A one-way membrane in spacetime. It sits at r = r_s = 2GM/c² — the radius where the escape velocity reaches c. Outside, every future path can still turn back out; at the surface, outgoing light freezes; inside, every future path leads to smaller r. Down the center, data flows: a mass and a radius go in, the metric decides, the verdict — can you leave? — comes out. The blue team builds and defends it; the red team tries to break it.
source K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsber. Preuss. Akad. Wiss. (1916) 189–196 — ADS 1916SPAW…189S; the horizon read as a one-way surface: D. Finkelstein, Phys. Rev. 110 (1958) 965. AMBER — historical print, no single stable open link. Rendered, not quoted.
The horizon is not a wall of matter; it is the radius where a purely kinematic condition flips:
F1 escape velocity v_esc(r) = √(2GM/r) equals c exactly at r_s = 2GM/c². F2 the radial outgoing light speed in Schwarzschild coordinates is c(1 − r_s/r): positive outside, zero at r_s, negative inside — the cone tips over. F3 the horizon area A = 4πr_s² scales as M² and, classically, never decreases.
For the current mass, live values at three radii:
| r / r_s | v_esc / c | outgoing light |
|---|
This sphere is the surface that lives at the length set by its neighbour: the-schwarzschild-radius gives the number r_s = 2GM/c² (the Sun's is ~2.95 km); the event horizon is the membrane that sits there.
And the freeze at the surface is exactly the-gravitational-time-dilation, dτ/dt = √(1 − r_s/r) → 0: a distant clock sees an infalling signal redshift without bound. Its area A ∝ M² never shrinks. Each sphere is the next one's premise.
The blue team's live check: recompute the horizon from the constants and confirm v_esc(r_s) = c, the one-way criterion across the boundary, and A ∝ M² non-decreasing. If red moves the surface, this badge is where it shows.
Two numbers fully fix the geometry of a non-rotating black hole: a mass M and how far out you stand, a radius r. From M alone the horizon radius is set: r_s = 2GM/c². Everything else — can you escape, how fast a distant clock sees yours tick, how much the signal reddens — is a function of the single ratio r / r_s.
| constant | value |
|---|---|
| G | 6.674×10⁻¹¹ m³·kg⁻¹·s⁻² |
| c | 299 792 458 m·s⁻¹ |
| M_sun | 1.98892×10³⁰ kg |
| AU | 1.4960×10¹¹ m |
Feed a mass and a radius into the panel below. The surface is not the input — it is derived from M, and that is the whole point.
Vertical = time (future up). Each cone is the local light cone; the dashed line is the horizon. Watch it tip inward as r → r_s. (cone drawing is schematic; the numbers are exact.)
Change mass or radius — every value is computed from r_s = 2GM/c² on the spot, never looked up.
What the machine proves, from the constants alone: the horizon sits at r_s = 2GM/c² (the Sun's is ~2.95 km); the Newtonian escape velocity equals c there and only there; outgoing light freezes at the surface and tips inward below it; the area A = 4πr_s² ∝ M² and merging two holes yields a larger area than either — it never shrinks. The current verdict is above; these invariants are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to move the surface.
And this is the uncharged, non-rotating idealisation. Real holes spin (Kerr): the horizon splits from the ergosphere and from the static limit, and r_s = 2GM/c² is no longer the surface. Add charge (Reissner–Nordström) and there can be two horizons. The clean sphere here is the simplest true case, not the general one.
"At the horizon gravity is so strong it crushes you." Cut. Tidal force at r_s scales as 1/M²; for a supermassive hole it is gentle at crossing. The horizon is a causal surface, not a force wall.
"Escape velocity = c is just Newtonian coincidence." Kept, corrected. The number matches Newton, but the meaning is GR: not "throw a rock hard enough" — no future-directed path, not even light, leads outward.
"A black hole's area can be reduced." Cut (classically). Hawking's area theorem: in classical GR the total horizon area never decreases. Only quantum evaporation shrinks it — outside this sphere's scope.
The red team's move: quietly relocate the horizon to r_s/2 — halfway inside the true surface. The blue team's witness (window 7) is watching.
At the false surface the escape velocity is c·√2, not c — the v_esc(horizon) = c criterion breaks. The witness recomputes, finds the surface no longer satisfies the metric, and turns red. Nothing is faked; the attack is real and it is caught.