Landau-Lifshitz stress-energy pseudotensor $t^{ik}$ is defined in such a way that, combined with matter stress-energy tensor $T^{ik}$, it leads to continuity equation: $$ \frac{\partial}{\partial x^k} \left[(-g)\left(t^{ik} + T^{ik}\right)\right] = 0 $$ which, under appropriate boundary condition, can be transformed into conservation law for $$ P^i = \int d^3x (-g)\left(t^{i0} + T^{i0}\right) $$ where $d^3x = dx^1dx^2dx^3$.

However, energy/momentum density of matter is given by $T^{i0}$, and infinitesimal element of physical volume is $\sqrt{-g}d^3x$, so, it seems to me, that the expression above for $P^i$ has an extra $\sqrt{-g}$, doesn't it? It should be $$ P^i = \int d^3x \sqrt{-g}\left(\textrm{something}^{i0} + T^{i0}\right) $$

So while $t^{ik}$ provides a conservation law, it's not actually conservation of matter+grav.field momentum?


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