Mass: a closed path hiding space, and what speed does to it

A photon is the smallest piece of light; this page follows one photon closed into a loop, the electron, and the space it hides. Light runs at c and hides the space behind it. Close its path into a loop and it hides a fixed amount of space every cycle; that hidden total is the inertial measure, mass. Seen from inside (panel 0) the loop is a path at c around a missing disk. Seen from outside (panels I to IV) the missing space cannot be seen at all: there is only a point, and every line of space that passes it leans in, the lines that would have crossed the missing disk meeting at the point and continuing. The electron sets the scale, ƛₑ = ħ/(mₑc) = 386.16 fm, the framework’s single calibration.

missing spacethe space the loop hides (A1)one photon, closed on itself, at c0 · the closed path, flat space, window ±2.5 ƛₑ (inside view)0.5 ƛₑ = 193 fm · loop radius ƛₑ, LΓ = 2πƛₑ

0 · the closed path, inside view. The electron: one photon on a closed path at c, hiding the space inside it. The grid is drawn straight; this view sets the object and the scale. S₀ = ħ on this loop is the single calibration; m = (σs/c²)·LΓ, LΓ = 2πƛₑ.

line at x₀ = 1.25 ƛₑclosest pass 290 fmRarc = 163 fm · κ = 6.1e-03 fm⁻¹line at x₀ = 2 ƛₑclosest pass 669 fmRarc = 2,007 fm · κ = 5.0e-04 fm⁻¹I · the merge point, window ±2.5 ƛₑ (outside view)0.5 ƛₑ = 193 fm

I · the merge point, outside view. The same electron seen from outside: the grid lines of space under the map. Lines that crossed the missing disk merge at the point and continue; passing lines bend inward. Two lines are annotated with their computed closest pass and their curvature there, where it is largest.

trace x₀ = 1.05 ƛₑclosest pass 124 fmRarc = 13 fmtrace x₀ = 1.01 ƛₑclosest pass 55 fmRarc = 1 fmtangent line x₀ = ƛₑ: touches the merge point and continuesII · vertex detail, window ±1.25 ƛₑ (2× closer)0.25 ƛₑ = 97 fm

II · vertex detail, 2× closer. Every line that crossed the missing disk passes through this one vertex: formerly parallel lines meeting, the geometry of an interaction. The tangent line touches and continues. Two dashed traces (computed, not part of the grid) show how fast the closest pass collapses just outside tangency.

line at x₀ = 1.5 ƛₑclosest pass 432 fmRarc = 540 fm · κ = 1.9e-03 fm⁻¹line at x₀ = 3 ƛₑclosest pass 1,092 fmRarc = 8,738 fm · κ = 1.1e-04 fm⁻¹III · zoom out ×2, window ±5 ƛₑ1 ƛₑ = 386 fm

III · zoom out ×2. Same law, half the magnification. The arc radii grow fast with distance: one grid step farther out bends on a circle several times larger.

line at x₀ = 25 ƛₑ, no visible bendVMS: δ = 1.600000e-03 · Rarc = 6.02 nmpure inverse square (ƛₑ/R)²: 1.602564e-03, gap 0.1603%line at x₀ = 112.5 ƛₑ, the farthest annotated lineVMS: δ = 7.901235e-05 · Rarc = 549.69 nmpure inverse square (ƛₑ/R)²: 7.901859e-05, gap 0.0079%mass centreIV · zoom out ×100, window ±125 ƛₑ (±48 pm)25 ƛₑ = 9.65 pm

IV · zoom out ×100. At this distance no line shows a visible bend; the numbers carry the physics. For each annotated line, the share of space missing at its closest pass, δ, is printed next to the pure inverse square (ƛₑ/R)² at the same seen radius. On the farthest line they agree to five decimals: from here outward a single particle’s missing space and the Newton form cannot be told apart, shown as arithmetic, not assertion.

Reading order. 0, the object itself in flat space, scale set. I, the object seen from outside, with measured bends. II, the vertex where every crossing line merges and continues. III, one zoom out, the bending dying as the square of the distance. IV, one hundred times out, nothing visibly bends and the printed arithmetic shows the missing-space profile and the pure inverse square agreeing to five decimals on the far line. No inverse-square law is used to draw any line; it appears in IV only as a printed number beside the computed value.

m ∝ ∮Γ Ad(s) ds,  m = (σs/c²)·LΓ
(the space a closed path hides per cycle is the inertial measure; for the electron LΓ = 2πƛₑ; Mathematical Bridge, section 3; Bridge Math Appendix, Inertial Mass)
δ(R) = ƛe²/(R² + ƛe²) → ƛe²/R²
(the share of space missing at seen radius R: 1 at the point, the inverse square far away; the drawing’s map of the Bridge’s “fixing the missing volume” and “Δg(r) ~ 1/r²”)

