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Gravity slows clocks

Space and time · level 8, lesson 2 of 6

A question to start

Two identical atomic clocks sit on a table, one 33 cm higher than the other. Do they keep the same time?

The idea

Use the equivalence principle. Picture a rocket accelerating upwards, with a light source on the floor and a detector on the ceiling. While the light travels up, the ceiling speeds up away from it, so the light arrives at a slightly lower frequency. By the equivalence principle, the same must happen in gravity: light climbing up loses frequency.

Clocks are counters of frequency. So a lower clock runs slow compared with a higher one. Near a surface with gravity \(g\), a height difference \(h\) changes the rate by

\[\frac{\Delta f}{f} \approx \frac{g\,h}{c^2}.\]

In 1960 Robert Pound and Glen Rebka sent gamma rays down, and up, a 74-foot (22.6 m) tower at Harvard and found the shift Einstein predicted, about \(2.5 \times 10^{-15}\). In 2010, NIST compared two aluminium-ion clocks just 33 cm apart in height, and saw the higher one run fast.

A clock 1 km higher gains, each day Mars Earth
Using \(g h / c^2\) about 3.6 ns about 9.4 ns
runs a little fasterruns a little slowerlight climbing:stretched

Worked examples

1. Pound and Rebka. \(\frac{9.8 \times 22.6}{(3 \times 10^8)^2} \approx 2.5 \times 10^{-15}\).

2. The 33 cm clocks. \(\frac{9.8 \times 0.33}{9 \times 10^{16}} \approx 3.6 \times 10^{-17}\). Over a 79-year life, about \(2.5 \times 10^9\) s, that adds up to about 90 billionths of a second.

3. On Mars. 1 km higher: \(\frac{3.73 \times 1{,}000}{8.99 \times 10^{16}} \approx 4.1 \times 10^{-14}\). Times 86,400 s: about 3.6 ns a day.

Try it

  1. Which runs faster: a clock in the attic or one in the cellar?
  2. Why does light climbing out of gravity lose frequency?
  3. Find the daily gain of a clock 2 km higher on Earth.
  4. Why is the effect smaller on Mars for the same height?
Answers
  1. The one in the attic.
  2. By the equivalence principle it acts like a detector speeding away from the light, so the light arrives at a lower frequency.
  3. \(2 \times 9.4 \approx 19\) ns a day.
  4. Mars's surface gravity is weaker, so \(g h\) is smaller.

Common mistake

"The clock's mechanism is affected by weight." Every clock, of any kind, runs slower lower down. It is time itself that passes at a different rate.

Where it leads