Energy is not conserved on the scale of the Universe. General Relativity has no energy conservation law (it has a stress-energy tensor conservation law). Two examples of this: expansion of the universe causes the total energy contained in radiation to decrease, and the total amount of dark energy to increase.
Noether's second theorem works fine in General Relativity (in fact, this was the historic context of her paper). So for any given time-like vector field, you'll get an energy conservation law. In case of Friedmann cosmology and chosing cosmological time as said vector field, you'll get a term proportional to H² which picks up the change in energy.
However, you won't be able to make this into a covariant expression: Gravitational energy-momentum can be expressed in terms of pseudo-tensors at best...
I have a related question, borne out of my ignorance as I am not a physicist nor a mathematician.
> Noether's first theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
Why is this not obvious? Are not symmetries necessarily the transformations which conserve quantities, and the types of symmetry equivalent to the type of conservation?
Because Noether got a lot of respect for her work despite working (in a time of great sexism, no less!), I know that either I have totally misunderstood or this is a Columbus’ Egg [0] — but I am curious which, and if it isn’t a Columbus’ Egg, what I’ve misunderstood (assuming that sort of thing can even fit into a HN-sized reply and I’m not asking something that would normally be the conclusion of a final year degree level physics module).
Are not symmetries necessarily the transformations which conserve quantities
A priori, a symmetry is a transformation that, when applied to any valid trajectory, yields another valid trajectory. It is not obvious to me why (certain types of) symmetries necessarily yield conserved quantities...
The symmetry defines that which is conserved, doesn’t it?
If I have a motion vector and a force vector, and if work done is ∫ force • displacement dx, and my transformation is one which conserves the scale of and relationship between both vectors (e.g. displacement), then is it not automatically true that work done is also conserved under that transformation?
That's not what Noether's theorem means! The "symmetries" referred to here are operations which leave invariant the action of the physical system (heuristically, the physical laws governing the system). To the extent that one can (at least in principle) write down the action of composite systems as the sum of their actions + interactions, these are taken to mean (again heuristically) the laws of physics writ large. Likewise, "conservation law" here means not specifically the particular result of an experiment (like the work integral you describe), but a more general notion of conservation, in the sense of there being universally invariant conserved quantities (i.e. things that cannot be created or destroyed).
To get a feel for it (and why it may not necessarily be intuitive) consider the following pairs of symmetries and conservation laws:
- The laws of physics are invariant under time translation (i.e. repeating an experiment at different times gives you the same result ceteris paribus). The corresponding conserved quantity is energy.
- The laws of physics are invariant under spatial translation (i.e. repeating an experiment at two different places gives you the same result ceteris paribus). The corresponding conserved quantity is momentum (this is a vector: one component of momentum for each possible direction of translation).
- The laws of physics are invariant under rotation (i.e. repeating an experiment under different orientations gives you the same result ceteris paribus). The corresponding conserved quantity is angular momentum (again a vector, since the rotation group SO(3) has three generators).
- Electromagnetism, constructed as a classical field, exhibits an internal symmetry in the field quantities. The corresponding conserved quantity is the electrostatic charge (actually a 4-vector current).
It is not a priori obvious from from the kind of arguments you've supplied why these should be the case. Demonstrating these would require access to the Lagrangian formalism (i.e. action principles) and how it behaves under these symmetry operations. I would say that, as you suspect, you are fundamentally misunderstanding the situation.
Thanks! I still don’t really get it, but telling me I need the Lagrangian formalism is helpful, and tells me what I need to study in order to really understand Noether's theorems.
I’ve heard of the Lagrangian before, but my knowledge of physics is probably somewhere around the level of someone who has only just finished the first year of their degree, at least judging by the modules my alma mater lists for their physics degree (I got Software Engineering from them 14 years ago, all my physics knowledge since then is personal geekery).
