A Meaning of the word "conserved" in relativity

[SiennaTheGr8]

If you have a system of equal mass particles, then in Newtonian physics conservation of momentum implies conservation of total velocity.

I'm reminded of something that I'd be curious to get some thoughts on.

There's a semantic issue with the word "conserved" that often flies under the radar: does the term refer to any quantity whose value remains the same before and after some process, or does it specifically refer to an additive quantity that fits that bill? Usage is inconsistent in the literature (I can dig up examples if I'm asked).

To make matters worse, there's also an inconsistency in what is meant by "the mass of a system"—some define it as the sum of the masses of the system's constituents, and some define it as the system's rest energy (expressed in mass units). The latter is more common, thankfully, but I'm sure I could provide an instance or two of the former.

When you put all these various usages together, you end up with a mess of possible answers to the question is mass conserved in SR? (and this is without even opening the "relativistic mass" can of worms!):

1. The word "conserved" simply isn't applicable to quantities that aren't additive.
2. The mass of an isolated system is not necessarily conserved because various internal processes can cause the sum of the constituent masses to change.
3. The mass of an isolated system is conserved as a trivial consequence of energy-conservation (i.e., energy is conserved for any frame, including the system's rest frame).

If one goes with #3, then one must admit that there's a sense in which velocity is conserved, too (finally connecting things back to your post here, @PeroK). And if that sounds silly then one might consider adopting a definition of "conserved" that applies only to additive quantities.

I once tried to point all this out to some knowledgeable people on another forum, but they were... not receptive.

 

[Nugatory]

If one goes with #3, then one must admit that there's a sense in which velocity is conserved, too

I'm not following this step?

 

[vis_insita]

There's a semantic issue with the word "conserved" that often flies under the radar: does the term refer to any quantity whose value remains the same before and after some process, or does it specifically refer to an additive quantity that fits that bill? Usage is inconsistent in the literature (I can dig up examples if I'm asked).

Recently, I had quite similar thoughts, when in a discussion the issue came up of whether mass is conserved in creation or annihilation processes. On the one hand it was argued, that mass cannot be conserved because if you count all the masses of the particles in the in-state they don't necessarily equal the masses in the out-state.

On the other hand, I think it could equally well be argued that this simply means that mass isn't an additive quantity, which should not be surprising since it is the Minkowski norm $m=\sqrt{\boldsymbol{p}\cdot\boldsymbol{ p}}$ of four momentum (which is additive). Also, the first point of view seems to make sense only in the context of scattering theory. Otherwise there aren't necessarily any particles whose masses to count.

When you put all these various usages together, you end up with a mess of possible answers to the question is mass conserved in SR? (and this is without even opening the "relativistic mass" can of worms!):

1. The word "conserved" simply isn't applicable to quantities that aren't additive.
2. The mass of an isolated system is not necessarily conserved because various internal processes can cause the sum of the constituent masses to change.
3. The mass of an isolated system is conserved as a trivial consequence of energy-conservation (i.e., energy is conserved for any frame, including the system's rest frame).

I'm inclined to the third option. It seems natural that a quantity that only depends on conserved quantities should also be conserved. This means that the conservation of mass is a consequence of energy-momentum-conservation.

If one goes with #3, then one must admit that there's a sense in which velocity is conserved, too (finally connecting things back to your post here, @PeroK).

Here I don't quite follow. Which velocity? Do you mean the velocity of the center of mass/energy?

 

[SiennaTheGr8]

1. The word "conserved" simply isn't applicable to quantities that aren't additive.
2. The mass of an isolated system is not necessarily conserved because various internal processes can cause the sum of the constituent masses to change.
3. The mass of an isolated system is conserved as a trivial consequence of energy-conservation (i.e., energy is conserved for any frame, including the system's rest frame).

If one goes with #3, then one must admit that there's a sense in which velocity is conserved, too (finally connecting things back to your post here, @PeroK). And if that sounds silly then one might consider adopting a definition of "conserved" that applies only to additive quantities.

I'm not following this step?

Here I don't quite follow. Which velocity? Do you mean the velocity of the center of mass/energy?

Sorry, I wasn't clear there.

I'm referring to the velocity of anything moving inertially, so yes, @vis_insita, the velocity of an isolated system's rest frame.

My point is that if you say that the mass of an isolated system is "conserved" in SR, then you're allowing non-additive quantities to qualify. I'm not arguing that that's bad, but a consequence of loosening the word to mean "any system quantity whose value doesn't change" is that the system's velocity now qualifies as a "conserved" quantity.

 

[SiennaTheGr8]

Ah, I found some links I was looking for.

Lev Okun talking about this:

https://books.google.com/books?id=OjgTS12V0VUC&pg=PA295
He references L&L:

https://books.google.com/books?id=HudbAwAAQBAJ&pg=PA26
https://books.google.com/books?id=bE-9tUH2J2wC&pg=PA13

 

[Herman Trivilino]

I think you're conflating two issues here. People who claim that the mass of a composite body consists of the sum of the masses of the constituents is not accepting the true meaning of the Einstein mass-energy equivalence. The invention of relativistic mass is an attempt to continue that failure to accept.

Once you dismiss this notion you no longer have, as I understand it, any kind of connection of being conserved with being additive.

 

[SiennaTheGr8]

Well, I raised both issues, but I don't think I've conflated them.

People who claim that the mass of a composite body consists of the sum of the masses of the constituents is not accepting the true meaning of the Einstein mass-energy equivalence.

I think it's more a semantic issue. For example, I'm confident that Griffiths understands the mass–energy equivalence, and that he's simply thinking of "the sum of the masses of the constituents" when he writes, "In every closed system, the total relativistic energy and momentum are conserved. Mass is not conserved[.]":

https://books.google.com/books?id=Kh4xDwAAQBAJ&pg=PA536

Once you dismiss this notion you no longer have, as I understand it, any kind of connection of being conserved with being additive.

Well, take it up with Landau & Lifshitz! (see my previous comment)