Gravity: falling without a force, and who agrees where

A photon is the smallest piece of light; this page follows one photon closed into a loop, standing in the missing-space profile of another loop, a mass. Nothing pulls it. Around a mass the lines of space lean in, and the space itself is not the same on the loop’s two sides: on the near side more of it is already missing. The loop’s photon still runs at c, but its route “leans slightly toward the region with more space already hidden and pays a tiny time premium to get past” (Bridge Narrative, step 4). That lean, every cycle, is gravity: “no pushes or pulls; cleaner route wins”. Far away and slow, it is Newton’s law exactly; the four classical tests of general relativity come out of the same route picture with one factor set once.

1. Why a loop falls

near side, at r = 2 ƛₑ from the mass: δ = 25.0% of space missing; each step here costs twice the far side’s premiumback side, at r = 4 ƛₑ: δ = 6.25% missing; over the whole far half the premium is 0.65 of the near half’sthe loop: one photon at c, radius ƛₑ, centre at r = 3 ƛₑ; drawn where it is seenthe massthe lean: routes turn toward higher n (ray equation); ∇n here points toward the mass

One loop of light near a mass, drawn on the same mapped space as the Mass page, and drawn where it is seen: through the mass’s map its circle appears elongated toward the mass (Bridge Math Appendix, 6A: “its apparent circle elongates toward the curvature source”). The photon runs at c the whole way round (axiom A2). The space under its two halves is not the same: at the near side, r = 2 ƛₑ from the mass, 25% of the space is missing; at the back, r = 4 ƛₑ, only 6.25%. In the Walk-Through’s cost map, n(x) = 1 − 2Φ/c² with Φ = −GM/r, a step near the mass costs more (“higher n means this step hides a bit more display-area than average”): the near edge’s premium is twice the far edge’s, and summed round the two halves the near half pays 1.53 times what the far half pays (computed, ∫(n − 1) ds; G cancels in the ratio). The colour runs by that cost per step: green where each step hides the most space, yellow where it hides the least. Routes turn toward higher n, that is the ray equation d/ds(n t) = ∇n, and ∇n at the loop points toward the mass; so every cycle the loop drifts that way. Only the size of the drift rests on G, an acceptance lock. As a seven-year-old put it: just like a Beyblade in a curved bowl, it wants to go toward the centre.

δ(r) = ƛe²/r²,  n(x) = 1 − 2Φ/c²,  dθ/ds ≈ ∇⊥ ln n
(the share of space missing at true distance r, the map; the Walk-Through’s cost map n, “higher n means this step hides a bit more display-area than average”, and the lean per step, its sideways slope; Mechanics Math Walk-Through, conventions and section 1)
a = −∇Φ = −GM/r²  (Φ = −GM/r)
(the cycle-averaged drift of the loop’s centre: the Newtonian limit, Bridge Math Appendix, Mass and Gravity 7)

2. Falling: same time between pictures

12345δ = 0.00592δ = 0.00444picture 1δ = 0.0969δ = 0.0368picture 5the mass: missing space, seen from outside as a pointequal-time snapshots of infall from rest under a = −GM/r² (the Newtonian limit the Bridge recovers); GM a visibility choice1 ƛₑ = 386 fm

Five equal-time snapshots of one loop released from rest, drawn where each is seen. The small arrows under pictures 1 and 5 print the share of space missing at the loop’s near and far edge: at picture 1, δ = 0.00592 against 0.00444; by picture 5, 0.0969 against 0.0368. Both arrows of a pair point toward the mass, since gravity has no handedness to flip a side, only depth. The near side is always deeper, and the pair grows as the loop falls. Each loop is painted by the cost map along its rim, n − 1 ∝ 1/r at each rim point’s distance from the mass: darkest green where each step hides the most space (the near side, drawn heavier), darkest yellow where it hides the least (the far side), pale between. The loop always swings toward its dark green.

