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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.

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    $\begingroup$ Comment to the question (v8): If $j^{\mu}$ with $\nabla_{\mu}j^{\mu}=0$ is supposed to be a (1,0) tensor, then there is no problem in constructing an integrated conserved quantity via a 4-dimensional divergence theorem. The link by John Baez, on the other hand, is talking about the stress-energy-momentum pseudotensor, not $j^{\mu}$. $\endgroup$ – Qmechanic Apr 13 '15 at 19:38
  • $\begingroup$ You are correct. I've replaced the vectors with tensors. $\endgroup$ – alphanzo Apr 13 '15 at 19:50
  • $\begingroup$ Could someone help me better understand what Baez is talking about with a concrete example? Yes. Send a 511keV photon into a black hole, and the black hole mass increases by 511keV/c². So energy is conserved in GR. What you think of as a blueshifted photon hasn't really gained any energy. In similar vein, in SR when you accelerate towards a light source you measure a blueshift, but the photons haven't changed, instead, you have. Baez articles are usually pretty good, but IMHO this one is unclear, and you should either forget about it or email Don Koks the PhysicsFAQ editor with a query. $\endgroup$ – John Duffield Apr 13 '15 at 22:03
  • $\begingroup$ I agree with you that it's a very sketchy article. He talks about gluing in the last few paragraphs. Unfortunately I can't let it go because the construction he is talking about is precisely one that I need. $\endgroup$ – alphanzo Apr 13 '15 at 23:30
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Well after beating my head against the wall, I just decided to email Baez. He's a great guy by the way and it was very nice of him to answer completely unsolicited emails from a confused grad student, but he did as much twice! so I thank him.

I'm convinced now that he had no intentions of constructing a globally conserved current using tetrads, or even parallel transport. This was just a story that has a nice imagery. In fact I'm sure he is just alluding to Noether's second theorem for constructing conserved currents from local symmetries, and just didn't want to go into what all that meant.

So the short answer is he was referring to standard methods using Noether's second theorem (for local symmetries instead of global symmetries, i.e. her first theorem), and the long answer is that I worked it out none-the-less using a very different method to build conserved quantities, and I will write it up once my dissertation is submitted, so stay tuned.

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