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This thesis investigates the dynamics of binary systems composed of spinning compact objects (such as white dwarfs, neutron stars, and black holes) in the context of general relativity. In particular, we use the method of Lyapunov exponents to determine whether such systems are chaotic. Compact binaries are promising sources of gravitational radiation for both ground- and space-based gravitational-wave detectors, and radiation from chaotic orbits would be difficult to detect and analyze. For chaotic orbits, the number of waveform templates needed to match a given gravitational-wave signal would grow exponentially with increasing detection sensitivity, rendering the preferred matched filter detection method computationally impractical. It is therefore urgent to understand whether the binary dynamics can be chaotic, and, if so, how prevalent this chaos is. We first consider the dynamics of a spinning compact object orbiting a much more massive rotating black hole, as modeled by the Papapetrou equations in Kerr spacetime. We find that many initial conditions lead to positive Lyapunov exponents, indicating chaotic dynamics. The Lyapunov exponents come in positive/negative pairs, a characteristic of Hamiltonian dynamical systems. Despite the formal existence of chaotic solutions, we find that chaos occurs only for physically unrealistic values of the small body's spin. As a result, chaos will not affect theoretical templates in the extreme mass-ratio limit for which the Papapetrou equations are valid. Chaos will therefore not affect the ability of space-based gravitational-wave detectors (such as LISA, the Laser Interferometer Space Antenna) to perform precision tests of general relativity using extreme mass-ratio inspirals. We next consider the dynamics of spinning black-hole binaries, as modeled by the post-Newtonian (PN) equations, which are valid for orbital velocities much smaller than the speed of light. We study thoroughly the special case of quasi-circular orbits with comparable mass ratios, which are particularly relevant from the perspective of gravitational wave generation for LIGO (the Laser Interferometer Gravitational-wave Observatory) and other ground-based interferometers. In this case, unlike the extreme mass-ratio case, we find chaotic solutions for physically realistic values of the spin. On the other hand, our survey shows that chaos occurs in a negligible fraction of possible configurations, and only for such small radii that the PN approximation is likely to be invalid. As a result, at least in the case of comparable mass black-hole binaries, theoretical templates will not be significantly affected by chaos. In a final, self-contained chapter, we discuss various methods for the calculation of Lyapunov exponents in systems of ordinary differential equations. We introduce several new techniques applicable to constrained dynamical systems, developed in the course of studying the dynamics of spinning compact binaries. Considering the Papapetrou and post-Newtonian systems together, our most important general conclusion is that we find no chaos in any relativistic binary system for orbits that clearly satisfy the approximations required for the equations of motion to be physically valid.

We investigate the dynamics of the Papapetrou equations in Kerr spacetime. These equations provide a model for the motion of a relativistic spinning test particle orbiting a rotating (Kerr) black hole. We perform a thorough parameter space search for signs of chaotic dynamics by calculating the Lyapunov exponents for a large variety of initial conditions. We find that the Papapetrou equations admit many chaotic solutions, with the strongest chaos occurring in the case of eccentric orbits with pericenters close to the limit of stability against plunge into a maximally spinning Kerr black hole. Despite the presence of these chaotic solutions, we show that physically realistic solutions to the Papapetrou equations are not chaotic; in all cases, the chaotic solutions either do not correspond to realistic astrophysical systems, or involve a breakdown of the test-particle approximation leading to the Papapetrou equations (or both). As a result, the gravitational radiation from bodies spiraling into much more massive black holes (as detectable, for example, by LISA, the Laser Interferometer Space Antenna) should not exhibit any signs of chaos., Comment: Submitted to Phys. Rev. D. Follow-up to gr-qc/0210042. Figures are low-resolution in order to satisfy archive size constraints; a high-resolution version is available at http://www.michaelhartl.com/papers/