Change in the Mass of Matter Due to the Conservation of the 4-momentum of Matter Together with the Gravitational Field

An obvious but unknown fact is presented: the generally accepted conservation of the 4-momentum of matter together with the gravitational field means a violation of the conservation law for matter under gravitational interaction. It is shown that the local conservation law for matter, which is expressed by the equality to zero of the partial divergence of the energy-momentum tensor when using Galilean coordinates, is expressed by the equality to zero of the covariant divergence of the tensor density of the energy-momentum of matter when using curvilinear coordinates. And this local law ensures the validity of the integral conservation law. However, in the presence of a gravitational field, this local conservation law leads to a violation of the integral conservation law for matter. The article calculates the variable mass of an isolated body, which changes with a change in the gravitational field, and presents an expression for the conserved 4-momentum of matter together with the gravitational field.