Momentum-current gravitational multipoles of hadrons

We study multipole expansion of the momentum currents in hadrons with three series $S^{(J)}, \tilde{T}^{(J)}$ and $T^{(J)}$ 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 the deeply virtual Compton scattering currently studied at the Jefferson Lab 12 GeV facility and future Electron-Ion Collider. We define the leading momentum-current multipoles [the "tensor monopole" $\tau$ (T0) and "scalar quadrupole" $\hat{\sigma} ^{ij}$ (S2) moments], relating the former to the so-called D term in the literature. We calculate the momentum-current distribution in the hydrogen atom and its monopole moment in the basic unit of $\tau_{0} = \hslash^{2} / 4m_{e}$, showing that the sign of the D term has little to do with mechanical stability. The momentum-current distribution also strongly modifies the static gravitational field inside hadrons