Momentum-Current Gravitational Multipoles of Hadrons

We study multipole expansion of the momentum currents in hadrons, with three series S^{(J)}S(J), \tilde T^{(J)}T~(J), and T^{(J)}T(J), in connection with the gravitational fields generated nearby. The momentum currents are related to their energy-momentum form factors, which in principle can be probed through processes like deeply-virtual Compton scattering currently studied at JLab 12 GeV facility and future Electron Ion Collider. We define the leading momentum-current multipoles, tensor monopole \tauτ (T0T0) and scalar quadrupole \hat \sigma^{ij}σ^ij (S2S2) moments, relating the former to the so-called DD-term in the literature. We calculate the momentum current distribution in hydrogen atom and its monopole moment in the basic unit of \tau_0 =\hbar^2/4Mτ0​=ℏ2/4M, showing that the sign of DD-term has little to do with mechanical stability. The momentum current distribution also strongly modifies the static gravitational field inside hadrons.