Hamiltonian approach to GR - Part 1: covariant theory of classical gravity

A challenging issue in General Relativity concerns the determination of the manifestly-covariant continuum Hamiltonian structure underlying the Einstein field equations and the related formulation of the corresponding covariant Hamilton-Jacobi theory. The task is achieved by adopting a synchronous variational principle requiring distinction between the prescribed deterministic metric tensor $\widehat{g}(r)\equiv \left\{ \widehat{g}_{\mu \nu }(r)\right\} $ solution of the Einstein field equations which determines the geometry of the background space-time and suitable variational fields $x\equiv \left\{ g,\pi \right\} $ obeying an appropriate set of continuum Hamilton equations, referred to here as GR-Hamilton equations$.$ It is shown that a prerequisite for reaching such a goal is that of casting the same equations in evolutionary form by means of a Lagrangian parametrization for a suitably-reduced canonical state. As a result, the corresponding Hamilton-Jacobi theory is established in manifestly-covariant form. Physical implications of the theory are discussed. These include the investigation of the structural stability of the GR-Hamilton equations with respect to vacuum solutions of the Einstein equations, assuming that wave-like perturbations are governed by the canonical evolution equations.