Newtonian comparison momentum and kinetic energy in a named inertial frame

Let $(A,V,\eta ,C^{+})$ be the oriented time-oriented affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), let $c>0$ with the SI conventions of SI units and coordinate conventions of the special-relativity framework, let $(o;e_0,e_1,e_2,e_3)$ be an inertial frame with inertial chart $x=(x^0,x^1,x^2,x^3)$ and spatial Euclidean structure $(S,\delta ,\mathrm{\mid }\cdot \mathrm{\mid })$ (Inertial frames and inertial charts of the affine Minkowski model, The Euclidean spatial structure and cross product of an inertial frame), and let $m>0$.

(a) For $w\in S$ define the Newtonian comparison momentum and the Newtonian comparison kinetic energy of the pair $(m,w)$ by $N_p(m,w):=mw,N_K(m,w):=\frac{1}{2}m\mathrm{\mid }w{\mathrm{\mid }}^2,$ where $mw$ is the scalar multiple of $w$ by the real number $m$ (Vector space over a field) and $\mathrm{\mid }w{\mathrm{\mid }}^2=\delta (w,w)$ (Integer powers $a^m$). With the SI conventions these carry the units kilogram metre per second and joule respectively.

(b) Let $I\subseteq \mathbb{R}$ be an interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $y:I\to A$ have inertial coordinate path $t\mapsto (ct,r(t))$; writing $r(t)=r^1(t)e_1+r^2(t)e_2+r^3(t)e_3$ with real functions $r^1,r^2,r^3$, at each parameter value $t$ where all three coordinate derivatives exist, define the coordinate velocity $w(t)\in S$ of the path by $w(t):=(r^1{)}^{\mathrm{\prime }}(t)e_1+(r^2{)}^{\mathrm{\prime }}(t)e_2+(r^3{)}^{\mathrm{\prime }}(t)e_3,$ the componentwise derivative of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. For a path differentiable at every parameter value these are functions on $I$; otherwise they are defined on its differentiability set. The comparison functionals along the path are $t\mapsto N_p(m,w(t))$ and $t\mapsto N_K(m,w(t))$.

(c) The functionals $N_p$ and $N_K$ are comparison expressions built from a named inertial frame, the mass parameter $m$ and the path's coordinate velocity. They are not defined by, and do not presuppose, a separately developed Newtonian theory, and this definition asserts no approximation, correspondence, error bound or physical claim; those are the content of Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities.

• Well-definedness. Clause (a) applies the scalar multiple of a vector of the real vector space $V$ (restricted to $S\subseteq V$) and the square of the norm $\mathrm{\mid }\cdot \mathrm{\mid }$ supplied by the frame's spatial structure, both total on $S$; clause (b) uses the componentwise derivative of a function $I\to {\mathbb{R}}^3$ and the unique expansion of a vector of $S$ in the ordered basis $(e_1,e_2,e_3)$ of The Euclidean spatial structure and cross product of an inertial frame. No division occurs and no choice principle is used.

• No correspondence is asserted here. The definition only names the two expressions that Specified low-speed momentum-energy correspondence between the relativistic measured quantities and the Newtonian comparison quantities compares with the model's adopted measured quantities, and the units are the ones fixed by SI units and coordinate conventions of the special-relativity framework. The derivative symbols in (b) are used only at points where the coordinate derivatives exist; no velocity is assigned to an arbitrary nondifferentiable path, and the physical correspondence consumers retain their stated differentiable future-timelike worldline hypotheses. The mass parameter $m>0$ is part of the definition's data; the definition itself attaches no dynamics, force or measurement to it.