Slow rotation of a localized Kalb-Ramond wormhole: Wald charges and fractional quadrupolar hair

RESUMO

We construct an action-consistent slow-rotation branch, through second order, for a traversable wormhole supported by a phantom scalar, a scalar-localized radial string sector, and a Kalb-Ramond two-form coupled nonminimally to both the Ricci scalar and Ricci tensor. For the regular benchmark branch, covariant asymptotic analysis shows a vanishing ADM mass, while the first-order axial mode carries finite Iyer-Wald angular momentum. The nonminimal contributions to the angular charge decay at infinity, so the covariant and metric definitions of angular momentum agree. At second order, no nonzero mass correction is resolved; at fixed asymptotic scalar charge, rotation increases the throat area and makes the throat oblate. The most distinctive result arises in the quadrupolar sector. A long-range polar Kalb-Ramond response couples to the localized background and generates a matter-supported metric tail that decays more slowly than the ordinary vacuum quadrupole. The same noninteger hierarchy appears in curvature and reduces the differentiability of the conformal completion at spatial infinity. Consequently, the Geroch-Hansen and asymptotically Cartesian mass-centered quadrupole constructions are not applicable in their standard smooth form to this branch, even though its global mass and angular-momentum charges remain well defined. This separates charge-level asymptotic flatness from smooth multipolar asymptotics and identifies fractional Kalb-Ramond hair as the leading rotational quadrupolar signature of the benchmark solution.