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© Copyright 1998, Jim Loy
How did Einstein come up with E=mc² (E=mc^2)? Did he just dream it up? What does it have to do with Relativity, which deals with time dilation and length contraction at great velocities (See Relativity 100)? Well, the equations involving time dilation and length contraction have consequences. One of these is that as velocity increases, mass increases.
Energy can be used to make an object accelerate. Some form of potential energy becomes kinetic energy, a simple conversion. But, the equations of relativity show that it becomes increasingly difficult to accelerate an object. Sorry, I won't show the algebraic details here. More and more energy produces less and less acceleration. There is an increasing resistance to acceleration. Resistance to acceleration is mass (inertial mass). The equations suggest that the mass is increasing, as the velocity increases. This means that part of the original energy is converted to kinetic energy, while some of it is converted to mass. And the equations show that the relationship between this energy and this mass is E=mc² (E=mc^2).
From there, Einstein deduced that mass is just another more compact form of energy, obeying the famous formula. And that, in turn, led to other consequences.
And, of course, E=mc² (E=mc^2) has been tested an enormous number of times. Subatomic particles do gain mass if accelerated. And nuclear reactions (including those used in nuclear weapons) involve conversion of mass to energy or vice versa.
Note: I hope to derive E=mc² (E=mc^2), in a future version of this article. For now, we must be satisfied with the above verbal version.
Note: Some WWW browsers cannot handle the little square character (²). So I have also used ^2 which is computer notation.