# Q & A: E=mc^2

Q:
In the relativity formula, E=mc^2, what is the significance of c^2? Energy and matter can be converted to each other, but what has the speed of light got to do with it? And why is it squared? I know c^2 would be a huge number, and "a tiny amount of" m = "a heckuva lot of" E, but I don't understand the importance of the speed of light to the formula other than being a really big number to sort of balance the equation. And the speed of light is not quite a constant, it travels faster in a vacuum or air than thru say a solid or a crystal (ie: diffraction and prisms). So does that mean that matter converted to energy in the vacuum of space creates more energy than matter converted in a pressurized nuclear containment vessel in a nuclear reactor here on earth? How about in the very dense center of a star? Seems like those hydrogen atoms wouldn't create so much energy under those conditions. -- Or is c^2 in the equation because Energy = Light in this case and the matter is converted to lots of energetic photons, X-rays, etc. [the matter is converted to Energy in the form of Light]? Is it squared because it scatters in all directions from the point of conversion? Why not c^3?
- Wizard (age 50)
Orange Park, FL, USA
A:

The "c" here is the fundamental constant, the speed of light in a vacuum. The various speeds of combinations of light and wiggles of electrons in materials are not relevant.

There is no "conversion of energy to matter" in the sense of energy being lost and replaced with something else. Energy is conserved. Some forms of it can be found in things at rest, and these go by the name of rest "mass". So energy can convert from one form to another, and some of these forms include more or less "matter".

In fundamental physics, we don't really think of inertial mass and energy as different things. They're exactly the same thing, measured in different traditional units. The traditional units for measuring energy are mass units times the square of velocity units.

Why is that? If you look at the energy of some mass m0 in motion as it moves it's:

m0/(1-v2/c2)1/2 = m0c2+m0v2/2 + (3/16)m0v4/c2....

The first term, the rest-mass energy, doesn't change when something moves. It's big, but because it doesn't typically change, people didn't keep track of it as "energy" but thought of it as something separate and kept track just of the mass, m0. The second term changes when something moves, so it was noticed and called "kinetic energy". Notice that its units are mass*speed2. What about the next term, and all the others I didn't write down? They're small, so long as v<< c, and weren't noticed. Fast moving things with non-zero m0 weren't studied until the last 100 years or so.

So traditionally two separate things were kept track of: m0  and m0v2/2. They picked up the names "mass" and "energy" and each got assigned its own units, for example kilograms and Joules. Once it was realized that they were really just parts of the same quantity, the conversion factor for the units was needed. It's c2, as in m0c2.

Mike W.

(published on 08/31/2013)