Abstract
We give a totally 3-covariant formulation of the gravitationally coupled electromagnetic field (with a source) in terms of physical 3-covariant variables, keeping the correspondence with the A.D.M. formulation of the problem. We introduce a 3-covariant time derivation which allows us to discuss the spreading of the transverse and longitudinal components of the electromagnetic field.Then we present the second-order propagation equation and the Hamiltonian 3-covariant equations in a complete electromagnetic gauge independent fashion using the well-known elliptic operator Δ(g ij ) and we also discuss the dynamical impossibility of disentangling the transverse and longitudinal modes.Thereafter we extend the 3-covariant formulation to the gravitational field and consider the problems of the initial conditions for the full system, as done by York for the vacuum case.Finally, we apply the formalism to the merostatic sourceless problem, showing the connection between the merostatic and the static cases.
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