I am currently working on A Depth Study for my year 12 Physics class, and am looking at relativity and time dilation of a lunar eclipse experience on the moon compared to as seen from on Earth. From what I understand there is no atomic clock data from the surface of the moon, and especially none relating to lengths of a lunar eclipse. However, I have decided that the best idea is to find a similar past project that doesn't directly show data for my investigation, but will be able to confirm that there is definently a time dilation. I have already made predictions for the time dilation of the moon over the length of a lunar eclipse seen from Earth using general and special relativity equations. If someone has an idea on how approach this, help would be much appreciated. Thanks.
A lunar eclipse as seen from the moon is actually a solar eclipse (by the earth). This should definitely be better to time than the lunar eclipse from the earth (which is a very 'fuzzy' event). Still, I doubt you could even come close to the microsecond accuracy needed here. Practically any other method would be much better. You could for instance just send a light signal of known duration from the earth and record its duration on the moon. That would give you a direct measure of time dilation after you remove additional effects like Doppler shift.