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General relativistic tidal work for Papapetrou, Weinberg and Goldberg pseudotensors

In 1998 Thorne claimed that all pseudotensors give the same tidal work as the Newtonian theory. In 1999, Purdue used the Landau-Lifshitz pseudotensor to calculate the tidal heating and the result matched with the Newtonian gravity. Soon after in 2001, Favata employed the same method to examine the Einstein, Bergmann-Thomson and Møller pseudotensors, all of them give the same result as Purdue did. Inspired by the work of Purdue and Favata, for the completeness, here we manipulate the tidal work for Papapetrou, Weinberg and Goldberg pseudotensors. We obtained the same tidal work as Purdue achieved. In addition, we emphasize that a suitable gravitational energy-momentum pseudotensor requires fulfill the inside matter condition and all of the classical pseudotensors pass this test except M$ø$ller. Moreover, we constructed a general pseudotesnor which is modified by 13 linear artificial higher order terms combination with Einstein pseudotensor. We find that the result agrees with Thorne's prediction, i.e., relativistic tidal work is pseudotensor independent.