After having commented an answer there: Relative potential energy vs Absolute potential energy, I realised that the energy concept may be much more subtle than what we usually believe, even if we restrict ourselves to the classical and special relativistic domains (non-quantum and non-General Relativistic). So I want to clear things up a bit.
I believe there are several subtle things that enters the energy concept, and we tend to forget or take as granted many of them. This may explain a lot of confusions in the physics community.
My question is quickly formulated, so it may sounds naive or badly formulated. Sorry about that (I may edit the question after some time, depending on the comments and answers I get). Please, lets restrict to classical and special relativistic theories, since it's much more complicated in quantum field theory, and especially in General Relativity (where energy may even be ill-defined, depending on the spacetime).
What are the "things" we need in physics to define any kind of energy, including "mass".
Of course, kinetic energy needs a reference frame relative to which the motion is defined.
When we want to add up the potential energy, we need to define vacuum relative to which some fields are defined, and as I see it we also need some boundary conditions (which may be arbitrary). We usually take for granted that a system's potential energy is 0 when the interacting particles are infinitely separated, which is a choice of gauge. What else?
Can a particle's "rest-mass" have an absolute meaning? Or is it still defined relative to something else, like boundaries (conditions at infinity, for example)? What is "rest-mass", really, above the relation $m_0 = E_0/c^2$? Can we give the potential energy an absolute meaning, or is it always a relative concept even in special relativity? (relative to what?)
As I see it, we need three very basic things to define any kind of energy (including rest-mass) :
- A reference frame,
- the local vacuum,
- some boundary conditions.
What else do we need? How would you make these things more precise, more rigorous or more specific?
