Is energy conservation necessary?

[atyy]

Must energy be conserved in a consistent Poincare invariant theory?

It's not necessary in Newtonian physics. Are things different in special relativity?

 

[PAllen]

Must energy be conserved in a consistent Poincare invariant theory?

It's not necessary in Newtonian physics. Are things different in special relativity?

It would be easier to think about some hypothetical Galilean invariant theory that included violation of energy conservation. It's not obvious to me what such a law or theory would look like. I assume you have something in mind. Then we could play with what would happen if tried to make it Poincare invariant.

 

[atyy]

It would be easier to think about some hypothetical Galilean invariant theory that included violation of energy conservation. It's not obvious to me what such a law or theory would look like. I assume you have something in mind. Then we could play with what would happen if tried to make it Poincare invariant.

Hmm, I guess I was thinking about inelastic collisions, but nothing more specific than yet.

Would Stoke's law work?

 

[PAllen]

Hmm, would Stoke's law work?

I guess I was thinking about inelastic collisions, but nothing more specific than that.

Is Stokes law (of viscous friction) Galilean invariant? That is, first must write it in a form that that is invariant for fluid flow and direction [and direction of gravity. edit: direction of gravity not relevant for simple form of the law]

What is your proposed law for inelastic collisions? I have noticed you often write extremely terse entries and expect others to guess what you mean. Since you are proposing the exercise, why not specify a working example?

For example:

- would conservation of momentum hold?
- would you propose that heating does not occur or is not energy for the purposes of this game?

 

[PeterDonis]

Hmm, I guess I was thinking about inelastic collisions

Inelastic collisions still conserve energy, don't they? They just don't conserve *kinetic* energy--some of the initial kinetic energy gets converted to heat (in the simplest case where there are no other internal energies involved).

[Edit: I see I hadn't fully understood the ground rules.]

 

[PAllen]

Anyway, I think Stoke's law is pretty easy to write in a suitable form with fluid velocity and particle velocity as 3-vectors (norm of difference of vectors). For inelastic collisions, assume heat does not exist, and conservation of 3-momentum holds (KE is simply lost).

So, then the question becomes, can one design a Poincare invariant object to use in place of a 3-vector that does not bring in energy?

I wonder if there are any 'no go' theorems related to this? For example, there are well known theorems that no law satisfying Newton's third law, and allowing objects to interact at a distance, can be Poincare invariant.

 

[atyy]

I wonder if there are any 'no go' theorems related to this? For example, there are well known theorems that no law satisfying Newton's third law, and allowing objects to interact at a distance, can be Poincare invariant.

Yes, something like that would answer the question.

To go off topic a bit, in the case of "action at a distance", I believe the no go theorems have a loophole, so that things can act at a distance if they have a Lagrangian formulation but no Hamiltonian formulation (not sure if I got that backwards) - which I think is how Feynman and Wheeler's theory slips through.

 

[PAllen]

Yes, something like that would answer the question.

To go off topic a bit, in the case of "action at a distance", I believe the no go theorems have a loophole, so that things can act at a distance if they have a Lagrangian formulation but no Hamiltonian formulation (not sure if I got that backwards) - which I think is how Feynman and Wheeler's theory slips through.

That would not satisfy Newton's third law. The delay would cause a violation. Specifically, this theory appears to reproduce Maxwell without a field. However, Maxwell interaction violates Newton's third law.

 

[atyy]

That would not satisfy Newton's third law. The delay would cause a violation.

Yes, that seems to be the case. Actually, I read your statement too quickly and thought you were talking about the Currie, Jordan, Sudarshan "no-interaction" theorem (which you weren't).

 

[Parlyne]

Must energy be conserved in a consistent Poincare invariant theory?

It's not necessary in Newtonian physics. Are things different in special relativity?

Poincare invariance implies conservation of energy (also, momentum, angular momentum, and the quantities which serve as the generators of boosts from a group theoretic perspective but which don't, to my knowledge, have any generally accepted name). Remember that Noether's theorem relates symmetries to conservation laws. Energy, in particular, is related to time-translation symmetry, which is the the time component of the translational part of Poincare symmetry.

 

[Dale]

Must energy be conserved in a consistent Poincare invariant theory?

If energy were not conserved and the theory were Poincare invariant then it could not be expressed in terms of a Lagrangian.

If a theory can be expressed in terms of a Lagrangian then Poincare invariance implies time translation invariance which implies energy conservation.

I believe that it is possible to come up with theories that cannot be expressed in terms of Lagrangians, but I have never actually studied that in depth.

 

[martinbn]

And what do you mean by "energy"? If energy is the thing that is conserved in a Lagrangian theory due to a specific symmetry, then your question is not meaningful. It seems that you are looking for an example in a different situation, so you need to say what energy is.

 

[PAllen]

If energy were not conserved and the theory were Poincare invariant then it could not be expressed in terms of a Lagrangian.

If a theory can be expressed in terms of a Lagrangian then Poincare invariance implies time translation invariance which implies energy conservation.

I believe that it is possible to come up with theories that cannot be expressed in terms of Lagrangians, but I have never actually studied that in depth.

Friction (that is not modeled microscopically) is often cited as a law that cannot be expressed with any action principle.

 

[PAllen]

Poincare invariance implies conservation of energy (also, momentum, angular momentum, and the quantities which serve as the generators of boosts from a group theoretic perspective but which don't, to my knowledge, have any generally accepted name). Remember that Noether's theorem relates symmetries to conservation laws. Energy, in particular, is related to time-translation symmetry, which is the the time component of the translational part of Poincare symmetry.

And this is consistent with the problem of friction in Galilean relativity. Modeled macroscopically, a friction law will not be time symmetric, thus violates energy conservation (unless heat is introduced into the laws).

Then, the issue for Poincare invariance is that, unlike Galilean relativity, a boost cannot be separated from time translation.

 

[atyy]

Thanks Parylne, Dale Spam, Matinbn and PAllen for the comments! Based on them, I googled "relativistic Brownian motion", and found http://arxiv.org/abs/0812.1996 and http://arxiv.org/abs/0902.4651 . I haven't read them yet, but it seems a place to look.