Adjusting chaotic indicators to curved spacetimes

Abstract

In this work, chaotic indicators, which have been established in the framework of classical mechanics, are reformulated in the framework of general relativity in such a way that they are invariant under coordinate transformation. For achieving this, the prescription for reformulating mLCE given by [Y. Sota, S. Suzuki, and K.-I. Maeda, Classical Quantum Gravity 13, 1241 (1996)] is adopted. Thus, the geodesic deviation vector approach is applied, and the proper time is utilized as measure of time. Following the aforementioned prescription, the chaotic indicators FLI, MEGNO, GALI, and APLE are reformulated. In fact, FLI has been reformulated by adapting other prescriptions in the past, but not by adapting the Sota et al. one. By using one of these previous reformulations of FLI, an approximative expression giving MEGNO as function of FLI has been applied on non-integrable curved spacetimes in a recent work. In the present work the reformulation of MEGNO is provided by adjusting the definition of the indicator to the Sota et al. prescription. GALI, and APLE are reformulated in the framework of general relativity for the first time. All the reformulated indicators by the Sota et al. prescription are tested and compared for their efficiency to discern order from chaos.

Recurrence analysis as a tool to study chaotic dynamics of extreme mass ratio inspiral in signal with noise

“…Although, the current work does not answer the given question, it provides a study of a particular statistical method that is applied to simulated data from a dynamical system that is a simplified model of EMRI. Namely, we use a statistical analysis of recurrences occurring in time series produced by a body moving in Manko, Sanabria-Gómez, Manko (MSM) spacetime, 22 since MSM provides chaotic geodesic orbits, 19,23,24 to study up to which noise level geodesic chaos can be detected even when dissipation is present. The MSM spacetime is an exact vacuum solution of the Einstein's field equations and describes the "exterior field of a charged, magnetized, spinning deformed mass".…”

On Geodesic Dynamics in Deformed Black-Hole Fields

“…Let us conclude by several suggestions of further possible work. One could certainly subject the system to still another methods and codes in order to check the results, but also to evaluate the methods/codes themselves and their practical features within general relativity; such analyses like [24] (on the usage of several indicators of orbital deviation in GR) will be helpful. For example, the Melnikov-integral method has already been employed several times in general relativity, as well as the study of the dimension of "basin boundaries" (boundaries between initial-condition sets which evolve to distinct end states), which are of particular appeal due to their coordinate-independent message.…”

“…on often queried (non-)invariance of the world-line deviation indicators. A thorough comparison of these quantities with other similar indicators has recently been given by [24] using the variational approach.…”

Chaotic photon orbits and shadows of a non-Kerr object described by the Hartle-Thorne spacetime

“…As a final note we should point out that in order to detect chaos in GR, one has to take into account the fact that time is now tied to the coordinate system and no longer absolute, as in the Newtonian case, and therefore coordinate-independent methods are needed [59]. As such methods, we will mainly employ the use of Poincare sections.…”