Squeeze any mass inside this radius and light itself cannot escape. The engine computes the horizon r_s = 2GM/c² live from real constants — the Sun's is about 2.95 km, Earth's about 8.87 mm. Rendered, not quoted.
source Schwarzschild, K. (1916), Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wiss. 7, 189–196 — ADS 1916SPAW.......189S
Outside a spherical mass M, the exact vacuum solution has a coordinate surface at
the event horizon. It is linear in mass: double M and the horizon doubles. At r_s the Newtonian escape velocity √(2GM/r) reaches exactly c — a coincidence GR makes exact.
The radius of no return. Schwarzschild 1916: r_s = 2GM/c², where the-escape-velocity reaches c — the surface the-event-horizon draws, and the well the-gravitational-time-dilation deepens without bound.
Time-dilation factor at r: √(1 − r_s/r) — clocks slow toward zero as r → r_s.
Live re-check of the coincidence that ties the horizon to escape velocity: v_esc(r_s) = c. Flips red if window 6 tampers with the law.
A mass M and a scale to compare. Drag to sweep from a mountain to a supermassive black hole.
Presets:
Ring = horizon at r_s. Inward paths (red) never climb back out; the outward path (green) launched at just under c still falls in from inside r_s.
The proven result for the live mass:
Linear in M, escape velocity = c at the surface, mean density ∝ 1/M².
wall r_s is a coordinate radius, not a measured tape-length — the proper radial distance to the horizon diverges. And r_s is NOT the physical size of a real star: the Sun's horizon (2.95 km) sits deep inside the Sun's actual 696,000 km body, where the vacuum solution does not apply. The horizon only exists once the mass is compressed within r_s.
"Nothing can ever cross the horizon — it takes infinite time to fall in."
→ Only for a distant observer's clock. The infalling object crosses r_s in finite proper time.
"A black hole is impossibly dense everywhere."
→ Mean density to form a horizon falls as 1/M². A supermassive hole's mean density can be below water's.
"r_s is where gravity becomes infinite."
→ Curvature at r_s is finite; the true singularity is at r=0. r_s is the one-way surface.
Drop the factor of 2 — use r_s = GM/c². The horizon halves and the escape-velocity coincidence breaks. Window 7 catches it live.