Energy Tensor of the Electromagnetic Field. Before the development of the theory of relativity it was known that the principles of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important conception, [Pg 50] that of the energy tensor, which is important for the further development of the theory of relativity. If in the expression for the 4-vector of force per unit volume, using the field equations (32), we express in terms of the field intensities, , we obtain, after some transformations and repeated application of the field equations (32) and (33), the expression where we have written[12] [12]To be summed for the indices and . The physical meaning of equation (47) becomes evident if in place of this equation we write, using a new notation, [Pg 51] or, on eliminating the imaginary, When expressed in the latter form, we see that the first three equations state the principle of momentum; ,..., are the Maxwell stresses in the electromagnetic field, and (, , ) is the vector momentum per unit volume of the field. The last of equations (47b) expresses the energy principle; is the vector flow of energy, and the energy per unit volume of the field. In fact, we get from (48) by introducing the well-known expressions for the components of the field intensity from electrodynamics, [Pg 52] We conclude from (48) that the energy tensor of the electromagnetic field is symmetrical; with this is connected the fact that the momentum per unit volume and the how of energy are equal to each other (relation between energy and inertia). We therefore conclude from these considerations that the energy per unit volume has the character of a tensor. This has been proved directly only for an electromagnetic field, although we may claim universal validity for it. Maxwell's equations determine the electromagnetic field when the distribution of electric charges and currents is known. But we do not know the laws which govern the currents and charges. We do know, indeed, that electricity consists of elementary particles (electrons, positive nuclei), but from a theoretical point of view we cannot comprehend this. We do not know the energy factors which determine the distribution of electricity in particles of definite size and charge, and all attempts to complete the theory in this direction have failed. If then we can build upon Maxwell's equations in general, the energy tensor of the electromagnetic field is known only outside the charged particles.[13] In these regions, outside of charged particles, the only regions in which we can believe that we have the complete expression for the energy tensor, we have, by (47),