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In this note, using the quasilocal formalism of Brown and York, the flow ofenergy through a closed surface containing a gravitating physical system iscalculated in a way that augments earlier results on the subject by Booth andCreighton. To this end, by performing a variation of the total gravitationalHamiltonian (bulk plus boundary part), it is shown that associated tidalheating and deformation effects generally are larger than expected. This isbecause this variation leads to previously unrecognized correction terms,including a bulk-to-boundary inflow term that does not appear in the originalcalculation of the time derivative of the Brown-York energy and leads tocorrective extensions of Einstein's quadrupole formula in the large spherelimit.

A remark on the quasilocal calculation of tidal heating: energy transfer through the quasilocal surface