Is time and space symmetric? Or am I misunderstanding symmetry in the physical sense? Also according to noether theorem you'll get corresponding conservation laws to the symmetries which prompts the question of what is symmetrical to the conservation of information law.
Some aspect of a system is symmetric if that aspect of the system remains invariant under some transformation. The most fundamental symmetry is CPT symmetry. Charge (C) - Does the system remain unchanged if we change the sign of the system's charge? Parity (P) - How does the system change if we change the sign of a single spatial coordinate? Time (T) - Does the system change if time were reversed? As far as we know, all systems obey CPT symmetry. To answer your question, we would need to see whether a system has some sort of symmetry associated with it. If a system remains invariant with respect to time, say, then that system has a symmetry associated with time. Noether's Theorem makes a 1-1 correspondence between conservation laws and symmetries. For example, energy conservation is due to the time invariance of systems. As for "conservation of information law". If this is the law that Dembski proposed, he is full of crap. His reasons for proposing such a law is his opposition to biological evolution.
One of the isometries of Minkowski space-time is a Lorentz boost, which "mixes" time and space coordinates. This is as close a thing that I can think of relating to what you said.
Thanks for the replies some really good patient people here. Though having said that I am confused once again. Isn't that law that Dembski proposed used in black hole theory I aint so sure but wasn't the black hole war fought over the whole loss of information thing?
So, Noether's theorem relates differential symmetries in the Lagrangian to conserved quantities. The fundamental Lagrangians that we know about all seem to have certain symmetries. One is that they do not change with shifts in time, this leads to conservation of energy. Another is that they do not change with shifts in space, which leads to conservation of momentum, and so forth. I am not sure what you mean by conservation of information. Are you referring to Liouvilles theorem?