Conformally invariant gauge conditions in electromagnetism and general relativity

The construction of conformally invariant gauge conditions for Maxwell and Einstein theories on a manifold M is found to involve two basic ingredients. First, covariant derivatives of a linear gauge (e.g. Lorenz or de Donder), completely contracted with the tensor field representing the metric on the vector bundle of the theory. Second, the addition of a compensating term, obtained by covariant differentiation of a suitable tensor field built from the geometric data of the problem. The existence theorem for such a gauge in gravitational theory is here proved when the manifold M is endowed with a m-dimensional positive-definite metric g. An application to a generally covariant integral formulation of the Einstein equations is also outlined.