How this calculator works
The Schwarzschild lapse compares a stationary clock at radius r with an ideal clock at infinity. Dividing two lapse factors compares stationary clocks at different radii, either around one body or in separately modelled isolated fields with the same infinity-time convention. Radii are measured from the body centre, not from its surface. Both clocks must be outside their horizons and outside the material body, whose physical radius is not entered. Rotation, orbital motion and tides are omitted. A negative reference-minus-local difference is possible if the reference is deeper in the field. G uses CODATA 2018.
Formula
rs = 2GM/c²; d tau(r)/dt_infinity = sqrt(1-rs/r). For two stationary clocks: tau_local/tau_ref = sqrt[(1-rs_local/r_local)/(1-rs_ref/r_ref)].
Worked example
For a synthetic central mass with Schwarzschild radius 2 m and a stationary clock at radius 8/3 m, the local rate is one half the rate at infinity. Ten seconds at infinity correspond to five seconds locally.
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Frequently asked questions
Can I put the clock at the horizon?
No. A stationary worldline at or inside the Schwarzschild horizon is not described by this exterior clock formula.
Is radius the height above ground?
No. Add the body physical radius to an altitude before using this input.
Does an orbiting clock have this exact rate?
No. Orbital motion adds special-relativistic effects, and a rotating body requires a more complete metric.