Synchronous Lagrangian variational principles in General Relativity

Abstract

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.

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Cited by in corpus (7)

• Hamiltonian approach to GR - Part 1: covariant theory of classical gravity
• Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity
• Manifest Covariant Hamiltonian Theory of General Relativity
• Quantum-wave equation and Heisenberg inequalities of covariant quantum gravity
• Generalized Lagrangian Path approach to manifestly-covariant quantum gravity theory
• Space-time second-quantization effects and the quantum origin of cosmological constant in covariant quantum gravity
• Loop quantum gravity with optimal control path integral, and application to black hole tunneling