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∑ The maths › The River · open the lab →

The River: the maths

The Gullstrand–Painlevé ('river') picture of a black hole: space flows inward at the Newtonian escape speed, and light moves at c through the flowing space. Exact for a non-rotating hole.

Equations

vflow = c √(rs / r)
The inward speed of space at radius r: light speed exactly at the horizon.
dr/dt = ±c − vflow
Light's speed over the ground, outward (+) or inward (−).
rate of a clock held still = √(1 − rs/r)
How fast a clock hovering at radius r ticks compared with a distant one.

Constants

rs2GM/c²the horizon radius (units of the lab)

Worked examples ✓ checked on every change

Flow speed at 4 horizon radii0.50 c√(1/4)
A held-still clock at 4 horizon radii86.6%√(1 − 1/4)

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

  • Top view of a slice; the flow is drawn at a playback speed.
  • Non-rotating, uncharged hole.

Where it breaks

Inside the horizon the picture holds right to the centre, where the singularity lies — there, general relativity itself gives out.

Sources: A. Hamilton & J. Lisle, Am. J. Phys. 76, 519 (2008); P. Painlevé (1921); A. Gullstrand (1922).

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