If the universe is heading for a big crunch, when the universe starts to collapse will entropy decrease and the arrow of time consequently reverse or not? I'm interested in the explanations, not just the answer.
The expansion of the universe have no influence at all on smaller scales (small in this case means the size of galaxies). The universe expands but galaxies are held together by their gravity.
If the universe would stop expanding and start to collapse the galaxies would keep on rotating, the stars would keep on turning nice, well ordered, nuclear enery into plain old heat, your metabolism would still turn the well ordered chemical energy in your food into heat and poop... Simply put, entropy will keep on increasing.
The only scenario where the arrow of time disappears is when all energy is converted to heat, uniformly spread throughout the universe. Then the entropy is maxed out and the arrow of time ceases to exist. Since nothing can happen in this universe the need to sort events chronologically also disappears so no one will miss the arrow.
1$\begingroup$ Roberth, what if the universe only begins to contract after the heat death. Can you explain why a collapse after a heat death would not lead to a decrease in entropy. $\endgroup$– KenshinNov 15, 2013 at 2:12
Total entropy of the universe is equal to the total area of the space boundary, according to holographic principle.
Our universe is expanding, asymptotically approaching de Sitter space. In de Sutter space the radius of the cosmic horizon is constant and equal to the Hubble radius - the distance at which cosmological red shift becomes infinite.
For expanding universe to start contracting, it should first stop expansing. Not expanding and not contracting universe is flat. This means it does not have the horizon, and its entropy is infinite.
In other words, before the universe to start to contract, its entropy should reach infinity.
2$\begingroup$ You should mention that the status of the holographic principle is, in general, that of a conjecture, and not a feature of e.g. classical general relativity. $\endgroup$– ACuriousMind ♦Nov 21, 2015 at 0:05
1$\begingroup$ @ACuriousMind all new theories are usually tested against holographic principle and if they contradict it, the theories are usually discarded. $\endgroup$– AnixxNov 21, 2015 at 0:06
2$\begingroup$ That people believe a good theory of quantum gravity should obey it does not mean you should present it as a fact. The currently accepted theories of nature (quantum field theory and classical general relativity) do not obey it. $\endgroup$– ACuriousMind ♦Nov 21, 2015 at 0:09