derivation 3.8A — Time-Dependent Transport and Coupled Sources
theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology
Purpose
Free carriers move through a changing geometry while contributing to its source. This derivation obtains their transport law, the causal weak-field response and an exact homogeneous source–field solution. A separate effective material law gives the finite-amplitude body example used in §3.8.
Starting assumptions
Retain the field and free-carrier coupling of Derivations 3.4A and 3.7A. Coordinate momenta are raised with \(g\); local momenta use the orthonormal frame. The future shell has \(p^0>0\) in the admissible time slicing. Nonzero null carriers are permitted, but their zero-momentum vertex is excluded. The material evolution described later uses a different, explicitly adopted effective matter law, not a sum of unsupported free dyads.
Derivation
Transport without stationarity
Variation of the free constrained carrier action gives, for a parameter \(\lambda\) and multiplier \(e\),
\[ \frac{dx^\mu}{d\lambda}=e g^{\mu\nu}p_\nu, \qquad \frac{dp_\mu}{d\lambda} =-\frac e2\partial_\mu g^{\alpha\beta}p_\alpha p_\beta, \qquad g^{\mu\nu}p_\mu p_\nu=-p_f^2. \]
Dividing by \(dx^0/d\lambda=e p^0\) gives §3.8’s transport equations and
\[ \frac{d(-p_0)}{dx^0} =\frac{1}{2p^0}\partial_0g^{\alpha\beta}p_\alpha p_\beta. \tag{1}\]
Stationarity conserves \(-p_0\) even with shift; zero shift additionally gives \(-p_0=NM\). Here \(M\) is the local core, while \(p^0\) and \(p_0\) are the coordinate contravariant and covariant temporal components.
For a distribution \(f\) on canonical spatial phase space, conservation along this Hamiltonian flow is
\[ \partial_0 f +\partial_i\left(f\frac{p^i}{p^0}\right) +\partial_{p_i}\left(f\frac{dp_i}{dx^0}\right)=0. \]
The phase and future-shell measures satisfy
\[ d^3x\,d^3p_{\mathrm{coord}} =d\Sigma\,d^3p_{\mathrm{local}}, \qquad \frac{d^3p_{\mathrm{coord}}}{\sqrt{-g}\,p^0} =\frac{d^3p_{\mathrm{local}}}{M}. \tag{2}\]
Taking moments of transport with decaying or compact momentum support gives \(\nabla_\mu J^{\mu\nu}=0\). Thus the matter equations and invariant measure supply the source balance required by the field equation.
Retarded fields and two tidal polarizations
The retarded Green function for the weak harmonic equation yields
\[ \theta^{\mu\nu}(t,\mathbf x) =\frac{G_N}{c^3}\int \frac{J^{\mu\nu}(t-R/c,\mathbf x')}{R}\,d^3x', \qquad R=|\mathbf x-\mathbf x'|. \]
For a conserved leading source and a convergent convolution, retarded homogeneous data select this sourced response. General initial data can also include free radiation.
For a linear vacuum example, put \(u=x^0-x^3\) and choose \(H_{11}(u)=-H_{22}(u)\), \(H_{12}(u)=H_{21}(u)\), with all longitudinal entries zero. Then
\[ \theta^{ij}=\frac14H_{ij}(u), \qquad \theta^{00}=\theta^{0i}=0, \qquad R^{(1)}_{0i0j}=-\frac12\frac{d^2H_{ij}}{du^2}. \tag{3}\]
The two independent functions solve the linear vacuum equation and constraints and produce nonzero tidal curvature. They describe freely propagating weak waves.
An exact nonlinear homogeneous kinetic solution
Use a flat periodic spatial domain or a local homogeneous patch, and
\[ g_{00}=-1,\qquad g_{0i}=0,\qquad g_{ij}=a(x^0)^2\delta_{ij},\qquad a>0. \]
Let \(p_{\mathrm c}=|\mathbf p_{\mathrm{coord}}|\) be the Euclidean magnitude of the canonical spatial covector in these homogeneous coordinates. For each species \(s\), choose a smooth nonnegative compact radial profile \(f_s(p_{\mathrm c})\); null profiles avoid \(p_{\mathrm c}=0\). Translation symmetry conserves \(p_i\), while local momentum is \(p_i/a\). Thus
\[ M_s(a,p_{\mathrm c})=\sqrt{p_{f,s}^2+p_{\mathrm c}^2/a^2}, \]
\[ \mathcal M(a)=4\pi a^{-3}\sum_s \int_0^\infty f_s(p_{\mathrm c})M_s p_{\mathrm c}^2\,dp_{\mathrm c}, \]
\[ \mathcal C_{\mathrm{iso}}(a)=\frac{4\pi}{3}a^{-5}\sum_s \int_0^\infty\frac{f_s(p_{\mathrm c})p_{\mathrm c}^4}{M_s}\,dp_{\mathrm c}, \qquad \mathcal C^{ij}=\mathcal C_{\mathrm{iso}}\delta^{ij}. \tag{4}\]
Primes in this subsection mean \(d/dx^0\), and \(H_a=a'/a\). Differentiation of the moment integral gives
\[ \mathcal M'=-3H_a(\mathcal M+\mathcal C_{\mathrm{iso}}). \]
The nonzero field equations reduce to
\[ 3H_a^2=\frac{8\pi G_N}{c^3}\mathcal M, \qquad H_a'=-\frac{4\pi G_N}{c^3} (\mathcal M+\mathcal C_{\mathrm{iso}}). \tag{5}\]
For a nonzero population the solution is specified by
\[ x^0-x^0_0 =\sigma\int_{a_0}^{a} \frac{d\alpha}{\alpha\sqrt{8\pi G_N\mathcal M(\alpha)/(3c^3)}}, \qquad \sigma=\pm1. \]
On a bounded positive-\(a\) interval the integrand is regular. Differentiating the first field equation and using the moment identity yields the second. Direct substitution also satisfies the spatial field equations; the momentum constraints vanish by homogeneity, and the distribution solves transport exactly. The same homogeneous construction accommodates mixtures of massive and massless species.
