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@ARTICLE{2026JGeod.100...46C,
       author = {{Chao}, Benjamin F. and {Gao}, Chunchun and {Li}, Zhen},
        title = "{Gravitational energy changes of planet earth under mass perturbations}",
      journal = {Journal of Geodesy},
     keywords = {Gravitational energy, Multipole expansion, Eulerian density anomaly, Lagrangian deformation, GRACE},
         year = 2026,
        month = jun,
       volume = {100},
       number = {6},
          eid = {46},
        pages = {46},
     abstract = "{The gravitational energy E$_{g}$ plays a deciding role in the
        thermodynamic evolution of planet Earth in the destination of
        seeking the lowest energy configuration. This paper examines
        E$_{g}$ and its changes {\ensuremath{\Delta}}E$_{g}$ due to
        geophysical mass perturbations. We derive the multipolar
        partitioning of E$_{g}$ based on the gravitational multipole
        expansion formalism, thereby study {\ensuremath{\Delta}}E$_{g}$,
        a quantity quadratic in dependence on density, due to
        perturbations of two distinct forms: (i) Eulerian density
        anomaly, and (ii) Lagrangian deformation. Perturbation (i)
        carries a negative {\ensuremath{\Delta}}E$_{g}$ relative to the
        laterally mean configuration from which the density anomaly is
        referenced. We calculate the surficial upper bound of this
        {\ensuremath{\Delta}}E$_{g}$ with GRACE-observed time-variable
        Stokes coefficients for the Earth, where we find a present-day
        \raisebox{-0.5ex}\textasciitilde 3 GW secular increase in
        Earth's (non-quadrupolar) E$_{g}$ superposed on seasonal and
        interannual undulations. Any monopolar
        {\ensuremath{\Delta}}E$_{g}$, however, is oblivious to external
        gravitational observations given the non-uniqueness of the
        gravitational inversion. Perturbation (ii) carries a positive
        {\ensuremath{\Delta}}E$_{g}$ relative to the unperturbed
        configuration of equilibrium, hence is by itself unfavored
        energy-wise. Case in point is the spin-induced ``oblating''
        process resulting in the planet's polar oblateness, which can
        happen spontaneously upon the accompanying, over-compensating
        decrease in the spin kinetic energy under the conservation of
        angular momentum.}",
          doi = {10.1007/s00190-026-02065-6},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2026JGeod.100...46C},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}