Derivation 3.6A — Reciprocal Sources and Static Feedback

Keywords

theoretical physics, foundations of physics, momentum conservation, canonical momentum, special relativity, stationary gravity, gravitoelectromagnetism, quantum mechanics, FLRW cosmology

Purpose

The particle momenta found in §3.5 supply the source in §3.6. This derivation obtains their local source contributions and the exact static temporal weighting. It then solves a supported continuum example and compares two scalar models of source feedback.

Starting assumptions

Use signature \((-+++)\), \(x^0=ct\) and the current all-upper momentum source. In the static zero-shift patch, write \(N=\phi^0{}_0>0\) and \(h_{ij}=g_{ij}\) as local calculation aliases. The retained equation is \(G^{\mu\nu}=(8\pi G_N/c^3)J^{\mu\nu}\) in matching frames. Its source includes the complete material stress for the model being solved.

The free-carrier example uses \(p_r^\mu=Np_{\mathrm{local}}^\mu\), \(M_r=NM\) and local physical-volume population weights. The prescribed scalar control later uses Euclidean volume instead; it is a different field-law problem. Neither the free-particle kernel nor the constant-density continuum resolves a singular particle self-field or supplies a microscopic internal-support law.

Derivation

From particle momentum to the source

For each massive carrier, the normalized four-velocity of §3.5 gives

\[ p_{\mathrm{local},a}^{\mu} =\frac{p_{f,a}}{c}\phi^\mu{}_\alpha u_a^\alpha, \qquad p_{r,a}^{\mu}=Np_{\mathrm{local},a}^{\mu}. \]

Consequently

\[ \begin{aligned} (j_a^{\mu\nu})_{\mathrm{local}} &=\frac{p_{\mathrm{local},a}^{\mu}p_{\mathrm{local},a}^{\nu}}{M_a}\\ &=\frac{p_{r,a}^{\mu}p_{r,a}^{\nu}}{NM_{r,a}}\\ &=\frac{p_{f,a}^2}{c^2M_a} \phi^\mu{}_\alpha\phi^\nu{}_\beta u_a^\alpha u_a^\beta. \end{aligned} \tag{1}\]

The first two forms also apply to nonzero null momenta; the massive proper-time parametrization in the last form does not. Form each product before taking the physical-volume moment. The blocks are

\[ \mathcal M=\left\langle\sum_a\frac{M_{r,a}}N\right\rangle, \qquad \mathcal P^i=\left\langle\sum_a\frac{p_{r,a}^i}N\right\rangle, \qquad \mathcal C^{ij}=\left\langle\sum_a \frac{p_{r,a}^ip_{r,a}^j}{NM_{r,a}}\right\rangle. \tag{2}\]

These are local source components expressed through reference-normalized inputs. Coordinate components require the separate congruence \(J_{\mathrm{coord}}=\phi^{-1}J_{\mathrm{local}}\phi^{-\mathsf T}\). Neither operation changes which physical content belongs in \(J\).

The static temporal equation

For

\[ ds^2=-N^2(dx^0)^2+h_{ij}dx^idx^j, \]

static curvature gives

\[ R_{00}=N D_iD^iN, \qquad D_iD^iN=\frac1{\sqrt{\det h}} \partial_i\left(\sqrt{\det h}\,h^{ij}\partial_jN\right). \]

Trace reversal of the retained equation, in the local orthonormal frame, gives

\[ R_{\mathrm{local},00} =\frac{4\pi G_N}{c^3} \left(\mathcal M+\delta_{ij}\mathcal C^{ij}\right). \]

Since \(R_{\mathrm{local},00}=R_{00}/N^2\), the exact result is

\[ D_iD^iN =\frac{4\pi G_N}{c^3}N \left(\mathcal M+\delta_{ij}\mathcal C^{ij}\right). \tag{3}\]

Trace reversal gives the positive spatial-trace term. This equation sources the temporal map \(N\); weak \(\theta^{00}\) alone is sourced by \(\mathcal M\).

