Prescribed constant-mass force law

For a named massive test body with constant primitive mass $m>0$ measured in kilograms, SI conversion constant $c>0$ in metres per second, and momentum $P=mU$ as defined in Mechanical momentum and observer energy, define the future unit timelike shell by

${\mathrm{\Sigma }}_c^{+}:=\{(p,V)\in TM:V\text{ future timelike},g_p(V,V)=-c^2\}\mathrm{.}$

Let $\mathcal{V}\subset TM$ be an open neighborhood of ${\mathrm{\Sigma }}_c^{+}$ contained in the future timelike velocity cone. For a massive body trajectory $x:I\to M$ that is $C^2$ and parameterized by proper time $\tau$, with $I\subseteq \mathbb{R}$ a nonempty open interval, put $U=dx\mathrm{/}d\tau$ and require $(x(\tau ),U(\tau ))\in {\mathrm{\Sigma }}_c^{+}\subseteq \mathcal{V}$ for every $\tau \in I$. Specify a smooth map $f:\mathcal{V}\to TM$ over the bundle projection $\pi :TM\to M$, meaning $\pi (f(p,V))=p$ (equivalently, $f(p,V)\in T_{p}M$), and require $g_p(V,f(p,V))=0$ for every $(p,V)\in \mathcal{V}$. One may adopt this force map and the equation

$D_{\tau }P=f(x,U)\mathrm{.}$

The orthogonality condition is part of this model's compatibility assumptions, imposed throughout $\mathcal{V}$; it preserves $g(U,U)=-c^2$ for constant $m$. Since $P=mU$, this equation is equivalent to $D_{\tau }U=f(x,U)\mathrm{/}m$ on the stated domain. It is not a law selecting a force for every particle or interaction. A local IVP follows only for data in $\mathcal{V}$, as in Smooth shell-tangent force initial-value problem. The force has SI units newtons when $\tau$ is in seconds and $P$ has units kg m s${}^{-1}$. No empirical force measurements, universal coupling, or global existence are asserted.