I address the question: “What is fixed on the boundary in the action principles of general relativity?” Four forms of the action are considered: the Einstein action, the Hilbert action, the first order action, and what may be called the cosmological action. The relationships and boundary data of these actions are described geometrically. Formal passage to the “Euclidean” forms of these actions is effected in detail.

"The last remnant of physical objectivity of space-time" is disclosed, beyond the Leibniz equivalence, in the case of a continuous family of spatially non-compact models of general relativity. The physical individuation of point-events is furnished by the intrinsic degrees of freedom of the gravitational field, (viz, the "Dirac observables") that represent - as it were - the "ontic" part of the metric field. The physical role of the "epistemic" part (viz. the "gauge" variables) is likewise clarified. At the end, a (...) peculiar four-dimensional "holistic and structuralist" view of space-time emerges which includes elements common to the tradition of both substantivalism and relationism. The observables of our models undergo real "temporal change" and thereby provide a counter-example to the thesis of the "frozen-time" picture of evolution. (shrink)