Closed timelike curves in the spin-2 gravity formalism Let's say we take some topologically trivial CTC spacetime, like the Gödel metric: 
$$ds^2 = -dt^2 - 2e^{\sqrt{2}\Omega y} dt dx - \frac{1}{2}e^{\sqrt{2}\Omega y} dx^2 + dy^2 + dz^2$$
And then I recast it as a spin-2 field over Minkowski space ($g_{ab} = \eta_{ab} + h_{ab}$): 
$$ h = -\left( \begin{array}{ccc}
0 & e^{\sqrt{2}\Omega y} & 0 & 0\\
e^{\sqrt{2}\Omega y} & \frac{1}{2}e^{\sqrt{2}\Omega y} + 1 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 \end{array} \right) $$
Mathematically, the two should be equivalent, but it now all takes place over Minkowski space. How would then the retrocausality effects be explained in this configuration? I vaguely suspect that the gravitational field might spill outside of its light cone perhaps, but how to check it then? Should I try to find out its stress energy tensor and check its dominant energy condition? Is there a simpler way of doing it? And if that is not the explanation, then what is? Could it perhaps even spill in the past light cone? And if possible, are there any papers on the topic? I haven't been able to find any.
Edit: Oh also I guess the interaction with matter could generate a weird mass term that might become negative? Maybe?
 A: If you want to make a flat space theory for a topologically trivial manifold you can do it in the standard ways if your metric was very very close to the Minkowski metric.
Yours is not. So for instance you aren't going to be able to ignore higher order terms in $h$ since in one of the $y$ directions your $h$ blows up.
You can still compute an $h$ field by taking the GR metric and subtracting the Minkowski metric. But then every single equation from GR that uses the metric will have to be replaced with an equation for SR that couples to the $h$ field.
But it will be totally different physics than SR physics. For instance in regular SR physics a force describes the rate of rotation (in Minkowski space) of the tangent to the world line of a particle, and since it is a combination of a spatial rotation and a boost it changes the direction and/or magnitude of the velocity respectively. Thus forces never make something exit their future light cone.
But when you write the geodesic equation as an interaction between the $h$ field and a particle you see that the interaction is not a force.
You also see that initial conditions that were physical in the GR version can be unphysical in the SR+$h$ version. For instance the metric you listed allows a world line with an initial condition with a tangent in entirely in the x direction. But in SR that corresponds to having an infinite initial velocity, and SR allows particles that are initially moving FTL to continue to move in such a fashion.
Since SR doesn't tell you how to get your initial conditions it won't tell you what is wrong. But one thing that is wrong is that we cont have such an $h$ field and we don't have such initial conditions and we don't have a way to evolve our universe to become like those either.
If you want to blame an energy condition, you can go beyond the test particle limit and give the CTC following object it's own mass and hence energy and then the initial condition of it moving FTL will violate some energy conditions when you include the object's momentum exceeding its energy.
But really its just that the SR version isn't at all like the GR version because the covariant derivative and the geodesic equations mean this non small $h$ field couples with all the other equations to give interactions that aren't forces and aren't the regular (e.g. Maxwell) equations.
