4.6 3D Einstein–Klein–Gordon system

The Einstein–Klein–Gordon (EKG) system can be used to simulate many interesting physical phenomena. In 1D, characteristic EKG codes have been used to simulate critical phenomena and the perturbation of black holes (see Section 3.1), and a Cauchy EKG code has been used to study boson star dynamics [216]. Extending these codes to 3D would open up a new range of possibilities, e.g., the possibility to study radiation from a boson star orbiting a black hole. A first step in that direction has been achieved with the construction of a 3D characteristic code by incorporating a massless scalar field into the PITT code [22]. Since the scalar and gravitational evolution equations have the same basic form, the same evolution algorithm could be utilized. The code was tested to be second order convergent and stable. It was applied to the fully nonlinear simulation of an asymmetric pulse of ingoing scalar radiation propagating toward a Schwarzschild black hole. The resulting scalar radiation and gravitational news backscattered to ℐ+ was computed. The amplitudes of the scalar and gravitational radiation modes exhibited the expected power law scaling with respect to the initial pulse amplitude. In addition, the computed ringdown frequencies agreed with the results from perturbative quasinormal mode calculations.

The LEO code [111Jump To The Next Citation Point] developed by Gómez et al. has been applied to the characteristic evolution of the coupled Einstein–Klein–Gordon fields, using the cubed-sphere coordinates. The long term plan is to simulate a boson star orbiting a black hole. In simulations of a scalar pulse incident on a Schwarzschild black hole, they find the interesting result that scalar energy flow into the black hole reaches a maximum at spherical harmonic index ℓ = 2, and then decreases for larger ℓ due to the centrifugal barrier preventing the harmonics from effective penetration. The efficient parallelization allows them to perform large simulations with resolution never achieved before. Characteristic evolution of such systems of astrophysical interest have been limited in the past by resolution. They note that at the finest resolution considered in [48Jump To The Next Citation Point], it would take 1.5 months on the fastest current (single) processor to track a star in close orbit around a black hole. This is so even though the grid in question is only 81 × 123 points, which is moderate by today’s standards.

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