Gravity as the Momentum of Absorbed Inflow

The central chapter of the volume takes the inflow rate Q from Chapter 1 as its only input and derives the gravitational field, the meaning of mass and of the gravitational constant, and the equivalence principle. The argument is a numbered chain, each step checked by simulation. Its predictions are degenerate with Newtonian gravity.

The model-independent facts of gravitation are few and familiar: unsupported bodies near the Earth accelerate downward at g≃9.81ms⁻² regardless of their composition; the planets trace closed, stable orbits about the Sun characterised by the heliocentric strength GM_(odot)=1.327×10²⁰m³s⁻²; and the orbital data obey an inverse-square force law. We will show that all three—the inverse-square law, the composition independence, and the identification of the source strength with GM—follow from a body being a sink of the vacuum medium, draining it at the rate Q of Chapter 1.

This is Chapter 3, the central chapter of the volume. It takes as its only body-specific input the inflow rate Q defined and tabulated in Chapter 1 (Q∝ M, with Q/M=1.7522×10²⁹quantas⁻¹kg⁻¹), and from it derives the gravitational field, the meaning of mass and of the gravitational constant, and the equivalence principle. Following the rules of Chapter 0, the argument is given as a numbered chain with no step left implicit, every non-trivial step is checked by an attached, reproducible simulation, and the strong-field results that belong to the physics volume are imported by citation rather than re-derived.

The raw facts to be explained

The model-independent facts of gravitation are few and familiar: unsupported bodies near the Earth accelerate downward at g≃9.81ms⁻² regardless of their composition; the planets trace closed, stable orbits about the Sun characterised by the heliocentric strength GM_(odot)=1.327×10²⁰m³s⁻²; and the orbital data obey an inverse-square force law. We will show that all three—the inverse-square law, the composition independence, and the identification of the source strength with GM—follow from a body being a sink of the vacuum medium, draining it at the rate Q of Chapter 1.

The inflow velocity field (steps 1–3)

We first establish the velocity field that a single sink sets up in the surrounding medium.

Depletion. A body of inflow rate Q₁ annihilates Q₁ vacuum quanta per second (Chapter 1). This continuously removes quanta from the region it occupies.
Refilling (no-void). By premise (P1) of Chapter 1 the medium is fully packed and admits no voids; the surrounding quanta therefore move inward to replace those annihilated. After transients this settles to a steady, spherically symmetric, radially inward flow.
Conservation fixes the profile. In steady state every quantum that is annihilated must first cross every sphere surrounding the body, so the inward flux through a sphere of radius r equals the annihilation rate:
$\begin{equation} \underbrace{4\pi r^{2}}_{\text{area}}\; \underbrace{n_{0}}_{\substack{\text{number}\\\text{density}}}\; \underbrace{v(r)}_{\text{inflow speed}} \;=\; Q_{1}, \end{equation}$

with n₀ the undisturbed number density of the medium (a universal constant). Solving,

$\begin{equation} \boxed{\,v(r)\;=\;\frac{Q_{1}}{4\pi n_{0}\,r^{2}}\;\propto\;\frac{Q_{1}}{r^{2}}.\,} \end{equation}$

The inflow speed falls as 1/r². This is the single field quantity from which the force will be built; note that nothing here yet refers to a second body.

The gravitational force as absorbed momentum (steps 4–6)

A second sink in the flow. Place a second body, of inflow rate Q₂, at distance r from the first. About itself the second body draws an isotropic inflow, which by symmetry exerts no net force on it. The only asymmetry it experiences is the inflow (vfield) set up by the first body, which at the second body's location carries the medium past it with velocity v(r) directed toward body 1.
Momentum delivered by absorption. The second body is a sink: it annihilates quanta at rate Q₂. Each quantum (mass m_q) that it absorbs from its surroundings is, on average, drifting toward body 1 with the local velocity v(r), hence carrying momentum m_qv(r) toward body 1. By momentum conservation the absorbing body acquires this momentum. The momentum delivered per unit time—i.e. the force—is therefore
$\begin{equation} F \;=\; Q_{2}\,\big(m_{q}\,v(r)\big) \;=\; Q_{2}\,m_{q}\,\frac{Q_{1}}{4\pi n_{0}r^{2}} \;=\; \frac{m_{q}}{4\pi n_{0}}\,\frac{Q_{1}Q_{2}}{r^{2}}, \end{equation}$

