We introduce the gravielectric (GE) and gravimagnetic (GM) fields in stationary spacetime using the Komar two-form and its dual. This opens the way to extend the Komar-Tomimatsu derivation of mass formulas to a more detailed picture in terms of the local lines of force. We show that Misner strings (MS) carry singular GE and GM fluxes connecting the horizon and the asymptotic zone. Moreover, MS are laterally transparent, so field lines can flow in and out of the bulk. This explains why the usual Komar mass integrals around the Misner strings in the Taub-NUT vacuum solution are negative: the pattern of field lines shows that they flow onto the string from the horizon, so it is necessary to calculate the incoming (positive) but not the outgoing Komar fluxes. This incoming flux is then turned back to the horizon through the Misner strings, realizing the closed circuit without sources. So Misner strings are massless empty tubes, but not rigid rods of negative mass. Similarly, GM field lines can connect positively and negatively charged regions of the horizon, generating, for example, the gravimagnetic dipole moment of the Kerr metric.
Gal’tsov et al. (Fri,) studied this question.