Let there be light

[krab]

Anton A. Ermolenko is correct in my view. I should add that by "correct", I mean by Occam's Razor: it results in a simpler description of reality.

In lower level physics courses in America, relativity is usually first taught in such a way that it is easier to compare with Newtonian mechanics. However, if you carry this to solving complicated problems relating to the dynamics of relativistic particles, it is way too cumbersome. One of its results is a distinction between "longitudinal" and "transverse" mass. Another is that pesky sqrt(1-v^2/c^2) factors appear everywhere.

I now believe it is the wrong way to teach relativity. My undergraduate education left me with the impression that relativity is very tricky, with many pitfalls. My later career turned out to be at an accelerator lab where it is essential to correctly describe trajectories of relativistic particles. To my surprise, I found the equations of motion are relatively (unintended pun) simple, and simple to derive, if one begins with 4-vectors and Hamiltonians. In 4-vector form, mass is rest mass; it is in a sense the norm of the 4-momentum. It is a useful concept because it is invariant. Defining mass in such a way that it depends on momentum only complicates life.

James Smith in his text Introduction to Special Relativity (1965) derives E=mc^2/sqrt(1-v^2/c^2) and then comments: "This raises a semantic question. Here is a new quantity that is the relativistic generalization of two classical quantities. Many treatments do just what we have temporarily done and call M=m/sqrt(1-v^2/c^2) the "mass", m the rest mass, and call Mc^2 the energy. But, if energy is always just c^2 times the "mass", this is not good economy with words or concepts. Furthermore, it seems desirable to have the word mass refer to an intrinsic property of the particle, and not to refer to a property of its motion. From now on this book will follow what is becoming quite general practice. The word energy will refer to the quantity mc^2/sqrt(1-v^2/c^2) . The word mass wil refer to m, the quantity many other treatments call the "rest mass"."

I am also quite close to the field of particle physics. No one says photons have mass. Everyone says photons are massless, without qualifying this by saying they are speaking of rest mass only.

 

[pmb]

Originally posted by krab
Anton A. Ermolenko is correct in my view. I should add that by "correct", I mean by Occam's Razor: it results in a simpler description of reality.

That's not a correct application of Ockham's razor. Ockham's razor really applies to explanations/theories. Not definitions. It states that

Ockham's Razor: Of two competing theories, the simplest explanation of an entity is to be preferred.

As to what a simple definition is - that's a matter of debate.

However, if you carry this to solving complicated problems relating to the dynamics of relativistic particles, it is way too cumbersome.

Not really. First off the ddefinition of mass is simply part of theory. And if you're refering to things like the subscript in equations like p = gamma*m_o*v then one can simply drop the subscript and write p = gamma*m*v. The "m" is then the proper mass.

This is exactly what Thorne and Blanchard do in their new text. E.g. See chapter one
http://www.pma.caltech.edu/Courses/ph136/yr2002/chap01/0201.2.pdf

One of its results is a distinction between "longitudinal" and "transverse" mass. Another is that pesky sqrt(1-v^2/c^2) factors appear everywhere.

For one, that was something Einstein used at first but which he soon gave up. It was a poor definition of force, F = ma, which led to that. Physicists today do not use that definition. In fact Newton didn't even define force like that. The correct definition of force is f = dp/dt. You cannot get rid of the gamma. Its there whether we like it or not. All you can do is shift it around.

I now believe it is the wrong way to teach relativity. My undergraduate education left me with the impression that relativity is very tricky, with many pitfalls.

I got the exact opposite impression. However I did find most if not all discussions of this subject very lacking.

To my surprise, I found the equations of motion are relatively (unintended pun) simple, and simple to derive, if one begins with 4-vectors and Hamiltonians.

Some people do prefer the geometric approach to the dynamic approach. However there are some bothersome things in that approach. The most simplest thing in relativity is the velocity of light. Its magnitude being indepedand of the frame of referance. Yet there is no 4-velocity for light.

Some people love to try to define "mass" as the magnitude of 4-momentum. Problem with that is the the time component, the free-particle energy E, is defined in terms of the particle's "mass." So such a definition is circular.

