Finding order in a sea of chaos

By Alejandro Cárdenas-Avendaño, Andrés F. Gutiérrez, Leonardo A. Pachón, and Nicolás Yunes

Hunting for constants of the motion in dynamical systems is hard. How can one find a combination of dynamical variables that remains unchanged during a complicated evolution? While it is true that answering this question is not trivial, symmetries can sometimes come to the rescue. The motion of test particles around a spinning (Kerr) black hole, for example, has a conserved mass, energy and angular momentum. Nevertheless, simple symmetries can only go so far. Given the complexity of the radial and polar sector of Kerr geodesics, it came as a complete surprise when Carter found, in 1968, a fourth constant of the motion, which was later found to be associated with the existence of a Killing tensor by Walker and Penrose. This fourth constant then allowed the complete separability of the geodesic equations, thus proving the integrability of the system, and as a consequence, that the motion of a test particle around a Kerr black hole is not chaotic in General Relativity (GR).


In modified theories of gravity, one has not been so lucky. Even if test particles follow geodesics in such theories, these geodesics need not be like those in GR, because black holes may not be described by the Kerr metric. For example, rotating black holes in dynamical Chern-Simons (dCS) gravity differ from Kerr black holes. This modified effective theory arises naturally from heterotic string theory, loop quantum gravity and effective field theories of inflation, enhancing parity violation in gravity spontaneously. Rotating black holes in this theory are sufficiently different from those in GR that for instance, the Carter constant is not conserved. Could there be another quantity that is conserved in the motion of test particles, which would then imply that geodesics are not chaotic? Intuitively, the complicated geometric structure of dCS black holes would suggest that chaos is present. But to our surprise, we found a large amount of numerical evidence to the contrary.

Evidence come from an extensive and comprehensive high-resolution numerical analysis that we performed to calculate and study the Poincaré surfaces of section and the rotation curve of a wide family of geodesics of dCS black holes in the search for chaos. As bound geodesics exist in a torus of phase space, the intersections of these geodesics on to a cross-section of this torus forms a Poincaré surface of section. The rotation curve, in turn, is a measure of the angle subtended between consecutive piercings of the cross-section. In dynamical systems theory, these quantities provide a way to quantify the regularity of these orbits and signal the presence of chaos independently of coordinate ambiguities.

The rotation curve of geodesics of dCS black holes indeed presented signatures of chaotic-like behaviour. However, the black hole background we were using was a low-order, approximate solution to the dCS field equations in the slow-rotation and small coupling approximation. When we added higher-order-in-rotation terms to the metric, we found that the features in the rotation number decreased and the sea of chaos essentially disappeared! This behaviour was consistent with what we found in the rotation curve of geodesics of the slow-rotation expansion of the Kerr metric. Clearly, these features seem to be an artefact of the slow-rotation expansion of the dCS metric. We found further evidence for this hypothesis when we changed the magnitude of the dCS coupling parameter and found the size of the features stayed roughly constant.

Our results are relevant because previous work had shown that a second-rank Killing tensor does not exist for slowly-rotating dCS black holes. Therefore, if geodesic orbits are regular, as our numerical findings seem to indicate, then there must exist a higher-rank Killing tensor that has not yet been discovered and which would yield a fourth constant of the motion. If so, geodesics of dCS black holes would be integrable, though not necessarily separable, which could be proved analytically if a closed-form solution for dCS black holes is found to all orders in spin. If this is the case, there must be a hidden symmetry in the theory.

Finding order in a sea of chaos_both_2018

Fig 1. (Left) Poincaré surfaces of section using the slow-rotation expansion of the Kerr metric to second order. The surfaces of section look regular, seemingly without any signatures of chaos, but this can be deceiving because features of chaos may be small and hard to resolve on this scale. (Right) Zoom of the left panel where Birkhoff islands were formed around a stable point.

Our results have an impact on future tests of GR with gravitational waves emitted by extreme mass-ratio inspirals (EMRIs). These waves are of too low frequency to be detectable by ground-based instruments, like advanced LIGO and Virgo, but they would be ideal for space-based detectors, such as the LISA. Tests of GR with such waves would require the construction of accurate models to describe the signal, an effort that could be incredibly challenging if chaos is present in the orbital dynamics. Our results indicate that such chaos may not be present in dCS gravity after all, thus opening a door toward the construction of accurate models for dCS waveforms in EMRIs. Moreover, our results suggest that it may be worthwhile to search for chaos in other well-motivated modified theories of gravity and determine the feasibility of the construction of accurate models to test GR.

The full CQG paper can be read here.

About the Authors (Left to right, top to bottom):

  • Alejandro Cárdenas-Avendaño is a graduate student at Montana State University and holds a junior researcher position at Fundación Universitaria Konrad Lorenz.
  • Andrés F. Gutierrez is a graduate student at Universidad de Antioquia.
  • Leonardo A. Pachón is an associate professor at Universidad de Antioquia.
  • Nicolás Yunes is an associate professor at Montana State University, and co-founder of the eXtreme Gravity Institute.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.