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Say there were a base on the Moon with people living/studying there. Would they age differently than those on earth due to a change in speed through space? Does the Moon move faster through space (relative to Earth), as it is orbiting the Earth? I feel like the answer would be yes, but very minor.

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The time dilation of the Moon relative to Earth is dominated by not being as deep in the gravitational field of Earth. Not so much the velocity of the Moon. Though the velocity effect isn't negligible, being about two orders of magnitude down from the gravity effect.

Time passes about 0.66 parts per billion faster on the Moon than on Earth, due to not being in as strong a gravity field. The dilation, including all the effects of being further from the Earth's gravity field, being in the Moon's gravity field, and the velocity of the Moon, could easily be measured in sum by comparing good atomic clocks on the Earth and the Moon.

The time dilation relative to some location in space not in a gravitational potential is given by:

$$1\over\sqrt{1-{2\mu\over r c^2}-{v^2\over c^2}}$$

where $v$ is the velocity of the object which, to keep it simple, is in a direction that does not change the gravitational potential (e.g. a circular orbit), $\mu$ is the $GM$ of a gravitational source, $r$ is the distance from the center of that source, and $c$ is the speed of light.

For these kinds of examples, the ratios are very small and so the above can be approximated:

$$1+{\mu\over r c^2}+{v^2\over 2c^2}$$

The small amount that that is above one is the fractional amount that time is passing slower in the gravitational field and moving at some velocity, as compared to the fixed point far away.

$${1\over c^2}\left({\mu\over r}+{v^2\over 2}\right)$$

We can put in the numbers for Earth gravity, radius, and surface rotational speed to get our time dilation here relative to a reference with zero velocity relative to Earth's center. For the Moon, we can put in the Moon's gravity, radius, and orbital velocity around the Earth to get the dilation there relative to the same reference point. Then dividing the dilation at the Moon by the dilation at the Earth, we get the dilation at the Moon relative to Earth. For the small ratios, this is just subtracting the fractional variations from one:

$${1\over c^2}\left({\mu_M\over r_M}+{v_M^2\over 2}-{\mu_E\over r_E}-{v_E^2\over 2}\right)$$

This gives a negative value, which is time contraction on the Moon relative to Earth. Note that $v_M$ is the velocity of the Moon about the Earth, whereas $v_E$ is the velocity of an object on the surface of the Earth. (Varies with latitude.)

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  • $\begingroup$ I would add that the medical question about aging differently is uncertain though. Time dilation doesn't change things, but the different gravity environment (etc.) could have an effect on people over many years. I don't think science has an answer for this yet; nobody has tried it. $\endgroup$ – Adam D. Ruppe Nov 19 '14 at 1:09
  • $\begingroup$ So, according to what Mark is saying, you would age faster on the Moon... $\endgroup$ – ElScorcho Nov 19 '14 at 1:43
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    $\begingroup$ I am saying that time will past a smidge faster. If that were the only effect, you would die about two seconds earlier on the Moon than if you had stayed on Earth (assuming ~100 year lifetime), as viewed from the Earth. However as Adam points out, environmental effects will grossly dominate over time dilation for how fast you will age biologically. $\endgroup$ – Mark Adler Nov 19 '14 at 1:52
  • $\begingroup$ You could use the adjustment for GPS time dilation for residents of the moon as well. The two components (Special and General relativity) work the same on the moon as for a GPS satellite, albeit with different parameter values. $\endgroup$ – dotancohen Nov 19 '14 at 10:37
  • $\begingroup$ Those are forms of the formula I used. $\endgroup$ – Mark Adler Nov 19 '14 at 15:51
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I disagree ,I think a person will age faster on the moon . Time in the human body is measured by biological reactions that happen which lead cells to age . Biological reactions depend on the kinetic energy of molecules. If gravity is less on the moon this will increase the inherent kinetic energy of molecules due to a lesser effect of pull or gravity on the molecules which will speed up biochemical reactions and aging. So the people on the moon will see each age faster.

T.Kyriakides

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    $\begingroup$ You are wrong, kinetic energy of molecules is not changed by gravity. The molecules move in all directions, not only parallel to the gravity field. $\endgroup$ – Uwe Nov 30 '18 at 22:06

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