@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
Frank Wappler