For resting free carriers, \(\mathcal M=np_f\) and \(\mathcal C^{ij}=0\), hence \(N\mathcal M=np_{f,r}\). The single factor of \(N\) converts the local fermic contribution to its reference-expressed value.

The same source must satisfy conservation. For static dust its spatial equation is

\[ \mathcal M\,\partial_i\ln N=0. \tag{4}\]

For positive density, this condition requires a spatially constant \(N\). A static body in a varying well therefore needs a supporting stress or force.

A complete static continuum example

Consider an incompressible continuum with constant total local core density, \(\mathcal M=\mathcal M_0\), and isotropic total spatial stress, \(\mathcal C^{ij}=\mathcal C_s(r)\delta^{ij}\). The supporting stress is included in \(\mathcal C_s\). Let \(r\) be areal radius and \(R_s\) the surface radius. Define

\[ r_s=\frac{8\pi G_N\mathcal M_0R_s^3}{3c^3}, \qquad F(r)=1-\frac{r_s r^2}{R_s^3}, \qquad F_s=1-\frac{r_s}{R_s}. \]

Inside the body,

\[ N=\frac{3\sqrt{F_s}-\sqrt F}{2}, \qquad \mathcal C_s=\mathcal M_0 \frac{\sqrt F-\sqrt{F_s}}{3\sqrt{F_s}-\sqrt F}, \tag{5}\]

and in areal Cartesian coordinates the spatial map is

\[ \phi^i{}_j=\delta^i{}_j+ \left(F^{-1/2}-1\right)\hat r^i\hat r_j. \]

The centre is defined by smooth extension. These expressions give \(ds^2=-N^2(dx^0)^2+dr^2/F+r^2d\Omega^2\). Outside, \(F=1-r_s/r\), \(N=\sqrt F\) and \(\mathcal C_s=0\).

The radial field and material equations can be written using the enclosed core parameter \(M_{\mathrm{enc}}\):

\[ \begin{aligned} \frac{dM_{\mathrm{enc}}}{dr}&=4\pi r^2\mathcal M_0,\\ \frac{N'}N&=\frac{G_N}{c^3} \frac{M_{\mathrm{enc}}+4\pi r^3\mathcal C_s} {r^2[1-2G_NM_{\mathrm{enc}}/(c^3r)]},\\ \mathcal C_s'&=-(\mathcal M_0+\mathcal C_s)\frac{N'}N. \end{aligned} \tag{6}\]

The displayed solution satisfies these equations and the tangential field equation. At \(R_s\), its lapse and radial geometry match the vacuum exterior and its stress vanishes, so no surface stress layer is required. Positive central lapse and finite central stress require \(0<r_s/R_s<8/9\).

Temporal weighting and the exterior source

For a regular isolated static spherical solution, integrating the temporal equation gives the asymptotic core parameter, in momentum units,

\[ M_\infty =\frac{c^3}{G_N}\lim_{r\to\infty}r^2\sqrt F\,N' =\int N\left(\mathcal M+\delta_{ij}\mathcal C^{ij}\right)d\Sigma. \tag{7}\]

For the supported uniform sphere, \(d\Sigma=4\pi r^2dr/\sqrt F\), and direct substitution gives

\[ \frac{N(\mathcal M_0+3\mathcal C_s)}{\sqrt F}=\mathcal M_0. \]

Thus \(M_\infty=4\pi\mathcal M_0R_s^3/3\). The proper integral of local cores alone is larger:

\[ \int\mathcal M_0\,d\Sigma =4\pi\mathcal M_0\int_0^{R_s}\frac{r^2}{\sqrt F}\,dr >M_\infty. \]

The exterior source depends on temporal weighting, supporting stress and physical volume together.

