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