directed toward body 1 (attractive).
The inverse-square law. Equation (force) is an attractive force proportional to the product of the two inflow rates and falling as 1/r². Writing κ' ≡ m_q/(4π n₀), a universal constant of the medium,
$\begin{equation} F \;=\; \kappa'\,\frac{Q_{1}Q_{2}}{r^{2}} . \end{equation}$

This has exactly Newton's form, with the role of “gravitational charge” played by the inflow rate Q.

Why 1/r² and not 1/r⁵ (step 7)

This step is the one most easily skipped and therefore, by the rule of Chapter 0, the one we make most explicit. A natural-seeming alternative would treat the gravitating body not as a sink that absorbs momentum, but as a parcel carried along by the inflow, feeling the flow's own (convective) acceleration. For the steady radial field (vfield) that acceleration is

\begin{equation} a_{\text{flow}} \;=\; (\mathbf{v}\!\cdot\!\nabla)\mathbf{v} \;=\; v\,\frac{dv}{dr} \;=\; \frac{Q_{1}}{4\pi n_{0}r^{2}}\cdot\!\left(-\frac{2\,Q_{1}}{4\pi n_{0}r^{3}}\right) \;=\; -\,2\!\left(\frac{Q_{1}}{4\pi n_{0}}\right)^{\!2}\frac{1}{r^{5}}\;\propto\;\frac{1}{r^{5}} . \end{equation}

The two candidate laws differ sharply: absorption of momentum gives 1/r² (Eq. (force)), whereas convective carry-along gives 1/r⁵ (Eq. (rfive)). Which is physically correct is decided by what the body actually is. A gravitating body annihilates the medium; it is a sink, not a passive tracer. A sink does not move with the local flow—it removes the flow—so the force it feels is the momentum it absorbs, Eq. (force), the 1/r² law. The convective expression (rfive) would apply to a neutrally buoyant tracer that neither emits nor absorbs the medium, which a massive body is not.

Stability confirms the selection.

The two laws are also distinguished observationally, because orbits under a central force F∝ r⁻ⁿ are stable only for n<3 (a circular orbit's small radial perturbations oscillate for n<3 and grow without bound for n≥3). Thus n=2 (absorption) permits the stable, long-lived planetary orbits we observe, while n=5 (carry-along) does not. Simulation A integrates a circular orbit given a 1% tangential-speed perturbation under each law:

def leapfrog(n_exp, perturb=0.01, T=60, N=600000, k=1.0):
    x=[1.,0.]; v=[0.,1.+perturb]; dt=T/N
    acc=lambda x:-k*hypot(x)**(-n_exp)*(x/hypot(x))
    # ... kick-drift-kick; record r(t) ...
n=2 : r stays in [1.00, 1.04]              -> BOUNDED (libration ~4%)
n=5 : r grows 1 -> 41 and keeps climbing   -> RUNAWAY (escape)

The result (Fig. (stab)) is unambiguous: under the n=2 absorption law the perturbed orbit librates within a few percent of its radius, exactly as the planets do; under the n=5 carry-along law the same orbit runs away by more than a factor of forty and continues to grow. Since planetary orbits are observed to be stable over billions of years, the gravitational mechanism must be momentum absorption (1/r²), not convective carry-along (1/r⁵). This is the “why this and not that” that an inverse-square claim requires.

