Foundations of the Theory of Gravity with a Constraint. Gravitational Energy of Macroscopic Bodies

This paper considers a set of equations describing the static isotropic gravitational field of a macroscopic body within the framework of the theory of gravity with a constraint. It is shown that for any static solid body, the energy of the gravitational field created by it is equal in magnitude to its rest energy, and the energy density defined in this theory is positive everywhere. A general approximate solution of the gravitational field equations is obtained. A nonsingular solution exists only at certain values for the three integration constants. The out-of-body metric coincides with the Schwarzschild metric, but unlike the general relativity theory (GR), the curvature tensor invariants have a certain finite value everywhere. It is claimed that a generally covariant theory of gravity, constructed on the basis of GR, is not physical since it violates the principle of material unity of the world.