this answer has been expanded at the end.
I am convinced that macroscopic wormholes are impossible because they would violate the energy conditions etc. so it is not a top priority to improve the consistency of semi-consistent stories. At the same moment, I also think that any form of time travel is impossible as well, so it's not surprising that one may encounter some puzzles when two probably impossible concepts are combined.
However, it is a genuinely confusing topic. You may pick Leonard Susskind's 2005 papers about wormholes and time travel:
http://arxiv.org/abs/gr-qc/0503097
http://arxiv.org/abs/gr-qc/0504039
Amusingly enough, for a top living theoretical physicist, the first paper has 3 citations now and the second one has 0 citations. The abstract of the second paper by Susskind says the following about the first paper by Susskind:
"In a recent paper on wormholes (gr-qc/0503097), the author of that paper demonstrated that he didn't know what he was talking about. In this paper I correct the author's naive erroneous misconceptions."
Very funny. The first paper, later debunked, claims that the local energy conservation and uncertainty principle for time and energy are violated by time travel via wormholes. The second paper circumvents the contradictions from the first one by some initial states etc. The discussion about the violation of the local energy conservation law in Susskind's paper is relevant for your question.
I think that if you allowed any configurations of the stress-energy tensor - or Einstein's tensor, to express any curvature - it would also be possible for one throat of an initial wormhole to be time-dilated - a gravity field that is only on one side - and such an asymmetry could gradually increase the time delay between the two spacetime points that are connected by the wormhole. For example, you may also move one endpoint of the wormhole along a circle almost by the speed of light. The wormhole itself will probably measure proper time on both sides, but the proper time on the circulating endpoint side is shortened by time dilation, which will allow you to modify the time delay between the two endpoints.
Whatever you try to do, if you get a spacetime that can't be foliated, it de facto proves that the procedure is physically impossible, anyway. Sorry that I don't have a full answer - but that's because I fundamentally believe that the only correct answer is that one can't allow wormholes that would depend on negative energy density, and once one allows them, then he pretty much allows anything and there are many semi-consistent ways to escape from the contradictions.
Expansion
Dear Julian,
I am afraid that you are trying to answer more detailed questions by classical general relativity than what it can answer. It is clearly possible to construct smooth spacetime manifolds such that a wormhole is connecting places X, Y whose time delay is small at the beginning but very large - and possibly, larger than the separation over $c$ - at the end. Just think about it.
You may cut two time-like-oriented solid cylinders from the Minkowski spacetime. Their disk-shaped bases in the past both occur at $t=0$ but their disked-shaped bases in the future appear at $t_1$ and $t_2$, respectively. I can easily take $c|t_1-t_2| > R$ where $R$ is the separation between the cylinders. Now, join the cylinders by a wormholes - a tube that goes in between them. In fact, I can make the wormhole's proper length decreasing as we go into the future. It seems pretty manifest that one may join these cylinders bya tube in such a way that the geometry will be locally smooth and Minkowski.
These manifolds are locally smooth and Minkowski, when it comes to their signature. You can calculate their Einstein's tensor - it will be a function of the manifold. If you allow any negative energy density etc. - and the very existence of wormholes more or less forces you to allow negative energy density - then you may simply postulate that there was an energy density and a stress-energy tensor that, when inserted to Einstein's equations, produced the particular geometry. So you can't possibly avoid the existence of spacetime geometries in which a wormhole produces a time machine sometime in the future just in classical general relativity without any constraints.
The only ways to avoid these - almost certainly pathological - configurations is to
postulate that the spacetime may be sliced in such a way that all separations on the slice are spacelike (or light-like at most) - this clearly rules "time traveling" configurations pretty much from the start
appreciate some kind of energy conditions that prohibits or the negative energy densities
impose other restrictions on the stress-energy tensor, e.g. that it comes from some matter that satisfies some equations of motion with extra properties
take some quantum mechanics - like Susskind - into account
If you don't do either, then wormholes will clearly be able to reconnect spacetime in any way they want. This statement boils down to the fact that the geometry where time-like links don't exist at the beginning but they do exist at the end may be constructed.
