Existing examples

Boosted BH with a complex scalar field example

This is an example in which a complex scalar field evolves around a black hole that is boosted in the x direction. It is essentially the set up used to measure the dynamical friction in the relativistic study here.

It includes diagnostics that measure the fluxes of energy and linear momentum at a given radius that can be reconciled to the dynamical friction effects and the gain in energy and momentum of the spacetime. These variables are calculated and output throughout the simulations and can be a very good consistency and resolution check. If the simulations are correct, the time-integrated quantities should agree, as shown in the plots below.

The conservation of linear momentum:

and of the energy:

These were plots were produced using the python script plot_conservation.py in the example folder, for the code run with the parameters as specified in the params.txt file.

Note that whilst energy is conserved due to the time symmetry of the metric, linear momentum is not because there is no translational symmetry of the metric in the x direction, this is why the latter component has an additional source term, ${\cal S}_x$, (that is in effect the dynamical friction).

Fixed Kerr BH with a real scalar example

The background is a stationary Kerr solution in Kerr Schild coordinates, and the matter content is a real scalar field that asymptotes to a constant density value (a spatially constant field with a time varying amplitude). It is similar to the set up described in this work.

It includes diagnostics that measure the fluxes of energy and angular momentum at a given radius that can be reconciled to the gain in energy and momentum of the spacetime. Since the background has symmetries in the time and angular directions, these are conserved in the matter sector, which provides a valuable check on the evolution.

As it is set up, this example requires a higher resolution than the Boosted BH case, in order to achieve good agreement between the time integrated densities and fluxes. This is due to that fact that given the chosen initial data here, the scalar field quickly dissipates away. This means that one would require high resolution not only at the centre, near the BH, but also further away where some of the quantities are measured, and where in this case the gradients of the field would become large for a time. Below we show as an example, the conservation of energy in this case, given the set-up in the params.txt file, at three different resolutions:

• Nx = Ny = 64 and Nz=32:

• Nx = Ny = 128 and Nz=64:

• Nx = Ny = 256 and Nz=128:

The python script to produce these plots is located in the KerrBHScalarField example folder.