For purely null matter with \(a(0)=1\), \(\mathcal M=\mathcal M_0a^{-4}\) and \(\mathcal C_{\mathrm{iso}}=\mathcal M/3\), so
\[ a^2=1+2\sigma\sqrt{\frac{8\pi G_N\mathcal M_0}{3c^3}}\,x^0. \]
The registered deformation is
\[ \theta^{00}=\frac{3(a^2-1)}8, \qquad \theta^{0i}=0, \qquad \theta^{ij}=\frac{1-a^2}{8}\delta^{ij}. \tag{6}\]
They reconstruct \(\phi=\operatorname{diag}(1,a,a,a)\) exactly. The sign \(\sigma\) allows expanding and contracting branches.
The finite-amplitude material illustration
The material example uses the self-bound fluid specified in §3.10 and Derivation 3.10A: sound-speed parameter \(s=1/3\), central pressure \(0.02\) times the surface energy density, and units \(G_N=c=1\) with unit surface energy and constituent densities. Its equilibrium radius is \(0.0909395683980\), and its fundamental linear period in the asymptotic time normalization is \(T_0=0.345685420215\).
A fixed-number, constraint-compatible initial velocity kick excites the radial mode. The full nonlinear material and current gravitational fields are evolved, with zero scalar signal. The separately reviewed finite-amplitude calculation uses \(\epsilon=0.001\) and \(0.005\) for \(0\leq t\leq2.0741125213\), six reference linear periods. The parameter \(\epsilon\) scales the prepared mode kick; the linear displacement is normalized by \(\xi(R_s)=R_s\) per unit amplitude.
For \(\epsilon=0.005\), successive maxima of the evolved surface give:
| Interval | Period in asymptotic time |
|---|---|
| 1 | 0.345751716735 |
| 2 | 0.345742850721 |
| 3 | 0.345730312911 |
| 4 | 0.345715330380 |
| 5 | 0.345699161567 |
The order-64 to order-96 period differences are no greater than \(2.31\times10^{-9}\). The independently evaluated surface-expansion residual decreases across orders 48, 64 and 96 from \(2.71\times10^{-5}\) to \(1.08\times10^{-5}\) to \(3.88\times10^{-6}\); it is finite, not exactly enforced at finite resolution. At order 96, maximum local constituent-label drift is \(6.39\times10^{-9}\) and the independent mass-work residual is \(3.84\times10^{-12}\).
The evolved body has cycle-dependent recurrence over the tested interval. This numerical result does not establish an exact periodic orbit or nonlinear stability. The \(\epsilon=0.01\) extension enters tension and worsens local and boundary diagnostics at higher order; it is outside the verified result.
Accumulated time on the moving surface
For the actual surface trajectory \(x_s^\mu(t)\),
\[ \Delta\tau_s =\int_{t_1}^{t_2} \sqrt{-g_{\mu\nu}(x_s(t)) \frac{dx_s^\mu}{c\,dt}\frac{dx_s^\nu}{c\,dt}}\,dt. \]
Independent integration on the evolved body gives successive proper intervals ranging from \(0.333259916533\) to \(0.333209250486\) in the same units. A fixed initial static multiplier differs by about \(2.01\times10^{-5}\) relatively. Both motion and the evolving geometry must stay in the time integral.
Result
The free coupling supplies time-dependent transport and its source balance. The retained weak field has retarded propagation and two tidal polarizations. A homogeneous distribution supplies an exact finite coupled example, while a separately adopted material law supplies a bounded nonlinear body evolution with measured changing recurrence.
Notes
The homogeneous kinetic solution and the material body use different source laws and boundary conditions. The body evolution has zero scalar signal; the weak outgoing-signal comparison and homogeneous finite-exchange solution in §3.10 are separate calculations. These results support §3.8.