# using tetrads to glue local currents into global currents

According to John Baez it is possible to take a locally conserved tensor

$\nabla_\mu\: T^{\mu\nu}(x)=0\ \ \ \ \ \mbox{(locally)}$

and convert it to a globally conserved tensor by "patching" together small regions of spacetime and gluing each local current together. The problem is that there is no unique way to do this. In fact there is one for each way to parallel transport a tensor in region $dV_1$ to a nearby region $dV_2$, i.e. the gluing depends on the choice of coordinates.

Could someone help me better understand what Baez is talking about with a concrete example? (ref: see last two paragraphs http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html)

Suppose we have an orthonormal tetrad $e^\mu_a(x)$ and a geodesic $\beta$ such that

$\frac{D e^\mu_a}{d\lambda}=0$ and $g^{\mu\nu}(x)=\eta^{ab}e^\mu_a(x)e^\nu_b(x)$.

How would I glue a locally conserved current $T_{ab}(x)=e^\mu_a(x)e^\nu_b(x)T_{\mu\nu}(x)$ in region $dV_1$ with $T_{ab}(x')=e^\mu_a(x')e^\nu_b(x')T_{\mu\nu}(x')$ in region $dV_2$ such that it is globally conserved?

I seem to recall that $De^\mu_a/d\lambda=0$ implies the parallel transport equation

$T^{(PT)}_{cd}(x')=e^\mu_c(x')e^\nu_d(x')e_\mu^a(x)e_\nu^b(x)T_{ab}(x)$

as long as $x'$ and $x$ are on the geodesic $\beta$. We can now glue this to $T_{ab}(x')$ by the gluing function $\phi^{cd}_{ab}$

$T_{ab}(x')=\phi^{cd}_{ab}\ T^{(PT)}_{cd}(x')$.

According to Baez this can be done in such a way that the covariant derivative becomes a partial derivative, i.e. such that $\Gamma^b_{ca}T^{ac}(x)=0$, but I don't see how to impose this condition in any kind of generic spacetime.