Originally Posted by

**Squashed**
In time dilation, accelerations have no effect on clocks - to understand an acceleration it must be broken down into a series of infinitely small velocity segments through integration to calculate the time dilation that occurs because of the acceleration. So an acceleration is just a constantly changing velocity and it is the __velocity__ that affects the clocks.

Only if you want to use special relativity to do the calculations. Of course, we usually *do* want to do that, because it's a lot easier than using general relativity!

Originally Posted by

**Squashed**
So somehow the __force__ of gravity slows clocks whereas the __force__ of an acceleration does not directly affect clocks.

No, if you're in a sealed room and experience 1 g of acceleration, you'll discover that clocks that are higher in the room run more quickly than clocks that are lower in the room, by exactly the same amount that they would if you were in a sealed room at rest on the Earth's surface. From your sealed room, you wouldn't be able to tell the difference.* For the accelerating room, you could be outside in an inertial reference frame and show how the observed time dilation results from differences in velocity between points that are higher or lower in the room. But for the Earth, you could do your calculations in a free-falling frame and also show that these differences in clock rates are from differences in velocity, rather than due to any force of gravity.

* Actually, there is a slight difference of course. That's because for the accelerating room, the gravitational force is constant, whereas for the Earth, the gravitational force gets weaker as you get farther away and points in a slightly different direction (toward the center of the Earth) as you move side to side. The equivalence principle really says that acceleration and gravity are the same in an infinitesimal region where the force can be considered constant, or for practical applications, where these effects will be small enough to be ignored for whatever it is you're doing.

Conserve energy. Commute with the Hamiltonian.