[moleperson]

> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow — a symmetry known as time translation symmetry, represented by the Lie group consisting of the real numbers — implies that the universe’s energy must be conserved, and vice versa. “I think, even now, it’s a very surprising result,” Alekseev said.

Maybe I’m misunderstanding the implication here but wouldn’t it be much more surprising if that weren’t the case?

[openasocket]

The surprising thing isn’t that physics remain the same from one day to another, it’s that that fact is the reason for conservation of energy. There are lots of different symmetries for the laws of physics: the laws don’t change from one day to another, they don’t change from one part of the universe to the next, and they don’t change based on angles (e.g. if you snapped your fingers and rotated the entire universe by 10 degrees around some arbitrary point, the universe would continue exactly the same as before, just 10 degrees rotated). From Noether’s theorem, you can take any symmetry on the laws of physics, and use that to derive a conservation law. In those examples, that gives you conservation of energy, conservation of momentum, and conservation of angular momentum, respectively.

[adrian_b]

It is surprising only when you are not aware of the right definition of energy.

The energy is a ratio between "action" and time, where "action" is a primitive quantity that does not depend on the system of coordinates.

While energy can be computed with various other formulae, like the product of force by length, all the other formulae obscure the meaning of energy, because they contain non-primitive quantities that depend themselves on time and length.

So energy depends directly on time, thus the properties of time transfer to properties of energy.

Similarly, the momentum is a ratio between "action" and length, so the symmetry properties of space transfer to properties of momentum, resulting in its conservation.

The same for the angular momentum, which is a ratio between "action" and phase (plane angle of rotation).

[vlovich123]

> For instance, the fact that the laws of physics are the same today as they were yesterday and will be tomorrow

Don’t we just commonly assume this axiomatically but there’s no evidence one way or the other? In fact, I thought we have observations that indicate that the physics of the early universe is different than it is today. At the very least there’s hints that “constants” are not and wouldn’t that count as changing physics.

[SpaceManNabs]

It is surprising that you can derive conversation laws entirely from the symmetry of lie groups, and that every conservation law can be tied to a symmetry.

[vintermann]

Are conversation laws the converse of conservation laws, or did autocorrect prank you? :)

[optimalsolver]

>the laws of physics are the same today as they were yesterday and will be tomorrow

We do not actually know that the current laws of physics will still hold tomorrow, we just assume they will. That's the entire problem of induction:

https://plato.stanford.edu/entries/induction-problem/

[free_bip]

It's funny you say that, because energy actually isn't conserved in general.

One somewhat trivial example is that light loses energy due to redshift since photon energy is proportional to frequency.

[pdonis]

What "loses energy" actually means here depends on what kind of redshift you're talking about.

If you're talking about gravitational redshift, because the light is climbing out of the gravity well of a planet or star, there actually is a conserved energy involved--but it's not the one you're thinking of. In this case, there is a time translation symmetry involved (at least if we consider the planet or star to be an isolated system), and the associated conserved energy, from Noether's Theorem, is called "energy at infinity". But, as the name implies, only an observer at rest at infinity will actually measure the light's energy to be that value. An observer at rest at a finite altitude will measure a different value, which decreases with altitude (and approaches the energy at infinity as a limit). So when we say the light "redshifts" in climbing out of the gravity well, what we actually mean is that observers at higher altitudes measure its energy (or frequency) to be lower. In other words, the "energy" that changes with altitude isn't a property of the light alone; it's a property of the interaction of the light with the observer and their measuring device.

If you're talking about cosmological redshifts, due to the expansion of the universe, here there's no time translation symmetry involved and therefore Noether's Theorem doesn't apply and there is indeed no conserved energy at all. But even in this case, the redshift is not a property of the light alone; it's a property of the interaction of the light with a particular reference class of observers (the "comoving" observers who always see the universe as homogeneous and isotropic).

[measurablefunc]

Where does the energy go then?

Edit: I just looked into this & there are a few explanations for what is going on. Both general relativity & quantum mechanics are incomplete theories but there are several explanations that account for the seeming losses that seem reasonable to me.

[pvitz]

That symmetries imply conservation laws is pretty fascinating (see the Noether theorem). I guess it seems only strange it you assume already that the conservation law holds.