On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance

Abstract

A new numerical method is introduced to study the problem of time evolution of generic non-linear dynamical systems in four-dimensional spacetimes. It is assumed that the time level surfaces are foliated by a one-parameter family of codimension two compact surfaces with no boundary and which are conformal to a Riemannian manifold C. The method is based on the use of a multipole expansion determined uniquely by the induced metric structure on C. The approach is fully spectral in the angular directions. The dynamics in the complementary 1+1 Lorentzian spacetime is followed by making use of a fourth order finite differencing scheme with adaptive mesh refinement. In checking the reliability 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 to be 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 crucial role in jet formation in black hole spacetimes while matter accretes onto the central object. Our findings appear to be on contrary to these claims as the angular dependence of a to be superradiant scattering of a massless scalar field does not show any preference of the axis of rotation. In addition, the process of superradiance, in case of a massless scalar field, was also investigated. On contrary to the general expectations no energy extraction from black hole was found even though the incident wave packets was fine tuned to be maximally superradiant. Instead of energy extraction the to be superradiant part of the incident wave packet fails to reach the ergoregion rather it suffers a total reflection which appears to be a new phenomenon.

49 pages, 11 figures

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• Intermediate behavior of Kerr tails
• Superradiance of a charged scalar field coupled to the Einstein-Maxwell equations
• Numerical evolutions of fields on the 2-sphere using a spectral method based on spin-weighted spherical harmonics
• Numerical investigation of the dynamics of linear spin $s$ fields on Kerr background I. Late time tails of spin $s=\pm 1,\pm 2$ fields
• Charged scalar perturbations around a regular magnetic black hole
• Noether currents for the Teukolsky Master Equation
• Numerical investigation of the dynamics of linear spin $s$ fields on a Kerr background II: Superradiant scattering
• On solving the constraints by integrating a strongly hyperbolic system
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• Superradiance or total reflection?
• Spectral approach to axisymmetric evolution of Einstein's equations
• A hyperboloidal method for numerical simulations of multidimensional nonlinear wave equations: nonlinear tails