Harmonic-gauge dipole metric perturbations for weak-field circular orbits in Schwarzschild spacetime

We calculate the harmonic-gauge even l=1 mode of the linear metric perturbation (MP) produced by a particle in a weak-field circular orbit around a Schwarzschild black hole (BH). We focus on the Newtonian limit, i.e. the limit in which the mass M of the central BH approaches zero (while fixing the orbital radius and the small-object mass), and obtain explicit expressions for the MP in this limit. We find that the MP are anomalous in this limit, namely, they do not approach their standard, Coulomb-like, flat-space values. Instead, the MP grows on approaching the BH, and this growth becomes worse as M decreases. This anomalous behavior leads to some pathologies which we briefly discuss. We also derive here the next-order correction (in the orbital frequency $\Omega$) to the MP.