Audacious Einstein

[Ben]

So the equality [math]E = mc^2[/math] is perhaps the most famous in all of science, and is certainly familiar to many non-scientists. But its derivation is truly audacious.

 

Don't get me wrong; I have no reason to doubt its truth, nor those of the Special Theory, whose mathematics is, as far as I can see, robust .

 

Here's Einstein (hugely paraphrased):

 

Consider a material body B with energy content [math] E_{\text{initial}}[/math]. Let B emit a "plane wave of light" (what's that, by the way?) for some fixed period of time [math] t[/math]. One easily sees that the energy content of B is reduced by [math]E_{\text{initial}} - E_{\text{final}}[/math], which depends only on [math]t[/math].

 

Let [math]E_{\text{initial}} - E_{\text{final}} = L[/math] i.e.the light energy "withdrawn" from B.

 

Now, says Einstein, consider the situation from the perspective some body moving uniformly at velocity [math]v [/math] with respect to B. Then, evidently, by Lorentz time dilation, [math]L'[/math] depends only on [math]t'[/math], which is [math] t(1 - \frac{v^2}{c^2})^{-\frac{1}{2}}[/math].

 

The difference between [math]L[/math] and [math]L'[/math] is simply [math]L' - L = L[(1 - (\frac{v^2}{c^2})^{-\frac{1}{2}} - 1].[/math]. By expanding [math](1 - \frac{v^2}{c^2})^{-\frac{1}{2}}[/math] as a Taylor series, and dropping terms of order higher than 2 in [math]v/c[/math], he finds that

 

[math]L(1 + \frac{v^2}{2c^2} - 1) = L\frac{v^2}{2c^2} = \frac{1}{2}(\frac{L}{c^2})v^2[/math].

 

With a flourishing hand-wave Einstein now says something like this: the above is an equation for the differential energy of bodies in relative motion; but so is [math]E = \frac{1}{2}mv^2[/math], the equation for kinetic energy - these can only differ by an irrelevant additive constant, so set

 

[math]\frac{L}{c^2} = m[/math] and so [math]L= mc^2[/math].

 

But, says he, [math]L[/math] is simply a "quantity" of energy, light in this case, that now depends only on [math]m[/math] and [math]c^2[/math] so......

 

[math]E = mc^2[/math].

 

Audacious or what?