Adopted free trajectories of massive particles and the clock hypothesis

Adopt as physical assumptions of this framework, extending the ideal-clock reading rule of Adopted ideal-clock and ideal-vacuum-light interpretation to the general worldline calculus of this page:

(i) Free trajectories. A massive test particle that is subject to no interaction (a free particle) has as its worldline a straight future timelike line of the affine model (Oriented time-oriented affine Minkowski space (the spacetime primitives)): there are an event $p\in A$ and a future timelike vector $v\in V$ with $x(s)=p+(s-s_0)v$ for $s\in I$, where $I\subseteq \mathbb{R}$ is a nondegenerate interval and $s_0\in I$ (Curves, regular causal curves and future timelike worldlines in affine Minkowski space). Interactions, forces and collisions are not part of this assumption: a particle that is not free may have any future timelike worldline admitted by its own dynamical model, and no force law is adopted here. The kinematical consequences of this shape for the four-velocity are derived, not assumed, in Four-velocity norm and four-acceleration orthogonality for the model's clocks.

(ii) Clock hypothesis. An ideal clock carried along a future timelike worldline of the model, including an accelerated one and a piecewise $C^1$ one with finitely many kinks, reads between any two parameter values the proper-time difference of Proper time along a future timelike worldline for $C^1$ histories and of Total proper time of a piecewise C^1 future timelike worldline for weak piecewise $C^1$ histories, with no additional term depending on acceleration, jerk or any higher derivative. A kink is an idealization of a finite short acceleration phase whose contribution is included through the integral on the smooth pieces, and the reading does not depend on an increasing regular change of parameter admitted by the curve definition — its derivatives on smooth pieces and all finite one-sided endpoint derivatives are positive, as in Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps and Proper time is invariant under regular increasing weak piecewise changes of parameter. Zero-tangent compositions are not worldlines of this model.

(iii) Scope and qualification. The assumptions concern ideal point particles, ideal clocks and flat classical spacetime. They are adopted, not proved; they are not consequences of the affine geometry, and no experiment is claimed here to establish them. They inherit the idealizations of Adopted ideal-clock and ideal-vacuum-light interpretation: real clocks have finite resolution, calibration and apparatus limitations, and real particles are not exactly free. The coordinate and unit conventions are those of SI units and coordinate conventions of the special-relativity framework, and inertial charts are those of Inertial frames and inertial charts of the affine Minkowski model.

The free-trajectory clause is deliberately silent about interactions and force laws, so it cannot be used to infer conservation laws, mass-shell relations or collision kinematics, which require the separate momentum postulate of the later energy-momentum page. The clock hypothesis is the assertion that ideal clocks have no acceleration-dependent term, so it must not be presented as following from the geometry, and its empirical content is not established here. The weak piecewise mathematical referent is supplied by Total proper time of a piecewise C^1 future timelike worldline and Improper integrability and endpoint increments of weak piecewise future causal curves, and regular increasing weak-piecewise parameter independence by Proper time is invariant under regular increasing weak piecewise changes of parameter; the $C^1$ case uses Proper time along a future timelike worldline and Length-integral invariance under monotone C^1 maps and proper-time invariance under regular increasing maps. This assumption repairs the formulation of the already adopted clock assumption; it does not derive clock behavior or extend its empirical scope.

The published sources describe smooth free-particle worldlines and proper-time clock readings (Carroll, Chapter 1 and Chapter 3; Reall, Sections 2.2-2.3; Crowell, Sections 3.5-3.7, with the signature and unit conversions recorded in the reference list). The adoption is stated conditionally: it is an assumption of this framework, and the source references document the smooth idealization in the literature, not a proof of it. The weak piecewise class with finite one-sided quotient tangents and no derivative limits is an explicit local extension of that adopted reading rule, supported mathematically by this page's integrability and regular-substitution lemmas; it is not claimed to be a verbatim regularity statement in those sources.