[obastani]

This is based on my limited physics education, but I believe there are two things at play:

1) Velocity doesn't affect time, acceleration does. So if your twin flies to Mars and back, you age more quickly. But if your twin flies to Mars, and you join him after a year (following the same trajectory), you age the same. This is special relativity.

2) Acceleration due to gravity doesn't count. In fact, the opposite is true -- by standing on the ground, you are accelerating up at 32 feet per second squared. This is general relativity.

So, to answer your question. On the one hand, yes, if I am accelerating (not counting gravity), then I observe time pass more slowly. On the other hand, most of the universe is dominated by gravitational forces, so most systems wouldn't notice these effects.

The fascinating thing is that both (1) and (2) happen because of a very basic physics principle: if I can't use an experiment to tell two frames of reference apart, then the physics in the two frames of reference are identical.

For (1), I observe the same speed of light as someone moving at velocity v relative to me. Einstein used this simple axiom to derive special relativity. For (2), I can't tell the difference between free fall and being at rest (alternatively, I can't tell the difference between being on Earth and being in an elevator in space). This is because the inertial mass equals gravitational mass, a "coincidence" that dates back to Newton's law of gravitation. From this (and a lot of math) Einstein derived general relativity. Beautiful!

[Retric]

That's not really correct, if you sit at the center of a large object you don't accelerate, but you experience time dilation.

Further, as you dig deeper your acceleration decreases but time dilation continues to increase.

[andrepd]

That's incorrect as well. You are mixing two relativistic effects: from the velocity of something relative to an observer, and from the gravity on that spot of the universe (which is what I assume you mean by acceleration).

Also wrong, as you dig deeper into a planet you experience less gravity, not more.

[Retric]

Velocity and acceleration are different things and Talking about less or more gravity is a poor descriptor.

At the center of the moon you would 'float' aka no acceleration relative to the moon. You would still be orbiting the earth, sun, etc.

However by being at the bottom of a gravity well you get time dilation relative to someone in the same orbit on the other side of the earth.

However, it's important to note LEO means high orbital velocity which counters being higher in the gravity field. Similarly, standing on the surface of the earth you have time dilation from the earths rotation which you would not have at the center.

PS: On way to think about it is at the center space time is pulled by all the mass around you which is a stronger pull than standing beside a planet. However the pull is in balance, like a tug of war game nobody is winning.