Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: arbitrary choice of zero of potential



John Denker wrote:

... the 4-momentum is defined as the invariant mass times
the 4-velocity, with no contribution from potential energy.

to which I responded:

... I think I hear you saying that the invariant mass
of a system is independent of any interactions between its
parts. Is that right? If not, please correct. If so, are you
saying, for instance, that the invariant mass of an "atom" of
positronium is precisely twice that of a single electron?

to which JD responded:

Nope.

Tempting me to ask, "'Nope' *what*?" but JD does go on to
partially elaborate saying:

... positronium consists of electron + positron + fields. The
fields contribute to the mass of the combined system, no
problem. ... Note that the fields are derivatives of the
potential.

So, John, I now have an inkling of your viewpoint, but no concrete
examples (within the stated context of relativistic systems,
momenergy four-vectors, and the Lorentz transformation) to test my
grasp of that view.

I hope we both agree that viewpoints in physics ought to be judged
on the basis primarily of 1) their ability to lead us to
conclusions about the physical world that are testable and
accurate, but also 2) their economy and ease of deployment. I am
pretty confident that my viewpoint (and that of Sherwood and
Chabay who are pretty sharp folks) satisfies the first criterion,
but I am always interested in learning about other viewpoints that
may better satisfy the second criterion.

To that end I'd like to request that you provide very short, very
specific answers (with no need whatsoever to "elaborate" via
appeals to authority or nonrelativistic analogies to things like
dropped bricks or model cars running around on high and low
shelves) to the following questions. I believe that those answers
will erase my remaining confusion about your viewpoint and also
allow me to judge 1) whether it results in conclusions that are
different from mine and 2) whether it is more economical or
efficient than my own.

Consider a pair of electrons each with mass m (in the generally
well understood sense of the "mass of an electron") that are
released at rest from an initial separation along the x-axis that
is specifically chosen so that each electron will approach an
asymptotic kinetic energy equal to m (in units where c = 1) as
they separate to large distances. For the purpose of this
problem, I would like to effectively "turn off" the EM radiation
that would accompany this process so that we need not concern
ourselves with accounting for it. I believe we can do this
without generating any unmanageable contradictions,
misunderstandings, or ambiguity, but do let me know if you
disagree and, of course, why.

1. What is the initial momenergy four-vector, P = [E,px,py,pz],
for either one of the two electrons?

P_1_o= [ _?_m, _?_m, _?_m, _?_m]

2. How did you obtain the answer given in 1?


3. What is the initial momenergy four-vector, P = [E,px,py,pz],
for the two electron system?

P_sys_o= [ _?_m, _?_m, _?_m, _?_m]

4. How did you obtain the answer given in 3?


5. What is the (asymptotically-approached) final momenergy
four-vector for the single electron that is moving in the +x
direction?

P_1_f= [ _?_m, _?_m, _?_m, _?_m]

6. How did you obtain the answer given in 5?


7. What is the (asymptotically-approached) final momenergy
four-vector for the two electron system?

P_sys_f= [ _?_m, _?_m, _?_m, _?_m]

8. How did you obtain the answer given in 7?


9. How much of the system energy given in the answer to 3 is
interaction energy?

Initial interaction energy = _?_m

10. How did you obtain the answer given in 9?


I will provide my own answers after your response. Thanks for
your help.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm