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So I have only just been introduced to the concept of rest mass in Special Relativity.

Do we assume that the rest mass of a fundamental particle is constant in all inertial reference frames? i.e. is the rest mass of an electron if it is travelling at constant velocity c/2 (relative to the distant stars) the same as the rest mass of the electron if it is travelling at velocity 0 relative to the distant stars?

Is the rest mass simply the "mass" of a particle in inertial frame of reference which has the particle at it's origin (assuming the particle is actually in an inertial reference frame)?

Qmechanic
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Adam Rubinson
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  • Related: http://physics.stackexchange.com/q/8610/2451 – Qmechanic May 18 '12 at 23:53
  • Okay, from what I gathered, it is invariance under change of reference frame. I am not "interested" in this, I just need to know for an exam in 2 days so I can't spend time researching this stuff. hence the question. – Adam Rubinson May 19 '12 at 00:12

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Rest mass (also known as the mass [1] ) is the Lorentz invariant absolute value of the particle's energy momentum 4-vector.

$$ m^2 = \mathbf{p}^2 = E^2 - \vec{p}^2 $$

If you don't use $c = 1$ units, that's $$m^2 c^4 = E^2 - (\vec{p}c)^2$$

Lorentz invariant means "the same in all inertial reference frames".

[1] Despite the continued use of the distinction between "rest mass" and "relativistic mass" in introductory texts particle physicists, cosmologist, and other professionals in relativity recognize only one mass--the one the intro tets call the "rest mass".

"Relativistic mass" represents a particular way to factor the energy of a moving particle, but is not otherwise particularly useful. We really, really prefer to use manifestly Lorentz invariant quantities.

dmckee --- ex-moderator kitten
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  • ah, I was thinking wiki meant it was form-invariant, but it says it is Lorentz invariant... – Adam Rubinson May 19 '12 at 00:34
  • @dmckee: As I said elsewhere, it is not reasonable to call rest-mass "mass" unless you already understand that "c=1" so that you can call the "relativistic mass" the "energy". Otherwise, you get confusions about the mass of an electron going fast in a circle in a magnetic field, which you can deflect by a force perpendicular to the circle, and find the Newtonian mass has increased. When you think mass and energy are different things, then you need different names for the energy when it is considered as energy, and when it is considered as mass. It is considered as mass more often than not. – Ron Maimon May 19 '12 at 02:35
  • I think it will be useful to add that the rest mass is the analogue of the length in a three dimensional vector. In three dimensions the length is invariant under translations and rotations, and similarly in these four dimensions the rest mass is invariant under Lorenz transformations. – anna v May 19 '12 at 03:55
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    @Ron: I disagree. "Relativistic mass" is a kludge to maintain the form of certain Newtonian equations, and its modest convenience is overshadowed by the confusion it engenders. Those electrons are not Newtonian, and it does not help to pretend you can fix it by treating $\gamma m$ as a property of the particle when it mixes a particle property with a frame of reference property. Better to simply say that $F = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}(\gamma m \vec{v})$ which gets the same result. – dmckee --- ex-moderator kitten May 19 '12 at 19:01
  • @dmckee: $\gamma m$ is a property of the particle--- it's the energy of the particle (in mass units). This property is additive over particles, because the energy is additive, and gives the total energy of a system. It isn't invariant because it's part of a four-vector with the momentum, but neither is Newtonian energy. This energy is what weighs on a scale, is the source of gravity, and is the measure of how hard it is to add a unit of momentum to the particle, so it corresponds to the intuitive concept of mass in many ways, but it isn't the "mass" of the particle, which is a scalar. – Ron Maimon May 19 '12 at 20:01
  • @Ron "[it] is the measure of how hard it is to add a unit of momentum to the particle" in a particular set of reference frames. It is as much a property of the frame as of the particle (and yes, that is true of energy in Newtonian mechanics, too, but not of mass). And calling $\gamma m$ "the source of gravity" is incomplete because it would lead to contradictions of the making a ball go fast enough to undergo collapse kind". – dmckee --- ex-moderator kitten May 19 '12 at 21:27
  • I know I'm coming across as dogmatic about this--I feel very strongly about it--so I will emphasize: the notion of "relativistic mass" is *not wrong*. You can construct and talk about a correct physics using it. It is simply unnecessary and pedagogically unhelpful. – dmckee --- ex-moderator kitten May 19 '12 at 21:40
  • @dmckee: I know, I know, I am also dogmatic about this, because it's purely an issue of clarity and pedagogy (we all agree on the physics). The fast moving particle will not collapse into a black hole (in its own frame), but it will gravitationally tug at an object it sweeps by more strongly. Further, the scattering of a highly boosted object does make it indistinguishable from a black hole (for unboosted probes), because the moment you probe such a fast moving thing with tons of energy, you collapse the environment around it. So in a certain postivistic sense, it is correct to deduce this. – Ron Maimon May 19 '12 at 22:55
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    I think this would be the point at which any further discussion of the issue goes to [chat]. – David Z May 20 '12 at 02:55