Re: [Phys-L] defining energy

[John Denker <jsd@av8n.com>]

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 books including Feynman, but it was 50 years out of date in
 1961 and it's 100 years out of date now.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