Re: [Phys-l] E = mc2

[John Denker <jsd@av8n.com>]

On 03/26/2007 07:45 AM, Anthony Lapinski wrote:
> My students know about Einstein's famous equation, and recently asked
> where it comes from. I know that it is a result of his special theory of
> relativity, but can someone point me to a reference which shows a "simple"
> derivation of this formula? And would it be easy for high school physics
> students to understand?

Far and away the easiest way to understand this, at any level from
high school to grad school or beyond, is to recognize the mass as
the invariant "length" of the [energy,momentum] 4-vector:

      m^2 = E^2 - ps^2                  [1]

where m is the mass of the particle (in any frame), E is the energy
(in some chosen reference frame), and ps is the spacelike part of the
momentum, i.e. the ordinary 3-momentum.

This is the spacetime approach, i.e. the modern (post-1908) approach.

This is a win/win situation, because this approach:
 a) reinforces what students already know about ordinary spacelike
  vectors, and
 b) uses that and extends it into the fourth dimension.


Here's a nifty illustration of how easy-to-use and how powerful
this approach is:
  http://www.av8n.com/physics/bevatron.htm

The same formula also crops up here:
  http://www.av8n.com/physics/spacetime-acceleration.htm#sec-p
and in gazillions of other applications.





It must be emphasized that the celebrated formula E=mc^2 is not valid
in all frames, but only in the rest frame of the particle, as you can
see from equation [1].  To say the same thing another way, mc^2 is the
rest energy (not the total energy) of the particle.