Taylor/Wheeler:
Spacetime Physics:
Does Einstein's statement that mass and energy are equivalent mean that energy is the same as mass? No. Value of energy depends upon inertial frame of reference from which the particle (or system of particles) is regarded. Value of rest mass is independent of inertial frame. Energy is only the time component of a 4-vector. The time component gives the magnitude of the 4-vector only in the special case in which that 4-vector has no space component; that is when the [3-]momentum of the particle (or the total [3-]momentum of the system of particles) is zero. Only in this special case does energy have the same value as rest mass.
[...]
The distinction between mass and energy is this: mass measures the magnitude of a 4-vector and energy measures the time component of the same vector. Any feature of any discussion that emphasizes this contrast is an aid to understanding. Any slurring of terminology that obscures this discinction is a potential source of error or confusion.
Perhaps the expression "mass-energy equivalence" should be numbered among such slurrings.
The famous equation E = mc
2 can be understood in two ways. Einstein's own preference was to treat mass here as the coordinate-independent kind, sometimes called rest mass; in that case, the equation says that the energy of a system is equal to its mass, times c squared, in a reference frame where the system has no 3-momentum, i.e. its rest frame (
http://physics.princeton.edu/~mcdonald/examples/EM/hecht_ajp_77_804_09.pdf ); this is the viewpoint recommended by Taylor and Wheeler who say it's best to dispense with the concept of relativistic mass. Others interpret the famous equation as saying the energy of a massive system equals its relativistic mass times c squared.
A more revealing equation is m
2=E/c
2-p
2, where m is (rest) mass, E total energy, and p the Euclidean norm (magnitude) of 3-momentum. This also applies to systems with no mass, so the energy of a photon is equal to the magnitude of its 3-momentum.