Does spacetime structure in GR break time symmetry? In Frederic Schuller's GR lectures, he states as a postulate of GR that spacetime comes equipped with a time orientation that distinguishes the "past" direction from the "future" direction, vaguely speaking. More precisely, the metric specifies two light-cones at each point, and the time orientation selects (at each point) which light cone is the "future" direction.
Two questions:


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*Why is this structure necessary, if at all?

*How does this structure preserve CPT symmetry, if at all?



It seems to me that we must pick between one of two problems:


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*This structure is necessary for some intrinsic reason that affects physics, in which case, CPT symmetry does not hold, or

*This structure is not actually necessary, in which case, Schuller's definition of spacetime is burdened with unnecessary structure.

 A: So first of all thank you for the link, Frederic Schuller comes from a tradition of careful German mathematical physicists, and I think his lectures can provide a different spin on a number of nuances people usually do not think of. 
The point is that by this postulate Schuller is implicitly restricting himself to time-orientable manifolds, that is, manifolds where such a globally non-degenerate vector field $T$ can be chosen at all. There are manifolds where the metric is globally non-degenerate yet no such $T$ can be constructed, an example is this "spinning time" manifold with the metric
$$ds^2 = \cos\varphi dx^2 + 2 \sin \varphi dx d\varphi - \cos\varphi d\varphi^2$$
living on the loop $\varphi \in (0,2\pi]$ and $x\in \mathbb{R}$. You can see that as you try to define a future direction and continuously extend it around the loop, you end up pointing in the opposite $x$ direction than at the beginning (while the determinant is always -1, so the metric is non-degenerate). 
It is true that for relativity a choice of a time direction is not needed, because it is (locally) time-reversible as long as the boundary conditions and matter-source dynamics are reversible. Also, the introduction of the vector $T$ carries too much specific information. The time orientation at any given point is more the equivalence class of vectors that can be continuously deformed into each other without changing the sign of their norm. And yes, time non-orientable manifolds are really more of a non-physical example that you typically do not have to care about.
On the other hand, it kind of highlights one of the things you never think of. For instance, people assume they can always define the law of entropy growth by taking an entropy current four-vector $\sigma^\mu$ and saying $\sigma^\mu_{\;;\mu}\geq 0$. But this would be bad if the vector was not future-oriented which also implies you already have a time orientation chosen. Similarly, a binary inspirals and merges due to gravitational waves only because we put time-asymmetric boundary conditions in - good to know when you are trying to compute your Green's function! And so on and so on. So you will basically always use a time orientation when doing physics with GR and Schuller is being upfront about it.
A: In the same way that a Mobius strip is not an orientable surface, not all spacetimes are time-orientable. This author is really postulating two things: (1) that we aren't interested in spacetimes that aren't orientable; (2) that the orientation is something we should fix and consider as a feature of the solution.

Why is this structure necessary, if at all?

These are reasonable things to say about any spacetime that is supposed to be a model of our universe. However, people can and sometimes do talk about cases where these assumptions don't hold.
Actually, we would probably like to have even tighter restrictions for realistic spacetimes: we would like them to be globally hyperbolic, meaning that we can specify initial conditions on a spacelike (Cauchy) surface and evolve the matter fields through time to get a unique solution. GR becomes a vacuous theory if we can't predict the behavior of the matter fields. You can take any spacetime geometry you like and put it in the field equations, and out will pop some stress-energy tensor. But if we can't say whether the stress-energy behaves correctly, then any geometry is possible, and GR has no predictive power.

How does this structure preserve CPT symmetry, if at all?

CPT is a local symmetry of the laws governing the matter fields. Also, GR locally becomes SR, which has PT symmetry. GR itself doesn't have a structure that allows us to talk about global symmetries. In a typical spacetime, there is not even any natural way to define what we would mean by something like a global time-reversal operator.
A: Suppose not -- that your spacetime does not have a coherent time orientation.  Now your manifold contains closed time-like loops.
In particular, there are two points $x$ and $y$ such that the forward lightcone of one intersects the past lightcone of the other and vice versa.  (You can find these pairs in every small enough neighborhood of the boundary between coherently oriented components.)  Worse, at each point of intersection, there is no choice of orientation -- either choice of orientation is incompatible with the choice of lightcone at $x$ or at $y$.  (In fact, if we pick a point of interesction, $p$, and pay attention only to the rays through $p$, the apparent forward and backward time directions are spacelike.  The two forward rays lie on one side of $p$ and the two backward rays lie on the other side.)
(A way to avoid this is to make the coherently oriented components be connected components.  But now you're not doing science because you are making claims about events in spacetime that are disconnected from your observable spacetime.  I.e., this escape makes the entire observable manifold coherently oriented, as postulated.)
