It is an often mentioned assumption in physics that in going from classical to relativistic spacetime the main difference is that the absolute time postulate holding in the former is "relaxed" or abandoned as a physical premise wich leads to generalizing the Galilean group. But I wonder how exactly is this implemented mathematically since I don't think that just going to an indefinite signature or to a non-compact group of rotations and boosts by itself is equivalent to abolishing absolute time, even if the simultaneity slicings are no longer unique when the limiting velocity c at each frame is no longer infinity. One can of course say that the simultaneity slices are now just a convention and that the absolute time that enters in the Einstein synchronization is purely conventional, but still operationally they are still there and physical consequences are derived from these conventions. So is there something else to abolishing absolute time mathematically?
I'll justify my question with the well known fact that there is a theory mathematically equivalent to SR, with the same transformations and giving the same predictions which was held by Lorentz himself (Lorentz ether theory) that uses a preferred frame and includes a non-observable ether with absolute time. I'm in no way trying to imply that it is the correct way to look at things, I'm just bringing it up to give an example of a theory that holds on to absolute time and is mathematically equivalent to SR, and uses the same trnasformations so they are not the element that mathematically prevents from having an absolute time.