A Reconciling units for the Einstein and Landau-Lifshitz pseudotensors

[Kostik]

TL;DR Summary

Examining a discrepancy in the energy-momentum conservation law expressed using the Einstein(-Dirac) and Landau-Lifshitz pseudotensors

The Einstein (or Einstein-Dirac) pseudotensor ${t_\mu}^\nu$ satisfies
$$\left[ \sqrt{-g}({t_\mu}^\nu + T_\mu^\nu) \right]_{,\nu}=0$$ (see Dirac, "General Theory of Relativity", eq. 31.2)). Similarly, the Landau-Lifshitz pseudotensor $t^{\mu\nu}$ satisfies
$$\left[ (-g)(t^{\mu\nu} + T^{\mu\nu}) \right]_{,\nu}=0$$ (see L-L, "Classical Theory of Fields" 4th Ed., eq. (96.10)).

In both cases, the authors deduce that the conservation equation implies that the quantity in square brackets represents the density of total energy and momentum of the matter-energy fields plus the gravitational field (curvature of space).

But surely the quantities in square brackets have different units. How can they both be energy-momentum density?

 

[Dale]

That depends very much on the conventions that one adopts. You need to look carefully into each author's conventions. They may not be the same or even compatible.

One possible convention is that $g$ is dimensionless. Then $t$ and $T$ could have the same units and both author's expressions would be consistent and compatible. However, that may not be the case, in which case the different authors will ascribe different units to the various quantities.

 

[Kostik]

That depends very much on the conventions that one adopts. You need to look carefully into each author's conventions. They may not be the same or even compatible.

One possible convention is that $g$ is dimensionless. Then $t$ and $T$ could have the same units and both author's expressions would be consistent and compatible. However, that may not be the case, in which case the different authors will ascribe different units to the various quantities.

I don't think that's the issue. There is no ambiguity in the metric $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$. There are some differences between Dirac and L-L; for example, Dirac assumes $G=c=1$ while L-L carries these constants throughout. Also, Dirac's Ricci tensor is the opposite of L-L's. But none of these enter into the derivation of their pseudo-tensors.

 

[anuttarasammyak]

Why don't you start with the familiar case that all
$$x^0,x^1,x^2,x^3$$
have dimension of length. All the metric tensor components, thus its determinant g also, are dimensionless.
Then you may investigate more general case, e.g., cylindrical or polar type coordinates, if you wish.