Energy–momentum tensors in the theory of electromagnetic fields admitting electric and magnetic charge distributions

Abstract

When the field tensor of an electromagnetic field admitting both electric and magnetic charge distributions is expressed in terms of a Clebsch representation, the extended Maxwell equations in the presence of a given gravitational field are derivable from an invariant variational principle in which the Clebsch potentials play the role usually assumed by the classical 4‐potentials. The corresponding Lagrange density gives rise in a unique manner to a symmetric tensor density Thj, which displays some of the properties normally associated with the energy–momentum tensor density of the electromagnetic field. However, this interpretation may be in conflict with the generally accepted expression for the modified Lorentz force. Accordingly an alternative energy–momentum tensor density ϑhj is derived which does not suffer from this drawback. However, when a generalized variational principle for the simultaneous determination of the behavior of both the electromagnetic and the dynamical gravitational fields is introduced, the resulting Euler–Lagrange equations give rise to extended Einstein–Maxwell equations which involve the density Thj. On the other hand, the alternative Einstein–Maxwell equations, obtained by the replacement of Thj by ϑhj, are not derivable from a variational principle. The solutions of the two Einstein–Maxwell equations, for the case of a spherically symmetric metric and static electromagnetic field, predict distinctly different effects of the magnetic charges on the gravitational field.