The idea is, you can transform a normal [0] wormhole that isn’t a time machine into one which is by:
1) accelerating one end to high speed relative to the other
2) keeping on end in a lower gravitational potential than the other
Why are either of these considered meaningful statements, never mind correct?
In the case of 2 in particular, isn’t GR supposed to require smooth values? So any time dilation effect would be almost identical on a pair of points +δ and -δ from the throat? Making it similar to the case of a gravitational potential without a wormhole?
And in the case of 1, the more I think about it, the less I understand the concept. What is being moved? An imaginary clock that would’ve been in the part of the wormhole at the far end? The apparent speed as measured going through the throat will be zero regardless of the apparent speed of the same as measured when going the long way around.
I'll answer 2. You are correct that clocks just to either side of the throat will stay synchronized. But it's important to remember that time dilation isn't some kind of absolute effect. The easiest way to talk about it is in terms of clocks following paths.
Imagine you have two spaceships in London. At the fire of a starting gun, they both take off, and fly around, each one taking a different path. Each ship has a clock that records the time elapsed since it took off. Eventually, they both return to the same landing pad in New York, landing at the same time. Thanks to time dilation, the readings on the clocks of the two ships might be different, even though they started and landed at the same places and times. Imagine we start both ships from a space station in deep space. One ship doesn't leave the station at all, it just stays in its docking bay, with its clock ticking along. The other flies down to the surface of Earth, sits there for a few years, and then flies back up to the space station. Thanks to the gravitational field of Earth, the ship that stayed home has more time elapsed on its clock than the ship that went to Earth.
Now suppose that each ship is carrying one end of a wormhole. The clocks on either end of the wormhole must stay synchronized. Someone sitting in the middle of the wormhole would be able to see the inside of one ship by looking to their left, and the inside of other ship by looking to their right. The clocks start out synchronized. As you point out, no matter how the ships move about, this does not change as the ships fly around. Anyone standing in the middle of the wormhole always sees that the clocks on the wall of each ship are in sync.
So: Entering the wormhole from ship A when ship A's clock reads X means you exit at ship B, at the time when ship B's clock reads X. And vice-versa for going from B to A. Now thanks to time dilation, ship B might arrive back at the station when its clock reads X, while ship A, which stayed behind, has a clock reading of (X + 20 minutes). If you are the station master, you can go into each ship to look at the clocks, and you will find that ship A's clock is always 20 minutes ahead of ship B's. But suppose that instead of walking between the ships through the station, you use the handy wormhole that connects them directly.
Suppose you enter ship B when its clock reads Y, and walk through the wormhole. You exit at ship A when its clock also reads Y. Then you step out of ship A, and walk through the station to ship B. It's clock reads (Y - 20 minutes), since according to people on the station, ship A's clock is still 20 minutes ahead of ship B's. When you originally entered ship B, its clock read Y. It now reads (Y - 20 minutes). Time travel. By retracing your path in the opposite direction, you can also travel 20 minutes into the future.
The idea is, you can transform a normal [0] wormhole that isn’t a time machine into one which is by:
1) accelerating one end to high speed relative to the other
2) keeping on end in a lower gravitational potential than the other
Why are either of these considered meaningful statements, never mind correct?
In the case of 2 in particular, isn’t GR supposed to require smooth values? So any time dilation effect would be almost identical on a pair of points +δ and -δ from the throat? Making it similar to the case of a gravitational potential without a wormhole?
And in the case of 1, the more I think about it, the less I understand the concept. What is being moved? An imaginary clock that would’ve been in the part of the wormhole at the far end? The apparent speed as measured going through the throat will be zero regardless of the apparent speed of the same as measured when going the long way around.
[0] yes, I know