3. How one electron sees another

1r = 26 ƛₑ · v = 0cγ = 1.000 · mass seen 1.000 mₑa = Gmₑ/r² = 6.03e-19 m/s²2r = 20.5 ƛₑ · v = 0.3cγ = 1.048 · mass seen 1.048 mₑa = Gmₑ/r² = 9.70e-19 m/s²3r = 15.5 ƛₑ · v = 0.5cγ = 1.155 · mass seen 1.155 mₑa = Gmₑ/r² = 1.70e-18 m/s²4r = 10.5 ƛₑ · v = 0.7cγ = 1.400 · mass seen 1.400 mₑa = Gmₑ/r² = 3.70e-18 m/s²5r = 6 ƛₑ · v = 0.87cγ = 2.028 · mass seen 2.028 mₑa = Gmₑ/r² = 1.13e-17 m/s²an electron: a point hiding πƛₑ² of spacefive separate situations, not one fall; the second electron is drawn at its own scale, its outline contracted to R/γ along the motion, and drawn where it is seen through the source’s map

The point on the right is an electron; its missing space is what bends the grid. The five shapes are a second electron at five distances and five speeds, five separate situations. At rest (1) it hides exactly the same space as the first, πƛₑ² each. Moving, its outline is contracted to R/γ along the motion and the mass seen is γmₑ (the Mass page). Under each, the acceleration between them in the Newtonian limit, a = Gmₑ/r²: about 10⁻¹⁸ m/s², which is why gravity between two electrons is never felt, yet the geometry drawn here is the same machinery that runs planets.

4. Who agrees where

Three descriptions, one ladder. Far and slow, everyone tells the same story. Where light bends, is delayed, and clocks drift, Newton falls behind and VMS lands on the tested numbers with general relativity, from one route picture and one factor set once. At the core of a single particle GR runs to infinity where VMS stops, because you cannot hide more space than there is.

1234A · far and slow: everyone agreesfall from rest, far from the mass; positions at equal ticks of a faraway clock:Newton, a = −GM/r²: x₄ = 9.335880 · GR, geodesic from rest: the same to O(r_s/r), 2.5 × 10⁻⁴⁶ for an electron sourceVMS: the loop deficit ∝ 1/r² → ∇²Φ = 4πGρ → a = −GM/r², with m = (σₛ/c²)L_Γ (Math Appendix 7)
B · the four tests: Newton falls behind, VMS and GR agreeeffectNewtonGR and VMSlight bending at the Sun’s limbnone *1.75 arcsec4GM/(Rc²); Mechanics Appendix 5Shapiro delay, Earth to Mars at conjunctionnone123.6 μs(2GM/c³) ln(4r₁r₂/b²); Appendix 6clock drift over a 22.5 m tower (Pound–Rebka)none2.455 × 10⁻¹⁵Δν/ν = gΔh/c²; Appendix 7Mercury’s perihelion advance, per century0 from the Sun43.0 arcsec6πGM/(a(1 − e²)c²); Appendix 8VMS: the same kernels for all four, one factor η₀ = 2 set once by the deflection (MechanicsWalk-Through 1 to 4). * a Newtonian corpuscle would bend half the GR value (textbook).
C · deep: light that circles, light that is capturedr_ph = 3GM/c²: closed null routes, the photon sphere (dashed)b_c = 3√3 GM/c²: the smallest impact parameter that still escapesr_s = 2GM/c²r_sFor a steep profile n(r) the same bending kernel α = ∫∇⊥ln n ds diverges as b → b_c⁺and captures for b < b_c; in the GR dictionary that is the photon sphere and the criticalimpact parameter above (Mechanics Walk-Through 1A). Newton’s gravity has no such limit;a Newtonian mass captures nothing that moves at c.
D · the core of one particle: GR and VMS part ways01 ƛₑ2 ƛₑ3 ƛₑ4 ƛₑ01VMS: δ = ƛₑ²/(R² + ƛₑ²), caps at 1 at the loopa point mass, (ƛₑ/R)² → ∞ at the centreA point mass runs off to infinity; the VMS deficit caps at 1 at the loop, since the missingspace is all the space there is to miss. For one electron GR’s strong field would begin atr_s = 2Gmₑ/c² = 1.4e-57 m; the VMS core at ƛₑ = 3.86e-13 m comes 3e+44 times sooner.

