/field studies  ·  [ fig. 16 ]  spacetime curvature

The funnel is
not the gravity

This is Flamm's paraboloid: a slice of space around a mass, bent into a third dimension so the stretched distances become visible. The dip is not a hill things roll down. It is a statement that a ruler laid near the mass measures more length than you would expect.

lawz(r) = 2√(rₛ(r − rₛ)) for the Schwarzschild exterior, with rₛ = 2GM/c². Clocks there run slow by √(1 − rₛ/r). tryDrag the probe ring in toward the shaded circle. noticeThe surface turns vertical at the horizon: radial distance diverges and the time factor falls to zero. Beyond it the embedding stops existing — which is the diagram's honest limit, not a rendering gap. in lifeGPS satellites gain about 38 microseconds a day from this; uncorrected, positions drift kilometres. found bySchwarzschild solved Einstein's equations in 1916 while serving at the front; Flamm drew the embedding that same year.
Time dilation at probe
event horizon — the throat of the funnel
probe — drag to read the local clock rate
drag empty space — orbit
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