Definition of four-momentum through energy-momentum field

To determine the covariant four-momentum of a physical system in curved spacetime, we introduce an energy-momentum field, whose four-potential and tensor are included in the Lagrangian density. From the principle of least action, we derive an equation for the metric, equations for the energy-momentum field, and the four-dimensional Euler-Lagrange equation. The equations of particle motion are derived in two ways: through the covariant derivative of the right-hand side of the equation for the metric, and by varying the action function with respect to the four-coordinates. These equations are equal when the invariant mass density of the particles is constant. The four-potential of the energy-momentum field makes it possible to find the four-momentum density and four-momentum of the system. The corresponding equations are derived by varying the action function under a four-shift of all particles of the system by a constant four-vector. The four-momentum density and the four-momentum of the system must satisfy these equations for the system's symmetry associated with such the four-shift to hold.