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→What the Equation Means: More. The website went down last night while I was trying to add this.
An interesting fact is that, normally, one considers only ''changes'' in potential energy; one doesn't need an absolute scale. A rock at the top of a hill has more potential energy than after it rolls to the bottom of the hill, but the energy at the bottom isn't necessarily zero. We could dig a hole and let it roll down farther, with its energy going negative. Only changes matter. Now it turns out that, once one accepts the implications of E=mc², one ''could'' assign an absolute potential energy to something—its mass times c², and changes in potential emergy would work out correctly because of the mass changes. But that isn't necessary, and, in any case, it would require accepting E=mc² and would therefore be getting ahead of the story.
An interesting fact is that, normally, one considers only ''changes'' in potential energy; one doesn't need an absolute scale. A rock at the top of a hill has more potential energy than after it rolls to the bottom of the hill, but the energy at the bottom isn't necessarily zero. We could dig a hole and let it roll down farther, with its energy going negative. Only changes matter. Now it turns out that, once one accepts the implications of E=mc², one ''could'' assign an absolute potential energy to something—its mass times c², and changes in potential emergy would work out correctly because of the mass changes. But that isn't necessary, and, in any case, it would require accepting E=mc² and would therefore be getting ahead of the story.
With those preliminaries out of the way, it is possible to give a concise explanation of what the equation means:
::'''Potential energy has mass.'''
That is, it weighs something. Whenever anything has potential energy of any kind in it, improbable as this may sound, it weighs more. The proportionality constant is 1/c<sup>2</sup>, or 1.11 x 10<sup>-17</sup> kilograms per Joule. A fresh battery weighs more than a spent one, a wound-up alarm clock weight more than a run-down one, etc.
Now that's way too small to measure for anything other than nuclear reactions, which is why it escaped everyone's notice for so long. But it has been measured and experimentally verified for nuclear transformations all across the periodic table.
:There's an interesting parallel with heat. Before the rise of thermodynamics, it was believed that heat was a "substance". That substance was called "caloric". When heat travels from one body to another, what was really happening was presumed to be a transfer of caloric. Much effort was put into measuring the mass of this mysterious "substance". It was always found to be zero, and we now know that what is actually being transferred is thermal energy. So it is not unheard-of to assign mass to intangible properties.
The nonzero mass of potential energy, and the equation E=mc², were determined on theoretical grounds, before any experimental observations were made. The logic of this follows from these assumptions:
#Galilean and Newtonian mechanics.
#Galilean relativity, that is, the notion that there is no absolute frame of reference.
#Conservation of energy.
#Conservation of momentum. (So far this is just classical physics.)
#The universality of the speed of light. (That is, special relativity.)
Keep in mind that, under special relativity, it's not just space and time that need to be redefined. The definitions of momentum and energy need to change also. This is necessary so that the '''conservation of energy and momentum will be absolutely precise in all circumstances.'''
For example, under classical Newtonian mechanics, the kinetic energy of a moving mass is
:<math>E = \frac{1}{2}mv^2\,</math>
But under special relativity is is
:<math>E = mc^2(\frac{1}{1 - v^2/c^2} - 1)\,</math>
One can verify that, in the non-relativistic limit, the second of those equations converges to the first.
It is this requirement, and some "gedanken experiments" involving conversion between potential and kinetic energy, that lead to E=mc².
==History of Experimental Verification==
==History of Experimental Verification==