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Re: arbitrary choice of zero of potential



On Wed, 17 Oct 2001, John S. Denker wrote:

At 09:19 AM 10/17/01 -0700, John Mallinckrodt wrote:

...Bruce Sherwood and Ruth Chabay ......

"It is customary to emphasize that only differences in potential
energy are physically meaningful, and therefore the zero of
potential energy is arbitrary. After 1905 we should have changed
our tune. When particles are at rest and very far apart, their
total energy must be equal to the sum of their rest energies,
which means that the potential energy of interacting particles
must go to zero at large separations. If an arbitrary constant is
added, energy and momentum will not transform correctly between
different reference frames. Recognizing the absolute nature of
potential energy has far-reaching pedagogical consequences, and we
have addressed the issues in a calculus-based introductory
textbook to appear this summer."

Wow. This seems to be an in-your-face assertion that gravitation is not
gauge invariant.

I'm not sure why you say that, but isn't it trivial to see that
Sherwood and Chabay are correct? Here's a simple example:

Take a system of two electrons each of mass m that are at rest and
separated by a large distance like, say, a lightyear. "Let" the
potential energy be zero. The components of the momenergy four
vector (E,px,py,pz) with c = 1 are

[2m,0,0,0]

and the invariant mass, (E^2-p^2)^1/2, is 2m.

Now use the Lorentz transformation to transform to a reference
frame moving at v = 4/5 in the +x direction. The new components of
the momenergy four vector are

[(10/3)m,-(8/3)m,0,0]

We find that the invariant mass is still 2m and, therefore, indeed
invariant. No surprise here, this is what the Lorentz
transformation *does*. We also find that the x-component of the
momentum, -(8/3)m, and the kinetic energy, (10/3)m - 2m = (4/3)m,
are both properly determined.

Now suppose that we "let" the potential energy be nonzero, say, m.
The components of the momenergy four vector in the initial rest
frame are

[3m,0,0,0]

and the invariant mass is 3m as expected (I suppose) since the
invariant mass *is* the rest energy of the system. (It is
peculiar, however, to think that the "invariant mass" of the
system should depend upon my choice of the zero for potential
energy. Nevertheless, push on...)

Now use the Lorentz transformation again to transform to the
reference frame moving at v = 4/5 in the +x direction. The new
components of the momenergy four vector are

[5m,-4m,0,0]

We find that the invariant mass is still 3m as expected since,
again, preserving the value of the norm is what the Lorentz
transformation *does*. But now both the x-component of the
momentum, -4m, and the kinetic energy, 3m, are simply wrong.

As Sherwood and Chabay point out, "If an arbitrary constant is
added, energy and momentum will not transform correctly between
different reference frames."

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