There's four simple-ish models to generate closed timelike curves :
- The timelike cylinder/torus (Minkowski space where $t = 0$ and $t = T$ are identified, and also possibly spatial dimensions)
- Misner space (Minkowski space identified along a boost)
- The Deutsch-Politzer spacetime (two spacelike cuts in Minkowski space identified)
- Thin-shell wormholes (spheres identified in Minkowski spacetime with a time-shift in between).
Those all have the benefit of being flat space (except for wormholes, where the shell itself has a mass distribution), so that everything can be done as in flat spacetime.
The simplest forms of matter to study causal loops are free point particles, in which case you only have to worry about geodesics on those spaces. Since this is just Minkowski space, this is basically just the study of straight lines on it. You can already find interesting results with this, such as for the Minkowski torus $S \times S$. If you have the identification
$$(x,t) \sim (x + L n, t) \sim (x, t + T m)$$
for $n,m \in \mathbb{Z}$, with $T$ the time period and $L$ the circumference of the spatial section, consider the solution of the geodesic equation starting at $(0,0)$, and parametrized by coordinate time, to simplify matters.
$$x^\mu(t) = u^\mu t$$
with $u^\mu = (1, v)$. You are required to additionally obey the condition $(T, v T) = (0, v T)$.
This means that, at any time (but in particular at $t = 0$), the curve will cross the spacelike hypersurface at several points, ie at $x = nvT$ for $n \in \mathbb{Z}$. But as the spacelike hypersurface is itself periodic, this means that the intersection at $t = 0$ will actually be at $x = nvT \mod L$.
If $vT/L \in \mathbb{Q}$, this is fine. If $vT/L = p/q$, then after $n = q$ cycles, the curve rejoins with itself (since $x = nvT \mod L= pL \mod L = 0$). But otherwise, the curve will never rejoin with itself, and the particle will be dense on the manifold (it will appear at every point). This is basically the problem of irrational geodesics on an torus with a spacetime.
The same reasoning can be done for scalar fields without too much troubles. For interacting fields though, where the problem gets more interesting, there is no "proper" physical system that can be done. Even the simplest non-linear systems (such as the Sine-Gordon model or even just charged point particles) haven't been done for such spacetimes. Here's a few examples of ideas that have been tried, at various levels of rigor (trigger warning : a lot of them involve guns) :
The earliest one, quite possibly, is Feynman and Wheeler's 1949 paper. While not actually about closed timelike curves (it's about advanced waves), the principle is fairly similar and was used for closed timelike curves by other authors :
To formulate the paradox acceptably, we have to eliminate human
intervention. We therefore introduce a mechanism which saves charge
$a$ from a blow at 6 p.m only if this particle performs the expected
movement at 8 a.m. (Fig. 1). Our dilemma now is this: Is $a$ hit in
the evening or is it not. If it is, then it suffered a premonitory
displacement at 8 a.m. which cut off the blow, so $a$ is not struck at
6 p.m.! If it is not bumped at 6 p.m. there is no morning movement to
cut off the blow and so in the evening $a$ is jolted!
Fig. 1. The paradox of advanced effects. Does the pellet strike $X$ at 6 p.m.? If so, the advanced field from $A$ sets $B$ in motion at 1
p.m., and $B$ moves $A$ at 8 a.m. Thereby the shutter $TS$ is set in
motion and the path of the pellet is blocked, so it cannot strike $X$
at 6 p.m. If it does not strike $X$ at 6 p.m., then its path is not
blocked at 5.59 p.m. via this chain of actions, and therefore the
pellet ought to strike $X$
To resolve, we divide the problem into two parts: effect of past of
$a$ upon its future, and of future upon past. The two corresponding
curves in Fig. 2 do not cross. We have no solution, because the action
of the shutter on the pellet, of the future on the past, has been
assumed discontinuous in character.
Fig. 2. Analysis and resolution of the paradox of advanced effects. The action of the shutter on the pellet—the interaction of past and
future — is continuous (dashed line in diagram) and the curves of
action and reaction cross. See text for physical description of
solution.
This was used a few times, with a few variants, such as Clarke's Time in general relativity :
To see how this is so, consider the case already cited of a person who
meets his former self in circumstances in which, if physics were
normal, he would be able to shoot him. Then, as a preliminary step in
the analysis, let us replace the complex human being by a simple
automaton which nonetheless exhibits the abnormal physics referred to.
This apparatus is to consist of a gun, a target, and a shutter so
arranged that the impact of a bullet on the target will trigger the
shutter so as to move in front of the gun. It pursues a
causality-violating curve in spacetime in such a way that two points
on the object's world line $A$ and $B$, with $B$ later in the object's
history than $A$, are physically contemporaneous and disposed as in
Figure 1 so that the gun at $B$ is aimed at the target at $A$ and the
shutter is initially up at $A$.
