Rest mass

[Kaeroll]

This may be in the wrong forum, but I'm told it's due to relativity, so this seems a decent place to put it...

 

Occasionally as a chemist I have to deal with - eek - maths, some of which involve electron rest mass (for example). My question is, how is this measured? It seems absurd to suggest somehow isolating and 'freezing' an electron in its tracks, so I can only assume it's an extrapolation. It's something I've been curious about for some time.

 

On a related note - why do we use 'rest' as a point of reference, if it is (as I assume above) strictly hypothetical? Why not the mass at a given energy?

 

Math welcome in answers, though commentary would be appreciated...

 

Kaeroll

[swansont]

You can measure it in a Penning trap

 

http://en.wikipedia.org/wiki/Penning_trap#How_it_works

 

You use rest as the common reference because any other measurement will depend on your choice of reference frame. But the rest mass is invariant, so the value will be the same in any inertial frame.

[timo]

Occasionally as a chemist I have to deal with - eek - maths, some of which involve electron rest mass (for example). My question is, how is this measured?

The simplest method probably is shooting a ray of electrons through a (perpendicular) magnetic field, visualizing the paths somehow (e.g. in a bubble chamber), measuring the resulting radii and calculating the mass from them.

 

On a related note - why do we use 'rest' as a point of reference, if it is (as I assume above) strictly hypothetical? Why not the mass at a given energy? Math welcome in answers, though commentary would be appreciated...

For simplicity, anything said below refers to single free particles. EDIT: Oh, and it also is just an attempt to explain the different meanings/definitions of mass in relativity, not the official physicists definition template or something like that.

 

In relativity you have four coordinates, i.e. four directions at each point. Often, and for this post, you chose one of them to be "time coordinate" and give it the label 0 or t and the other three to be "space" with labels 1,2,3 or x,y,z. The (4-)momentum [think of the momentum as the property determining the motion] of the particle can be described by four numbers, [math]p^0, p^1, p^2, p^3[/math]. Assuming the most common choice for the coordinate system then apart from factors of c [which only determine the unit a quantity is measured in] these four values are interpreted as follows: [math]p^0[/math] is the (relativistic) energy E of the free particle, [math]\vec p = \left( \begin{array}{c} p^1 \\ p^2 \\ p^3 \end{array} \right)[/math] is the non-relativistic momentum or 3-momentum.

 

The values for [math]p^0 \dots p^3[/math] depend on the choice of the coordinate system. However, there are properties of the momentum of that are not dependent on how you did chose your coordinate system [*]. One such property is the (pseudo-)magnitude [math]m_i[/math] of the momentum. In the coordinates I used above it is [math]m_i^2 = \left( p^0 \right) - \vec p^2 [/math].

 

Ok, now for some different choices of the term mass:

- [math] m_i [/math] is the "invariant mass" (or "inertial mass" or "proper mass"). It is independent of a chosen coordinate system and only a property of the particle type. This then allows statements like "an electron has a mass of 511 keV".

-[math] m_\gamma := p^0 [/math] (the energy) is the "relativistic mass". Some people seem to like this definition of mass for its property that it equals the energy [math]m_\gamma = p^0 = E[/math] (with canonic factors: [math] E=mc^2[/math]).[**] This then allows statements like "according to Einsteins famous E=mc², everything ultimately consists of pure energy."

- In the special case that [math]\vec p = \vec 0 [/math], you can easily verify that [math] m_\gamma = m_i[/math]. So if you want "the best of both worlds" you can say that you define mass to be the energy of a particle for which [math] \vec p = 0[/math]. Under the constraints I put in the introduction and the additional constraint that [math]m_i > 0[/math], [math]\vec p=\vec 0[/math] means that the particle must be at rest. So you get the "rest mass" [math]m_0[/math].

- There are other possible meanings of mass that are related to other fields.

 

So what is the "rest mass"? Perhaps think of it as a definition of mass that has the nice property of the invariant mass (i.e. it is a property of the particle, not the way you look at it), does not have the redundancy of "relativistic mass" and is perhaps easier to define for someone with little to no background in relativity. Note that when the momentum of a free particle is negligible compared to its mass, then the three definitions of mass give the same result.

 

Footnotes:

 

[*]: Starting from this insight you can even go as far as to demand that no physics at all should depend on the choice of a coordinate system. This leads to a very elegant formulations within relativity (note that "very elegant" does not need to mean "very useful" )

[**]: I dislike it for pretty much for the same reason.

Edited February 27, 2009 by timo

[Kaeroll]

Thank you both for your replies - they're both very informative, and I really appreciate you taking the time to boil that down to idiot's (chemist's?) terms for me, Atheist.

 

Makes much more sense now. What more can I say?