Proper-time derivative of a vector field along a worldline and the four-force notation

Throughout, $c>0$ and $(A,V,\eta ,C^{+})$ is the oriented time-oriented affine Minkowski model of Oriented time-oriented affine Minkowski space (the spacetime primitives), with displacement space $V$; $I\subseteq \mathbb{R}$ is an interval with at least two elements (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and $x:I\to A$ is a $C^1$ future timelike worldline (Curves, regular causal curves and future timelike worldlines in affine Minkowski space) with proper-time function $\tau :I\to J$ and inverse $\rho :J\to I$ (Proper time along a future timelike worldline, The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse). By that lemma $\tau$ is $C^1$ with

${\tau }^{\mathrm{\prime }}(s)=\frac{1}{c}\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}>0(s\in I),$

$\tau$ is strictly increasing, $J=\tau [I]$ is an interval with at least two elements, and $\rho :J\to I$ is $C^1$ with ${\rho }^{\mathrm{\prime }}(\tau (s))=1\mathrm{/}{\tau }^{\mathrm{\prime }}(s)>0$ for every $s\in I$.

(a) The proper-time derivative. Let $P:I\to V$ be a $C^1$ map, its derivative $P^{\mathrm{\prime }}$ being the derivative of a $V$-valued function of one real variable, read componentwise in any ordered basis of $V$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). The proper-time derivative of $P$ is the map $\dot{P}:I\to V$ defined by

$\dot{P}(s):=\frac{P^{\mathrm{\prime }}(s)}{{\tau }^{\mathrm{\prime }}(s)}\mathrm{.}$

This is well defined because ${\tau }^{\mathrm{\prime }}(s)>0$ for every $s\in I$, so the scalar division by ${\tau }^{\mathrm{\prime }}(s)$ is a legitimate vector-space operation on $V$. The definition does not depend on the basis of $V$ in which $P^{\mathrm{\prime }}$ is read: a change of basis acts on the coordinates of $P^{\mathrm{\prime }}(s)$, and hence on those of $\dot{P}(s)$, by the same constant invertible linear map, which commutes with multiplication by the scalar $1\mathrm{/}{\tau }^{\mathrm{\prime }}(s)$. Since $P^{\mathrm{\prime }}$ and $1\mathrm{/}{\tau }^{\mathrm{\prime }}$ are continuous, $\dot{P}$ is continuous; equivalently, the proper-time derivative of a $C^1$ field is a continuous field.

The proper-time reading. Since $\tau$ is a strictly increasing bijection $I\to J$ with inverse $\rho$, one has $\rho (\tau (s))=s$ and therefore $(P\circ \rho )\circ \tau =P$ on $I$. Fix $s\in I$; the components of $P$ in a basis of $V$ are differentiable at $s$ and $\rho$ is differentiable at $\tau (s)$, so the scalar chain rule (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$) applied to each component of $P\circ \rho$ gives

$P^{\mathrm{\prime }}(s)=(P\circ \rho {)}^{\mathrm{\prime }}(\tau (s)){\tau }^{\mathrm{\prime }}(s),\text{equivalently}\dot{P}(s)=(P\circ \rho {)}^{\mathrm{\prime }}(\tau (s))\mathrm{.}$

At an endpoint of $I$ this is the one-sided reading of the derivative, as in The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral. Thus $\dot{P}$ is exactly the derivative with respect to the proper-time parameter of the field $P$ read along the parametrisation $s\mapsto \tau (s)$; for every $u\in J$ the chain rule on $J$ gives $(P\circ \rho {)}^{\mathrm{\prime }}(u)=P^{\mathrm{\prime }}(\rho (u)){\rho }^{\mathrm{\prime }}(u)=\dot{P}(\rho (u))$.

(b) The four-force notation. Let $m>0$ be constant and suppose the four-velocity $U:I\to V$ of $x$ is $C^1$ (Four-velocity and four-acceleration of a future timelike worldline); by that definition $U$ is always continuous, and $U$ is $C^1$ whenever the worldline $x$ is $C^2$. Then $P:=mU$ is a $C^1$ map $I\to V$ and $F:=\dot{P}$ is called the four-force on the particle. In an inertial frame the components of $F$ are the derivatives of the components of the four-momentum with respect to the proper-time parameter: writing $P^a(u)$ for the components of $P$ as a function of the proper-time parameter $u=\tau (s)$, one has $F^a(s)=\frac{d}{du}{P}^a(u){\mid }_{u=\tau (s)}$.

(c) Scope. This item introduces a derivative and a name only. It asserts no force law, no field equation, no equation of motion and no existence or uniqueness of solutions of any equation of motion; in particular the hypothesis that $U$ be $C^1$ is a regularity hypothesis on the given worldline, not a dynamical statement. The identities satisfied by $F$ for a stipulated four-momentum field are proved in Four-force identities and the work-energy relation in an inertial frame, and a physical force law requires the separate model premise Adopted massive-particle momentum, observer measurement and isolated-collision conservation. The names four-momentum and four-force, and the units in which the components of $P$ and $F$ are measured, belong to the framework's coordinate and unit conventions (SI units and coordinate conventions of the special-relativity framework); the mathematical content asserted here is the derivative construction (a) and the notation (b) alone. No choice principle is used.