Nobody can deny the usefulness of proper quantities like rest mass. But just because a property is not invariant it does not mean that it is not physically meaningful. Length, time, density etc are all of thast nature. The two should never be confused. For example: The lifetime of a free neutron is about 15 minutes. You will almost always seen this defined in this manner. However the actually physically measured lifetime of the particle is a frame-dependant quantity. The lifetime really should be called the proper lifetime - but it never is. Just like the proton mass being 938 MeV/c^2 when it really should be called the proper proton mass or proton rest mass.

However if I were writing a paper/article/text on particle physics calculations I'd say once that m is rest mass and call it mass and be done with it. The reason being that particle physics is about the inherant properties of particles. E.g. A particle physicist doesn't care about the difference between proper time and time (aka coordinate time).

Smith makes the same bogus arguement that many do. Its a very poor explanation to say that there is a poor economy of words when in fact this is not "poor economy of words" but is physics.

For example: m = mass is defined such that mv is a conserved quantity. E = Free-particle energy is defined as the sum of rest energy and kinetic energy. It is then *proven* than E = mc^2.

What you later call these terms after its all fleshed out is a matter of convinience.

Furthermore, it seems desirable to have the word mass refer to an intrinsic property of the particle, and not to refer to a property of its motion.

That is purely a matter of opinion though. And it can only apply to particles. It cannot apply as a general usage. For example: the most general thing mass can be is a tensor. A second rank tensor to be exact. This tensor describes bodies which are composed of continuous matter. One should take the most general qauntity and define terms in that manner. Then specialze to simplier things like matter of a point like nature. If you recall the derivation that MTW give to show that the stress-energy tensor is symetric you will see that this is exactly what they do. The can't use rest mass in that since not all stress-energy tensors describe matter which has a non-zero rest mass so that wouldn't work.

I am also quite close to the field of particle physics. No one says photons have mass.

That is incorrect. If they say that then they are refering to proper mass. And I know of at least one particle physicist who will tell you that light has mass. And that's Alan Guth. I even have that in writing since it's in his class lecture notes.

And we haven't even touched on gravity. Do you think there should be different definitions of mass for each different use?

Besides - nobody can be sure on what "everybody" does or says unless they exact "everybody" what they think and say. And that would just take far too long! :-)

And if you work in particle physics then that is probably the only exposure that you get to physicists correct? How many cosmologists do you see on a day to day basis? And not all relativists are either particle physicits not are they all cosmologists.

Rindler had a nifty problem about the mass of a charged capacitor. If you're truly interested in this subject then I highly suggest reading it. The article is

"A simple relativistic paradox about electrostatic energy," Wolfgang Rindler and Jack Denur, Am. J. Phys. 56(9), Sep 1988

Pete

 

[pmb]

Originally posted by Anton A. Ermolenko
I like it. After Pauli's book everyone should termed the mass... kinda "Oh! yes, of course, I quite forgot (in Pauli's terms, not Einstein) the rest mass..."
Just say me, you really use those terms when you read or use the QFT??? There are no the relativistic or rest mass and total and rest energy, but there are energy and mass. Isn't it? Dude

I don't know QFT. But I imagine that QFT only uses invariant (and thus intrinsic) properties like proper mass and not inertial mass.

Tell me. It's said that the lifetime of a neutron is 15 minutes. That is the "proper lifetime" d(tau) i.e. the lifetime as measured in the rest frame of the neutron. As measured in a frame of referance moving relative to the neutron the lifetime is dt = gamma*d(tau). You won't see dt mentioned in tables of the lifetime or particles. Why do you think that is? Do you think this time dilation relation wrong? Or do you think that it is a matter of definition of the term "lifetime"?

Pete

 

[pmb]

Originally posted by Anton A. Ermolenko

In general, I think, that we already have understood each other.

I appoligize Anton. I didn't see this comment at first. I was unware that you didn't want to discuss this.

I had posted one last comment about lifetime but I will cease on this since you believe that we understand each other.

Pete

 

[Anton A. Ermolenko]

Originally posted by pmb
I appoligize Anton. I didn't see this comment at first. I was unware that you didn't want to discuss this.

Pete

Don't mention it, Pete. Thank's for discussion.

 

[pmb]

Originally posted by Anton A. Ermolenko
Don't mention it, Pete. Thank's for discussion.

You're welcome.

Just one last note. There are many people who claim that mass, m, always means 'rest mass' and that is what the entire physics community means when they use this term. I've composed a list of counter examples to dispel this myth. See

http://www.geocities.com/physics_world/relativistic_mass.htm

Pete