Post-Newtonian corrections to the motion of spinning bodies in nonrelativistic general relativity

In this paper we include spin and multipole moment effects in the formalism used to describe the motion of extended objects recently introduced in hep-th/0409156. A suitable description for spinning bodies is developed and spin-orbit, spin-spin, and quadrupole-spin Hamiltonians are found at leading order. The existence of tidal as well as self-induced finite size effects is shown, and the contribution to the Hamiltonian is calculated in the latter. It is shown that tidal deformations start formally at $\mathcal{O}(v^6)$ and $\mathcal{O}(v^{10})$ for maximally rotating general and compact objects, respectively, whereas self-induced effects can show up at leading order. Agreement is found for the cases where the results are known.

Figure 1

Leading order mass vertex. The dashed line represents a potential graviton, whereas the wavy line stands for the full graviton propagator.

Figure 2

Leading order spin-graviton vertex interaction. The blow represents a spin insertion.

Figure 3

Leading order spin-orbit interaction. Diagram (a) takes into account a $v^1$ mass insertion. Diagram (b) corresponds to the $v^3$ spin-graviton vertex.

Figure 4

Leading order spin-spin interaction.

Figure 5

Leading order quadrupole-spin one graviton exchange. The black square represents a quadrupole insertion.

Figure 6

Leading order diagram whose imaginary part gives rise to the quadrupole radiation power spectrum. The black boxes are quadrupole insertions and the curly propagator a radiation graviton.

Figure 7

Two graviton contributions to the one point function in effective action with a single spin insertion.

Figure 8

Three graviton contribution to the one point function in the effective action with two spin insertions.

Figure 9

A typical contribution to the one point function in the effective action coming from leading order mass and spin insertions.