Mass is the inflow rate; G is a property of the medium (step 8)

Identifying GM. Comparing the force (Fnewton) with Newton's F=GM₁M₂/r² shows that the gravitational mass of a body is proportional to its inflow rate, MproptoQ. Equivalently, the source strength is
$\begin{equation} GM_{1}\;=\;\kappa\,Q_{1},\qquad \kappa\equiv\frac{\kappa'}{\beta}, \end{equation}$

where β is the constant relating inertia to inflow rate (next section) and κ is a universal medium constant. We fix κ once, by matching the Sun:

$\begin{equation} \kappa=\frac{GM_{\odot}}{Q_{\odot}} =\frac{1.327\times10^{20}\,\mathrm{m^{3}s^{-2}}}{3.485\times10^{59}\,\mathrm{s^{-1}}} =3.81\times10^{-40}\ \mathrm{m^{3}\,s^{-1}} . \end{equation}$

Every other body's source strength then follows from its tabulated Q (Chapter 1, Table 1) with no further input. As a consistency check, κQ_(⊕) =3.81×10⁻⁴⁰×1.046×10⁵⁴=3.98×10¹⁴m³s⁻² =GM_(⊕), recovering the measured terrestrial value.

The mass/G split is conventional.

Only the product kappaQ(=GM) enters any orbit (Eq. (GMkappa) feeds the acceleration a=kappaQ₁/r² of the next section). Rescaling Q→ cQ and κ→κ/c leaves kappaQ unchanged and the orbit identical. Simulation C verifies this to machine precision:

(kappa, Q)  vs  (kappa/1e6, Q*1e6)   [same product kappa*Q = GM]
   max |trajectory difference| over an orbit = 2.7e-13   ->  identical

Hence the framework fixes the product; how one chooses to call part of it “G” and part of it “mass” is a matter of units. Adopting the measured G recovers the conventional mass scale, and the inflow rate Q is then simply the mass expressed in annihilation-rate units.

The equivalence principle, as a theorem (step 9)

Inertia is also the inflow rate. The same sink property that makes Q the gravitational charge also makes it the inertia: resisting acceleration means rearranging the quanta one is absorbing, so a body's inertial mass is m=βQ with the same universal β for all matter. The acceleration of body 2 in body 1's field is then
$\begin{equation} a_{2}=\frac{F}{m_{2}} =\frac{\kappa'\,Q_{1}Q_{2}/r^{2}}{\beta\,Q_{2}} =\frac{\kappa'}{\beta}\,\frac{Q_{1}}{r^{2}} =\kappa\,\frac{Q_{1}}{r^{2}} . \end{equation}$

The test body's own rate Q₂ appears in the force (proptoQ₂) and in its inertia (proptoQ₂) and therefore cancels. The acceleration depends only on the source, not on the composition or amount of the falling body.

This is the (weak) equivalence principle—universality of free fall—obtained here as a theorem rather than assumed as a postulate, because gravitational coupling and inertia are literally the same physical quantity, the inflow rate. Simulation B drops two test bodies whose inflow rates differ by a factor 10⁴ in the same source field:

Q_a   vs   Q_b = 1e4 * Q_a   in the same field  a = kappa*Q_source / r^2
   max |trajectory_a - trajectory_b| over an orbit = 1.9e-14  ->  identical

The trajectories coincide to machine precision, as the cancellation (EP) requires.

What is deferred to the physics volume

The derivation above is the leading-order, weak- and far-field result, and it is exactly Newtonian gravity. The strong-field completion is established in the physics volume and is not re-derived here; we record its results in one paragraph and import them by citation. There the field is two-channel: a geometric channel g_(geom)=GM/R² that is not capped, plus a contact (restoring) channel g_(restore)=min(gₚₒₜ,g_(*)) that saturates at a yield value g_(*)=c²Ψ_(yield) — these, with the four-wall status below, are the genuine physics-volume imports. The strong-field consequences, by contrast, are owned by this volume (ownership corrected in v2; the physics volume does not carry them): the inflow reaching the wave speed at Rₛ=2GM/c² gives the river/Painlev\'e picture of black holes developed in Chapter 9, and the resulting effective optical index n=1+Rₛ/r reproduces light bending δ=1.752” (the Eddington value; derived in Chapter 9 from the same inflow). The physics volume (v0.6.0) has additionally opened a recovery program R1–R4 for the historical full-physics magnitude claim (its §17.4.4). One honest consequence of that structure must be flagged here: because the restoring channel saturates, the absolute value of G (equivalently, of g=9.81ms⁻²) is tied to a yield cap, and a first-principles micro-derivation of that absolute value is not yet complete—the “four-wall” no-go discussed in the physics volume. In the present volume G therefore enters only through the empirically fixed product κ of Eq. (kappaval); we never claim to derive its absolute value here, and we mark this as open in the ledger (Chapter 16).

