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
Typically solving the full Einstein equations is rather difficult, so to calculate stuff about gravitational waves people typically use the following approximation
$$ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} $$
That is, they approximate the full metric $g_{\mu\nu}$ as some perturbation of flat Minkowski spacetime. This approximation is called 'linearized gravity', as one uses a linear approximation of the full Einstein equations to calculate the dynamics of these small perturbations $h_{\mu\nu}$. See e.g. chapter 7 in Carroll for this. Working out the Einstein equations in this regime one typically finds as a solution for $h_{\mu\nu}$ of the following form for a `gravitational' wave propagating in the $z$ direction:
$$ h_{\mu\nu} = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & h_+ & h_\times & 0 \\
0 & h_\times & h_+ & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}$$
Since the metric gives us the distance between two points $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ we see that such a gravitational wave really causes behavior that in any experiment would be identical to stretching the $x$ and $y$ directions of our space-time.
Just like an electromagnetic wave has two polarization directions perpendicular to its momentum, a gravitational wave moving in the $z$ direction also has two polarizations, here indicated as $h_+$ and $h_\times$. The $h_+$ polarization of the gravitational wave (moving in the $z$ direction) stretches and squeezes the spacetime in the $x$ and $y$ directions. The $h_\times$ polarization stretches and squeezes the spacetime diagonally in the $x$ and $y$ directions
This is a linear approximation to the Einstein equations which means that we are neglecting the energy carried by the gravitational wave that would itself distort the space-time curvature and cause more gravitational waves. When we neglect that gravitational waves actually carry energy themselves, react with each other and create more gravitational waves. This is a legitimate approximation as gravitational waves with a small amplitude (and of relatively large wavelengths) carry very little energy indeed. We find that General Relativity in this 'linear regime' looks very much like the Maxwell equations (i.e. electromagnetic waves also pass right through one another without interacting, we say their equations are 'linear').
Gravitational waves with large amplitudes and very short wavelengths (high frequencies) carry a lot of energy, and we can no longer neglect the interactions between gravitational waves of this sort (or the interactions between them and the additional gravitational waves they could emit themselves). This self coupling of gravitational waves, that becomes more and more important at higher energies, makes the Einstein equations so hard to solve and is what gives rise to all sorts of complications, both in classical GR as the quantization of this theory. However, studying one gravitational wave on its own in the limit that it has a long wavelength (and therefore carries relatively little energy) is a very legitimate approximation of the theory.
Best Answer
A quick answer, sorry if you know this, but I am just writing to remember for myself. Bear with me, and please check all this in any GR textbook.
Angular momentum is preserved, so when a star shrinks (after the core stops producing radiation), it's a.m. will still be present, so my point is that a non rotating Schwarzchild black hole would be very rare. But for teaching purposes, it's handy to make all the assumptions you can and be left with as few variables as possible.
You then need the place holding functions A, B and C (which can be reduced to just A and B). This will be of the form $$ds^2 = Adtdt - Bdrdr$$ ignoring the other angular terms, which are irrelevant as you have chosen them to be so.
You are left with the standard Schwarzchild metric anstanz, after assuming no angular dependence, no cross terms say $drd\theta $, and invariance under time reversal, and using the vacuum solution ($T_{\mu\upsilon}=0$). Also drop the cosmological constant.
Then you need to connect this place holding metric with $R_{\mu\upsilon}$, which you do by writing A and B as exponential terms, so they are always positive, and then, depending on your book, you either use forms or the $\Gamma $ symbols to get the variables that make up A and B.
Then, as you only have two variables to deal with, you can draw an embedding diagram illustrating the cross section of the wormhole.
Source: Wormhole embedding diagram
How you go from one step to another depends on the author you are following, but all this hoopla is just to show you the (no offence), baby steps in deriving a metric from the EFE and then using it to show how the curved wormhole is produced. Without choosing your physical situation to allow you to suppress variables, it would be very and (totally unnecessarily) difficult to get to the wormhole drawing stage.
We don't have the metric at the start, what we do have is this horror below, the Riemann tensor elements (all of them based on the metric):
$${\displaystyle R^{\rho }{}_{\sigma \mu \nu }=\partial _{\mu }\Gamma ^{\rho }{}_{\nu \sigma }-\partial _{\nu }\Gamma ^{\rho }{}_{\mu \sigma }+\Gamma ^{\rho }{}_{\mu \lambda }\Gamma ^{\lambda }{}_{\nu \sigma }-\Gamma ^{\rho }{}_{\nu \lambda }\Gamma ^{\lambda }{}_{\mu \sigma }}$$
The R. tensor is a set of second order, coupled, nonlinear partial differential equations, (as I am pretty sure you know already, sorry).
The holding functions are no value in themselves, they are literally just letters, they help remind us of the real functions we need to find (and we can manipulate them to reduce their number). In order to discover the metric and then use that to draw the wormhole, they are handy place holders, that's all. In flat space we don't need them, they are equal to one, but they are one of the tricks you need to use to find the curved space metric, as the functions are not equal to one in places where there is a gravitational mass.