> Mass is Lorentz-invariant, energy is not.

That's not completely accurate. Chemical energy and binding energy are also Lorentz-invariant.

The only type of energy is not Lorentz-invariant is velocity energy, so you are putting your distinction in the wrong place.

On top of that, there are other violated invariants.

The weight of a lump of iron near a magnetar is greater than the weight of the same lump of iron near an identically massing neutron start.

This is because the potential energy of the iron is greater near the magnetic field, so its mass (as seem by the magnetar) is greater, and so is the gravitational attraction between them.

This means you can't just say "No velocity, the mass is identical", it's not - the extra potential energy means extra mass.

Or in other words there is no such thing as mass as distinguishable from energy.

I haven't thought about it carefully, but since magnetic fields don't do work, my initial reaction to your example of iron near a magnetar is surprise. But, I also expect that the iron magnetizes in a strong field. Wouldn't that be lowering its energy (compared to an unmagnetized lump in a background field), however?

It's not a conservation of energy question, obviously you would have to deal with the energy of getting it there in the first place.

The question is once it's there: Do the magnetar and the neutron star see different masses for the lump of iron?

Imagine the magnetar and the neutron star at the tips of an L, and the iron at the vertex.

Certainly, but both are encoded in the energy-momentum tensor and for better or worse that's usually considered to be what generates the metric (or perturbations thereof, if you want to do things that way).

In Newtonian mechanics, kinetic energy is rotationally but not Galilean invariant, sure. But in GR in a local inertial frame the pressure T^{ij}, i=j, i!=0 sure looks like kinetic energy \gamma m(v^i)^2. [1]

Should we really strongly distinguish between the pressure and T^00 just because in a local inertial frame the latter looks like \gamma mc^2 ? Sharpen the question by considering vastly different frames of reference.

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[1] https://en.wikipedia.org/wiki/Kinetic_theory_of_gases#Pressu...