Adjusting chaotic indicators to curved spacetimes

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.