What Is Space?

To a first order approximation (e.g. in the FLRW perfect fluid model), a Big Crunch would resemble a time-reversal of the standard big bang cosmology.

In our universe, time reversal results in galaxies appearing at a comoving observer's horizon, and everything inside the horizon becoming denser and hotter.

Ultimately the density and heat is expected to result in beyond-the-Standard Model physics in the matter sector, and (hopefully) new physics in the gravitational sector.

We don't really know what those physics will be. Everyone hopes for something that prevents a gravitational singularity from forming, but so far what we have in terms of possible stabilizers are conjectural at best.

The time-reversals of Big Crunch and the usual-forward-time big bang cosmology with structure formation relate to the BH information loss problem. For a non-eternal BH when we time-reverse from i+ we have a gas of Hawking radiation that collapses into a time-reversed BH which eventually emits matter with much less (Boltzmann) entropy -- ions, molecules, dust, stars and even the things orbiting them. (A time-reversed stellar BH likewise will eventually spit out a white dwarf or neutron star with their respective complex layerings). How does a time-reversed BH "know" how to spit out whole stellar-mass objects when only time-reversed Hawking radiation fell into it? Most quantum gravity programmes hope that strong gravity[1] produces an environment where this sort of (time-reversed) structure formation is likely.

Flipping the arrow(s)-of-time is extremely useful for reasoning in a setting in which strong gravity is important, and extra-underlines the question of why we have the arrow(s)-of-time we do in the first place. "Boundary conditions did it" is pretty unsatisfying.

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[1] i.e., where the radius of curvature is on the order of the Planck length; this mostly comes from the non-renormalizability of perturbatively quantized gravity (where we quantize perturbations of a background metric) and from work in finding the effective field theory limit of semiclassical gravity; generally it's very close to a gravitational singularity and -- depending on the censorship conjecture -- always invisible to outside observers.

Not really. If you have two bodies in hydrostatic equilibrium arranged so that you can extend the rotational axis of one such that it overlaps completely with the rotational axis of the other, how do you come up with "absolute rotation"?

You could consider the case where one is exactly spherical and the other is highly oblate. While you're free to choose a system of coordinates which keeps the highly oblate one non-rotating with respect to the coordinates, you have to appeal to fictitious forces to explain the oblateness of the coordinate-stationary body and the exact spherical symmetry of the coordinate-rotating body. Physics typically takes a simpler form if you decide the spherical body should be non-rotating against a set of coordinates that covers both bodies. (On the other hand, if these are large bodies -- planets, for example -- and you can put a lab on the surface of each, you get the Special Relativistic form of physics for practically all possible experiments wholly contained within each lab).

However, a more typical case is that neither body is exactly spherically symmetrical, in which case you might want to appeal to the view of the motions of deep sky objects observable on each body. One might be ultra-Machian and say that "absolute rotation is determined by the movement of fixed stars", but really you probably should be more interested about whether you need to resort to pseudotensors in your write-down of local physical behaviours, i.e., rely upon the principle of covariance instead, and ignore arguments motivated by less-formal arguments (sometimes only allegedly) related to any number of things Mach said.

The principle of covariance only allows a picking-out of "absolute rotation" in specific types of universe, and that picking-out is not very useful in a universe like ours.

> angular momentum is quantized

Spin is quantized in the Standard Model. Although one can call spin an intrinsic angular momentum, it's not the same thing as the angular momentum of a rotating macroscopic body like a planet. In particular, intrinsic spin survives changes of coordinates, whereas rotation does not (as discussed above).

There is a (very) technical and quite beautiful overview of the difference here:

http://www.askamathematician.com/2011/10/q-what-is-spin-in-p...

Ain't nobody got time for that.