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I agree with the concept of "invariant mass" WRT imputed mass changes
which depend on the vantage point of the observer, but haven't we
already given some status to the
variable-mass-under-the-influence-of-external-fields camp when we
talk about, and explain to our students, how the "mass defect"
(terrible term) comes about in nuclei? I don't understand how the
measurable difference in mass that occurs when one combines a proton
and neutron into a deuteron, under the influence of the strong
interaction, differs in concept from the unmeasurable difference
(because it is many orders of magnitude too small to be detectable)
in mass that occurs when an object is moved from a high elevation to
a low one under the influence of the gravitational interaction.
Granted that in the case of the deuteron, we are not part of the
system whose total energy is changing and that in the case of the
gravitational change, we are, but how does that change outlook we
take to mass being dependent upon an object's location within field
that it responds to? Perhaps we can apportion the mass change among
the two objects involved, which in the case of the earth and a box,
would make the box's mass change even smaller than it already is by
about 20 orders of magnitude, but two gravitationally interacting
objects (a double star system, for example) ought to show, if we
could determine it, the same sort of "mass defect."