Kostik
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- 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?
$$\left[ \sqrt{-g}({t_\mu}^\nu + T_\mu^\nu) \right]_{,\nu}=0$$ (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?