First I provide some history of how the equation arose, establish what “mass” means in the context of this relation, and present some aspects of how the relation can be understood. Then I address the question, Does mean that one can “convert mass into energy” and vice versa?
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The word inertia denotes an object’s (or a system’s) reluctance to undergo a change in velocity. Inertia is distinct from momentum. For example, a golf ball exhibits a “reluctance to undergo a change in velocity” even when at rest on the tee. That’s why one has to whack it with a golf club to send it down the fairway.
Inertia can be given an operational definition. For example, if a mass spectrometer uses a “velocity filter” in the sense that only particles of a specific velocity make it through a region of crossed electric and magnetic fields and then magnetic deflection at low speed, the outcome is a measurement of the particle’s rest mass, that is, its inertia when it is accelerated from rest. Thus inertia can be given an operational meaning and one that is independent of the notion of energy.
A calculation that one person views as showing that inertia and energy are necessarily proportional another person may view as indicating that inertia and energy are identical. The difference in viewpoints may not be as great as it seems. The two camps would agree, I believe, that—most fundamentally—the connection between inertia and energy follows from the stress-energy tensor.
Rest energies figure prominently in the discussion that follows, and so a detailed example is in order. Consider a hydrogen atom whose center of mass is at rest. The atom’s rest energy consists of the electrostatic potential energy (of the electron-proton interaction), the kinetic energy of motions relative to the center of mass (primarily the electron’s kinetic energy), and the rest energies of the electron and proton. According to the standard model, the electron’s rest energy cannot be dissected into distinct contributions. The proton’s rest energy could be described in terms of the rest energies of the constituent quarks, their motion (internal to the proton), and their interaction energy.
The relativistic mass is defined as the proportionality factor between momentum and velocity . The relativistic mass may depend on the object’s speed.