A scalar feedback model

Now prescribe constant \(n\) per Euclidean volume, retain only \(\theta^{00}\), and add the field-law postulate

\[ \Delta_{\mathrm{flat}}\theta^{00} =-\frac{4\pi G_Nnp_f}{c^3}\sqrt{1-2\theta^{00}} \quad (r<R_s). \]

Require a regular centre, decay at infinity and continuous value and first radial derivative at the surface. With

\[ x=\frac r{R_s},\qquad \alpha=\frac{4\pi G_Nnp_fR_s^2}{c^3},\qquad y=1-2\theta^{00}=N^2, \]

the boundary-value problem becomes

\[ y''+\frac2x y'=2\alpha\sqrt y, \qquad y'(0)=0, \qquad y(1)+y'(1)=1. \tag{8}\]

Primes now mean \(d/dx\). Outside, \(y=1-b/x\) with \(b=y'(1)\). The reduced-to-unreduced source fraction is

\[ S=3\int_0^1x^2\sqrt y\,dx, \qquad b=\frac{2\alpha}{3}S. \]

Solving without expanding the square root gives:

\(\alpha\)	Central \(N\)	Surface \(N\)	Source fraction \(S\)
0.1	0.950833949	0.967443049	0.960809196
1	0.581376331	0.740308770	0.677914387
2	0.302614700	0.596052040	0.483541474
4	0.019726127	0.465118871	0.293874163

The tabulated values are numerical solutions of this scalar boundary-value problem, checked by shooting, collocation and the source–flux identity. They are not solutions of all tensor field equations. The flat operator and Euclidean density are constitutive choices; a physical spatial metric and reciprocal material dynamics cannot be inferred from this scalar equation alone.

Giulini’s homogeneous scalar sphere

Giulini’s account of Einstein’s Prague field equation, §§2–4, supplies a related exact scalar example. For a uniform bare mass \(m_b\) inside \(R_s\), his normalized scalar \(\Psi=\sqrt{\Phi/c^2}\) satisfies

\[ \Delta_{\mathrm{flat}}\Psi=\kappa^2\Psi, \qquad \kappa^2=\frac{3G_Nm_b}{2c^2R_s^3}, \qquad \Psi(\infty)=1. \]

Put \(z=\kappa R_s\). Regularity and matching give

\[ \Psi_{\mathrm{in}}(r) =\frac{\sinh(\kappa r)}{\kappa r\cosh z}, \qquad \Psi_{\mathrm{out}}(r)=1-\frac{R_g}{r}, \]

\[ R_g=R_s\left(1-\frac{\tanh z}{z}\right) =\frac{G_Nm_g}{2c^2}, \qquad \frac{m_g}{m_b} =\frac3{z^2}\left(1-\frac{\tanh z}{z}\right). \tag{9}\]

The active mass \(m_g\) includes the scalar field’s contribution in that theory. At \(z=1\), \(m_g/m_b=0.715217532\); its matter and field energy fractions are respectively \(0.512429722\) and \(0.202787811\).

Identifying the bare mass density with \(np_f/c\) gives \(z^2=\alpha/2\) in Giulini’s model. A flat-spatial version of the retained resting-lapse equation would instead have \(z^2=\alpha\), while the nonlinear-\(\theta\) problem above has a different field equation again. These comparisons relate their source parameters; each model retains its own force law and source accounting.

Result

The reference shell enters the kinetic source through \(j_a^{\mu\nu}=p_{r,a}^\mu p_{r,a}^\nu/(NM_{r,a})\). The exact static temporal equation weights a resting contribution as \(np_{f,r}\). The supported continuum solution shows how pressure and curved volume enter the complete source, while the two scalar examples describe feedback under their own field-law postulates.

Notes

The supported uniform sphere is the interior Schwarzschild incompressible continuum in momentum units. Its stress is specified at the continuum level; internal particle support and causal material evolution require further physical laws. Giulini’s formulas used here are his equations (24), (32)–(33) and (37). The calculations support §3.6.