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.
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.