Chaotic Behavior in Relativistic Motion

Abstract

Our objective in this talk is, in technical terms, to employ the so – called “Melnikov Method” [1] to show that the geodesic motion of massive test particles near the unstable circular orbit surrounding a Schwarzschild Black Hole becomes chaotic under most gravitational perturbations of the metric. Our motivation is, beyond the intrinsic interest of this result, to provide a concrete example of the application of the Melnikov Method to problems issued from General Relativity. Indeed, while chaotic behavior is widespread in relativistic problems (e. g., the presence of chaos in the geodesic flow on compact spaces of negative curvature), the methods available for the detection of chaos (such as numerical computation of Liapunov exponents) have not allowed so far a systematic, wide range analysis of the implications of chaotic behavior for Einstein’s Theory. The Melnikov Method is an important acquisition for our theoretical toolbox, for, it being purely analytical and relatively simple to use, it can provide a quick check on the presence of chaos in a class of problems, which can then be studied by more sophisticated methods.

Work done in collaboration with Luca Bombelli, RGGR, ULB (Belgium).