How is the direction of time determined in general relativity?

Question

In special relativity every frame has its own unique time axis, represented in Minkowski diagrams by a fan-out of time vectors that grows infinitely dense as you approach the surface of the light cone for a given rest frame. Viewed from any other frame, these same time vectors shift about exactly the same fan-out topology, so no one frame perspective is any more "correct" than any other one.

In general relativity this does not quite seem to be the case. If nothing else, all those worldlines extend over billions of years and so presumably end up having a quite precise average time orientation for any given local region of space, even if you select a complicated overall topology. Conversely (and I assume equivalently), if space is viewed interpreted as manifold, the idea of an average "flatness" that is orthogonal to this long-term time axis appears to be assumed, since it would look quite odd to see the entire universe slanted relative to its own time axis, yes? So:

1. Am I understanding this correctly? Does GR applied over eons necessarily end up defining and distinguishing a unique and well-defined overall time axis?

2. If so, how is this unique time axis ascertained in experimental practice? Is it taken to be some sort of very large center-of-momentum average, as the billions-of-years bundle analogy would seem to suggest?

3. If a distinguished time arrow does exist in GR, does the existence of that axis imply any experimentally meaningful implications for its associated SR frame?

Answer 1

There is a reasonably well-defined unique time in our universe - the proper time of comoving observers, those who have no peculiar velocity relative to the Hubble flow. You can look at a large number of randomly chosen distant galaxies and ask if their radial velocities toward/away from you average out to 0, in which case your rest frame coincides with this special frame.

Alternatively, you can examine the CMB and ask if you seem to be moving with respect to it. If you are, there will be a blueshift in front of you and a redshift behind. In fact, we do observe a small dipole anisotropy, even after correcting for the motion of the Earth around the Sun and the Sun around the galaxy. Thus our time axis is slightly "tilted" with respect to the universe as a whole.

Of course, this distinction is only possible in a universe with matter distributed in such a way that spacetime can be foliated into homogeneous 3-surfaces.¹ And the selected frame is only "unique" in the sense of being the only isotropic one. Any timelike path through the manifold is equally valid for defining some universal clock and for foliating space; it's just that any other path will have an observer who sees an anisotropic universe, possibly with a time-dependent anisotropy.

This won't have any impact on local (read: SR) physics. If you restrict yourself to a small patch of spacetime, then we still have that all Lorentz transformation-accessible frames are equally valid.

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¹ I'm pretty sure such homogeneity, if present on a single such Cauchy surface, is preserved under the evolutionary equation(s) for spacetime (e.g. the ADM equations).