# What exactly is Einstein's famous E = MC^2

1. May 3, 2010

With the famous news of the LHC, I had did a little research about how it works. Apparently, it works via energy mass equivalence.

After dancing around the net, I had a brief idea of how E = MC^2 came about. But still, it isn't very clear.

Exactly how does the speed of light come into play?

I know C meant constant, and speed of light just happens to be that constant, but how did we arrive on that constant? What makes it so sure that this C^2 is the relationship?

Would appreciate a layman answer, preferably with less maths.

2. May 3, 2010

3. May 3, 2010

### bcrowell

Staff Emeritus
FAQ: Where does E=mc2 come from?

Einstein found this result in a 1905 paper, titled "Does the inertia of a body depend upon its energy content?" This paper is very short and readable, and is available online. A summary of the argument is as follows. Define a frame of reference A, and let an object O, initially at rest in this frame, emit two flashes of light in opposite directions. Now define another frame of reference B, in motion relative to A along the same axis as the one along which the light was emitted. Then in order to preserve conservation of energy in both frames, we are forced to attribute a different inertial mass to O before and after it emits the light. The interpretation is that mass and energy are equivalent. By giving up a quantity of energy E, the object has reduced its mass by an amount E/c2, where c is the speed of light.

Why does c occur in the equation? Although Einstein's original derivation happens to involve the speed of light, E=mc2 can be derived without talking about light at all. One can derive the Lorentz transformations using a set of postulates that don't say anything about light (see, e.g., Rindler 1979). The constant c is then interpreted simply as the maximum speed of causality, not necessarily the speed of light. We construct the momentum four-vector of a particle in the obvious way, by multiplying its mass by its four-velocity. (This construction is unique in the sense that there is no other rank-1 tensor with units of momentum that can be formed from m and v. The only way to form any other candidate is to bring in other quantities, such a constant with units of mass, or the acceleration vector. Such possibilities have physically unacceptable properties, such as violating additivity or causality.) We find that this four-vector's norm equals E2-p2c2, and can be interpreted as m2c4, where m is the particle's rest mass. In the case where the particle is at rest, p=0, and we recover E=mc2.

A. Einstein, Annalen der Physik. 18 (1905) 639, available online at http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51