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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:

  1. Why is this structure necessary, if at all?
  2. How does this structure preserve CPT symmetry, if at all?

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

  1. This structure is necessary for some intrinsic reason that affects physics, in which case, CPT symmetry does not hold, or
  2. This structure is not actually necessary, in which case, Schuller's definition of spacetime is burdened with unnecessary structure.
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    $\begingroup$ This question has nothing to do with GR, it’s just the usual thing about time reversal: it holds microscopically (assuming a theory with T symmetry) but not macroscopically, because of thermodynamic considerations. Schuller is putting in the macroscopic arrow of time by hand. So neither of your problems appears. $\endgroup$ – knzhou Nov 4 '19 at 22:29
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    $\begingroup$ @knzhou I don't think it's so simple--Schuller is sort of a theoretical purist and otherwise avoids adding any structures that aren't fundamentally necessary, so I doubt he's just trying to throw in the statistical arrow of time. $\endgroup$ – WillG Nov 4 '19 at 22:43
  • $\begingroup$ Poisson & Will's textbook Gravity (2014, $\S12.1.4$, 12.5) states that "radiating systems... necessarily break the time-reversal invariance of the underlying theory." This is for both electromagnetic and gravitational waves, under post-Coulombian or post-Newtonian approximation schemes. "In each case the fundamental theory is time-reversal invariant, but the selected solutions specify a direction for the arrow of time." The cause is odd powers of $c^{-1}$. $\endgroup$ – Colin MacLaurin Nov 6 '19 at 1:51
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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.

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  • $\begingroup$ "an inspiraling and merging binary happens only because we put time-asymmetric boundary conditions" - What are these conditions? Isn't the in-spiraling embedded in the Schwarzschild metric that is in tself time symmetric? E.g. would stuff not spiral-out, because this would require specific gravitational waves coming in? But how is this different from irreversibility of a particle decay into multiple parts? +1 $\endgroup$ – safesphere Nov 5 '19 at 1:55
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    $\begingroup$ @safesphere So first of all sorry for being imprecise, I had the particular case of binaries inspiraling due to gravitational waves in mind. There the boundary conditions allow gravitational waves to leave through future null infinity, but you do not supply any through past null infinity. But in principle, you should be able to "despiral" a binary by flipping the boundary conditions and sending in very precisely selected gravitational waves through past null infinity. $\endgroup$ – Void Nov 5 '19 at 17:38
  • $\begingroup$ On geodesics in Schwarzschild: physically speaking, the global Schwarzschild solution only approximates a late-time field after a very time-nonreversible collapse of matter. So the far past of Schwarzschild $t\to -\infty$, in particular the white hole part of the solution, is not physical. So it is true that for every inspiraling (future-oriented) geodesic in global Schwarzschild there is a precisely matching (future-oriented) outspiralling one. But the point is that the outspiralling geodesic starts near the white hole and goes through $t \to -\infty$ at horizon, so it isn't physical. $\endgroup$ – Void Nov 5 '19 at 17:46
  • $\begingroup$ On the last point, would it not be true that we can reverse the velocity of any Schwarzschild in-spiraling object (neglecting gravitational waves of course) and make it out-spiral to return to the same initial position? $\endgroup$ – safesphere Nov 5 '19 at 21:29
  • $\begingroup$ @safesphere So the Schwarzschild solution is approximately valid from some time some matter collapsed to create the black hole, let's call that time $t_{\rm c}$. We can only use the solution for $t>t_{\rm c}$, otherwise it is unphysical. Now, any future-oriented orbit that went from $r>2M$ to $r<2M$ had to pass through $t=+\infty$. Similarly, any future-oriented orbit that goes from $r<2M$ to $r>2M$ has to go through $t=-\infty<t_{c}$. In other words, future-oriented orbits outspiraling from the black hole cannot do so, because they would emerge from a region where the solution is not valid. $\endgroup$ – Void Nov 5 '19 at 21:47
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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.

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  • $\begingroup$ I'm curious about your last paragraph. Why can't we extend local time reversal into a global time reversal in GR and look at whether matter fields still obey the same laws? If global time reversal makes sense in SR, I don't see why it can't make sense in GR. $\endgroup$ – WillG Nov 5 '19 at 16:04
  • $\begingroup$ @WillG: A time reversal operator on a spacetime would be a function that takes events to events. How would you determine what events were mapped to what events? Note that you can't just say $t\rightarrow -t$, because there is no preferred time coordinate. $\endgroup$ – user4552 Nov 5 '19 at 20:09
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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.)

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  • $\begingroup$ Picking a local time orientation does not prevent you from getting closed time-like loops. It just prevents the even more problematic situation where you pass the same point forwards and backwards. Take an empty spacetime and roll time up the time coordinate, i.e. take $\mathbb{R}/\mathbb{Z} \times \mathbb{R}^3$ with the generic diagonal Minkowski metric. Then at each point I can consistently define the future cone as the one for which $t$ increases. Yet there are many closed loops and in fact the future light cone of any point intersects the past lightcone of any other and vice versa. $\endgroup$ – mlk Nov 5 '19 at 8:21
  • $\begingroup$ @mlk : At this level of discourse, I have not seen the term "closed time-like loop" used to describe the boring case of one enforced by global quotient. But I can waste words being less clear to a larger audience. Editing... $\endgroup$ – Eric Towers Nov 5 '19 at 8:27

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