### 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
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 [111] 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 , and then decreases for larger 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 [48], 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 points, which is moderate by today’s
standards.