This feature stems from the fact that the gravitational energy-momentum “tensor” is indeed a pseudo-tensor, i.e. it behaves “nicely” under affine transformations but not under diffeomorphisms. This puzzling feature of the General Relativity, first pointed out by Landau and Lifshitz, becomes particularly dramatic in view of the correct definition of the gravitational radiation.

https://www.ralphkjones.com/book/gravitational-momentum/