Lagrangian formalism in the theory of relativistic vector fields

The Lagrangian formalism is used to derive covariant equations that are suitable for use in continuously distributed matter in curved spacetime. Special attention is given to theoretical representation, in which the Lagrangian and its derivatives are directly involved. The obtained results, including equations for metric, equations of motion, and equations for fields, are applied to purely vector fields. As a consequence, formulas are determined for calculating the basic quantities necessary to describe physical systems. In this case, not only the pressure field and acceleration field are taken into account, but also the electromagnetic and gravitational fields outside the matter, which contribute to the four-momentum and to the four-dimensional angular momentum pseudotensor of each system. Each of the presented fields, including the gravitational field, has its own four-potential and its own tensor, which allows all calculations to be performed in a covariant way. In particular, when applying the principle of least action, the optimal approach is one in which the four-currents, field four-potentials and metric tensor depend on the observation point and vary independently from each other. In addition to the Euler–Lagrange equation, another equation of motion is derived containing the density of generalized four-force and the time derivative of the volume density of the generalized four-momentum. In order to uniquely calibrate the energy, which is a scalar quantity, the cosmological constant included in the Lagrangian is used. This leads to the fact that both the scalar curvature and the cosmological constant disappear in expression for the four-momentum of the system. In addition, the equation for the metric is simplified. The peculiarity of the equation for the metric is that the total stress-energy tensor of the physical system, presented in the right part of the equation for the metric, is associated only with field tensors and does not depend on particle four-velocities. Moreover, the trace of stress-energy tensor is zero. To calculate the stress-energy tensor, the functional derivative of Lagrangian density with respect to the metric tensor is used. The covariant derivative of stress-energy tensor leads to the equation of particle motion under the action of fields, and to the generalized Poynting theorem. The contractions of the field tensors with the Ricci tensor are equal to zero, so that each field makes its own contribution to the curvature of spacetime in the system. The inertial mass of a system is defined as the value connecting the four-momentum and four-velocity of the center of momentum of the system, and can be calculated using the square of the four-momentum. It is shown that the canonical representation of the angular momentum pseudotensor is its representation with covariant indices. The radius-vector of the center of momentum of a physical system is determined in covariant form.