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Wormholes: the maths

The complete (maximally extended) Schwarzschild black hole in Kruskal–Szekeres coordinates, where light travels at 45°. Exact.

Equations

U² − V² = (r/r_s − 1) e^r/r_s

Kruskal coordinates U (space-like) and V (time-like), related to the radius r.

(1 − r/r_s) e^r/r_s = V²

On the slice at time V, the throat of the bridge (U = 0) has this radius: widest (r = r_s) at V = 0, pinching to zero at V = ±1.

Constants

rs

2GM/c²

the horizon radius (units of the lab)

Worked examples ✓ checked on every change

Throat radius at V = 0	1.00 r_s	widest: one horizon radius
Throat radius at V = 0.91	0.495 r_s	solving (1 − r) eʳ = V²

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• An eternal black hole with a second universe: a real black hole formed by collapse has no bridge.
• The bridge picture on the right is schematic; the throat radius is exact.

Where it breaks

At the singularity (V² − U² = 1), and for real, collapsing black holes.

Sources: M. Kruskal, Phys. Rev. 119, 1743 (1960); G. Szekeres (1960); R. Fuller & J. Wheeler, Phys. Rev. 128, 919 (1962).