I’ll just answer the physics question, since I don’t know anything about cellular automata.
When you say time-reversal symmetry, do you mean that t → T-t is a symmetry for any T?
If so, the composition of two such transformations is a time-translation, so we automatically get time-translation symmetry, which implies the 1st law.
If not, then the 1st law needn’t hold. E.g. take any time-dependent hamiltonian satisfying H(t) = H(-t). This has time-reversal symmetry about t=0, but H is not conserved.
“Time-Symmetric” and “reversible” mean the same thing to me: if you look at the system with reversed time, it obeys the same law. But apparently they don’t mean the same to OP, and I notice I am confused. In any event, as Mr Drori points out, symmetry/reversibility implies symmetry under time translation. If, further, the system can be described by a Hamiltonian (like all physical systems) then Noether’s Theorem applies, and energy is conserved.
From my perspective, time reversibility means that you can always simulate the laws of physics backward in a well defined way, and time symmetry means the laws of physics remain the same if you simulate it backward in time.
I didn’t see an obvious way of showing that “simulating the laws of physics backwards in a well defined way” which is reversibility, always implied symmetry, which means that the laws of physics remain unchanged when running the simulation backwards.
Also, he showed this:
When you say time-reversal symmetry, do you mean that t → T-t is a symmetry for any T?
If so, the composition of two such transformations is a time-translation, so we automatically get time-translation symmetry, which implies the 1st law.
If not, then the 1st law needn’t hold. E.g. take any time-dependent Hamiltonian satisfying H(t) = H(-t). This has time-reversal symmetry about t=0, but H is not conserved.
Which I’m pretty sure doesn’t mean that all physical systems that are described by a Hamiltonian satisfies Noether’s theorem, only the ones that have time-symmetry for all T/any T.
For some context on the question, I’m not focused on our physical universe, but about hypothetical universes where time-reversible laws may not always imply time-symmetric laws, nor does such a time-symmetric universe have symmetry for all T.
I’m essentially asking if these are able to be picked apart in the general case, not the specific case like in our universe.
I was right to seperate the 2 concepts, since @ProgramCrafter has provided an example where a time-reversible law doesn’t imply a time symmetry, possibly avoiding the first law of thermodynamics in the process. Here’s the link:
Hm, I’m talking about time reversible physical laws, not necessarily time symmetric physical laws, so my question is do you always get time-symmetric physical laws that are symmetric for any T, out of time-reversible physical laws?
See also this question in another comment:
For the physics case, I’m asking essentially whether there can be a physical rule (at least in a hypothetical universe different than our real one) that is time-reversible, but not having time shift symmetry, and thus not implying conservation of energy, or if time-reversible physical rules always imply time shift symmetry/time symmetry.
Another way to say it is I’m asking if there are hypothetical time-reversible rules that don’t have the first law of conservation of energy due to not implying time symmetry for any T.
I have edited the question to clarify what exactly I was asking.
I’ll just answer the physics question, since I don’t know anything about cellular automata.
When you say time-reversal symmetry, do you mean that t → T-t is a symmetry for any T?
If so, the composition of two such transformations is a time-translation, so we automatically get time-translation symmetry, which implies the 1st law.
If not, then the 1st law needn’t hold. E.g. take any time-dependent hamiltonian satisfying H(t) = H(-t). This has time-reversal symmetry about t=0, but H is not conserved.
“Time-Symmetric” and “reversible” mean the same thing to me: if you look at the system with reversed time, it obeys the same law. But apparently they don’t mean the same to OP, and I notice I am confused. In any event, as Mr Drori points out, symmetry/reversibility implies symmetry under time translation. If, further, the system can be described by a Hamiltonian (like all physical systems) then Noether’s Theorem applies, and energy is conserved.
From my perspective, time reversibility means that you can always simulate the laws of physics backward in a well defined way, and time symmetry means the laws of physics remain the same if you simulate it backward in time.
I didn’t see an obvious way of showing that “simulating the laws of physics backwards in a well defined way” which is reversibility, always implied symmetry, which means that the laws of physics remain unchanged when running the simulation backwards.
Also, he showed this:
Which I’m pretty sure doesn’t mean that all physical systems that are described by a Hamiltonian satisfies Noether’s theorem, only the ones that have time-symmetry for all T/any T.
For some context on the question, I’m not focused on our physical universe, but about hypothetical universes where time-reversible laws may not always imply time-symmetric laws, nor does such a time-symmetric universe have symmetry for all T.
I’m essentially asking if these are able to be picked apart in the general case, not the specific case like in our universe.
I was right to seperate the 2 concepts, since @ProgramCrafter has provided an example where a time-reversible law doesn’t imply a time symmetry, possibly avoiding the first law of thermodynamics in the process. Here’s the link:
https://www.lesswrong.com/posts/NakKdgW4BJKdwXAe8/#yStS9fff4HzrtPZAC
Hm, I’m talking about time reversible physical laws, not necessarily time symmetric physical laws, so my question is do you always get time-symmetric physical laws that are symmetric for any T, out of time-reversible physical laws?
See also this question in another comment:
I have edited the question to clarify what exactly I was asking.