This thesis focusses on three main aspects of the foundations of any theory of gravity where the gravitational field admits a geometric interpretation: (a) the principles of equivalence; (b) their role as selection rules in the landscape of extended theories of gravity; and (c) the possible modifications of the spacetime structure at a "mesoscopic" scale, due to underlying, microscopic-level, quantum-gravitational effects.
The first result of the work is the introduction of a formal definition of the Gravitational Weak Equivalence Principle, which expresses the universality of free fall of test objects with non-negligible self-gravity, in a matter-free environment. This principle extends the Galilean universality of free-fall world-lines for test bodies with negligible self-gravity (Weak Equivalence Principle).
Second, we use the Gravitational Weak Equivalence Principle to build a sieve for some classes of extended theories of gravity, to rule out all models yielding non-universal free-fall motion for self-gravitating test bodies. When applied to metric theories of gravity in four spacetime dimensions, the method singles out General Relativity (both with and without the cosmological constant term), whereas in higher-dimensional scenarios the whole class of Lanczos--Lovelock gravity theories also passes the test.
Finally, we focus on the traditional, manifold-based model of spacetime, and on how it could be modified, at a "mesoscopic" (experimentally attainable) level, by the presence of an underlying, sub-Planckian quantum regime. The possible modifications are examined in terms of their consequences on the hypotheses at the basis of von Ignatowski's derivation of the Lorentz transformations. It results that either such modifications affect sectors already tightly constrained (e.g. violations of the principle of relativity and/or of spatial isotropy), or they demand a radical breakdown of the operative interpretation of the coordinates as readings of clocks and rods.
