Differentiated SSC momentum–velocity identity

Assume Countable Choice as inherited by the supplied Lorentzian smooth-structure and connection results. Let $I\subseteq \mathbb{R}$ be a nonempty open interval and let $z:I\to M$ be a $C^2$ future timelike worldline in a time-oriented smooth Lorentzian four-manifold $(M,g,N)$, parametrized by proper time in the convention of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. Set $U:=dz\mathrm{/}d\tau$, so $g(U,U)=-c^2$. For finite regularity, a $C^1$ tensor field along $z$ means a tensor in the fibre over $z(\tau )$ whose components in every smooth chart are $C^1$ functions of $\tau$. Define its derivative in that chart by differentiating each component and adding $+{\mathrm{\Gamma }}^a{}_{bc}(z)U^b$ on each upper index and $-{\mathrm{\Gamma }}^c{}_{ba}(z)U^b$ on each lower index. In particular, $(D_{\tau }V{)}^a={\dot{V}}^a+{\mathrm{\Gamma }}^a{}_{bc}(z)U^b{V}^c,(D_{\tau }n{)}_a={\dot{n}}_a-{\mathrm{\Gamma }}^c{}_{ba}(z)U^b{n}_c,$ $(D_{\tau }S{)}^{ab}={\dot{S}}^{ab}+{\mathrm{\Gamma }}^a{}_{cd}(z)U^c{S}^{db}+{\mathrm{\Gamma }}^b{}_{cd}(z)U^c{S}^{ad}\mathrm{.}$ These formulas extend the smooth-curve derivative of Covariant derivative along a curve to the stated $C^2\mathrm{/}{C}^1$ data. Let $p$ be a $C^1$ vector field along $z$, let $S^{ab}=-S^{ba}$ and ${\mathcal{T}}^{ab}=-{\mathcal{T}}^{ab}$ be $C^1$ bivector fields along $z$, and let $n$ be a $C^1$ future unit timelike field along $z$. Suppose $S^{ab}{n}_b=0,D_{\tau }{S}^{ab}=p^a{U}^b-p^b{U}^a+{\mathcal{T}}^{ab}\mathrm{.}$ Define $\gamma :=-g(n,U)\mathrm{/}c$ and $E_n:=-cg(p,n)$. Then $\gamma >0$, and $c\gamma p^a=\frac{E_n}{c}{U}^a+S^{ab}{D}_{\tau }{n}_b+{\mathcal{T}}^{ab}{n}_b\mathrm{.}$ Here $\mathcal{T}$ denotes an arbitrary antisymmetric torque; setting it to zero gives the torque-free case.

This is tensor algebra along one specified curve. It does not select an SSC, prove closure of the MPD equations, or identify $p$ with $mU$.