**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

There is a subtle but important distinction you might want to consider which may help you form a better conceptual picture of what happens.

When we talk of time dilation, it is a geometric effect- it means that the length of paths between two points in time can vary. To explain what I mean by analogy, imagine that you and I are standing at a corner of a large square. You walk diagonally across to the opposite corner while I walk around the edge of the square to meet you there. The distances we each walk are quite different because we have followed different paths. If we were carrying pedometers, they would show we had walked different distances, and you would consider that entirely natural- you would not feel you had to make up an explanation about your pedometer being 'pace dilated' relative to mine.

Now, carry that idea over into special relativity and consider time dilation in that context. If you move between two clocks in my frame of reference, and during your journey 4 second pass on your watch, while my clocks show a time difference of 5 seconds between the start and end of your journey, it is not because something has caused your watch to run slow, but that the path through time you have taken between the two points was only 4 seconds long, and your watch has correctly recorded it as such, running at its usual rate to tick off the seconds faithfully.

In general relativity, the mathematics are much more complicated, but the conceptual idea still applies. If you follow one curved path through time its length (in seconds) can be more or less than another curved path through time, so clocks taking the different paths will show different elapsed times. It is not because they are somehow running slow or fast, in the sense of not faithfully recording a true time, but because they are accurately showing real differences in the lengths of their paths through time.

## Best Answer

The Earth's surface is curved and this can be observed via the vast number of pictures of the Earth from space that now exist.

However, the surface curvature can also be "seen" via measurements on the surface itself.

For example, if one were start at the North Pole and travel in a "straight line" (a great circle) to the equator, then move east along the equator for a quarter of the circumference, and then move North (always along a great circle), one would eventually reach the starting point at the North Pole.

But look, one would have formed a "triangle" with the interior angles adding up to 270 degrees! This is one way that

intrinsiccurvature is measured.Simply put, intrinsic curvature is mathematically characterized by the Riemann Curvature Tensor and

observedvia geodesic deviation.