One difficulty is that books on special
relativity tend to be limited to rest mass,
kinetic energy, photon exchange and mass changes
in nuclear decay. The more general problem of
what to do with potential energy is often
ignored. One way this question is dealt with is
to say that if the system is chosen to be a
single object then E = (moc2 + sum of PEs) / (1-v2/c2)1/2.
If the object is at rest in the chosen reference
frame this becomes E = moc2 + sum of PEs. If the
reference frame is chosen so PE(grav) = mgh = 0,
then E = moc2 + 1/2kx2. If m is taken as a
measure of inertia and inertia depends on more
than rest mass, then you have new mass = m = mo + ½ kx2/c2.
Does anyone know if current technology allows a
mass change that small to be measured? Perhaps
the Discovery Channel Mythbusters could figure
out a way to compress a monster spring.