Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework

Adopt as physical assumptions of the classical gravitational framework, in addition to Adopted spacetime model and classical domain of the gravitational framework and the matter-content assumptions of the framework, the following rules by which ideal clocks, test bodies and light couple to the metric $g$ of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model.

(i) Ideal clocks. An ideal clock is an idealized physical device carried along a future timelike worldline (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold) whose readings are assumed to reproduce the proper time of that worldline: the difference of two readings taken at two events of the worldline is the proper-time increment between those events as defined by the metric, and it is independent of the construction of the clock. Two ideal clocks moving along the same worldline differ only by an additive constant (their choice of zero); different worldlines generally accrue different proper times between the same pair of events. The physical constant $c$ and the unit of duration are those of SI unit and coordinate conventions of the classical gravitational framework.

(ii) Free fall of massive test bodies. A test body is an uncharged idealized body whose internal structure is small compared with the scale over which the geometry of the model varies and whose gravitational self-field is negligible within the model. An ideal freely falling massive test body is assumed to have a worldline that is a future timelike geodesic of the metric-selected connection (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Geodesic of an affine connection), traversed with physical four-velocity $U=d\gamma \mathrm{/}d\tau$ satisfying $g_{\gamma (\tau )}(U,U)=-c^2$; proper time is an affine parameter of the worldline (Proper time is well defined and reparametrization invariant; the four-velocity is its proper-time tangent). The worldline is determined by the event at which the body is released together with its four-velocity there, by the uniqueness of the geodesic initial-value problem (The geodesic initial-value problem and the exponential map of the metric-selected connection); no force, charge, or rest mass of the body enters this rule.

(iii) Light rays. In the geometric-optics approximation an ideal light ray propagating in the model is assumed to be a future null geodesic of the metric-selected connection (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Geodesic of an affine connection). A null worldline carries no proper time and no four-velocity in the sense of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold; within the approximation the propagation of an ideal ray is independent of its frequency, and only the ray's direction in the tangent space is fixed, not a scale along it.

(iv) Scope and exclusions. The three rules are adopted in the test-field regime of the smooth classical model: the clocks, bodies and rays do not back-react on the geometry, and the geometry varies smoothly on the scales relevant to them. Configurations requiring the body's own gravitational field, radiation reaction, an electromagnetic or other self-force, finite-size or tidal couplings beyond the test-body idealization, non-geometric-optics propagation, dispersion or dissipation of light, or quantum effects are outside these assumptions and require additional model content. The assumptions do not assert that any real clock, body or light ray is exactly ideal, nor do they attach measured readings to any model.

• Adopted, not derived. Clauses (i)-(iii) are physical assumptions of the framework. Nothing in the mathematical items on this page or its prerequisites produces them: the geodesic initial-value theorem and the first-variation lemma identify geodesics as the stationary curves of the length functional (First variation of the length of a timelike curve: stationarity is the geodesic equation) and construct the exponential map, but they do not assert that any physical body, clock or light ray follows those curves. The equivalence principle and the structure of the nonrelativistic limit motivate these rules; neither proves them.

• Empirical content is downstream. The framework's empirical significance lies in predictions of fully specified models — a geometry, a matter content and a field law — compared with reported observations under their apparatus, calibration and uncertainty qualifications. Agreement of such a comparison is evidence about the combined model and apparatus assumptions, not a proof of clauses (i)-(iv), and a disagreement can expose an auxiliary assumption, an apparatus effect or an analysis error rather than a foundational postulate.

• Relation to the flat model. The corresponding assumptions of flat special relativity are separate physical postulates with Minkowski scope; they are not used here as suppliers of curved-spacetime content. The flat model appears on this track only as a comparison and limiting case.