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Here's my favorite way of thinking about Noether's theorem: In physics, we have to assume the laws of nature are the same everywhere and for all time. Otherwis

Here's my favorite way of thinking about Noether's theorem: In physics, we have to assume the laws of nature are the same everywhere and for all time. Otherwise, we're wasting our time. Because of Noether's theorem, this means there are some conserved quantities. Whatever these happen to be is what we _call_ momentum and energy. If the laws of physics change (they are still the same everywhere, just modified based on an improved theory), then we have to change our definitions of what is conserved. If you define kinetic energy as mv^2/2, then when you take into account relativity, you realize that's not conserved any more. However, because the physical law that relatively predicts still does not change over time, then _something_ must be conserved. So we call that energy instead. Until the next iteration, of course.

lucozade 11y ago
I actually think it may be a little more prosaic than that. In order for us to have a scientifically verifiable theory, or at least a consequence of a theory that is scientifically verifiable, that theory/consequence must be invariant under translations in space, time and rotation. Otherwise there would be no repeatability of any related experiment. As such, by Noether's theorem, that theory/consequence must conserve momentum, energy and angular momentum. At the very least. So any theory that didn't conserve these quantities couldn't be scientifically verifiable. So it's sort of a tautology that scientific theories conserve these quantities.
- drjesusphd 11y ago
  Yes, I think that's what I was trying to say, but you did a better job.
- cbd1984 11y ago
  ... and therefore, anyone peddling a perpetual motion machine must be scientifically illiterate or believe that their audience is, because what they claim to have goes against all of the physical theories which get verified every time we launch a rocket, build a bridge, or throw a damned baseball. I think we should teach a lot more things using the basic framework of "baloney detection": Use what you know to prove that this claim is either bogus or potentially true. Show your work.
- Florin_Andrei 11y ago
  Or maybe what the theorem really tries to say is that the universe is self-consistent in time and space - it does not consist of multiple, fractured domains. Conversely, a violation of NT might indicate just such a fracture.
- tjradcliffe 11y ago
  It is nothing of the kind. It is a matter of perfectly ordinary empirical fact that we live in a universe that, so far as we can tell, is translationally, temporally and rotationally invariant. There is absolutely nothing in Noether's theorem that forbids us from having theories that violate momentum, energy or angular momentum conservation if they happen to describe a universe that has aspects that violate those invariance conditions. There have been theories that do so, like Dirac's weird large number thing. These are completely legitimate theories, and are entirely subject to absolutely ordinary observational, experimental and inferential verifiability (which is why we know Dirac's large number thing is likely false.)
  - lucozade 11y ago
    Yes indeed. However, for us to be able to scientifically verify those theories, something about them must be invariant in space, time and rotation. That's pretty much the definition of a scientifically verifiable theory. My point isn't that there can't be theories without those properties nor that the universe necessarily has them (or not). Just that, if it doesn't, we couldn't verify it scientifically.
    - abecedarius 11y ago
      According to Feynman there once were people saying that since experiments need to be repeatable, to be scientific the laws of physics must be deterministic, by pure logic. Oh, quantum mechanics? Oops.
- nabla9 11y ago
  Note that Noether's theorem does not apply to the universe as a whole. Expanding universe is not invariant under shifts in time. Energy is not conserved in expanding universe. https://www.reddit.com/r/askscience/comments/1m43o0/if_noethers_theorem_says_that_conservation_of/cc5kwst https://www.reddit.com/r/askscience/comments/1m43o0/if_noeth... http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ http://www.preposterousuniverse.com/blog/2010/02/22/energy-i...

lucozade 11y ago

I actually think it may be a little more prosaic than that. In order for us to have a scientifically verifiable theory, or at least a consequence of a theory that is scientifically verifiable, that theory/consequence must be invariant under translations in space, time and rotation. Otherwise there would be no repeatability of any related experiment. As such, by Noether's theorem, that theory/consequence must conserve momentum, energy and angular momentum. At the very least. So any theory that didn't conserve these quantities couldn't be scientifically verifiable. So it's sort of a tautology that scientific theories conserve these quantities.

drjesusphd 11y ago
Yes, I think that's what I was trying to say, but you did a better job.
cbd1984 11y ago
... and therefore, anyone peddling a perpetual motion machine must be scientifically illiterate or believe that their audience is, because what they claim to have goes against all of the physical theories which get verified every time we launch a rocket, build a bridge, or throw a damned baseball. I think we should teach a lot more things using the basic framework of "baloney detection": Use what you know to prove that this claim is either bogus or potentially true. Show your work.
Florin_Andrei 11y ago
Or maybe what the theorem really tries to say is that the universe is self-consistent in time and space - it does not consist of multiple, fractured domains. Conversely, a violation of NT might indicate just such a fracture.
tjradcliffe 11y ago
It is nothing of the kind. It is a matter of perfectly ordinary empirical fact that we live in a universe that, so far as we can tell, is translationally, temporally and rotationally invariant. There is absolutely nothing in Noether's theorem that forbids us from having theories that violate momentum, energy or angular momentum conservation if they happen to describe a universe that has aspects that violate those invariance conditions. There have been theories that do so, like Dirac's weird large number thing. These are completely legitimate theories, and are entirely subject to absolutely ordinary observational, experimental and inferential verifiability (which is why we know Dirac's large number thing is likely false.)
- lucozade 11y ago
  Yes indeed. However, for us to be able to scientifically verify those theories, something about them must be invariant in space, time and rotation. That's pretty much the definition of a scientifically verifiable theory. My point isn't that there can't be theories without those properties nor that the universe necessarily has them (or not). Just that, if it doesn't, we couldn't verify it scientifically.
  - abecedarius 11y ago
    According to Feynman there once were people saying that since experiments need to be repeatable, to be scientific the laws of physics must be deterministic, by pure logic. Oh, quantum mechanics? Oops.
nabla9 11y ago
Note that Noether's theorem does not apply to the universe as a whole. Expanding universe is not invariant under shifts in time. Energy is not conserved in expanding universe. https://www.reddit.com/r/askscience/comments/1m43o0/if_noethers_theorem_says_that_conservation_of/cc5kwst https://www.reddit.com/r/askscience/comments/1m43o0/if_noeth... http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ http://www.preposterousuniverse.com/blog/2010/02/22/energy-i...