Under the hood
The maths
Every lab runs real equations. Here is each one's model: the equations with every symbol defined, the constants and where they come from, what's simplified, where the model stops working, and worked examples.
The worked examples are also the app's tests. On every change, the test suite sets each lab to the example, reads the lab's own answer, and checks it against the value here — worked out independently, from the equation or from the published figure.
∑ Clock LabSpecial relativity for the light clock; the weak-field approximation of general relativity for clocks in circular orbit.∑ Spacetime diagramSpecial relativity in one space dimension: Lorentz transformations between inertial observers.∑ Expanding universeA smooth, uniform universe (the Friedmann equations) with matter, radiation and dark energy as a cosmological constant.∑ Black holes evaporateHawking's formulas for a non-rotating, uncharged black hole radiating as a black body, photons only.∑ Earth's energy budgetA one-layer greenhouse model: Earth absorbs sunlight and radiates as a black body, with one atmospheric layer absorbing a share ε of the outgoing infrared.∑ Mission clocksThe weak-field approximation of general relativity for clocks in circular orbit (the Moon base adds the Moon's own gravity).∑ Talking to MarsEarth and Mars on circular orbits around the Sun; radio travels at light speed.∑ The 1 g voyageSpecial relativity for a ship with constant proper acceleration (what the crew feels), accelerating to halfway and braking the rest.∑ The Field OceanThe Klein–Gordon equation (a relativistic field) on a lattice of points, integrated step by step.∑ Delayed choiceAn ideal Mach–Zehnder interferometer sent one photon at a time; whether the second splitter is in can be chosen after the photon passes the first.∑ The frozen universeThe Page–Wootters picture: a 12-reading clock entangled with a spin-½, in one stationary state of the whole.∑ Cosmic horizonsThe same smooth ΛCDM universe as the Expanding-universe lab, drawn in conformal time so light travels at 45°.∑ Boot a UniverseTegmark's map of space and time dimensions, as a checklist of published results; and one planet orbiting a star with gravity generalised to n space dimensions, integrated step by step (velocity Verlet).∑ WormholesThe complete (maximally extended) Schwarzschild black hole in Kruskal–Szekeres coordinates, where light travels at 45°.∑ Time loopsGödel's rotating universe (1949), an exact solution of Einstein's equations.∑ The RiverThe Gullstrand–Painlevé ('river') picture of a black hole: space flows inward at the Newtonian escape speed, and light moves at c through the flowing space.∑ Entropy box240 soft discs under Newton's laws, integrated in exact integer arithmetic so reversing every velocity retraces the past exactly.∑ Two FilmsTwo exact solutions of a wave equation with two time directions, utt + uss − uxx = 0, chosen to agree completely on the starting slice.∑ Cosmic timelineMostly published data: each event's date is taken from the literature, not computed here.∑ The Janus pointA swarm of 60 stars under Newtonian gravity (softened), started at the Janus point — its moment of least size — and integrated both forwards and backwards in time.∑ Stars forge the elementsMostly published data: which elements a star of each mass makes, how long it lives, and where each element in the Solar System came from.∑ FlatlandExact geometry: slices of spheres, cones and cubes by a lower-dimensional space, projections, and ray casting for A Square's view.