@JProl

A bit more concise, a lot less "cislunar astronauts"-model-dependent, and manifestly independent of coordinates, timestamps, particular "types of clocks", or particular "designs of #LunarTime":

For participant \(M\) being held and carried along by (some representative patch of) the moon's surface
and any two distinct events \(\varepsilon_{(M ~ P)}\) and \(\varepsilon_{(M ~ Q)}\) in which \(M\) had taken part (having been met and passed by participant \(P\), or by participant \(Q\), resp.),

and participant \(E\) being held and carried along by (some representative patch of) Earth's surface
and any two distinct events \(\varepsilon_{(E ~ J)}\) and \(\varepsilon_{(E ~ K)}\) in which \(E\) had taken part (having been met and passed by participant \(J\), or by participant \(K\), resp.),

the ratio between \(M\)'s duration from having passed \(P\) until having passed \(Q\) and \(E\)'s duration from having passed \(J\) until having passed \(K\) is

\[ \left( \frac{\tau M[ ~ \_ P, \_ Q ~ ]}{\tau E[ ~ \_ J, \_ K ~ ]} \right) \approx \]

\[ \left( \frac{\ell[ ~ \varepsilon_{(M ~ P)}, \varepsilon_{(M ~ Q)} ~ ]}{\ell[ ~ \varepsilon_{(E ~ J)}, \varepsilon_{(E ~ J)} ~ ]} \right) ~ \times ~ \left(1 + \frac{58.7 * 10^{-6}}{86400}\right) \approx \]

\[ \left( \frac{\ell[ ~ \varepsilon_{(M ~ P)}, \varepsilon_{(M ~ Q)} ~ ]}{\ell[ ~ \varepsilon_{(E ~ J)}, \varepsilon_{(E ~ J)} ~ ]} \right) ~ \times ~ (1 + 6.8 * 10^{-10}), \]

where the values \(\ell\) are (without loss of generality) the non-zero values of #LorentzianDistance for pairs of (suitably ordered) timelike separated events (https://www.google.com/search?q=%22Lorentzian+distance%22+%22wikipedia%22).

#LTC #LunarTime #Duration #Rate #Clock #Relativity #Spacetime

@JProl

It's not hard to snicker about amateurs. I've given you a boost for having been so keen to point out Will Shanklin's <em>"apparent gain"</em> instead of <a href="https://www.whitehouse.gov/wp-content/uploads/2024/04/Celestial-Time-Standardization-Policy.pdf">Arati Prabhakar's <em>"apparent loss"</em> ... and for keeping the attention, too.

Coming up with some (presumably) correct + still reasonably short description seems more challenging (to me). My best attempt so far:

<blockquote>
<b>If</b> two astronauts had met "somewhere in (cislunar) space", and subsequently separated from each other,

- with one astronaut venturing on to land on the lunar surface, and

- the other astronaut returning to the Earth's surface,

such that (as may happen in selected trials)

- it takes both astronauts exactly equally long, resp., from separating until reaching (halting on) the Moon, or on Earth,

after some (not further specified) while, either astronaut perhaps being prompted by suitable prearranged signals,

- both again take off from Earth, and from the Moon, resp., and they meet again "somewhere in (cislunar) space", where again (trials must be selected such that)

- the duration of one astronaut from her take-off until the re-union meeting

- happens to be exactly equal to the duration of the other astronaut from his take-off until being together again

<b>then/therefore</b>

the astronaut who had stayed on the lunar surface had remained there
(pretty much) <b>exactly</b>
\[\left(1 + \frac{58.7 * 10^{-6}}{86400}\right) \approx (1 + 6.8 * 10^{-10})\]
<b>times as long as</b>
the astronaut who had stayed on the the surface of the Earth had remained there.

</blockquote>

So: Good luck, #NASA ! ...

#LTC #LunarTime #Duration #Rate #Clock #Relativity #Spacetime

@saila

[How to determine whether an atomic clock had "ticked at the Moon's natural pace"? (Nature 614, 13-14 (2023))](https://physics.stackexchange.com/questions/748237/how-to-detetermine-whether-an-atomic-clock-had-ticked-at-the-moons-natural-pac)

#time #LunarTime #UTC #relativity #clock #clockrate #rate #duration