About Relativistic time in different planets and moons

Question

Since the planets and moons in the Solar System has diferent gravitational fields, What happen to the relativistic course of time if we compare it to an absolute Newtonian time or in other words measuring time with equivalences between astros? What will happen to humans whom gets older? I have the intuition that being for example the Earth's Moon a less masive object and giving un a less gravitational field in the surface a person which experiments a 5-Earthyears on Moon will have a bigger oldering that a person who has stay a 5-Earthyears time on Earth. And if that person return to Earth will be significantly older than other person who born in the same years and stay all life on Earth... I am wrong?

Answer 1

It is the difference in gravitational potential that alters the relative rates at which clocks run. There is no direct connection with the gravitational acceleration at the position of the clock.

An astronaut on the moon is at a higher gravitational potential than a clock placed on the Earth. Therefore, the astronaut's watch would run faster than the clock on the Earth's surface, if that were the only effect in action.

However, the Moon orbits the Earth and the Earth spins with respect to the surface of the Moon. Both of these give the clocks a relative velocity and this relative velocity leads to relativistic time dilation that would be in the opposite sense.

To see which one wins, you have to do a calculation. It turns out that the "gravitational time dilation" of the Earth clock being within the Earth's potential is the biggest factor; the astronaut being in the Moon's potential is second biggest and the special relativistic time dilation due to relative motion is smaller.

The calculation is done here and amounts to a difference of 56 microseconds per day ($\pm 2.2\mu$s because of the varying distance between the Earth and the Moon). i.e. The astronaut's watch gains time at the rate of 56 microseconds for every day they spend on the Moon compared with an Earth-based clock. So, after 5 years, they have aged more quickly than an Earth-bound person and as a result are older by 0.10 seconds.

The calculations would have to be done individually to make a comparison with clocks placed on other bodies in the Solar System - it depends on their masses, radii, distance from the Sun, and their speeds relative to a clock on Earth.

An interesting aside is that you could also consider the special relativistic time dilation experienced by an astronaut travelling to the body in question and then returning to the Earth in order to make the comparison with an Earth-based clock (the twin paradox effect). This effect always results in the astronauts clock running slower than the clock on Earth, but I think it will always be negligible compared to the effects discussed above if the astronaut stays on the other body for years, for any sensible return trip within the Solar System.

Answer 2

You question has some problems:

1. The premiss is wrong: Local gravitational field doesn't matter for time dilation.

(Since you directly mention the gravitational field of the moon).

What matters is the difference in gravitational potential between two clocks... at least in the weak field limit, which the solar system meets.

2. You forgot about the Sun: The sun is 330,000 times more massive than the Earth, and 333K earth radii is 14 A.U.: The sun's gravity dominates the planets..at least the ones you can stand on.

If you have two points in the solar systems, you can estimate gravitational time dilation by compute the difference in Newtonian potential, e.g.:

$$ U = \frac{GMm} R $$

and replace the kinetic energy term

$$ T = \frac 1 2 m v^2 $$

in the Lorentz factor:

$$ \gamma = \frac 1 {\sqrt{1-\frac{2T}{mc^2}}}$$

with $U$, and work out some cases for yourself, and then you can decided what is significant or not.

Now if you're doing deep-space navigation, or instrument timing of gas giant fly-by's--it's a big deal.