What speed does: the same loop, seen moving

Nothing about the loop changes and the light still runs at c. Seen from a laboratory it is moving past, its cycle takes γ times longer, the photon’s path opens into the curve drawn, and the loop’s outline is squeezed to R/γ along the motion. In the loop’s own frame the path is the closed circle and head meets tail at the same place every cycle; from the laboratory the point where head meets tail is itself moving, so the path never rejoins where it started (the small pinch marks). The mass seen is γ times the rest mass. In the textbook’s letters this is E = γmc², and the first two terms of that are the rest energy mc² and the kinetic energy ½mv².

head meets tail here, every cycle1 · v = 0c, γ = 1.000mass seen: 1.000 mone loop period; the loop advances 2πβγR = 0.00 Routline at one instant (dashed): the circle of radius RR = ƛₑ

1 · at rest. The photon’s path is the circle; the loop hides one disk of space per cycle. Mass seen: m.

the closure point, where head meets tail, moves with the loop: 2πβγR = 3.63 R per cycle2 · v = 0.5c, γ = 1.155mass seen: 1.155 mone loop period; the loop advances 2πβγR = 3.63 Routline at one instant (dashed): R/γ = 0.866 R along the motion, R across itR = ƛₑ

2 · v = 0.5c, γ = 1.155. The path over one loop period, the loop advancing 2πβγ = 3.63 loop radii per cycle; the outline at one instant is contracted to R/γ along the motion. Mass seen: 1.155 m.

the closure point, where head meets tail, moves with the loop: 2πβγR = 12.97 R per cycle3 · v = 0.9c, γ = 2.294mass seen: 2.294 mone loop period; the loop advances 2πβγR = 12.97 Routline at one instant (dashed): R/γ = 0.436 R along the motion, R across itR = ƛₑ

3 · v = 0.9c, γ = 2.294. The loop advances 12.97 loop radii per cycle and its outline is squeezed to 0.436 R. Mass seen: 2.294 m. The loop can never reach c: a loop at v = c could not close.

E = γmc² ≈ mc² + ½mv² + ⅜mv⁴/c² + …,  γ = 1/√(1 − v²/c²)
(mass seen γm; the first term is the rest energy E = mc², the second the textbook’s kinetic energy ½mv²; Mechanics Math Appendix, section 3, where these relativistic relations are stated and the Newtonian limit recovered)
p = γmv,  E² = p²c² + m²c⁴
(the same section; a loop cannot exceed c because a loop at v ≥ c would fail to close, Bridge Math Appendix, Deflection, Elongation, and Observed Mass Increase)
Mathematical Bridge, section 3 (m ∝ ∮Ad ds; the missing volume; Δg ~ 1/r²), Bridge Narrative, step 3 (mass as missing space), Bridge Math Appendix, Mass and Gravity, sections 1 to 4 and 6A, Mechanics Math Appendix, section 3 (E = γmc²), Particle Mechanics Math Appendix (S₀ = ħ, ƛₑ).

Every line computed. Panels 0 to IV: the disk of area πƛₑ² is the space the loop hides (axiom A1); from outside it is not seen, and every point of space at true distance r from the loop centre is seen at R(r) = √(r² − ƛₑ²), points inside the disk all at the centre point. This map is the drawing’s construction of two lines of the Mathematical Bridge, section 3: the loop “removes a volume of available transverse area from the surrounding space, fixing the missing volume”, and “this deficit propagates outward with profile Δg(r) ~ 1/r²”; it removes exactly πƛₑ² from every circle around the point, and its missing share δ = ƛₑ²/r² is exactly the inverse square. Every grid line is the image of a straight line under that map; lines that crossed the disk are drawn through the point and out the other side. Annotated values: the closest pass of a line is its seen radius √(x₀² − ƛₑ²); the curvature κ is evaluated where the line crosses the centre line, its maximum; Rarc = 1/κ. Panel IV prints δ at each line’s true offset x₀ beside (ƛₑ/R)² at its seen radius. Speed panels: the photon’s path is the standard relativistic transformation of the rest-frame circle (Mechanics Math Appendix, section 3): x = γR cos θ + βγRθ, y = R sin θ, its speed c at every point (axiom A2), the loop period γ times longer, the outline contracted to R/γ along the motion; γ values and the mass ratio computed, nothing else drawn.