density $\langle \rho^\mathrm{split}\rangle_\omega$ of the Dirac field in that state. For notational convenience, we decompose $\langle \rho^\mathrm{split}\rangle_\omega$ into the operator $(T^\mathrm{split}_{00'})^{AB}$ acting on ${\mathcal{W}}_{\omega AB} - {\,}^\psi
H_{1 AB}^{(+)}$ where $(T^\mathrm{split}_{00'})^{AB}$ is given by $$\begin{aligned}
(T^\mathrm{split}_{00'})^{AB} = \frac{i}{2}\delta^{AB}\bigg(\mathbbm{1} \otimes
e_0\cdot\nabla - e_0\cdot\nabla\otimes\mathbbm{1}   \bigg)+ \Theta^{AB}\end{aligned}$$ and $\Theta^{AB}$ is a term which depends only on the spin-connection. The precise form of $\Theta^{AB}$ may be found in [@fewster; @verch] (eqn. (3.10) of that reference) but does not affect our discussion due to a useful result[^4] which shows it is identically zero on a timelike worldline under certain conditions which we shall now motivate: Let $\gamma:{\mathbb{R}}\mapsto {\mathcal{M}}$ be a timelike worldline with unit tangent vector $\dot\gamma$. Pick a point $x$ on $\gamma\subset{\mathcal{M}}$ and construct a local frame $\{ e^a_\alpha  \}_{\alpha=0,1,2,3}$ subject to $e^a_0=\dot\gamma^a$ at $x$. As we shall only be concerned with averaging along a compact subset of $\gamma$ we shall fix a closed interval $I\subset\gamma({\mathbb{R}})$ such that $x\in I$. One may utilise Fermi-Walker transport to move $\{ e^a_\alpha \}_{\alpha=0,1,2,3}$ along $I$ keeping $e^a_0|_\gamma=\dot\gamma^a$. The salient feature of Fermi-Walker transport is that it preserves angles, i.e. $\{
e^a_\alpha\}_{\alpha=0,1,2,3}$ remains an orthonormal family along $I$. Next, at each point $\gamma(s)$ along $I$ consider the convex 