Fast spinning pulsars as probes of massive black holes’ gravity

Dwarf galaxies and globular clusters may contain intermediate-mass black holes (103–105 M⊙) in their cores. Estimates of ∼103 neutron stars in the central parsec of the Galaxy and similar numbers in small elliptical galaxies and globular clusters along with an estimated high probability of millisecond (ms)-pulsar formation in those environments have led many workers to propose the use of ms-pulsar timing to measure the mass and spin of intermediate-mass black holes. Models of pulsar motion around a rotating black hole generally assume geodesic motion of a ‘test’ particle in the Kerr metric. These approaches account for well-known effects like de Sitter precession and the Lense–Thirring effect but they do not account for the non-linear effect of the pulsar's stress–energy tensor on the space–time metric. Here we model the motion of a pulsar near a black hole with the Mathisson–Papapetrou–Dixon (MPD) equations. Numerical integration of the MPD equations for black holes of masses 2 × 106, 105 and 103 M⊙ shows that the pulsar will not remain in an orbital plane with motion vertical to the plane being largest relative to the orbit's radial dimensions for the lower mass black holes. The pulsar's out-of-plane motion will lead to timing variations that are up to ∼ 10 μs different from those predicted by planar orbit models. Such variations might be detectable in long-term observations of ms pulsars. If pulsar signals are used to measure the mass and spin of intermediate-mass black holes on the basis of dynamical models of the received pulsar signal, then the out-of-plane motion of the pulsar should be part of that model.