Reversibility of the arrow of time I often read in physics vulgarisation books about how paradoxal it is that the time seems to go only one direction, as entropy grows with time, and nobody has ever seen a broken cup repair itself and jump back from the table it just fell, but I find it pretty confusing.
Surely if time was running in reverse, our brains would too, and it would make no difference to us if the time was running forward or backward, time could go back on forth and we wouldn't be the wiser.
What am I missing? Apologies if my question seems too vague.
 A: It is a convention that we take the direction of increasing time as the direction of increasing entropy. We could reverse the convention. But in either case the thermodynamic arrow of time must align with the perceptual arrow of time because it is not possible to use an observed state in the present (a memory or other record) to infer the details of a state of higher entropy. So our perceptual past must always be in the direction of lower entropy.
A: Generally, I think what these popular treatments are trying to get at is the Loschmidt paradox. Roughly, this asks the following question: why, if the underlying laws of physics are symmetric, is there any preferred direction of time at all? Why is one direction of time (the one we call the past) different from the other (the one we call the future)?
There have been many proposed solutions to this. People have been arguing about it since Boltzmann's time and still do - it's really not trivial at all. But I think probably the most widely accepted solution is that time is asymmetric because the universe's boundary conditions are asymmetric - entropy is lower in the past because the big bang is in the past. It's similar to the way a room is warmer close to a fire, even though the laws of physics are the same in all parts of the room - the fire breaks the symmetry.
If that's right then what you say makes sense. If the big bang was in the future instead of the past then we wouldn't notice. We'd just call that direction the past and the other one the future. Our arrows of time would be inverted but we'd experience everything just the way we experience it now.
But if there were no big bang, or if there were big bangs at both ends of time and we were roughly in the middle, then there would be no reason for the past and the future to be different from each other at all. If we lived in such a universe we might well see objects spontaneously reassembling themselves, because there's be no reason to expect that the object and the brain of the person observing it would have the same arrow of time. (Though more realistically, such a universe would probably just be a gas in equilibrium without any people or objects in it at all.)
In the end, the fact that all parts of our universe seem to have the same arrow of time is a really non-trivial observation that needs a very non-obvious explanation, and that makes the question "why doesn't a broken cup repair itself and jump back from the table it just fell" actually rather a good one.
A: You are not missing anything, what you said is correct.
Physical theories are time symmetric, e.g. Maxwell's equations.
There is one exception and that's cosmology, however that's more of a problem for our cosmological models and doesn't show that the arrow of time is not reversible.
See also this question Does time symmetry cause the matter/antimatter asymmetry?
A: See here: https://en.wikipedia.org/wiki/G%C3%B6del_metric
The Gödel metric is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles (dust solution), and the second associated with a nonzero cosmological constant (see lambdavacuum solution). It is also known as the Gödel solution or Gödel universe.
This solution has many unusual properties—in particular, the existence of closed timelike curves that would allow time travel in a universe described by the solution. Its definition is somewhat artificial in that the value of the cosmological constant must be carefully chosen to match the density of the dust grains, but this spacetime is an important pedagogical example.
A: Entropic time can only be reversed if all fields, real matter/energy ones and the virtual interaction fields, are momentum reversed. To make this happen for the whole universe would require quite something!
And you would surely notice! You would first be aware of the world and then see it. You would hear a reversed thunder stroke and then see a lighting flash, very shortly before the electrons actually discharged the clouds to produce the flash of lightning. In fact, there would travel photons from your eyes to miraculously meet electrons that were traveling towards higher potentials. The photons would even be the cause of the reverse traveling. All emergent photons in the normal situation would form the cause for the time-reversed situation.
The difference with the the normal situation is that there is no begin situation, except for the moment you reverse momentum universally. And the momenta on that moment depend on the begin state of the universe, which is one of low entropy. The very process of changing all momenta is purely a process that can only happen in the mind. Any attempt to realize even a small part of momenta to reverse would involve irreversible processes which enlarges the total universal
entropy (entropy of the universe).
There could be processes though, on small scales, that involve time-reversal, but in that case it's not entropic time that reverses, though that's needed to measure it.
