Our source is a finite volume in which intrinsic processes occur so that the homogeneous elastic equation of motion is violated; is the stress tensor calculated from with a linear and time-invariant law. The violation may be restricted to part of the source volume, to a crack or several cracks inside this volume, etc. The action of the source on the surrounding medium can be replaced by stresses acting on the boundary of a cavity that has the same shape as the source. If either these stresses or the associated displacements at the boundary are known, the seismic deformation of the outer medium is uniquely determined, and so are the displacements resp. stresses that were not known a priori.
The seismic moment tensor is not the moment tensor of forces acting at the boundary of a cavity; it is the moment tensor of those forces that must be applied to the same mathematical surface in an elastic full medium to produce the observed deformation outside the source. (This is, of course, just one of many equivalent definitions of the moment tensor.) I take it for granted by general theory that a unique relationship exists between the moment tensor and the far-field deformation of the outer medium. In order to model the near-field correctly, one would have to include higher-order moments as coefficients in a suitable series expansion of the wavefield.
Using the concept of forces acting on a mathematical boundary in a full space, we can split the forces and their moments into one contribution that deforms the outer medium and one that deforms the interior of the source (not necessarily in the same way as in the real source). For this purpose we consider the boundary as a double surface, one surface forming the inner boundary of the outer medium, the other vice versa. We let act forces on the outer medium so that it is properly deformed, and on the inner medium so that the gap between the two surfaces is closed. Both sets of forces are uniquely determined by the deformation of the boundary, and the same holds for their moments.