Growth of resonances and chaos for a spinning test particle in the Schwarzschild background

Abstract

Inspirals of stellar mass compact objects into supermassive black holes are known as extreme mass ratio inspirals. In the simplest approximation, the motion of the compact object is modeled as a geodesic in the space-time of the massive black hole with the orbit decaying due to radiated energy and angular momentum, thus yielding a highly regular inspiral. However, once the spin of the secondary compact body is taken into account, integrability is broken and prolonged resonances along with chaotic motion appear.We numerically integrate the motion of a spinning test body in the field of a non-spinning black hole and analyse it using various methods. We show for the first time that resonances and chaos can be found even for astrophysical values of spin. On the other hand, we devise a method to analyse the growth of the resonances, and we conclude that the resonances we observe are only caused by terms quadratic in spin and will generally stay very small in the small-mass-ratio limit. Last but not least, we compute gravitational waveforms by solving numerically the Teukolsky equations in the time-domain and establish that they carry information on the motion's dynamics. In particular, we show that the time series of the gravitational wave strain can be used to discern regular from chaotic motion of the source.

Complete set of quasi-conserved quantities for spinning particles around Kerr

“…As detailed e.g. in [25], the spin term in the MPT equations for such a background spacetime scales as…”

“…The result applies in particular to Kerr spacetime, where it was shown that under the assumption of stationarity and axisymmetry only trivial mixed-symmetry Killing tensors exist which lead to products of known quasi-invariants. The absence of complete integrability in the sense of Liouville prevents us to rule out chaos at linear order in the spin, which is nevertheless suggested from the conclusions of [25,27,28]. An open question is how many of the two independent quasi-conserved quantities at linear order in the spin found by Rüdiger [22,23] admit a generalization that is conserved at quadratic order.…”

“…The unanswered question of the existence of other independent quasi-conserved quantities for the MPT equations led to an undetermined status of the role of integrability and by opposition, chaos, in the dynamics of spinning particles around Kerr. While chaos has been established to appear at second order in the spin [25], numerical simulations suggest that no chaos occurs at linear order in the spin [25][26][27]. From the solution of the Hamilton-Jacobi equations at linear order in the spin, one can infer that chaotic motion at linear order is negligible [28].…”

Relativistic scattering of a fast spinning neutron star by a massive black hole

“…While analytical analysis of the dynamics of spinning binary tend to restrict the orbits to equatorial plane, where the spin is aligned or anti-aligned with the orbital angular momentum, the orbits can be complicated and even chaotic (see e.g. Suzuki & Maeda 1997;Hartl 2003a,b;Kao & Cho 2005;Zelenka et al 2019;Witzany et al 2019;Zelenka et al 2020) when the spin of the secondary is not aligned with the orbital angular momentum. The rich dynamical features of non-equatorial motions are studied by Singh, Wu & Sarty (2014); Han & Cheng (2017); Witzany (2019); Li, Wu & Singh (2019); Kimpson, Wu & Zane (2019); Keresztes & Mikóczi (2019); Keresztes & Mikoczi (2020), etc.…”

Adiabatic equatorial inspirals of a spinning body into a Kerr black hole

“…This reasoning holds away from the resonances, since the resonances are governed by the O S 2[26], which implies a contribution to the phase of order of radians.…”