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Synchronous Lagrangian variational principles in General Relativity

The problem of formulating synchronous variational principles in the context of General Relativity is discussed. Based on the analogy with classical relativistic particle dynamics, the existence of variational principles is pointed out in relativistic classical field theory which are either asynchronous or synchronous. The historical Einstein-Hilbert and Palatini variational formulations are found to belong to the first category. Nevertheless, it is shown that an alternative route exists which permits one to cast these principles in terms of equivalent synchronous Lagrangian variational formulations. The advantage is twofold. First, synchronous approaches allow one to overcome the lack of gauge symmetry of the asynchronous principles. Second, the property of manifest covariance of the theory is also restored at all levels, including the symbolic Euler-Lagrange equations, with the variational Lagrangian density being now identified with a $4-$scalar. As an application, a joint synchronous variational principle holding both for the non-vacuum Einstein and Maxwell equations is displayed, with the matter source being described by means of a Vlasov kinetic treatment.