On the use of multipole expansion in time evolution of nonlinear dynamical systems and some surprises related to superradiance

摘要

A new numerical method is introduced to study the problem of time evolution of generic nonlinear dynamical systems in four-dimensional spacetimes. It is assumed that the time level surfaces are foliated by a one-parameter family of codimension-2 compact surfaces with no boundary and which are conformal to a Riemannian manifold ?. The method is based on the use of a multipole expansion determined uniquely by the induced metric structure on ?. The approach is fully spectral-i.e. it avoids pointwise evaluations of the basic variables-in the angular directions. Instead, Gaunt coefficients as matrix elements are used to evaluate multilinear expressions. The dynamics in the complementary 1+1 Lorentzian spacetime is followed by making use of a fourth-order finite differencing scheme. In handling the pertinent 1+1 transverse degrees of freedom, the techniques of adaptive mesh refinement (AMR) is also applied. In checking the reliability and effectiveness of the introduced new method, the evolution of a massless scalar field on a fixed Kerr spacetime is investigated. In particular, the angular distribution of the evolving field in superradiant scattering is studied. The primary aim was to check the validity of some of the recent arguments claiming that the Penrose process, or its field theoretical correspondence-superradiance-does play a crucial role in jet formation in black hole spacetimes while matter accretes onto the central object. Our findings appear to be contrary to these claims as the angular dependence of superradiant scattering of massless scalar fields does not show any preference of the axis of rotation. In addition, the characteristic properties of superradiance in the case of a massless scalar field were also investigated. Contrary to the general expectations, we found that by an incident wave packet, which had been tuned to be maximally superradiant, the acquired extra energy in the scattering processmust be less than 0.1% of the energy sent in. It was found that instead of the occurrence of an anticipated scale of energy extraction from black hole, the to be superradiant part of the incident wave packet fails to reach the ergoregion; rather it suffers a nearly perfect reflection which appears to be an interesting phenomenon.