Abstract
The leading-order spin-orbit coupling is included in a post-Newtonian Lagrangian formulation of spinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and 2PN non-spin terms and 2PN spin-spin coupling. This makes a 3PN spin-spin coupling occur in the derived Hamiltonian. The spin-spin couplings are mainly responsible for chaos in the Hamiltonians. However, the 3PN spin-spin Hamiltonian is small and has different signs, compared with the 2PN spin-spin Hamiltonian equivalent to the 2PN spin-spin Lagrangian. As a result, the probability of the occurrence of chaos in the Lagrangian formulation without the spin-orbit coupling is larger than that in the Lagrangian formulation with the spin-orbit coupling. Numerical evidences support the claim.
Highlights
On February 11, 2016, the LIGO Scientific Collaboration and Virgo Collaboration announced the detection of gravitational wave signals (GW150914), sent out from the inspiral and merger of a pair of black holes with masses 36M and 29M [1]
Considering that the 2PN spin–spin coupling H2ss is equivalent to L2ss and the 3PN spin–spin coupling H3ss associated to L1.5so plays an important role in the onset of chaos in the Hamiltonians, we should focus on differences between the H3 and H4 dynamics
A globally stable orbit is chaotic if its fast Lyapunov indicator (FLI) is larger than the threshold, but ordered if its FLI is smaller than the threshold
Summary
On February 11, 2016, the LIGO Scientific Collaboration and Virgo Collaboration announced the detection of gravitational wave signals (GW150914), sent out from the inspiral and merger of a pair of black holes with masses 36M and 29M [1]. For the sake of a true description of the relation between the PN Lagrangian and Hamiltonian systems, the four difference sources should be avoided An example satisfying these requirements is a special two-black hole system with two bodies spinning, whose ADM Lagrangian formulation includes the Newtonian term and the 1.5PN spin–orbit coupling. This is because the Hamiltonian formalism can exactly provide the equations of motion and some constants of motion, and it has many advantages on the properties of a canonical system.3 Another important result of [30] is that for a lower-order PN Lagrangian formulation with Euler–Lagrange equations to an infinite PN order there always exists a formally equivalent PN Hamiltonian at the infinite order from a theoretical point of view or a certain finite order from a numerical point of view.
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