B Questions about Noether's theorem -- Conservation of Energy and Mass

[Agent Smith]

Very basic question ... Heard someone say that Noether's theorem talks about, among other things, invariance (under transformation). Further, possibilities were discussed:
1. Invariance in time
2. Invariance in space

3. Conservation of energy (kinetic?)
4. Conservation of mass

I forgot which goes with which. Is conservation of energy about invariance in time and conservation of mass about invariance in space or is it the opposite?

Please help.

 

[jbriggs444]

I forgot which goes with which. Is conservation of energy about invariance in time and conservation of mass about invariance in space or is it the opposite?

Energy goes with time invariance. Momentum goes with spatial invariance (translations). Angular momentum goes with invariance under rotations.

 

[renormalize]

I forgot which goes with which. Is conservation of energy about invariance in time and conservation of mass about invariance in space or is it the opposite?

Please help.

Invariance and Noether's theorem have the implications:

• Invariance under translations in time $\Rightarrow$ conservation of total energy.
• Invariance under translations in space $\Rightarrow$ conservation of total linear momentum.
• Invariance under rotations $\Rightarrow$ conservation of total angular momentum.
• BONUS: invariance under electromagnetic gauge transformations $\Rightarrow$ conservation of total electric charge.

But there is no "conservation of mass"; e.g., massive electrons and positrons can annihilate to create pairs of massless photons.

 

[Agent Smith]

@jbriggs444 and @renormalize muchas gracias.

What does Noether's theorem actually do? Does it provide a mathematical "model" from which the conservation laws emerge/follow? I can imagine creating a/using a preexisting mathematical "object" that can parse transformations and then mapping the trasnformations and properties that are invariant under those transformations to physics entities like passage of time/movement through space and energy/momentum.

 

[haushofer]

Very basic question ... Heard someone say that Noether's theorem talks about, among other things, invariance (under transformation). Further, possibilities were discussed:
1. Invariance in time
2. Invariance in space

3. Conservation of energy (kinetic?)
4. Conservation of mass

I forgot which goes with which. Is conservation of energy about invariance in time and conservation of mass about invariance in space or is it the opposite?

Please help.

This is a good question. No, conservation of mass is in the non-relativistic case not described by a symmetry of space or time. In the so-called Bargmann algebra it is described by a central extension of the Galilei group, and one cannot assign spacetime transformations to it. See also chapter 3 of

https://pure.rug.nl/ws/portalfiles/portal/34926446/Complete_thesis.pdf

One can reinterpret this transformation as a spatial symmetry by extending space with an extra dimension, but I can't find the paper anymore and I don't remember the details anymore (and I guess it's a bit contrived for what you ask for).

By the way, the Galilei boosts also have conserved charges along with them, but they aren't very illuminating. See e.g.

https://physics.stackexchange.com/q...ariant-associated-with-the-symmetry-of-boosts

 

[haushofer]

@jbriggs444 and @renormalize muchas gracias.

What does Noether's theorem actually do? Does it provide a mathematical "model" from which the conservation laws emerge/follow? I can imagine creating a/using a preexisting mathematical "object" that can parse transformations and then mapping the trasnformations and properties that are invariant under those transformations to physics entities like passage of time/movement through space and energy/momentum.

In modern physics, one often takes symmetries as a starting point. The continuous symmetries then generate conservation laws, which you calculate by using Noether's theorem. So your question is answered "yes", although I'd rather talk of a procedure or algorithm instead of "model" ;)

 

[Agent Smith]

symmetries

For faults that are my own, I'm fixated on reflection symmetry. It seems that symmetry has a more general meaning e.g. in rotational symmetry, there's invariance: we're unable to distinguish, post-rotation, the preimage from the image. Do you have the time to explain symmetry to me? Gracias

 

[pbuk]

For faults that are my own, I'm fixated on reflection symmetry.

What do you think reflection symmetry means in the context of spacetime geometry? Do you think that physical objects can undergo reflective transformations?

It seems that symmetry has a more general meaning e.g. in rotational symmetry, there's invariance: we're unable to distinguish, post-rotation, the preimage from the image.

[Edit: this paragraph doesn't really say what I intended to say, so please don't nitpick it, but I will let it stand because the intention is to make @Agent Smith think about the differences between the symmetries of an object and of a geometry]. This is also the case for translational and rotational symmetry. However it is not the case for reflection symmetry. Does this affect what you think about the questions above?

Do you have the time to explain symmetry to me?

No, but Wikipedia does:
https://en.wikipedia.org/wiki/Symmetry_(physics)
https://en.wikipedia.org/wiki/Symmetry_(geometry)

 

[Vanadium 50]

Heard someone sa

Here we go again. Do the PF guidelines on sources mean nothing to you?

 

[haushofer]

For faults that are my own, I'm fixated on reflection symmetry. It seems that symmetry has a more general meaning e.g. in rotational symmetry, there's invariance: we're unable to distinguish, post-rotation, the preimage from the image. Do you have the time to explain symmetry to me? Gracias

An expression has a symmetry if a change of coordinates leaves the expression invariant. E.g. the inner product between 2 vectors is invariant under a rotation. That's it, I guess