BMT transport constraint and restricted-ansatz algebra

Assume Countable Choice for the canonical tangent-bundle smooth structure supplied by the GR items below. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold with its metric-compatible connection $\mathrm{\nabla }$, let $c>0$, and let $I\subseteq \mathbb{R}$ be a nonempty open interval of proper-time values. Let $z:I\to M$ be a $C^2$ future timelike curve and set $U:=dz\mathrm{/}d\tau$, with $g(U,U)=-c^2$. For a $C^1$ vector field $V$ along $z$, define its covariant derivative by the pullback connection $D_{\tau }V:=(z^{\ast }\mathrm{\nabla }{)}_{\mathrm{\partial }\mathrm{/}\mathrm{\partial }\tau }V$; in a chart this means $(D_{\tau }V{)}^a=d{V}^a\mathrm{/}d\tau +{\mathrm{\Gamma }}^a{}_{bc}(z(\tau ))U^b{V}^c$. Let $F_{\tau }$ be a continuous alternating two-form along $z$. Define $F_{\tau }^{\mathrm{♯}}$ by $g(F_{\tau }^{\mathrm{♯}}(v),w)=F_{\tau }(w,v)\mathrm{.}$ Then $F_{\tau }^{\mathrm{♯}}$ is skew-adjoint with respect to $g$.

For constants $\alpha ,a,b\in \mathbb{R}$, consider a $C^1$ vector field $s$ along $z$ satisfying the restricted linear transport equations $D_{\tau }U=\alpha F^{\mathrm{♯}}(U),D_{\tau }s=\alpha (a{F}^{\mathrm{♯}}(s)+\frac{b}{c^2}U\kappa ),\kappa :=g(s,F^{\mathrm{♯}}(U))\mathrm{.}$ For every ${\tau }_0\in I$ and initial value $s({\tau }_0)$ there is a unique solution on each compact subinterval of $I$ containing ${\tau }_0$. Along any such solution, $\frac{d}{d\tau }g(U,s)=\alpha (1-a-b)\kappa \mathrm{.}$ Consequently, the orthogonality constraint $g(U,s)=0$ is preserved when $a+b=1$, and on that constraint $g(s,s)$ is constant. For the coefficient comparison, fix a parameter value $\tau$ and put $n:=U(\tau )\mathrm{/}c$. Then $n$ is an observer at $z(\tau )$ and its rest space is the three-dimensional positive-definite subspace $E_{\tau }:=n^{\perp }=U(\tau {)}^{\perp }$ (the observer and projection conventions are those of Observers, rest spaces and the spatial projection at an event and The observer rest-space decomposition and the spatial projection). Choose an orientation of $E_{\tau }$ and a positively oriented orthonormal basis $(e_1,e_2,e_3)$; set $e_0:=n$. For $v\in E_{\tau }$, write $\mathbf{v}=(g(v,e_1),g(v,e_2),g(v,e_3))$, use $(\mathbf{u}\times \mathbf{v}{)}_i={ϵ}_{ijk}{u}_j{v}_k$, and define $\mathbf{B}$ by $F_{ij}:=F(e_i,e_j)={ϵ}_{ijk}{B}_k$. Let $P_U(v):=v+g(U,v)U\mathrm{/}{c}^2$, the projection onto $E_{\tau }$, and fix a target coefficient $G\in \mathbb{R}$. The rest-space matching condition means the pointwise equality $(P_U(D_{\tau }s){)}^i=\alpha G(\mathbf{s}\times \mathbf{B}{)}_i(i=1,2,3)$ for every $\mathbf{s},\mathbf{B}\in {\mathbb{R}}^3$, when $s(\tau )\in E_{\tau }$. Thus it compares spatial components of the covariant derivative at the event, not the derivative of components in an unspecified moving tetrad. For $\alpha \mathrm{\ne }0$, if the coefficients both preserve $g(U,s)=0$ for every initially orthogonal datum and every allowed alternating field and satisfy this matching condition, then $a=G$ and $b=1-G$. Conversely, these coefficients satisfy both requirements. This uniqueness is only within the displayed two-term ansatz; no uniqueness among all covariant spin-transport laws is asserted. If $\alpha =0$, the rest-space matching condition does not identify $a$ or $b$.

The proof is a conditional calculation for an already specified smooth connection and transport equation. The Countable Choice requirement is inherited through the supplied GR tangent-bundle and connection construction; no choice is used in the coefficient algebra.