Abstract
In this note, using the quasilocal formalism of Brown and York, the flow of energy through a closed surface containing a gravitating physical system is calculated in a way that augments earlier results on the subject by Booth and Creighton. To this end, by performing a variation of the total gravitational Hamiltonian (bulk plus boundary part), it is shown that associated tidal heating and deformation effects generally are larger than expected. This is because this variation leads to previously unrecognized correction terms, including a bulk-to-boundary inflow term that does not appear in the original calculation of the time derivative of the Brown-York energy and leads to corrective extensions of Einstein's quadrupole formula in the large sphere limit.
Highlights
The influence of tidal deformation and heating effects on a nearly isolated gravitating systems due to external fields was successfully treated and described by Booth and Creighton in [1]
As will be shown in the second section of this work, with respect to the latter application, the analysis of Booth and Creighton can be extended in one particular respect: The mass-energy transfer through the quasilocal surface can alternatively be calculated by varying the total gravitational Hamiltonian, which, in contrast to the results originally obtained by the authors, leads to the emergence of correction terms; that is, a confined stress-energy term and a bulk-toboundary inflow term that combines the dynamical degrees of freedom of the bulk with those of the boundary
Previous results by Booth and Creighton on quasilocal tidal heating were extended in that the mass-energy transfer through the quasilocal surface was calculated in a different manner, namely by varying the total gravitational Hamiltonian and the boundary part
Summary
The influence of tidal deformation and heating effects on a nearly isolated gravitating systems due to external fields was successfully treated and described by Booth and Creighton in [1]. As will be shown in the second section of this work (after a brief overview of the essentials of the considered quasilocal geometric framework in the first section), with respect to the latter application, the analysis of Booth and Creighton can be extended in one particular respect: The mass-energy transfer through the quasilocal surface can alternatively be calculated by varying the total gravitational Hamiltonian (bulk plus boundary parts), which, in contrast to the results originally obtained by the authors, leads to the emergence of correction terms; that is, a confined stress-energy term and a bulk-toboundary inflow term that combines the dynamical degrees of freedom of the bulk with those of the boundary. It is argued that corrections to the time derivative of the BrownYork mass occur and lead to corrections to the Einstein quadrupole formula in the large sphere limit; corrections that should find application in Einstein-Hilbert gravity in the special cases discussed in this paper, but in many cases of interest, including the description of, for example, tidal deformation and tidal heating effects caused by accretion phenomena or merger processes in relativistic N-body systems
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