Worldline

Worldline is a browser sandbox for relativistic many-body physics. You spawn charged, spinning particles, choose the laws they obey, and watch orbits precess, charges radiate, black holes evaporate, and scalar fields ripple, then tilt the plane into spacetime to see the world lines they trace, with live conservation readouts to check the numbers.

Open it · part of a9l.im

What's in it

The simulation is two-dimensional, in natural units (c = G = ħ = 1). Particles carry proper velocity, so nothing outruns light, and a Boris integrator with adaptive substepping advances them.

• Forces. Newtonian gravity, Coulomb, magnetic dipole and Lorentz forces, gravitomagnetism (frame dragging), tidal locking, Yukawa (a screened nuclear force with a massive pion mediator), spin-curvature coupling, and uniform external gravitational, electric, and magnetic fields. Forces are summed pairwise, and by a Barnes–Hut tree, chosen automatically, once an open Newtonian scene has enough bodies.
• Relativity. Signal delay: forces come from where sources were when their signal left, read off each body's recorded world line. Charges interact through their exact Liénard–Wiechert fields; gravity and the other forces use their static laws at the delayed position with a scalar aberration factor, a toy model of finite-speed propagation. First post-Newtonian corrections (Einstein–Infeld–Hoffmann for gravity, Bažański cross terms, scalar Breit for Yukawa) give the relativistic perihelion advance.
• Spacetime and frames. The plane tilts into spacetime, where every body's world line climbs to the present, beaded by its proper time, with the selected body's past light cone below it. The view can be redrawn for a moving observer, the zero-momentum frame, the selected body's rest frame, a rotating frame, or free fall, by slicing every recorded world line on the observer's surface of simultaneity; the lab's rulers contract with the observer.
• Black holes. Kerr–Newman horizons set by mass, spin, and charge (super-extremal inputs are clamped to an effective radius), a Paczyński–Wiita gravity well that swallows bodies crossing the horizon, ergosphere rendering, Hawking evaporation, Schwinger discharge that sheds charge one lepton at a time, extremal charge shedding, and axion superradiance that spins a hole down until it saturates.
• Radiation and bosons. Larmor radiation with Landau–Lifshitz reaction, electromagnetic and gravitational-wave quadrupole losses, and pion emission. Photons, pions, and leptons are real particles: they carry energy and momentum, other bodies absorb them, and pions decay. With boson interaction on they are lensed by masses, gravitate and repel among themselves, annihilate in opposite-charge pairs, and collapse into a new particle when enough energy gathers in one place (a kugelblitz). Matter meeting antimatter annihilates into photon pairs.
• Scalar fields. A Higgs field with a Mexican-hat potential that sets particle masses, and an axion field that modulates the electromagnetic coupling and, with Yukawa on, couples with opposite sign to matter and antimatter (a Peccei–Quinn toy). Both evolve on a grid and are excited by collisions. With gravity on they evolve in a curved metric sourced by their own energy and the particles' mass, so field oscillations slow near heavy bodies.
• Proper time. Every body carries a clock, slowed by speed and by gravity, and pion decay, Hawking evaporation, Schwinger discharge, superradiance, and mass transfer run on it: a fast pion beam lives longer in the lab, and a moving black hole takes longer to evaporate. Every body's outline is its clock, swept heavier once per 16 units of its proper time.
• The spoke. Each body is drawn through the Lorentz transformation of its motion, with one spoke marking its orientation: moving bodies contract, a spinning, moving wheel's spoke bends (relativity of simultaneity), and the spoke precesses on orbits (Thomas, geodetic, and Lense–Thirring precession).
• Charge quantization. Every charge is a multiple of 0.1, and emission, absorption, decay, and discharge move charge in whole units.
• Space. Open edges, walls, or the plane glued into a torus, a Klein bottle, or the real projective plane, each chosen by its identification diagram.
• Lensing. With gravity on, the rulers are a backdrop seen through the bodies as a thin gravitational lens, with Einstein rings, and a backdrop star whose magnification traces a microlensing light curve.
• Cosmology. With expansion on, the periodic domain is a slice of an expanding FRW universe: an imposed history or one set by the contents through the Friedmann equation, redshift, Hubble friction on the fields, the Hubble radius and event horizon, and an energy ledger for what the expansion takes. Presets for axion misalignment and Kibble–Zurek domain walls.
• Analysis. Energy, momentum, and angular-momentum totals with their drift over the run; the selected body's clock, speed and the forces on it; the fields drawn as a textbook figure prints them (contours, field lines, ⊙ and ⊗); and a phase-space plot and an effective-potential curve for the selected body against the heaviest other.
• Figures. Nine curated figures in five chapters: the twin paradox, a relativistic wheel and the Rindler horizon; aberration and apparent superluminal motion; cyclotron drift and bremsstrahlung; microlensing; and Kibble–Zurek walls. Each caption names the figure with its live values, and its dotted numbers are knobs you drag.
• Save and load. Quick save to the browser, or download and upload the state as JSON (domain, particles with their clocks, laws, settings, and camera). Each figure sets its own domain size; the window is a camera onto it.
• Reference. Hovering a law's formula, or a section's ?, opens a short explanation; Shift+click opens a longer reference page with the equations. The interface is in English and Japanese.

Controls

• Mouse. Click to spawn a particle; drag to launch it. Right-click spawns antimatter. Clicking a particle of the button's kind (left for matter, right for antimatter) selects it; clicking one of the other kind deletes it. Scroll to zoom, middle-drag to pan, hover for a readout.
• Touch. Tap to spawn, drag to launch, pinch with two fingers to zoom and pan. The antimatter button switches taps to antimatter.
• Keyboard. Space pauses, / steps, R resets, , and . change speed, 1–6 load the figures, X toggles antimatter mode, T the theme, S the margin, W tilts into spacetime. Ctrl+S / Ctrl+L quick save and load. B cycles observers, L returns to the lab, O switches coordinates, and G the rulers and key. ? opens the about panel with the full list.

Running locally

The project has no build step of its own; it loads the shared design system from the parent site by absolute path. From a checkout of a9lim.github.io with submodules:

npm install && npm run build
python3 -m http.server --directory dist 8000   # http://localhost:8000/worldline/

Worldline needs WebGPU. tests/physics-validation.html checks the physics against analytic results in the browser.

How it's built

Vanilla JavaScript modules on WebGPU. The physics runs as WGSL compute shaders (up to 4096 bodies, with trees built from sorted Morton codes and 128 × 128 field grids) and draws with instanced rendering. Shared controls, theme, and typography come from the parent site, and KaTeX renders the reference pages. AGENTS.md maps the code.

License

AGPL-3.0