5. The far field: the missing-space profile meets the Newton form

10.386 pm103.86 pm10038.6 pm10⁰10⁻110⁻210⁻310⁻4R = 2 ƛₑ: gap 0.25R = 5 ƛₑ: gap 0.04R = 10 ƛₑ: gap 0.01R = 30 ƛₑ: gap 0.0011R / ƛₑ (log); real distance belowδ (log)VMS δ(R) = ƛₑ²/(R² + ƛₑ²), solid(ƛₑ/R)², the Newton/GR point-mass form, dashed

Solid: δ(R) = ƛₑ²/(R² + ƛₑ²), the quantity every line above is drawn from. Dashed: the pure inverse square (ƛₑ/R)², the point-mass form Newton’s gravity and weak-field GR require, normalised in the far field. Near the core they disagree, VMS keeps the scale ƛₑ, a point mass has none. The gap closes as the square of the distance: by 10 ƛₑ it is 1%; by the Bohr radius, 5 × 10⁻³ %. For a single particle the two are indistinguishable from there outward.

F = G mΓ mΓ′ / r²
(two loops, the probe’s route computed in the geometry the source shapes; the weak-field, large-distance limit; Mathematical Bridge, section 3, step 6)
α = 4GM/(bc²),  Δt = (2GM/c³) ln(4r₁r₂/b²),  Δν/ν = gΔh/c²,  Δω = 6πGM/(a(1 − e²)c²)
(bending, delay, clock drift, perihelion advance: the same kernels on the cost map n(x), |η₀| = 2 set once by the deflection; Mechanics Math Walk-Through 1 to 4, Math Appendix 5 to 8)
F = GMm/r²,  g = GM/r²,  U = −GMm/r
(the same line in the textbook’s letters: Newton’s law of gravitation, the field g, the potential energy; G is an acceptance lock, κ = 8πG/c⁴ “fixed by comparison”, not an input)
Mathematical Bridge, section 3, Bridge Narrative, step 4, Mechanics Narrative, Mechanics Math Walk-Through (the cost map, the ray equation, the three kernels, photon sphere), Mechanics Math Appendix (sections 4 to 8, the worked numbers), Bridge Math Appendix, Mass and Gravity (sections 5 to 7 and 6A).

Every line computed. Grids: the map R(r) = √(r² − ƛₑ²) about the mass, the Mass page’s construction of the Bridge’s fixed missing volume and Δg ~ 1/r². Loops: radius ƛₑ, their rims pushed through the same map, so each is drawn where it is seen, elongated toward the mass. Positions in sections 2 and 4A: equal-time snapshots of infall from rest under a = −GM/r² (GM a visibility choice; the spacing pattern is scale free). Gauges: δ = ƛₑ²/r² at each edge’s true distance, arrow lengths √δ on one common scale (declared for visibility), values printed exact. Hue: the Walk-Through’s cost per step along the rim, n − 1 = 2GM/(rc²) ∝ 1/r at each rim point’s true distance, normalised per loop; the near-to-far premium ratio printed in section 1 is ∫(n − 1) ds over the two halves, G cancelling. The lean arrow in section 1 is the direction of ∇n at the loop centre, computed; its length is drawn. Section 3: outlines contracted to R/γ along the motion (Mechanics Math Appendix, section 3), then mapped; accelerations Gmₑ/r² with CODATA G, mₑ, ƛₑ. Section 4B: the four numbers are the Mechanics Math Appendix’s own worked examples (1.75 arcsec, 123.6 μs, 2.455 × 10⁻¹⁵, 43.0 arcsec per century). 4C: rph = 3GM/c² and bc = 3√3 GM/c² from Walk-Through 1A, drawn to scale in units of GM/c². 4D: δ(R) from the map against (ƛₑ/R)²; rs = 2Gmₑ/c² computed. The probe’s own dent in space is omitted for clarity. “VMS lands with GR” means: through the Bridge’s coupling Gμν = κTμν and the Walk-Through’s matched kernels, as published.