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Been reading a lot lately about how space agencies are talking about setting up timekeeping on the Moon and sync that with Earth's system. Nearly every article I read cites: time on the Moon runs faster because of relativistic effects. The total effect of both gravitation time dilation and Lorentz dilation are cited as "56 microseconds a day gain by lunar clocks". But my numbers keep working out to only 2.2 microseconds a day. Here's the numbers I got from online calculators:

Gravitational time dilation: Δt' = 1.00000000003152 seconds (clock runs faster) Lorentz time dilation: Δt' = 0.99999999999419 seconds (clock runs slower) Combined dilation effects: Δt' = 1.00000000002571 seconds (clock runs faster) = 2.2 μs per Earth-day

I gotta be missing something obvious to be that far off. Any insights most appreciated.

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If you plug the moon's mass, radius, etc. into these formulas, you'll get time dilation factors relative to a stationary object at infinity. Time dilation relative to the earth's surface is that divided by the corresponding values for the earth. The result is dominated by the earth's gravitational time dilation, which is $\displaystyle \frac{GM_\text{earth}}{r_\text{earth}c^2} \approx 60\text{ μs/day}$.

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  • $\begingroup$ Thanks, I was expecting something fundamental like that. So presumably, the diff between your 60 μs and the oft-cited 56 μs must be due to the lunar gravitational and Lorentz dilations. I'll run the numbers and report back if they don't work out. But I fully expect they will. $\endgroup$
    – Heimhenge
    Commented Mar 12, 2023 at 0:03
  • $\begingroup$ Check! Gravitational time dilation from Earth: Δt' = 1.00000000068664 seconds (clock runs faster) Gravitational time dilation from Moon: Δt' = 1.00000000003152 seconds (clock runs slower) Lorentz time dilation from Moon's speed: Δt' = 0.99999999999419 seconds (clock runs slower) Combined dilation effects: Δt' = 1.0000000006493 seconds (clock runs faster) = 56 μs per Earth-day $\endgroup$
    – Heimhenge
    Commented Mar 12, 2023 at 1:40

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