Adopted massive-particle momentum, observer measurement and isolated-collision conservation

Adopt as physical assumptions of the relativistic particle model, on top of the flat spacetime model of Oriented time-oriented affine Minkowski space (the spacetime primitives) and the SI unit conventions of SI units and coordinate conventions of the special-relativity framework, together with the framework's idealisations of point particles, ideal clocks and ideal light signals:

(i) Mass and momentum. Every massive classical particle of the model carries a positive constant invariant mass parameter $m>0$ and has four-momentum $P=mU$, where $U$ is the four-velocity of its future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition); $P$ is a vector of $V$ whose SI unit is kilogram metre per second. The formula is adopted only for $m>0$; the case $m=0$ is not defined by it and is the separate premise Adopted momentum model for an ideal massless particle. In particular $\eta (P,P)=-m^2{c}^2$ is a consequence of the four-velocity normalisation, not a separate assumption.

(ii) Observer measurement identification. Relative to a future unit observer $n$, the model identifies the measured energy of the particle with $E_n(P)$, its measured spatial momentum with $q_n(P)$, its kinetic energy with $T_n(P)=E_n(P)-m{c}^2$ and its rest energy with $m{c}^2$, where $E_n,q_n,T_n$ are the quantities of Observer energy, spatial momentum, invariant mass, rest energy and kinetic energy of a four-momentum; the units are the joule, kilogram metre per second and kilogram of SI units and coordinate conventions of the special-relativity framework. This clause is a formulation convention identifying mathematical quantities with the quantities the model calls measured; it is not a measurement report and no experiment is claimed.

(iii) Isolated collision balance. An isolated collision is represented by a mass-specified on-shell collision datum at an event, with finite nonempty incoming and outgoing lists of the massive particles of (i) and/or the ideal massless particles of Adopted momentum model for an ideal massless particle, and with no external four-momentum transfer or omitted leg. The interaction is idealized as confined to that event. The sum of the incoming four-momenta equals the sum of the outgoing four-momenta, in the sense of the collision data of Collision data and four-momentum conservation: the model adopts $Q_{\mathrm{i}\mathrm{n}}=Q_{\mathrm{o}\mathrm{u}\mathrm{t}}$ for such an isolated datum. No particular collision is asserted to occur, and no mechanism, probability, cross section or amplitude is supplied by this clause.

(iv) Equation of motion for a stipulated force. Put $H_c^{+}:=\{U\in C^{+}:\eta (U,U)=-c^2\}$. A stipulated force law is a given map $F:D\to V$ on a given subset $D\subseteq A\times H_c^{+}$, with force components in $\mathrm{k}\mathrm{g}\mathrm{m}\mathrm{/}{\mathrm{s}}^2$. No regularity away from the specified worldline is assumed. Along a given $C^2$ massive worldline $x:I\to A$ with $(x(s),U(s))\in D$ for all $s\in I$, the model's particle equation is $mdU\mathrm{/}d\tau =F(x,U)$, with $dU\mathrm{/}d\tau$ the proper-time derivative of Proper-time derivative of a vector field along a worldline and the four-force notation, so that the stipulated $F$ is the four-force of the motion. A particular force law, and the existence, uniqueness and domain of solutions of the resulting equation, are not adopted here and belong to a separate model.

Empirical support for these assumptions is a nonlogical relation of the framework, not a proof; the idealisations of point particles, ideal clocks, ideal light signals and flat spacetime are inherited and travel to every consumer of this premise.