Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses

Adopt as physical assumptions of the framework the following statements about particles and their motion, in the spacetime of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure and the frames of Inertial frames, stationary observers and the adopted Galilean transformations between frames.

(i) Particle labels and worldlines. A model of the framework has particle labels $i=1,\dots ,N$ for some $N\ge 1$. To each label and each inertial frame of the model — an adapted chart $(o,u,(e_1,e_2,e_3))$ with coordinates $(t,x)$ in the sense of Inertial frames, stationary observers and the adopted Galilean transformations between frames — there is assigned a worldline ${\gamma }_i:I\to A$ described in that frame, where $I\subseteq \mathbb{R}$ is a nondegenerate open interval — that is, an interval of the form $(a,b)$ with $-\mathrm{\infty }\le a<b\le +\mathrm{\infty }$ and at least two points — the curve ${\gamma }_i$ is of class $C^2$ in the adapted coordinates of the frame, and the parameter condition

$\tau ({\gamma }_i(t)-o)=t(t\in I)$

holds: the curve parameter is the elapsed time from the origin of the frame, so in the frame's coordinates ${\gamma }_i(t)=(t,x_i(t))$ with $x_i:I\to {\mathbb{R}}^3$ of class $C^2$. The parameter condition replaces the causal conditions of relativistic kinematics: no timelike condition on the velocity and no upper bound on its Euclidean norm is imposed, and the parameter is not free up to reparametrization.

(ii) Description in another frame. The same motion described in a second inertial frame is the curve obtained by the time reparametrization of the two frames: if $(t^{\mathrm{\prime }},x^{\mathrm{\prime }})$ are the coordinates of the second frame and $t^{\mathrm{\prime }}=t+b$ is the time component of the transformation between the two frames (Inertial frames, stationary observers and the adopted Galilean transformations between frames, Transformation between adapted charts and its kinematic consequences), then the description of the motion in the second frame is the curve ${\gamma }_{i,b}(t^{\mathrm{\prime }}):={\gamma }_i(t^{\mathrm{\prime }}-b)$, defined on $I+b$, and it again satisfies the parameter condition $\tau ({\gamma }_{i,b}(t^{\mathrm{\prime }})-o^{\mathrm{\prime }})=t^{\mathrm{\prime }}$ with respect to the second frame's origin $o^{\mathrm{\prime }}$. This is a consistency requirement of the model's frame language; the accompanying spatial transformation of the coordinate curve is the affine law of Transformation between adapted charts and its kinematic consequences, and its kinematic consequences are proved in Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes.

(iii) Regularity is frame-independent. The class of the worldline is independent of the frame used to describe it: by the transformation law of Transformation between adapted charts and its kinematic consequences, the coordinates of the same event in two inertial frames differ by $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Rx+vt+a$ with constant $b,a,v$ and $R\in SO(3)$, so a coordinate curve is twice continuously differentiable in one inertial frame exactly when its reparametrized description is twice continuously differentiable in the other. The assumptions of this postulate are therefore read in any one inertial frame without loss.

(iv) Velocity and acceleration. In a fixed inertial frame the quantities $w_i:=d{x}_i\mathrm{/}dt$ and $a_i:=d^2{x}_i\mathrm{/}d{t}^2$ are the coordinate velocity and coordinate acceleration of the label, of dimensions $\mathrm{\ell }{\tau }^{-1}$ and $\mathrm{\ell }{\tau }^{-2}$ by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; they are defined precisely, and their transformation under a change of inertial frame is recorded, in Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors.

(v) Fixed inertial mass. Each particle label $i$ carries a mass $m_i>0$, a primitive parameter of the model with dimension $\mu$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The mass is fixed: it is the same positive number at all times and in all inertial frames of this model; the index $i$ identifies a label within that model and asserts no cross-model identification or equality of mass parameters. It is not a function of the time coordinate, of the position or of the velocity, and no variable-mass model is a model of this clause. The mass enters the dynamics only through its later appearance in the equation of motion; this postulate attaches no dynamics to it. Distinct labels may carry distinct masses, and the masses are model data, not measured values.

(vi) Point-particle idealization. The model retains of each label only its position and its fixed mass, together with the label index: size, shape, spin, deformation, temperature and internal degrees of freedom are omitted, and the quantity $x_i(t)$ is the single point representing the label at time $t$. An application of the model must state when this idealization is adequate for the system at hand; for a body whose rotation, deformation or internal structure matters for the question asked, the point-particle model is not adequate and a richer model must be specified. The framework makes no claim that every physical body is a point particle.

(vii) Calibration caveat. An apparatus calibration may estimate a mass parameter of a concrete body under a stated measurement model, with its assumptions, sampling and uncertainty; such an estimate neither defines the mass parameter of a label nor verifies the equation of motion in which it occurs, and it is not deduced from this postulate. No measured value of any mass is asserted by this item.

• Adopted, not deduced. Clauses (i)-(vi) fix the model's particle ontology; the parameter condition and the fixed-mass clause are part of that ontology, not theorems. Clause (vii) delimits the interface with measurement. Nothing here asserts that a concrete object satisfies the idealization, and no empirical comparison is made.

• No dynamics. Neither a force law nor an equation of motion is asserted. In particular the model does not yet say what $a_i$ is; the dynamics is adopted on the later pages that introduce force, initial data and the equations of motion. Consequently no conservation statement and no prediction follows from this postulate alone.

• Domain of intended use. The non-relativistic regime of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure; particles moving at speeds small compared with the speed of light, in situations where the point-particle idealization and the classical description of the apparatus are adequate. The framework sets no numerical cutoff and records no measurement.