Local conservation
It is clear that the energy of matter is not conserved. When an object radiates light it loses energy. However, the energy lost is possibly describable in some other form, say in the light. Therefore the theory of the conservation of energy is incomplete without a consideration of the energy which is associated with the light or, in general, with the electromagnetic field. We take up now the law of conservation of energy and, also, of momentum for the fields. Certainly, we cannot treat one without the other, because in the relativity theory they are different aspects of the same four-vector. Very early in Volume I, we discussed the conservation of energy; we said then merely that the total energy in the world is constant. Now we want to extend the idea of the energy conservation law in an important way—in a way that says something in detail about how energy is conserved. The new law will say that if energy goes away from a region, it is because it flows away through the boundaries of that region. It is a somewhat stronger law than the conservation of energy without such a restriction. To see what the statement means, let’s look at how the law of the conservation of charge works. We described the conservation of charge by saying that there is a current density $\FLPj$ and a charge density $\rho$, and that when the charge decreases at some place there must be a flow of charge away from that place. We call that the conservation of charge. The mathematical form of the conservation law is \begin{equation} \label{Eq:II:27:1} \FLPdiv{\FLPj}=-\ddp{\rho}{t}. \end{equation}  This law has the consequence that the total charge in the world is always constant—there is never any net gain or loss of charge. However, the total charge in the world could be constant in another way. Suppose that there is some charge $Q_1$ near some point $(1)$ while there is no charge near some point $(2)$ some distance away (Fig. 27–1). Now suppose that, as time goes on, the charge $Q_1$ were to gradually fade away and that simultaneously with the decrease of $Q_1$ some charge $Q_2$ would appear near point $(2)$, and in such a way that at every instant the sum of $Q_1$ and $Q_2$ was a constant. In other words, at any intermediate state the amount of charge lost by $Q_1$ would be added to $Q_2$. Then the total amount of charge in the world would be conserved. That’s a “world-wide” conservation, but not what we will call a “local” conservation, because in order for the charge to get from $(1)$ to $(2)$, it didn’t have to appear anywhere in the space between point $(1)$ and point $(2)$. Locally, the charge was just “lost.” There is a difficulty with such a “world-wide” conservation law in the theory of relativity. The concept of “simultaneous moments” at distant points is one which is not equivalent in different systems. Two events that are simultaneous in one system are not simultaneous for another system moving past. For “world-wide” conservation of the kind described, it is necessary that the charge lost from $Q_1$ should appear simultaneously in $Q_2$. Otherwise there would be some moments when the charge was not conserved. There seems to be no way to make the law of charge conservation relativistically invariant without making it a “local” conservation law. As a matter of fact, the requirement of the Lorentz relativistic invariance seems to restrict the possible laws of nature in surprising ways. In modern quantum field theory, for example, people have often wanted to alter the theory by allowing what we call a “nonlocal” interaction—where something here has a direct effect on something there—but we get in trouble with the relativity principle. “Local” conservation involves another idea. It says that a charge can get from one place to another only if there is something happening in the space between. To describe the law we need not only the density of charge, $\rho$, but also another kind of quantity, namely $\FLPj$, a vector giving the rate of flow of charge across a surface. Then the flow is related to the rate of change of the density by Eq. (27.1). This is the more extreme kind of a conservation law. It says that charge is conserved in a special way—conserved “locally.” It turns out that energy conservation is also a local process. There is not only an energy density in a given region of space but also a vector to represent the rate of flow of the energy through a surface. For example, when a light source radiates, we can find the light energy moving out from the source. If we imagine some mathematical surface surrounding the light source, the energy lost from inside the surface is equal to the energy that flows out through the surface.