Suppose now that the machine "shoots its former self" : the gun at $B$
is fired, either by an automatic timing mechanism or by the
intervention of a human being making a conscious decision. If the
shutter in $B$ were still up, the bullet would strike the target at
$A$, which would cause the shutter in $B$ to be down, a contradiction.
But if the shutter were down in $B$, then the bullet would be stopped,
the target $A$ would not be hit, and the shutter in $B$ should still
be up: the shutter is up if, and only if, it is down; the situation is
logically impossible.
and Kriele's "Spacetime : Foundations of general relativity and differential geometry" :
At a first glance, the possibility of "free will" seems to be at the
center of the issue. However, following (Wheeler and Feynman 1949)
Clarke (1977) has re-formulated the thought experiment in terms of a
simple machine and has argued that the thought experiment is
fallacious : Assume that there is a gun directed at a target in
spacetime. This target is connected with a shutter which, if closed,
blocks off the path between the gun and the target : If the gun is
triggered, the bullet will hit the target which in turn will cause the
shutter to fall. A second shot will now be blocked by the shutter and
therefore cannot hit the target (c.f. Fig. 8.1.3). Now assume that the
configuration is located in a region with causality violation such
that the shutter falls along a closed timelike curve so that it blocks
the bullet before the gun has been triggered. Again we seem
to arrive at a contradiction : If the shutter is open the bullet can
hit the target. But the target closes the shutter which in turn blocks
the path of the bullet.
As you mention in your post, a lot of the interacting examples are elastic billiard balls, as they're the closest we can get to a rigorous system. The system in question is a set of $n$ point-particles which obey geodesic motion (in the spacetimes considered, almost entirely straight lines), except where they intersect, where the conservation of four-momentum will dictate their behaviour.
There's three different types of outcomes (for any systems, but in particular here) : either there's no self-consistent solutions given a system of billiard balls with some initial velocity, or there's exactly one solution (the system is called benign, and we can simply predict what happens), or there is more than one solution. Behaviours vary a lot for billiard balls. From Earman, there are always multiple solutions on the timelike cylinder. The Gödel spacetime is benign (this is because only accelerated curves are closed, and this isn't the case for elastic collisions). Thin-shell wormholes seem to only have infinitely many solutions for most non-trivial systems, there doesn't seem to be any inconsistent solutions.
The paper you list does have fairly simple mathematical results for this, since, in the non-relativistic regime and moslty ignoring the traversal of the wormhole, everything is just classical mechanics. While the calculations are fairly long, there's nothing too physically involved once you just admit the basic result for how the momentum is conserved upon traversal of the wormhole : momentum is conserved, $v_{\text{out}} = v_{\text{in}}$, and if the particle has angle $\psi_{\text{in}}$ compared to the line connecting the two wormholes, and enters the throat at the angle $\phi$, then it will exit at angle $\psi_{\text{out}} = \psi_{\text{in}} - 2\phi$. After that, everything is just classical mechanics.
Macroscopic mechanical systems are also used for such examples, as they're easier to deal with, although less rigorous obviously. Novikov did a fair bit of them, such as in Time machine and self-consistent evolution in problems with self-interaction, where he used the following few systems :
First, a system of tubes connected to the wormhole, such that a piston may travel back and block itself from going forward. Due to the lack of freedom to move sideways, this one does contain initial conditions where there is no consistent development.
Then, to avoid any issue of the past object replacing the future object, he also considered a few bomb-related systems. The first system is a ball rigged to explode when touching any other object.
The initial data are arranged in such a way that the ball enters mouth
$B$, emerges from mouth $A$ in the past, continues the motion and
arrives at the point $Z$ just in time to collide with the "younger"
version itself. This encounter leads to the explosion. We did not take
into account the influence of the future on the past before the ball
entered the mouth $B$, and this is the reason for the "paradox."
But there is a self-consistent evolution, as shown in Fig. 6. The
initial data are the same as in Fig. 5, but before reaching the point
Z it meets the fragment of the explosion of itself. This fragment hits
the ball and it is the cause of the explosion, the fragments of the
ball fly in all directions with velocities much larger than the
velocity of the ball. Some of them fly into mouth Band emerge from
mouth $A$ in the past. One can show that they will continue to fly in
practically all directions from mouth A, because they have different
impact parameters when they few into mouth $B$. One of the fragments
from mouth $A$ crosses the trajectory of the ball at the point $Z'$
exactly at that moment when the ball arrives at the same point $Z'$.
This fragment is the cause of the explosion of the ball. The
consequence of the explosion (the fragment) is the cause of the
explosion.
Fig. 5. The self-inconsistent evolution in the problem of a ball with a bomb.
Fig. 6. The self-consistent evolution in the same problem as Fig. 5.
A bit less rigorous than before, since we're making fairly grand hypothesis about the trajectory of fragments.
There's quite a variety of such toy models, both in those papers and others that you might find in their bibliography.
