The Hamiltonian of the General Theory of Relativity with Electromagnetic Field

Abstract

In this paper we have given a specific example of a Hamiltonian of a non-linear field theory, a Hamiltonian density completely free of time derivatives. In accordance with the general theory developed previously, this Hamiltonian is one of the constraints between the canonical variables and, therefore, vanishes everywhere. To obtain this function, we have developed methods that will also permit the construction of Hamiltonian densities in any field theory in which the Lagrangian density is quadratic in the first derivatives. Our Hamiltonian differs from the one obtained by Schild and Pirani in that they use Dirac's method to derive a Hamiltonian that is invariant but contains velocities, so that their canonical field equations cannot be solved with respect to the time derivatives of all canonical variables. In our formalism, the canonical equations contain no time derivatives on the right-hand sides, but the adoption of a particular Hamiltonian is equivalent to the adoption of a particular coordinate condition and gauge condition. However, once we have obtained any one Hamiltonian density, we can readily obtain any other one (and thus go over to arbitrary coordinate and gauge conditions) by combination with the other constraints of the theory_in question.