Why does everything fall alike?

1 · It explains gravity with gravity

The heavy ball only presses the dent in because the Earth pulls it down, and the small ball only rolls in for the same reason. Take the Earth away from under the sheet and see what is left of the model.

2 · What really pulls: curved time

In Einstein’s theory, clocks close to a mass run more slowly. On Earth the effect is tiny: a clock one metre higher runs faster by 1.1 × 10⁻¹⁶, about 3.4 billionths of a second per year. In 2010, atomic clocks measured it for a height difference of just 33 centimetres.

Animation: 100 m of height, the difference in rate exaggerated about five trillion times.

So over 80 years the head of a person 1.70 m tall ages about half a millionth of a second more than their feet.

3 · The path through spacetime

Throw a ball straight up: after 2 seconds it is back down. In a picture of height against time its path is an arch. According to Einstein, a body flying freely between throw and landing takes the path on which its own clock shows the most time. Clocks run faster higher up, so it pays to fly upward. But going too high means flying faster, and fast clocks run slower. Try out different arches.

Too low: going a little higher would gain the clock more than the faster flight costs.

For slow bodies – balls, planets, you – gravity comes almost entirely from curved time. The curvature of space, which the sheet shows, is much weaker for them – it only shows up in subtle effects such as Mercury’s 43″. Light, however, feels both in equal parts. That is why Einstein’s deflection of light (1.75″) is twice Newton’s (0.87″).

For the curious: the calculation

• Rate difference of two clocks whose heights differ by h: Δf/f ≈ g·h/c² = 9.81 m/s² · 1 m / (2.998 · 10⁸ m/s)² ≈ 1.09 · 10⁻¹⁶; over a year (3.16 · 10⁷ s) that is 3.4 · 10⁻⁹ s.
• Time on a clock along a path h(t), for slow motion in a weak field: τ ≈ ∫ (1 + g·h/c² − v²/(2c²)) dt.
• For arches h(t) = k·t·(T − t) with T = 2 s, the gain over lying still is Δτ = T³/6 · (g·k − k²)/c². It is largest for k = g/2: exactly a throw that falls with g, peak height g·T²/8 = 4.9 m, gain 3.6 · 10⁻¹⁶ s.
• Light deflection: the warping of time alone gives 0.87″, the warping of space adds the same amount again, 1.75″ in total.