Observable Properties of Orbits in Exact Bumpy Spacetimes

TL;DR

This work probes how non-Kerr multipole structure in exact MN bumpy spacetimes affects EMRI geodesics and gravitational-wave observables. By leveraging an exact vacuum solution that deviates from Kerr in higher moments (via the parameter $q$) while remaining close to Kerr at large radii, the authors analyze geodesic integrals, ergodicity, ISCO structure, and precession frequencies. They show most geodesics retain an approximate fourth constant of motion and are tri-periodic, but oblate perturbations ($q>0$) can induce ergodic regions near the central object; ISCO behavior can shift to vertical instabilities, and precession/divergence near ISCO offers potential smoking-gun signatures of non-Kerr spacetimes. The results suggest that, although chaotic regions exist mathematically, astrophysical EMRIs are unlikely to populate them, making the primary observational discriminants the three fundamental frequencies, harmonic content, and the ISCO’s nature and frequency, which could be probed with future waveform models and Fisher analyses.

Abstract

We explore the properties of test-particle orbits in "bumpy" spacetimes - stationary, reflection-symmetric, asymptotically flat solutions of Einstein equations that have a non-Kerr (anomalous) higher-order multipole-moment structure but can be tuned arbitrarily close to the Kerr metric. Future detectors should observe gravitational waves generated during inspirals of compact objects into supermassive central bodies. If the central body deviates from the Kerr metric, this will manifest itself in the emitted waves. Here, we explore some of the features of orbits in non-Kerr spacetimes that might lead to observable signatures. As a basis for this analysis, we use a family of exact solutions proposed by Manko & Novikov which deviate from the Kerr metric in the quadrupole and higher moments, but we also compare our results to other work in the literature. We examine isolating integrals of the orbits and find that the majority of geodesic orbits have an approximate fourth constant of the motion (in addition to the energy, angular momentum and rest mass) and the resulting orbits are tri-periodic to high precision. We also find that this fourth integral can be lost for certain orbits in some oblately deformed Manko-Novikov spacetimes. However, compact objects will probably not end up on these chaotic orbits in nature. We compute the location of the innermost stable circular orbit (ISCO) and find that the behavior of orbtis near the ISCO can be qualitatively different depending on whether the ISCO is determined by the onset of an instability in the radial or vertical direction. Finally, we compute periapsis and orbital-plane precessions for nearly circular and nearly equatorial orbits in both the strong and weak field, and discuss weak-field precessions for eccentric equatorial orbits.

Figures (20)

• Figure 1: Spacetime structure for $\chi=0.9$. The upper row shows zeros of $g_{tt}$ for $q=-1$ (left column), $q=0$ (middle column) and $q=1$ (right column). This defines the boundary of the ergoregion of the spacetime. The region with $g_{tt} >0$ is shaded. The bottom row shows points where $g_{\phi\phi}$ changes sign for the same values of $q$, and the region where $g_{\phi\phi} <0$ is shaded. This defines the region where closed timelike curves exist. The middle bottom panel is empty since there is no such region in the Kerr spacetime. The shape of the two boundaries is qualitatively the same for other values of $q$ with the same sign, although both regions grow as $|q|$ is increased.
• Figure 2: The fractional errors in energy $E$ (solid line), angular momentum $L_z$ (dashed line), and the quantity $g_{\alpha\beta} \dot{x}^\alpha \dot{x}^\beta$ (dotted line) accumulated over $1700$ orbits of a geodesic with $E=0.92$ and $L_z=2.5M$ in a spacetime with spin $\chi=0.9$ and anomalous quadrupole moment $q=0.95$.
• Figure 3: Effective potential for geodesic motion around a Kerr black hole, with $E=0.95$, $L_z=3 M$ and $\chi=0.9$. The curves indicate zeros of the effective potential. Allowed orbits are found in the small region around $\rho=0$, $z=0$ (rising and plunging orbits) or in the region containing $\rho =10$, $z=0$ (bound orbits).
• Figure 4: Effective potential for geodesic motion around a bumpy black hole with $\chi=0.9$, $q=0.95$, $E=0.95$, and $L_z=3 M$. The thick dotted curves indicate zeros of the effective potential. The trajectory of a typical geodesic in the outer region is shown by a thin curve. The regular pattern of self-intersections of the geodesic projection onto the $\rho-z$ plane indicates (nearly) regular dynamics.
• Figure 5: Poincaré map showing ${\rm d} \rho/{\rm d} \tau$ vs $\rho$ for crossings of the $z=0$ plane for a sequence of orbits in the outer allowed region of the Kerr spacetime with $E=0.95$, $L_z=3 M$ and $\chi=0.9$. The closed curves indicates the presence of a fourth isolating integral, which we know to be the Carter constant.