I don’t know how rigorous it is, but a complete definition is given in the paper as “the relative gravitational acceleration of two test particles in free fall.” (I didn’t make up that definition.)Originally Posted byGuitarist

I think you need two test particles. Let them float in close proximity to each other, initially at rest with respect to each other. If they stay in lockstep (or negligibly move relative to each other), then the gravitational field is uniform in that region. If they don’t, it’s nonuniform, and the particles are accelerating relative to each other. The degree of their relative acceleration indicates the degree of the tidal force there.But this seems unsatisfactory. If it is a property of the field, which surely it is(?), then it should be possible to explain it with reference to a single particle. For surely, any force that requires two bodies to detect it (cf. Coulomb), implies an interaction between the bodies, which clearly is not intended in the case of tidal force?

The degree of tidal force at any point (small region) is the curvature of spacetime at that point. Those are synonymous. Fig. 4 in the paper shows the theorized curvature of spacetime (degree of tidal force) along a segment of a radius.I can see, intuitively, that it has something to do with the "shape" of the gravitational field, but that's already given by the curvature tensor in the general theory.