How Did Einstein Formulate E=mc²?

[narrator]

I've been Googling to find out how Einstein worked out his famous equation. Some texts (that may or may not be factual) suggest it was an educated guess and not even expressed as the equation initially. Wiki http://en.wikipedia.org/wiki/E=mc2#Einstein:_mass.E2.80.93energy_equivalence" that Planck was critical of the initial formula. Others go into a more dry discourse on historical moments that led to it.

Does anyone have a link to something reliably factual that leads you along the trail of his discovery in a way that a layman could understand?

One aspect that I've often found difficult to follow is how light speed came to be connected with it. From the outside, the connection isn't readily apparent. I read here somewhere that it has some connection to F=ma.

 

[phyzguy]

This paper, " DOES THE INERTIA OF A BODY DEPEND
UPON ITS ENERGY-CONTENT?" is simple to understand, requiring only algebra. Here is a link:


http://www.fourmilab.ch/etexts/einstein/E_mc2/www/

 

[Jimmy Snyder]

Some texts (that may or may not be factual) suggest it was an educated guess and not even expressed as the equation initially.

According to Feynmann, making an educated guess is the correct first step in the scientific process. However, I don't think e = mc^2 falls into that category. Rather, it was derived mathematically from an educated guess that he made earlier that same year and which concerned the constancy of the speed of light in a vacuum. You are correct, however, that the expression was something other than an equation (sort of). Following the link given by phyzguy, we see:

If a body gives off the energy L in the form of radiation, its mass diminishes by L/c²

That's how he expressed it. It is an equation, but it's not dressed up like one.

 

[collinsmark]

Einstein's big 1905 paper on [what we now call] special relativity, "On the Electrodynamics of Moving Bodies," A. Einstein, Annelen der Physik, June 30, 1905, Einstein calculates the kinetic energy (energy of motion), W, of a particle to be

W = \frac{1}{\sqrt{1 - v^2/c^2}}mc^2 - mc^2

And using somewhat more modern notation, and defining \gamma,

\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}},

W = \gamma mc^2 - mc^2 .

Of course, that's just the kinetic energy only. There's nothing about total energy, or even rest energy in that equation. But it might make one scratch one's head, "where are all these mc² terms coming from? It seems important." Although a bit of a jump, it turns out that \gamma mc^2 is the total energy of body, and mc^2 is the rest energy. The total minus rest energy is the kinetic energy. And it's interesting that the above formula for W reduces to the Newtonian W ≈ ½mv² for v near 0, using the Taylor series expansion. But "On the Electrodynamics of Moving Bodies" doesn't take it that far. That paper doesn't make such assumptions about total energy and rest energy.

A few months later Einstein released his "Does the Inertia of a Body Depend Upon Its Energy Content?" paper, that Phyzguy mentioned above. This paper uses the Doppler shift, combined with conservation of energy to show that if a body releases energy E, its mass must decrease by a corresponding amount. And that amount is E/c². (And like Jimmy Snyder mentioned, the notation and variables are different than what are shown in this post, but with just a change in variables and a little algebra, you can get the E = mc².)

 

[zoobyshoe]

Great post, collinsmark!