What should we understand by the four-momentum of physical system?

TL;DR

This paper proposes a new covariant formalism for defining four-momentum in curved spacetime, explicitly including all particles and fields, and applies it to relativistic systems with electromagnetic and gravitational fields.

Contribution

It introduces a novel representation of four-momentum as a sum of two nonlocal four-vectors, addressing limitations of existing definitions in curved spacetime.

Findings

Standard definitions only work for free particles.

The new formalism accounts for all fields and particles.

Integral vectors are not four-vectors and do not equal four-momentum.

Abstract

It is shown that in curved spacetime none of the known definitions of four-momentum correspond to the definition, in which all the system particles and fields, including fields outside matter, make an explicit contribution to the four-momentum. This drawback can be eliminated under the assumption that the primary representation of four-momentum is the sum of two nonlocal four-vectors of the integral type with covariant indices. The first of these four-vectors is the generalized four-momentum, found with the help of Lagrangian density. The second four-vector is the four-momentum of fields themselves, and its time component is related to the energy given by tensor invariants. The standard approach makes it possible to find the four-momentum in covariant form only for a free point particle. In contrast, the obtained formulas for calculating the four-momentum components are applied to a stationary and moving relativistic uniform system, consisting of many particles. In this case, the main fields of the system under consideration are taken into account, including the electromagnetic and gravitational fields, the acceleration field and the pressure field. The formalism used includes the principle of least action, charged and neutral four-currents, corresponding four-potentials and field tensors, which ensures unification and the possibility of combining fields into a single interaction. The calculation of the integral vector components shows that the so-called integral vector is not equal to four-momentum and is not a four-vector at all, although it is conserved in a closed system. Thus, the four-momentum cannot be found with the help of an integral vector and components of the system stress-energy tensor, in contrast to how it is assumed in the general theory of relativity.