On the classic geometrodynamics of a spherically-symmetric configuration of gravitational and electromagnetic fields

Abstract

Analytical aspects of the classical geometrodynamics for the spherically-symmetric configuration of the electromagnetic and gravitational fields in GR are considered. The feature of such configurations is that they admit two motion integrals – the total mass and charge. The Einstein-Hilbert action for the configuration, after dimensional reduction, by means of the Legendre transformation is reduced to the Hamiltonian action. Using the conservation laws for the mass and charge, as well as the Hamiltonian constraint, the momenta are found as functions of configuration variables. The set of equations, which associate momenta and functional derivatives of the action in the configuration space (CS) is integrable. This allows us to obtain the action functional as a solution of the Einstein-Hamilton-Jacobi equation in functional derivatives. Variations of the action functional with respect to mass M and charge Q of the configuration lead to the motion trajectories in the CS. We note that the minisuperspace metric, which is induced by the kinetic part of the Lagrangian, does not coincide with the CS metric that arises when the function of lapse N is excluded from the action. The space-time metric for which the indicated metrics coincide in the T-region up to a coefficient are considered. The metric of CS is constructed and its geometry is studied. Under the trivial embedding of hypersurfaces of the foliation into a dynamical T-region, the CS is flat. It allows introducing pseudo-Cartesian coordinates in which the CS metric takes the Lorentz form.