# E=mc^2

*Most recent answer: 10/22/2007*

Q:

Why did Einstein use the speed of light squared in his E=mc^2 equation? I know that nothing can go faster than light, but why did he use THAT and why did he square it? Thanks.

- Margie (age 56)

Bayonne, NJ

- Margie (age 56)

Bayonne, NJ

A:

Nice question. I’ll give an answer which is a sort of compromise
between being reasonably short and reasonably complete. By coincidence,
a more complete account can be found in one of my lecture notes from
this semester:

. Go to Lecture 11.

Here’s one argument. Imagine a star radiating light away. If you look at it from a reference frame in which the star is at rest, the star isn’t going to accelerate off somewhere because the light is shooting out equally in all directions. So if you look at it from a frame in which the star is moving, it’s also true that the star won’t accelerate. Now if you set things up right (see my notes) you can simplify the problem so that only two beams of light come off the star, in opposite directions if you say the star is at rest. From a reference frame that says the star is moving, these beams seem to point a bit in the direction the star is going. Oddly, that light is still travelling at the same speed, c, in each frame, because that speed is set by basic laws (Maxwell’s equations) and the principle of relativity says that the same basic laws work for each frame. They carry away not only energy, but also momentum. J. C. Maxwell showed, around 1870, that for any electromagnetic wave traveling in one direction, there’s a relation between the magnitude of the momentum p and the energy E: E=pc. The fraction of the momentum in the two beams that is forward, i.e. doesn’t cancel, is just v/c, where v is the star’s speed.

So if the star loses energy E, its momentum (mv, where m is mass) must go down by the same amount as the light momentum increased, if we assume that total momentum is conserved. That amount is (v/c)(E/c). So we need that when an energy E is lost,

a mass E/c^2 is also lost, or else momentum wouldn’t be conserved. (You can check the algebra.)

So those are the big ingredients in the argument:

1. Light travels at the same speed, c, in each frame.

2. Classically, E=pc for light.

3. Total E is conserved,

4. Total momentum is conserved.

5. Things that don’t accelerate in one frame don’t accelerate in another frame moving at a steady rate with respect to the first.

Mike W.

. Go to Lecture 11.

Here’s one argument. Imagine a star radiating light away. If you look at it from a reference frame in which the star is at rest, the star isn’t going to accelerate off somewhere because the light is shooting out equally in all directions. So if you look at it from a frame in which the star is moving, it’s also true that the star won’t accelerate. Now if you set things up right (see my notes) you can simplify the problem so that only two beams of light come off the star, in opposite directions if you say the star is at rest. From a reference frame that says the star is moving, these beams seem to point a bit in the direction the star is going. Oddly, that light is still travelling at the same speed, c, in each frame, because that speed is set by basic laws (Maxwell’s equations) and the principle of relativity says that the same basic laws work for each frame. They carry away not only energy, but also momentum. J. C. Maxwell showed, around 1870, that for any electromagnetic wave traveling in one direction, there’s a relation between the magnitude of the momentum p and the energy E: E=pc. The fraction of the momentum in the two beams that is forward, i.e. doesn’t cancel, is just v/c, where v is the star’s speed.

So if the star loses energy E, its momentum (mv, where m is mass) must go down by the same amount as the light momentum increased, if we assume that total momentum is conserved. That amount is (v/c)(E/c). So we need that when an energy E is lost,

a mass E/c^2 is also lost, or else momentum wouldn’t be conserved. (You can check the algebra.)

So those are the big ingredients in the argument:

1. Light travels at the same speed, c, in each frame.

2. Classically, E=pc for light.

3. Total E is conserved,

4. Total momentum is conserved.

5. Things that don’t accelerate in one frame don’t accelerate in another frame moving at a steady rate with respect to the first.

Mike W.

*(published on 10/22/2007)*