Numerical Relativity · 1+1D

Choptuik Critical Collapse Lab

A live integration of Einstein's equations coupled to a massless scalar field, in spherical symmetry, the exact system where Choptuik found the echoes in 1993. Nothing below is pre-rendered: your device is solving the field equations.

γ ≈ 0.374 · mass exponent
Δ ≈ 3.4453 · echo period
eΔ ≈ 31.3 · shrink factor
The Equations01

Geometry · polar-areal gauge

Two metric functions, both fixed on each time slice by constraints. No gravitational waves in spherical symmetry: geometry is a slave to the matter.

Matter · massless scalar field

First-order variables: Φ is the field's slope, Π its momentum.

Evolution · a wave equation on curved spacetime

Constraints · geometry responds instantly

Radial ODEs solved fresh on every slice. The Φ²+Π² source is the field's energy; note it appears in both: energy curves space and warps time.

Diagnostics

The mass inside radius r, and the horizon signal: a black hole is forming where 2m/r → 1.

Choptuik's discoveries

Tune any parameter p of the initial data to the threshold p*. Just above it, the black hole mass obeys a power law with a universal exponent:
And the critical solution echoes, discretely self-similar. In log time τ and scale-free coordinate x, it repeats exactly, each echo 31× smaller and faster:
The two numbers are linked: the mass law carries a tiny periodic wiggle with period Δ/(2γ) ≈ 4.61 in ln|p−p*|, the echoes' fingerprint on the scaling law.
Live Collapse02
field energy rings = equal steps of real distance: they crowd where space stretches darkening = time slowing (lapse) horizon
the well: embedding cross-section of space itself, computed live from the metric: depth = ∫√(a²−1) dr
φ(r) field 2m/r horizon signal 2m/r = 1
φ(0,t), the field at the origin. Near-critical runs oscillate here: that's the first echo.
0.0708
Ready. Grid: 800 points, polar-areal gauge, RK3, CFL 0.4. Pick an amplitude and run, or let the bisector hunt for p* below.
Hunt for p*03

This is Choptuik's actual method: bisect on the amplitude. Each run either disperses or forms a horizon; the boundary between them is the critical point. Every line in the log is a full solve of the field equations.

p* (this grid)
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bracket width
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— lab log idle —
Phase Space05

The deepest view of criticality: plot the field at the origin against its velocity. Every run traces a trajectory in this plane, and the critical solution is a saddle, an orbit with exactly one unstable direction. Near-critical runs shadow it, loop with it, then get thrown off: green to dispersal, red to collapse. The exponent from the power law is hiding here too: γ = 1/λ, the inverse growth rate of the saddle's single unstable mode.

Launches eight solves straddling p* at |p−p*|/p* = 3×10⁻² … 3×10⁻⁵. The tighter the pair, the longer they travel together before the unstable mode separates them: that shared stretch is the critical solution. This is the renormalization-group picture in miniature: the critical solution is a fixed point of the zoom, its echoes a limit cycle, and γ a critical exponent in the same sense as boiling water. If p* hasn't been found yet, a quick bisection runs first.
What You're Seeing06

The knife edge. Below p*, gravity loses: the pulse implodes through the origin and re-explodes to infinity, leaving flat space. Above p*, gravity wins: 2m/r pins to 1 and a horizon forms. Exactly at p* sits the critical solution, a universal attractor that has forgotten everything about your initial data.

The echoes. Watch φ(0,t) on a near-critical run: the field rings at the origin, each oscillation ~31× faster and smaller than the last. In the variables (x, τ) that's an exact period-Δ repetition: a fractal falling straight out of Einstein's equations. A uniform grid resolves the first echo or two; Choptuik's adaptive code marched through dozens, which is precisely why the discovery needed him.

Why it matters for the blank zone. The critical solution is a naked, self-similar spacetime region where curvature climbs without bound: classically it cascades through every scale on the ruler, from the star all the way down toward Planck. It's the cleanest mathematical thread we have that runs continuously into the unmapped basement. Whatever cuts the cascade off is, by definition, quantum gravity.

Same math as boiling water. γ is a critical exponent in the exact sense of phase transitions; the black-hole threshold is a second-order transition with M as the order parameter, and the echoes are a discrete version of the scale-invariance at a critical point. The universe reuses the pattern.