First recalling here that physical time is not time as understood by our Sensability & Intuition - the proper sense of time; we also recall that famously Newtonian or Einsteinian Physics do not have an arrow of time. The theory is time reversable. It was an urgent question to establish an arrow.
It is thermodynamics that provides such an arrow by using probability via entropy. However there is still no now; and other questions of time remain open.
Now, if no matter was available to universe, that is the universe was empty, one is somehow still convinced that time will flow in the same direction. This of course is a Gedanken experiment; it cannot be carried out ever.
One resolution is to embed thermodynamics in spacetime altogether by atomising spacetime and applying ideas of entropy to the atoms of space & time. Then one expects an arrow of time to emerge from the simple existence of the space-time manifold.
Does this solve the arrow of time question with respect to the Gedanken experiment described above?
Notably one attempt at doing this is through causal set theory where spacetime is taken as atomistic and relational.