All the best
Lubos
It's true that traversable wormholes would theoretically allow for travel into the past according to general relativity (though this may be ruled out by quantum effects in whatever theory of quantum gravity eventually replaces general relativity, see the chronology protection conjecture). But the only way this would happen is if one mouth of the wormhole experiences time dilation relative to the other, either because it's taken on a journey at relativistic speed relative to the other mouth, or because it's moved closer to a source of gravity where it experiences gravitational time dilation. If the two mouths have clocks moving alongside them which were initially synchronized, then if some time dilation has accumulated, when the two mouths are brought back together an outside observer looking at them side-by-side will see the clocks showing different times. But, as explained by physicist Kip Thorne (who discovered traversable wormholes as a theoretical possibility in general relativity) in his book Black Holes and Time Warps, time threads differently through the wormhole, in such a way that if you look through one mouth at the clock which is alongside the other mouth, and compare it with the clock alongside the mouth you're looking through, then the clocks will still be synchronized (assuming the time for light to travel through the wormhole from the clock to your eyes is negligible). This means for example that if a side-by-side comparison by an outside observer (one who was not looking through either wormhole) showed mouth #1's clock to read 2015 and mouth #2's to read 2010, then if you jumped through mouth #2 you would exit mouth #1 when its clock also read 2010, not 2015. In some cases this would make it so that when you jump through a wormhole, you could end up in a region of spacetime where it would be possible to send a signal to your own younger self, which would be received at an earlier time then it was sent according to a clock you carried along with you (i.e. in terms of your own proper time).
But you don't mention anything about time dilation--if one mouth of the wormhole is moved slowly from A to B, so that no time difference accumulates, then there will be no time travel in your scenario. One thing to keep in mind is that in relativity, simultaneity is not generally defined in terms of when you see the light from events--for example, in an inertial frame in special relativity, if I see the light from an explosion 10 light-years away in the year 2020 (in the space and time coordinates of that frame), I subtract out the light travel time and say the event actually took place in 2010 in my frame. And wormhole spacetimes can be asymptotically flat, meaning to a good approximation you can treat them as localized distortions of spacetime moving around in an otherwise flat spacetime, so you can still set up something very close to an inertial frame in the region outside the wormholes. In this case, we might say that in a frame where A and B are at rest, the event of the clocks at A and B each reading 12:00:00 AM on Jan 1. 2020 are simultaneous. Then if I jump into mouth B at that moment, and exit from mouth A 4 seconds later according to the time coordinates of this frame, we can predict that I will see the clock at A reading 12:00:04 AM, Jan. 1 2020. If I immediately look back through the wormhole I will see the clock at B reading the same time of 12:00:04 AM, Jan. 1 2020, but if I immediately look at B through normal space using a telescope, due to the 10-light-year distance I will see the clock at B reading 12:00:04 AM, Jan 1. 2010. Assuming I had been living near B until I jumped through that mouth in 2020, looking through my telescope will also allow me to see myself when I was ten years younger.
But the main thing to realize is that there's no way I can actually send a signal to reach my younger self in the past in this scenario, as was possible in the other scenario where time dilation had created a time difference between clocks next to each mouth. If I send it through the wormhole, then (assuming a negligible travel time for a light signal) it exits mouth B 12:00:04 AM Jan. 1 2020, whereas I had jumped through mouth B 4 seconds earlier at 12:00:00 AM Jan. 1 2020. Meanwhile, if I send a light signal through normal space at 12:00:04 AM Jan 1 2020, in my frame it takes 10 years to get there, and another 10 years for the light from the event of someone receiving it at B to get back to me. This means I won't see anyone at B receive it (at least not if I am looking through normal space) until my clock reads 12:00:04 AM Jan 1 2040, and I will see the clock at B reading 12:00:04 AM Jan. 1 2030 when someone next to it receives my signal. So regardless of whether I send a signal through normal space or back through the wormhole, in this scenario the signal will arrive at B when the clock there shows a later time than the moment I jumped through the wormhole to travel from B to A.
Best Answer
This seems to be called the eternal-time-machine spacetime, and I believe the original paper was Morris 1988, which is available online and not paywalled. On p. 1447, they claim:
The question says:
I'm not aware that this aspect of the idea is controversial at all. In your notation, the paper is saying that entering y and emerging at x sends you back in time. If there's some source that says it's the other way around, please tell us what the source is.
Here's a diagram showing what I understand them to be saying.
The wormhole is created before the diagram begins. The left-hand black line is the world-line of one mouth (x), and the other black line is the world-line of the other mouth (y). An observer's world-line ABCD is shown in red. B is the point where the observer enters y. C is the point where he exits x. The observer's world-line contains a closed, timelike curve. In the paper, they label the wormhole's mouths with time coordinates, which I think are the readings on a clock that's inside the wormhole. Since the wormhole can be assumed to be internally short, one can synchronize these clocks without any of the usual ambiguities -- even relativistically, one can absolutely synchronize two clocks that are at the same location. So B and C are simultaneous according to a synchronization process that's carried out inside the wormhole.
Morris, Thorne, and Yurtsever, "Wormholes, time machines, and the weak energy condition," Phys Rev Lett 61 (1988) 1446; http://authors.library.caltech.edu/9262/