lunartime.org

Why clocks run faster on the Moon

Key facts

  • A clock at rest on the Moon's selenoid gains 56.02 µs per day on an identical clock at rest on Earth's geoid. [1]
  • The Moon's surface gravity is about one sixth of Earth's, so its gravitational well is much shallower. [2]Official
  • At that rate the lunar clock gains a full second on the Earth clock about every 48.9 years, in our model. [3]
  • The figure is the sum of two opposing terms, one from gravity and one from motion, not a single effect. [1]

Clocks on the Moon do run faster than clocks on Earth. The reason is not that lunar hardware is different. It is that general relativity makes the rate of a clock depend on where the clock sits and how it moves.

Two effects act at once, and they pull in opposite directions.

The short answer

A clock at rest on the Moon’s selenoid — the lunar equivalent of sea level — ticks faster than an identical clock at rest on Earth’s geoid. The published secular rate is 56.02 µs per day [1]. NIST gives the same result rounded, about 56 microseconds per day [4]Official.

Note what is being compared: two clocks, both standing still on a surface, one on each body. Change either clock and the number changes. That is the whole subject of the two rate figures in circulation.

Effect one: gravitational time dilation

A clock deep in a gravitational well runs slow compared with a clock higher up. Earth’s well is the deeper of the two: the Moon’s surface gravity is about one sixth of Earth’s [2]Official.

So the Earth clock is the slow one, and the lunar clock gains on it. On its own this effect would make the lunar clock faster by more than the figure we actually observe.

Effect two: motion

A moving clock also runs slow, by the second-order Doppler effect. Both clocks move: Earth’s clock is carried around by Earth’s rotation, and the lunar clock travels with the Moon along its orbit.

The velocity terms therefore push the other way and subtract from the gravitational gain [1]. Neither effect can be dropped: a calculation that keeps only gravity gives the wrong answer.

Putting the two together

The published figure is what remains after both terms are added. The framework behind it also lists the orbital eccentricity term and station-dependent terms for particular sites on the lunar surface [5]Official.

Gravity wells of the Earth and the Moon comparedTwo funnels drawn side by side. The Earth funnel is the deeper one, with the geoid marked at 6.26 × 10⁷ m²/s²; the Moon funnel is the shallower one, with the selenoid marked at 2.82 × 10⁶ m²/s². An arrow between the two levels is labelled plus 56.02 microseconds per day: the rate by which a clock resting at the selenoid gains on a clock resting at the geoid.gravitational potential → 0 (far field)EarthMoongeoid6.26 × 10⁷ m²/s²selenoid2.82 × 10⁶ m²/s²+56.02 µs/daya clock at the selenoid gainson a clock at the geoid
Both bodies sit in a gravitational well, and Earth's is far deeper. The arrow marks the rate difference that a clock feels between the two floors: 56.02 µs per day in favour of the lunar clock.

The same paper reports a periodic term of about 0.108 µs per day on top of the secular rate, varying over the lunar orbit [5]Official. Our clock model keeps the secular part only, and the methodology page says so explicitly.

For scale, an independent treatment in Metrologia reaches the same surface-to-surface figure of about 56 µs per day [6]. Two groups, two methods, one answer.

Why “faster” needs two clocks named

“The Moon runs faster” is not a statement about the Moon. It is a statement about a pair of clocks — one on the Moon, one on Earth — and both of them have to be named.

Say instead: a clock on the lunar surface gains 56.02 µs per day on a clock on Earth’s geoid [1]. Now the sentence can be checked, and it stops being confused with the other figure in circulation, which compares a lunar coordinate time with Terrestrial Time.

The same discipline applies to familiar numbers. GPS satellites are often described as running about 38 µs per day faster than clocks on the ground [7]. Ground clocks, not the Moon, and not each other.

How much is 56 microseconds?

Small per day, awkward per decade. The table accumulates the surface-to-surface rate; every row compares the same two clocks.

Interval Accumulated difference Source
One day 56.02 µs [1]
One year of 365 days 20 447 µs [3]
About 49 years 1 s [3]

The exact figure for one second of accumulated difference is 48.9 years at this rate [3]. Our lunar clock shows the running total as a model value, because no clock on the Moon has ever measured it.

A related question people ask in the same breath is how long a day on the Moon lasts. That one is about sunlight, not about clock rates, and the answer is unrelated. Both sit in the Learn hub with the rest of our explanations.

Sources

  1. A Relativistic Framework to Estimate Clock Rates on the Moon — Neil Ashby and Bijunath R. Patla, NIST — The Astronomical Journal 168:112, . Peer-reviewed. Verified .
  2. Moon Facts — NASA Science, . Official document. Verified .
  3. Our calculation: the lunartime.org clock model — Lunartime Editorial, . Our calculation. Verified .
  4. What Time Is It on the Moon? — National Institute of Standards and Technology, . Official document. Verified .
  5. A Relativistic Framework to Estimate Clock Rates on the Moon (NIST copy) — NIST Time and Frequency Division, . Official document. Verified .
  6. Lunar reference timescale — A Bourgoin, P Defraigne, F Meynadier — Metrologia 63(1) 015003, . Peer-reviewed. Verified .
  7. A Time Standard for the Moon—Thanks to General Relativity — APS Physics (Research News), Physics 17, 140, . Press. Verified .

Last verified