Surprisingly, the writers themselves understand that the worlds with incompatible arrows of time would be logically inconsistent. They use this whole theme as a method to entertain the readers. The arrow of time is very real, very powerful, very universal, and it influences everything about the real world that has anything to do with incomplete information, with our knowledge (and the way how we formulate problems), or with a macroscopic description of events.
Imagine two subsystems of the physical Universe that would experience the opposite arrows of time. Is that possible? How would it look like? Of course, it is possible to draw the world lines of these objects and T-reversed objects into the same spacetime. At this level, there is no problem. Also, you can read a story backwards. But could the Universes with the contradictory arrows of time ever follow some physical laws similar to those we know? Could these systems with the opposite arrows of time meaningfully interact with each other?
I am convinced that the answer is No.
In the article about Lewis Carroll, Sean Carroll asks the question about the interaction of the objects worshipping incompatible arrows of time. He answers as follows:
Actually that one’s not hard to answer. If two systems with incompatible arrows were to noticeably interact, the one with more degrees of freedom would swamp the other one and quickly “correct” its arrow of time. No being that “remembered the future” would survive very long in the real world.That may sound logical but it is actually a meaningless verbal construct. What does it mean for two objects that disagree about the arrow of time to "quickly" settle their disagreement? If the battle starts at noon, will the argument be over around 11 am or 1 pm?
The two objects can't even agree in what time direction the battle about the domination over the arrow of time should take place. ;-)
Every calculation of probabilities in physics and science in general is based on the knowledge of some initial conditions and the application of the laws of physics to deduce the probabilities of various values of certain quantities in the future. The initial states and "final states" can't really be combined easily.
A "black hole final state" was once proposed as a mechanism how the information gets away from the black hole. Such acausal mechanisms have serious logical problems once you try to describe the interaction of the degrees of freedom with incompatible arrows of time (and this interaction must surely exist if the objects occupy the same region of space). While we might artificially design a set of formulae to calculate some probabilities that assume something both about the initial state as well as the final state, the resulting probabilities can't be interpreted as probabilities extracted from a unitary evolution.
If you impose inequivalent conditions upon the initial state and the final state, you break unitarity because the probability of the initial state evolving into something - the sum over all possible outcomes - won't be equal to one if you simultaneously constrain the final state.
If you try to fix this problem and redefine the formulae for your probabilities so that the probabilities always seem to add up to one, you will violate locality. Such a "normalization" of probabilities will have to be made according to the normalization factors involving properties of the whole Universe which will allow distant subsystems to influence the local ones. And there won't even be any reason to think that such an influence is "small" in any sense.
A Möbius background
You can also discuss the theoretical question how physics works on backgrounds with the Minkowski signature where the arrow of time cannot be defined globally. For example, define the following orbifold. Identify the point
(t,x,y,z) with (-t,x+L,y,z).If you move by L to the right, the time goes backwards. I deliberately chose a nonzero shift L so that the map has no fixed points (but x is periodic with a period of 2L). Such a spacetime reminds you of an infinitely wide Möbius strip but the direction that flips is time-like.
Locally, it is a perfectly ordinary Minkowski spacetime. At least seemingly.
Can you define sensible dynamical laws, e.g. any physical quantities associated with the evolution in such a spacetime? For example, try to define a scattering S-matrix. Normally, the S-matrix informs you about the amplitudes that an initial state, conventionally described by the momenta of particles at t=-infinity, evolves into another state described by momenta at t=+infinity. Can you do it in our Möbius spacetime?
It is not hard to see that the configurations in the past and the future must be essentially identical (up to some phases), due to the orbifold projection. You are only allowed to consider external states in which the particles have essentially the same momenta. In this sense, the number of particles and their energies cannot really change in such a spacetime. Try to define any kind of physical framework similar to one that we use in particle physics. I think you will fail.
The only example in which you won't fail will involve local field equations of motion. Because the spacetime is locally healthy, you may think that everything works just fine. But quantum physics is not about the equations of motion only. It also requires the logical framework, namely a method to connect the initial and final states with the "bulk" of spacetime governed by the evolution that respects the laws of Nature. This logical portion of the physical laws simply won't work even though the local equations of motion seem consistent.
I presented the problem as a problem of quantum mechanics but it already occurs in classical physics. The orbifold constraint described above would force the classical initial state to coincide with the classical final state, up to the translation above. For particular Hamiltonians, it may be a nontrivial task to find out whether states satisfying such a constraint exist at all and if they do, what they are.
When you try to solve this task, you will see that you are imposing non-local and acausal constraints on the allowed states. There is no Hilbert space of the initial states freely generated by various allowed eigenvalues of local and similar operators. And that's too bad because such an act is almost guaranteed to be incompatible with locality and causality. The tolerable orbifolds must act locally and causally at least in the orbifolded understanding of these words.
You are only allowed to identify points in spacetime that can be considered close to one another: identical things simply must be close. But the identification of points in the distant past with points in the distant future can't obey this requirement: they are always far from each other, separated by the effectively infinite evolution encoded in the S-matrix.
So even though locally, the field equations of motion seem unchanged, the whole physical framework becomes dramatically different from the usual one. The non-local and acausal identifications destroy the locality and causality of the theory. It will become impossible to talk about any history that implicitly includes the arrow of time or the assumption about spatial decoupling of two or more subsystems. To be very specific, I talked about the Möbius spacetime but any spacetime that doesn't allow us to define the arrow of time globally will obviously suffer from the same problems.
The world simply can't work like that and I recommend Sean Carroll to join me and Lewis Carroll in our understanding that the worlds with incompatible, spatially dependent arrows of times are just a matter of fiction and entertainment, not a consistent starting point to develop a physical theory and certainly not a prediction of any known serious theory of physics.
And that's the memo.