Is Rest Mass Conserved by the Conservation of Energy and Momentum?

[DrGreg]

An isolated system of particles is in some senses "equivalent" to a single particle. Think of it being located at the "centre of mass" (although that concept isn't entirely well defined in relativity).

The equivalent particle's momentum is $\textbf{P}=\sum_n \textbf{p}_n$

The equivalent particle's energy is $E=\sum_n e_n$

The equivalent particle's mass is given by $Mc^2 = \sqrt{E^2 - |\textbf{P}|^2c^2}$, i.e. the "system mass" or "invariant mass of the system" or "rest mass of the system". (In the special case where $\textbf{P}=0$, that simplifies to $Mc^2 = E$.)

The system momentum, system energy and system mass are all conserved (remain constant over time).

The main point of confusion is that the system mass is not the sum of the individual particle's rest masses; that sum is not (in general) conserved.

The phrases "conservation of mass" or "conservation of rest mass" etc are liable to be misunderstood (as this thread proves), so I think it's better to refer to "system mass", or better still, explain what you mean when you talk about mass conservation.

 

[Sam Gralla]

The only mass you can define unambiguously is the invariant mass, and as we have seen this is NOT always the rest mass.

Huh? We're talking about particle collisions here, right? In that case the four-momentum vector of each particle is well-defined. The rest mass of each particle is the (square root of minus the) norm of its four-momentum vector. So it's perfectly well-defined. The "invariant mass", I gather, is the norm of the four-momentum vector of the whole system (also perfectly well defined). Personally, in both doing and teaching SR particle collisions, I've never heard of this concept nor found it useful; but if it really is useful for something I'd certainly advocate giving it a less misleading name, such as "system mass" as others have suggested.

 

[tom.stoer]

OK, simple question: what is the rest mass of a pair of colliding particles?

Of course the rest mass of each particle is well defined, that's not the question, but what about the mass of the system of these two particles? Provided that the two particles have rest mass m, momentum p and -p, and energy E² = m²+p². Of course the rest mass of the system is not simply the sum of the two rest masses 2m. And of course we can easily calculate its invariant mass: its total energy-momentum 4-vector is (2E, 0) from which we get the invariant mass (squared) M² = 4E².

Would you call this M² = 4E² the rest mass? Or would you prefer to call it invariant mass and explain its origin (two particles with rst mass, energy E, ...)? The two colliding particles are far apart, they are not in a bound state, they are not at rest, neither w.r.t. each other nor wr.t. the lab frame.

Personally I would never talk about the rest mass of the system but I would always use the term invariant mass. That's my point.

 

[Sam Gralla]

Okay good, we're in agreement then... just a matter of words.