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Re: [Phys-L] defining energy



APPLAUSE

The below is essentially SR in a single email, it doesn't get much more succinct than the below.

Admittedly, some background knowledge is necessary to appreciate the succinctness. Primarily a working knowledge of what a frame of reference entails, and the mathematics of Vector arithmetic/algebra with a signature -1 metric; as well as basic kinematic concepts.

I'm getting ready to start my SR unit in the "modern physics" class I'm teaching this term and I plan to plagiarize from this email, in particular the spirit and attitude. I chose Townshend's new book, which doesn't have SR chapters (only an appendix), so I'll have an easier time not showing "dirty laundry" to the class.

Joel Rauber

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of John Denker
Sent: Wednesday, October 30, 2013 7:51 AM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] defining energy

On 10/30/2013 01:12 AM, Savinainen Antti wrote:

I have thought that kinetic energy does *not* add to the rest energy
in the relativistic perspective as the rest mass is defined in the
rest frame. Rotational energy can be viewed as the sum of kinetic
energies of "small" parts of the wheel. Hence, it would not add to the
rest mass...or just mass as the qualifier "rest" is unnecessary as
eplained by Taylor & Wheeler in their splendid book Spacetime physics
(1992).

1) Yes, Taylor and Wheeler is indeed splendid. It has been very influential.

Actually, from a student's point of view, the book is more complicated than it needs to be. That's because the book tries to do two things
a) Explain spacetime physics to students, and
b) explain it to teachers who are accustomed to thinking
in non-spacetime terms.

Part (a) is the easy part. Students do *not* need to learn about the pre-1908 way of doing things. The less said about that stuff, the better.

Part (b) is important ... and is more difficult than part (a).
Teachers do need to see how the modern (post-1908) way of doing things can be reconciled with the other way of doing things. This needs to be done, but it is not helpful to the students.

I've tried to separate the two parts of the story:
http://www.av8n.com/physics/spacetime-welcome.htm
http://www.av8n.com/physics/spacetime-dirty-laundry.htm

Also Misner, Thorne and Wheeler _Gravitation_ is a good reference that takes an uncompromisingly modern view of relativity. The nominal topic of the book is general relativity, but about 10% of the book is spent reviewing special relativity. The whole thing is astonishingly well done.

2) Yes, the term "rest mass" is a misnomer. It is misleading.
It should be banished. I know the term can be found in various books including Feynman, but it was 50 years out of date in
1961 and it's 100 years out of date now.

3) All problems of this ilk can be handled using the 4-dimensional momentum p.

In any chosen frame F, the momentum p can be expanded in terms of components:
p = momentum
= [E, p_x, p_y, p_z]@F

In the chosen frame, the 19th-century «momentum» is the spatial part of p, namely
p_s = 19th-century «momentum»
= spatial part of p
= [p_x, p_y, p_z]@F

I write «momentum» in scare quotes to distinguish it from the modern, 4-dimensional, actual momentum p.

So we can say that p is the [energy, «momentum»] 4-vector.

4) If you have a collection of particles, the momentum (p) of the whole collection can be found by summing the momenta of the individual particles. It's as simple as that. This is required by conservation of momentum ... which includes conservation of energy, since energy is the timelike component of p.

5) The case of a rotating wheel is included in item (4). For present purposes, the wheel is just a bunch of particles moving in a particular pattern.

6) For any two 4-vectors A and B, the dot product A•B is a Lorentz scalar. It is the same in any frame.

7) Since p is a 4-vector (not just some random collection of four numbers), p•p is a Lorentz scalar. We define "mass" to be
m = √(-p•p) [6]

For a particle with nonzero mass moving at 4-velocity u, it turns out that
p = m u [7]
which is nice and simple and consistent with classical notions, but we should not take p/u as the definition of mass, because it would be unnecessarily ugly when applied to massless particles.

8) In any frame, we can expand equation [6] in terms of components:
m^2 = E^2 - p_x^2 - p_y^2 - p_z^2 [8]
= E^2 - p_s•p_s
= E^2 - p_s^2
or, sticking in explicit factors of c,
m^2 c^4 = E^2 - p_s^2 c^2

Since this is true in any frame, we don't need to be tooo specific about which frame we are talking about. However, in principle, equation [8] doesn't make sense unless we choose some frame.
The LHS is frame-independent, but the stuff on the RHS is not even definable except by reference to some frame.

9) We can define the /rest energy/ E(0) to be the energy in the frame where p_s = 0 i.e. the rest-frame of the particle. Applying equation [8] we find
E(0) = m c^2
which is a rather famous result. Note that it is OK to talk about rest energy ... but not OK to talk about rest mass. Mass is just mass. Mass is Lorentz-invariant.

10) On the other hand, mass is /not/ a linear function of energy or momentum. If you have a collection of particles, the momentum p is given by a linear sum over particles, but the mass is not.

Suppose we have three boxes. We restrict attention to motion in one spatial dimension.
-- Box A contains a photon moving to the right.
p = [1, 1]
m = 0
-- Box B contains a photon moving to the left.
p = [1, -1]
m = 0
-- Box C contains two photons, one moving in each direction.
p = [1, 1] + [1, -1]
= [2, 0]
m = 2

The picture, and some additional discussion, can be found at
http://www.av8n.com/physics/spacetime-welcome.htm#sec-invariance-conservation

The rotating wheel is essentially just a fancier version of box C. There are various pairs of particles moving in opposite directions.

11) We can re-express this result in terms of kinetic energy.
We define the KE as the total energy minus the rest energy:
KE = E - m c^2 (in some chosen frame)
This is a very nonlinear function. In particular, the kinetic energy of box C as a whole cannot be found by summing over the kinetic energies of the individual particles.

This is true even in classical non-relativistic low-speed
mechanics: The KE of a whole collection is not the sum of the individual KEs. People throw around terminology such as "total KE" but there are at least two things the term could mean. Sometimes it takes a lot of head-scratching to figure out what they're talking about.

Total E makes sense. E is conserved. That means we *can* find the total E by summing over all the individual particles.
However, "total KE" is sketchy. KE is a very nonlinear function of E, and we cannot find the KE of the whole just by summing.

=======================

Bottom line: Four-vectors are Good Things. All kinds of problems just melt away when formulated in terms of 4-vectors.
http://www.av8n.com/physics/spacetime-welcome.htm

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