What can be shown here: gate physics (saturation and a critical radius)

Two of those imported results—that the restoring channel saturates at g_(*), and that the inflow reaching the wave speed defines a critical radius—need not be taken purely on loan. Their mechanism follows from one minimal ingredient, a finite microscopic transport speed, and can be shown directly (Fig. (gate)); only the absolute value of the cap remains a physics-volume matter (DOI \href{https://doi.org/10.5281/zenodo.17932566}{10.5281/zenodo.17932566}). (This is the volume's stance throughout: we do not prove the foundations, we show their consequences.)

The flux bound.

If the active phase cannot move faster than c (|v|≤ c), then the energy flux is bounded by the active energy density it carries:

\begin{equation} \|\mathbf{S}\|\le c\,e_{\mathrm{a}} \qquad\Longrightarrow\qquad \|\mathbf{u}\|:=\frac{\|\mathbf{S}\|}{e_{\mathrm{a}}}\le c . \end{equation}

The mean transport velocity therefore has a ceiling. (The same bound caps the second moment, trT≤ c²eₐ, which limits any isotropic closure to κ_T≤ c²/3.)

Saturation.

The active→stored conversion rate is a maximum rate times a dimensionless, bounded gain,

\begin{equation} \Gamma=\Gamma_{\max}\,g(e_{\mathrm{a}}), \qquad g:[0,1]\to[0,1],\quad g(0)=0, \end{equation}

for example the Hill form g(eₐ)=eₐⁿ/(K_(Γ)ⁿ+eₐⁿ). Since g≤1, the rate cannot exceed Γ_(max): the restoring channel saturates. This is the microphysical shape of the cap g_(*)—why a contact restoring force levels off rather than growing without bound.

An emergent critical radius.

Under an inverse-square drive |F_r|=K_F/r², the demanded transport velocity grows inward as u_(dem)=K_F/(B r²) and meets the ceiling c at

\begin{equation} r_{\mathrm{ch}}=\sqrt{\frac{K_{F}}{B\,c}}, \end{equation}

inside which the flow is choked: the transport velocity is pinned at c and the choking ratio χ_S=u_(dem)/c≥1. A horizon-like critical radius—a surface where the inflow reaches the wave speed—thus emerges solely from a flux-capacity bound, with no extra postulate. (Its coefficient is drive-dependent; the strong-field black-hole horizon Rₛ=2GM/c² is the geometric-channel instance, Chapter 9.)

What the simulation shows.

ch3_gate.py evaluates the three relations (Fig. (gate)): the gain g(eₐ) climbing from 0 to its ceiling 1 (the conversion rate saturating at Γ_(max)); the inward demand u_(dem)∝1/r² crossing the cap c at the analytic r_(ch)=√K_F/(Bc) (measured 1.001 in units where r_(ch)=1); and the resulting choked interior in which u=min(u_(dem),c) saturates at c.

Status of the gate claim.

Evidence (Gate 1): mechanism, not measurement. A finite transport speed does produce both a saturating restoring rate and a horizon-like critical radius. The units are toy and c,K_F,B,K_(Γ),n are inputs; the simulation does not derive the absolute value of G/g_(*) (the “four-wall” question stays open), nor the Schwarzschild coefficient Rₛ=2GM/c² (whose linear-in-M scaling comes from the uncapped geometric channel, not the inverse-square choke radius, which scales as √K_F).
Label (Gate 2): degenerate in outcome. The existence of a saturating cap and of a surface where the inflow reaches c is degenerate with a capped restoring force and the standard horizon; the distinguishing content is the mechanism—a throughput choke with a finite, non-singular interior (Chapter 9).
Scope (Gate 3): present-day. These are statements about how gravity saturates and how a horizon forms now; no origin or cosmic-history claim.

