Physics

General Relativity

Mass changes the measure of time and of distance. Hang two clocks above a black hole, a neutron star or the Earth and the low one runs slow, so its light arrives stretched. Lay a ruler between two circles and it comes out longer than the circles allow: space itself is curved.

clockslightclock carried downruler

Drag a clock up or down. The bow-ties are light cones: light from that point can only go inside them, and near the mass they flatten, so light barely climbs.

Low clock, timed by the high clock—

—

Show

Mass

Lower radius r₁ 1.25 rₛ

Upper radius r₂ 6.00 rₛ

Numbers

Mass—

Schwarzschild radius r_s = 2GM/c²—

r₁—

r₂—

Clock rate at r₁, dτ/dt—

Clock rate at r₂, dτ/dt—

Low clock falls behind, per day—

Δr = (C₂ − C₁)/2π—

Ruler distance r₁ to r₂—

Radar echo, late against 2Δr/c—

What to observe

Above a black hole the low clock ticks at about half the rate of the high one. Every flash takes the same time to climb, so they arrive two high ticks apart: light from below is redshifted.
Carry a clock down, let it wait by the low clock and bring it back. It returns younger than the clock that stayed up, though it ticked normally the whole way.
In Space, a ruler laid between two circles needs more rungs than the gap their circumferences give. No flat sheet can do that: space is curved too, not just time.
On Earth, a clock at GPS height gains 45.7 µs a day on the ground from gravity alone. Its orbital speed takes 7.2 back; uncorrected, the net 38.5 µs would shift a GPS fix by 11 km a day.

Shortcuts:space run/pause · r reset · f fullscreen