Black hole's tidal heating and angular momentum

In 1985 Thorne and Hartle used the Landau-Lifshitz pseudotensor to demonstrate the tidal heating and angular momentum flux for a black hole. Later in 2004, Poisson used the gravitational perturbation method to study a black hole and obtained the same result. Poisson proposed a new idea, that the mass quardupole moment and current quadrupole moment can be written as the rate of change of the tidal gravitational field. Inspired by these two papers, we use the method of Thorne and Hartle to study other classical pseudotensors: Einstein, Bergmann-Thomson, Papapetrou and Weinberg. Moreover, we also constructed a general expression pseudotensor. We find that for (i) tidal heating: other classical pseudotensors give the same result as the Landau-Lifshitz contribution. (ii) angular momentum flux: except for the Einstein pseudotensor, all of them give the same value as the Landau-Lifshitz pseudotensor.