A black hole, forged on the knife edge.
Choptuik's 1993 result, running live on your device. A pulse of scalar field falls together under its own gravity. Tuned just right, it hovers on the exact boundary between dispersing back to flat space and collapsing to a black hole, and on that boundary Einstein's equations do something nobody expected: they echo, and the mass obeys a power law with a universal exponent. Nothing below is pre-rendered. Your phone is integrating the field equations.
One HTML file. An 800-point radial grid in the polar-areal gauge, SSP-RK3 in time, the two constraint ODEs re-solved fresh on every slice with Kreiss–Oliger dissipation for stability. Every line in the lab log is a genuine solve of Einstein's equations coupled to a massless scalar field, not a lookup. KaTeX is vendored locally, so the whole instrument runs with no network at all.
Two numbers fall out of the edge
The mass exponent
How big is the black hole?
Cross the threshold by a hair and the hole that forms has mass M ∝ |p − p*|^γ. The exponent is universal: change the pulse shape, change the field, and you land on the same 0.374. It is a critical exponent in the exact sense that boiling water has one.
The echo period
What does the edge look like?
Sitting exactly on the edge, the solution repeats itself as it shrinks, each copy about 31 times smaller and faster than the last, without end. Δ is the period of that repetition in log time. A fractal falling straight out of Einstein's equations, and secretly it fixes γ.
Three instruments on one solver
The same field solver drives all three panels. First you watch a single collapse by hand, then you let the machine bisect its way to the threshold, then you make it measure the exponent that lives there. Each step only needs the one before it.
The collapse
Pick an amplitude and press Run. The amber field φ(r) implodes toward the origin while the cyan 2m/r curve climbs underneath it. Below threshold the pulse passes through r = 0 and escapes to infinity; above it, 2m/r pins to 1 and a horizon snaps shut.
The hunt for p*
This is Choptuik's actual method: bisect on the amplitude. Each trial either disperses or forms a horizon, and the boundary between the two verdicts is the critical point. Twenty-six halvings pin it to eight digits, and every digit costs one full solve of the field equations.
The power law
Fire supercritical runs at log-spaced distances above p* and fit ln M against ln(p − p*). The slope is γ. On a uniform grid you get the ballpark, not four digits: the real measurement needed adaptive mesh refinement, which is exactly why the discovery took a supercomputer and a thesis.
Run it yourself
- 1Drag the amplitude low and hit Run. The pulse implodes through the origin and escapes, the status turns green, and flat space is all that remains.
- 2Push the amplitude high and run again. This time 2m/r climbs to 1, the status turns red, and a black hole has formed with a definite mass.
- 3Now let the machine hunt. Press Bisect to p* and watch the bracket halve, SUB against SUPER, until the two verdicts disagree only in the eighth decimal place.
- 4It parks the slider just above p* for you. Run that and watch the lower panel: φ(0,t) rings at the origin, each oscillation faster and smaller than the one before. That ringing is the first echo.
- 5Finally run the mass-scaling study and read γ off the fit line. Compare your slope to the cyan 0.374 reference. On a laptop grid, landing anywhere near it is the whole miracle.
Why a knife edge matters
The critical solution is a naked, self-similar region of spacetime where curvature climbs without bound. Classically it cascades through every scale on the ruler, from the size of a star all the way down toward the Planck length. It is the cleanest mathematical thread we have that runs continuously into the part of physics no theory has mapped, and whatever finally cuts the cascade off is, by definition, quantum gravity.
It is also the same mathematics as boiling water. γ is a critical exponent in the literal sense of phase transitions, the black-hole threshold is a second-order transition with the mass as its order parameter, and the echoes are a discrete version of the scale invariance that shows up at any critical point. The universe reuses the pattern, and here it is running on the metal.
It is one file and no build step. Read the source, change the pulse, and watch a different threshold assemble itself out of the same equations. More single-file instruments live back on the projects index.