Anticipated objections

“This is Le Sage gravity, which is known to fail.”

Classical Le Sage models post a flux of particles streaming through space and founder on the drag and heating such particles would produce. The present mechanism differs in kind. (i) The leading-order law is exactly Newton's 1/r² (Eq. (Fnewton)), with no leading drag term, because the force is the momentum of the absorbed medium, not of a transverse particle flux. (ii) Aberration of the inflow produces only a v/c-order secular effect, i.e.\ an extraordinarily slow orbital decay, not a first-order drag. (iii) The energy carried in by absorption is not a fatal heating paradox but is reinterpreted as the power source of central engines (active galactic nuclei and their jets), a point developed in Chapters 6 and 8. The medium here is the vacuum itself, and the relevant transport is its inflow momentum.

“This is just Newton relabelled.”

At Solar-System scale, this is correct, and we say so without hedging: the predictions are identical to Newton's, so this chapter and the next are degenerate with standard gravity and constitute a consistency check, not evidence of superiority. The framework's distinct, non-degenerate content appears where the same inflow law, combined with the cosmic background inflow, departs from Newton—galactic rotation and the derived acceleration scale a₀=cH₀/2π (Chapter 6). The value of the re-description is twofold: economy (a single quantity, the inflow rate Q, replaces the separate notions of gravitational charge, inertial mass, and the equivalence principle, which are here one thing and one theorem); and extensibility to regimes where it ceases to be degenerate.

Status of this chapter

Derived (internal success). The inverse-square force, the identification of mass with the inflow rate, the identification of G with a medium constant, and the equivalence principle. The two non-trivial selections—1/r² over 1/r⁵ by stability, and the exact cancellation of the test body's Q—are confirmed by Simulations A and B.
Degenerate. All weak-field/orbital predictions coincide with Newtonian gravity (quantified body-by-body in Chapter 4). This is consistency and economy, not a discriminating result.
Open / partly shown. The mechanism of the saturation and of the critical radius is now shown by simulation (§(gate), gate physics): a single finite transport speed yields both the cap g_(*) and a choke radius r_(ch). What remains open is the absolute value of G/g_(*) (the “four-wall” question), still fixed empirically through κ; aberration-induced secular decay and the absorbed-energy/central-engine link remain qualitative.

With the inverse-square law, the meaning of mass and G, and the equivalence principle in hand—all from the single inflow rate of Chapter 1—we turn in Chapter 4 to the quantitative consistency test: reproducing the measured orbital speeds and periods of the Solar System to sub-percent accuracy.

Reproducibility

All three simulations are pure-Python (NumPy), use a symplectic kick–drift–kick integrator, and report the numbers quoted above. Units are normalised (r₀=1, circular speed =1 for Sim A; GM=4π², i.e. years/AU, for Sims B and C).

Sim A — stability (1/r² vs 1/r⁵). Integrate a 1%-perturbed circular orbit under a=-kr⁻ⁿ r, k=1. Output: n=2 keeps r∈[1.00,1.04] (bounded); n=5 grows r:1→41 and rising (runaway). Generates Fig. (stab).
Sim B — equivalence principle. Field a=kappaQₛ/r² with kappaQₛ=4π²; integrate two bodies with Q differing by 10⁴ (force proptoQ, inertia proptoQ, ratio independent of Q). Output: max trajectory difference 1.9×10⁻¹⁴.
Sim C — mass/G degeneracy. Integrate under (κ,Qₛ) and (κ/10⁶,10⁶Qₛ) (same product kappaQₛ=GM). Output: max trajectory difference 2.7×10⁻¹³.

Next stage: Chapter 4 (the Solar System as a consistency test), then Chapter 5 (axial spin and tidal locking). Foundational results continue to be imported from the physics volume, DOI \href{https://doi.org/10.5281/zenodo.17932566}{10.5281/zenodo.17932566}.