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By Root 937 0

a revised law for the conservation of energy. If we imagine a system of particles all jiggling about, then we have just figured out that adding together the kinetic energy of all the particles plus the mass of all the particles multiplied by c squared we get something that is unchanging. Now, the Victorians would have been happy with the assertion that the sum of kinetic energies should be unchanging, and they would also have been happy with the assertion that the sum of the masses should be unchanging (multiplying by c squared is irrelevant when we’re thinking about what is unchanging). Our new law is consistent with that being the case, but it is much more than that. As it stands there is nothing at all preventing some of the mass from being converted into kinetic energy and vice versa, as long as the sum of these two things is always conserved. We have discovered that mass and energy are potentially interchangeable and the amount of energy we can extract from a mass m at rest (γ is equal to one in that case) is captured by the equation E = mc2.

Our friend Thales of Miletus can at last achieve complete enchantment. He rises from his bath, dripping asses’ milk onto the floor, and welcomes his concubines into his magnificent presence.

Let’s recap: We wanted to look for an object in spacetime that did the job of momentum in three-dimensional space, because momentum is a conserved quantity and therefore useful. We were able to find such an object by building it only out of things that everyone agrees upon, namely the distance in spacetime, the universal speed limit, and the mass. The spacetime momentum vector that we constructed turned out to be very interesting. By looking at the part that points along the space direction, we rediscovered the old law of momentum conservation, with a tweak for things moving close to the speed of light. But the real gold came from looking at the part of the vector that points along the time direction. This gave us an entirely new version of the law of conservation of energy. The old-fashioned kinetic energy,mυ2, was there, but a totally new piece appeared: mc2. Thus, even if an object is standing still, it has energy associated with it, and that energy is given by Einstein’s famous equation: E = mc2.

What does it all mean? We have established that energy is an interesting quantity because it is conserved: “You can increase energy over here provided you lower it over there.” Moreover, we have established that the raw mass of an object provides a potential source of energy. We can imagine taking a blob of matter, say 1 kilogram of “stuff ” (it doesn’t matter what) and “doing something to it” so that afterward there is no 1 kilogram of stuff anymore. And by that we don’t mean the 1 kilogram has been smashed up into tiny bits, we mean that it has vanished. In fact, we can imagine an extreme scenario where all of the original mass gets used up. In its place must be 1 kilogram worth of energy (plus any energy we might have put in when we did the “doing something to it”). That energy could itself be in the form of mass, for example a few hundred grams of new “stuff ” might be created, and the remaining energy could be in the form of kinetic energy: the new stuff could be whizzing about with speed. Of course, we just made all of that up; it was an imaginary scenario. The point to appreciate is that this is the kind of thing that could be allowed by Einstein’s theory. Before Einstein, no one had dreamed that mass could be destroyed and converted into energy because mass and energy seemed to be entirely disconnected entities. After Einstein, everyone had to accept that they are different manifestations of the same type of thing. This is because we have discovered that energy, mass, and momentum must all be combined into a single spacetime object that we have been referring to as the spacetime momentum vector. Actually, its more usual name in physics circles is the energy-momentum four-vector. Just as we discovered that space and time should no longer be thought of as separate entities, so we