[Physics] Clocks on the moon

[Tom Hollings]

When Einstein published his paper the General Theory of Relativity in 1915, one of the things that it predicted was that gravity affects time. The lower the gravity, the faster time flows, the higher the gravity, the slower it flows. It would appear that this prediction has been proved correct, but has it?
Here is one article which says it does :- https://www.dailymail.co.uk/…/Scientists-prove-time-really-… https://l.facebook.com/l.php?u=https%3A%2F%2Fwww.dailymail.co.uk%2Fsciencetech%2Farticle-1314656%2FScientists-prove-time-really-does-pass-quicker-higher-altitude.html%3Ffbclid%3DIwAR22N2BgMt7UuriRI790QKZQM5hJre6Tgwisb6nAKyuL_zK3dkXtBO7PTA0&h=AT38JrxbY7JprZDDe4oB6VwEcI382U-DL3nJfez10ALKG0HeThftIhWR-aLgMMvDzG-MWRljoPbZwyQd4RW3aeMmo7KTja-dgEXb4HKRiWquj4f95eRS80dP4p6TIE9UFZ7wVqQtY_a0F2bl4WjspL6ww9hiBKpm

There is another school of thought, and one which I subscribe to. Time is not running faster in lower gravity (ie at the top of a mountain as in the example), it is quite simply the atomic clock which is running faster.
This may be seen by some to be an outrageous claim. How can anyone dare to contradict the great Albert Einstein?
Here is a thought experiment which proves my point, and it could be very easily (but perhaps not cheaply) performed in actuality.
Place an atomic clock on the moon alongside a pendulum clock. A pendulum clock keeps time according to the formula T = 2π * √(L/g) . The only two variables are the length of the pendulum (L) and gravity (g = 9.83 m/s^2 here on Earth, and 1.6 m/s^2 on the moon). The gravity on the moon is therefore 1/6 of that on Earth, so a pendulum clock with a pendulum length of 0.1 meter (100mm) would run at one complete oscillation of the pendulum in 0.634 seconds here on Earth, but on the moon it would take 1.561 seconds. Here is the maths for those interested.

P = period of pendulum = 2 * pi * sqrt( L / g )
L = length of pendulum
g = gravitation force

Earth : P     = 2 * 3.1416 * sqrt( 0.1 / 9.83)   = 6.2832 * sqrt(0.0102)   = 6.2832 * 0.1009   = 0.634 seconds

Moon : P    = 2 * 3.1416 * sqrt( 0.1 / 1.62)   = 6.2832 * sqrt(0.0617)     = 6.2832 * 0.2484   = 1.561 seconds

This of course is assuming that all other factors are the same, probably the most important being temperature, but even a large temperature change would be nowhere near the effect of the gravity change. For example, if the temperature of the pendulum is increased by 200 degrees centigrade, which is quite possible in direct sunlight on the moon, its length would increase by around 10%, and the period would increase by a small amount.

Moon @ 200 degrees C : 

 P = 2 * 3.1416 * sqrt( 0.11 / 1.62)   = 6.2832 * sqrt(0.0679)   = 6.2832 * 0.2606   = 1.6373 seconds

This is an increase of 0.0763 seconds. The effect of lower gravity on the moon accounts for an increase of 0.927 seconds, so the heat effect is small but noticeable.
Because of the reduced gravity, the atomic clock will run faster on the moon by about 1/95 of a second per year (this was the only reference to the atomic clock speed change that I could find). If Einstein is correct, and it is time itself which has speeded up, the pendulum clock will speed up and agree with the atomic clock. This claim is absurd, as the pendulum clock will run as defined by gravity and pendulum length alone, so the two clocks will disagree.

Which clock is correct? The answer is neither is correct. Both clocks are affected by gravity, the atomic clock is of course affected to a far less extent than the pendulum clock, and in the opposite sense. Gravity is affecting the rate of both clocks, not altering the passage of time.

The pendulum length of 100mm has been chosen as a clock this size can easily be stowed on the moon lander and kept in a shock proof container until the lander is on the moon. Also, the duration needed to notice the difference in timekeeping is very short, just a matter of a few minutes, so the time check can be done many times during a three day stay on the moon.

I am here assuming that the moon lander has an atomic clock on board, but have not been able to find any firm evidence if this is the case. If not, an atomic clock would have to be taken as well as the pendulum clock.
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