Integral electromagnetic theorems in general relativity

Abstract

In classical physics Gauss’s theorem, connecting normal flux of intensity with enclosed mass or charge, has one single form for gravitation and for electrostatics: it is, in fact, a direct consequence of the inverse square law. In the general theory of relativity gravitation and electricity play very different parts, and one might expect a divergence between an extension of Gauss’s gravitational theorem to general relativity and an extension of his electrostatic theorem. Whittaker, and Ruse, have developed the gravitational extension, and Whittaker has indicated the electrical extension. It is the purpose of the present paper to complete the electrical extension. It is found that for an electromagnetic field in general relativity there exists a theorem which is expressed naturally in precisely the same form as the classical theorem of Gauss, but which admits a more general interpretation. Other integral theorems are also obtained by systematic application of the three Green-Stokes theorems available in four-dimen­sional space-time. The fact that the fundamental form of space-time is not positive-definite does not affect certain results, such as (2.1) below. But when we seek physical interpretations, this fact must be taken into consideration, as the presence of the indicators in (3.13), (3.15), (3.28) shows. It is through careful attention to this detail that the results obtained differ from those given by Pauli. The utility of the classical formulae of Green and Stokes depends to a great extent on the fact that they are expressed in forms involving essentially positive elements of area and volume: it would seem that the value of their extensions to space-time depends similarly on the use of essentially positive elements of area, volume, and 4-volume, with explicit attention to the indefinite character of the funda­mental form.