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What does E=mc2 really mean?
The title of Albert Einstein’s 1905 paper in which this formula is first
introduced in the modern sense provides a perfect answer:
“Does the inertia of a body depend upon its energy-content?”
Einstein frames the title in the form of a question but the paper answers it
affirmatively. The inertia, i.e., the inertial mass of a body is its internal
energy-content. Mass and energy are the same thing.
Now the “internal” part has caused a lot of confusion. Obviously, kinetic
energy (the energy of motion) is not internal, since it depends on the observer
(for a traveler on a train, another traveler on the same train has no kinetic
energy. But for a person standing on the station platform, the train and
everything, everyone on it will have plenty of kinetic energy.) But sometimes,
kinetic energy is internal (e.g., the vibration of atoms in a warm object) and
thus part of the energy-content of that body. Long story short, the
energy-content of a body is its energy as measured in its own center-of-mass
reference frame, not in the reference frame of some other observer.
The fact that we measure mass and energy using grossly different units is
another source of confusion. But here is the thing: suppose we decided to
measure time and length using compatible units: For instance, seconds and
light-seconds (the distance light travels in one second). Then, the speed of
light would just be unity:
c
=
1.
The famous formula of mass-energy equivalence?
E
=
m
.
Now for reasons that have to do with both history and convenience, we do not
use such compatible units, so the speed of light has a value other than one. But
think of it is a mere conversion factor, nothing more. And thus, in
E
=
m
c
2
,
the mass-energy equivalence relationship, the
c
2
part is just there to convert between units that we choose to use not because
Nature dictates it so, but because of human tradition and convenience.