# Landau Lifshitz pseudotensor — energy density

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?