Does mass affect time dilation?

Question

Take this example.

Say the ISS is mass m. Time dilation there is T due to Earth gravity.
For simplicity, let's just say 1 second on ISS is 10 seconds on Earth.

Now say we make ISS-2 orbiting on the other side on same orbit and speed as ISS. It has mass 2m.
What is time dilation on ISS-2? Is it still 1 second to 10 seconds?

This is confusing me, because, as I understand, Earth's mass is defining its gravitational force on other objects. But, so is the other object's mass.
Obviously if another Earth was right next to Earth they wouldn't cause time dilation to each other. Do they?

Answer 1

For weak gravitational fields, which means anywhere that isn't near a black hole, the gravitational time dilation can be related to the gravitational potential energy by the equation:

$$ \frac{\tau}{t_\infty} = \sqrt{1 - \frac{2\Phi}{c^2}} $$

In this equation $\tau$ is the time that passes inside the gravitational field, $t_\infty$ is the time measured far away where the gravitational field is zero, and $\Phi$ is the Newtonian gravitational potential energy. This is known as the weak field approximation and it is an excellent approximation for the Earth's gravitational field.

Since the right hand side of the equation is less than one we can immediately see that $\tau < t_\infty$ i.e. time passes more slowly in the gravitational field, and this is what we mean by the gravitational time dilation.

Suppose the ISS and ISS2 orbit at a distance $r$ from the centre of the Earth. The Newtonian gravitational potential energy is $U = GM/r$, and substituting this into our equation gives:

$$ \frac{\tau}{t_\infty} = \sqrt{1 - \frac{2GM}{c^2r}} $$

And in fact this is an exact result that is the same as predicted by general relativity, though this is due to the way the $r$ coordinate is defined in GR so we shouldn't read too much into this.

Anyhow, if the masses of ISS and ISS2 can be ignored they will have the same gravitational potential energy since they are the same distance from the Earth, and therefore the gravitational time dilation will be the same.

If you include the masses of the space stations then their GPEs will be slightly different. A clock in ISS2 as a slightly greater GPE because the $2m$ mass of ISS2 will increase the GPE more than the $m$ mass of ISS1. As a result a clock in ISS2 will run slightly slower than a clock in ISS1, though both clocks will run faster than a clock on the surface of the Earth.

Actually calculating the difference would be complicated as it would depend on the exact shape of the space station and where in the space station the clocks were located. In any case the difference is going to be immeasurably small since the gravitational fields produced by the space stations' masses are tiny compared to the gravitational field of the Earth.

One last comment: an object orbiting the Earth experiences time dilation due to both the gravitational time dilation and the Lorentz time dilation due to its velocity. However in this case it won't affect the comparison between the two space stations since both will have the same orbital velocity.

Answer 2

SHORT ANSWER: It is not the gravitational force, it is the gravitational potential that causes the time dilatation due to gravitation. The contributions of both stations to the gravitational potential inside them are negligible compared to the potential caused by the Earth, and, hence, the difference in gravitational time dilatation is also negligible.

Answer 3

Allow me to clarify one quick thing: The difference in time dilation between Earth and the ISS is MUCH less than 1 second versus 10 seconds. At an altitude of 10,000 km, the difference is only 1 second per century. There is no place in the universe where time runs 10 times as fast as on Earth.

As others have mentioned, the time dilation on either ISS1 or ISS2 would be so extremely small that it would not be measured in any testing. So we can just ignore that.

Yes, another large mass (like the moon or the sun) will have its own time dilation and this absolutely does impact on gravity on Earth. Think of the ocean tides. These tides are caused by the moon (about 80%) and the sun (about 20%)

Now let me explain gravity more fully. You have probably heard that gravity is caused by the warping of spacetime. This is correct. What this means is that both the warping of space and the warping of time together cause things to fall. On a small planet like Earth, 99.9999% of this is time dilation. It is only when you get to a black hole that 50% of the gravity is caused by the warping of space. You have to get into spacelike and timelike calculations of Schwarzschild to prove this. But what is means is that on Earth gravity does not cause time dilation; rather gravity is time dilation. Nobel Lauriat Kip Thorne referred to what he calls "Einstein's Law of Time Warps." Thorne said "Things like to live where they age the most slowly.... The Earth's mass warps time according to Einstein. It slows time near the surface of the Earth. And this time warp is what produces gravity."