General Covariance & CTCs Consider Minkowski space defined in coordinates $(t,x,y,z)$. Let us equip the manifold with a time orientation $$T = \frac{\partial}{\partial t}$$ such that $g(T,T)<0$, in  $(-+++)$ signature, this will tell us whether or not a trajectory $v_{\gamma}$ is future-directed or not by checking $g(T,v_{\gamma})<0$.  Now define coordinates $(t',x',y',z')$ in terms of the unprimed coordinates such that a particle with trajectory $(t, 0, 0, 0)$ travels backwards in $t'$ for some time, then forwards in $t'$ again to form a CTC. The time-orientation vector field representation in such coordinates will transform to point in the -ve $t'$ direction for the backwards leg, and to point in the +ve direction for the forward leg, thus the particle remains future-directed.
So now because of wonky coordinates, we have built a Closed Timelike Curve. How do we establish causality, if we could choose any coordinates that we desire? It seems as if one could establish a CTC in any spacetime by just choosing weird coordinates.
 A: I think the problem with the transformation you propose is that it could not be a diffeomorphism: its Jacobian, acting on tangent vectors, would belong to different disconnected sets of Lorentz transformations at different points, so you could not smoothly join the regions in which you flip the time direction to those in which you do not.
Edit:
The reason I was referring to Lorentz transformations in the (admittedly imprecise) first formulation of this answer is because of the fact that at each point the tangent space to the manifold will be isomorphic to Minkowski spacetime (see, for example, pages 49-51 of Carroll's notes).
I will try to give a more formal argument using this fact.
The norm of the velocity is a scalar, so it will stay timelike under any transformation: $g(v_\gamma, v_\gamma) < 0$ is conserved.
Timelike vectors, as you mention, can be either future- or past-oriented, and these regions can be understood as the halves of a cone in Minkowski; they are only joined at the point $v_\gamma = 0$.
This is a singular point in the parameterization; besides since the metric is bilinear we will have $g(0,0)=0$, breaking the condition of the curve being timelike at that point.
So, a transformation can leave a timelike trajectory future-oriented or make it past-oriented (by setting $t \to -t$, for example) but the temporal orientation is a global characteristic of the curve, it cannot be changed partway through.
As a final aside, you can have CTCs in flat spacetime if you change the topology - an example of this is Misner space.
A: A change of coordinates has no effect on topology. Your original curve has the topology of the line, so a change of coordinates can't make it into a closed curve. All you've done is to change the time orientation for half of the curve, and you can do that by fiat, without even bothering to change coordinates. However, that doesn't make it a closed curve, and you will also have a problem because the time orientation will have a discontinuity, which is not allowed.
The answer by Jacopo Tissino points out that you haven't defined your coordinate transformation as a diffeomorphism, but I think that's a side issue, because the coordinates aren't even relevant.
