Numerical experiments using harmonic evolution

For details of this scheme see the gr-qc paper Numerical Relativity Using a Generalized Harmonic Decomposition

Binary boson star merger

The following animations are from the merger of an equal mass binary boson star system, using plain harmonic coordinates (i.e. the source functions are zero). The initial profile for each boson star is a best-fit Gaussian approximation to a static solution (hence the oscillations in the movies of phi below), and then Lorentz boosted to the orbital velocity for a circular, Keplerian orbit. Initially, 95% of the mass M of each boson star is contained within a radius of roughly 29M (|phi|>3e-3 initially ... notice the color coding on the movies), hence the densities are much higher than white dwarf densities, but not quite that of a typical neutron star. The initial separation is about 230M. The time on the animations has been scaled so that M equals one solar mass.

The complex scalar field amplitude on the z=0 slice of the simulation (which is the orbital plane), linear color map phi_lin.mpg (2.4MB), and logarithmic color map phi_log.mpg (5.5MB) to better show the scalar field "atmosphere". The height of the surface in both cases is also a measure of the magnitude of the field. Note that this is a very early result, and convergence testing still needs to be done. In particular, I suspect that the rapid merger (in less than three orbits) is in part due to poor resolution, as is some of the "spreading" of the boson stars. Also, that the final rotating object is slightly off-center is also due to numerical error.

The Newman-Penrose scalar Psi_4 multiplied by distance r from the source, along the z=0 slice psi4_r_z_log.mpg (14MB) and x=0 psi4_r_x_log.mpg (14MB) slice of the simulation (the black edges around the different AMR levels are a visualization artifact, though show where the refinement boundaries are). Far from the source Psi_4 is a measure of the gravitational radiation emitted by the system. Even though in this wave zone the resolution of the simulation is quite coarse, there are several interesting things to note from these animations:

The initial burst of radiation is in part due to the radiation present in the initial data (which is a conformally flat, maximal slice), and partly due to mesh refinement "effects". Though notice that this radiation rather quickly disperses, and soon afterwards one can identify radiation coming from the binary system.
The spatial compactification causes the waves to become square shaped, and to slow down, as they approach the outer boundary. Also, the coordinate wavelength of the waves decrease as they near the outer boundary, eventually causing them to be very poorly resolved on the mesh (and we do not let the refinement track the waves once they're a certain distance away from the center; to try to do so would make the problem computationally intractable with this kind of coordinate system). There is then some concern that such poorly resolved waves may cause instabilities in the code, or reflect back to the interior. Numerical dissipation helps control the instabilities, and these results suggest that reflections are not too significant. Therefore, compactification to spatial infinity may be a very useful option for numerical relativity codes.

Some simulation details:

AMR with 8 levels of 2:1 refinement, base grid resolution = 33^3 (outer boundary of coarsest two levels are at spatial infinity due to the compactification used). This zoom-in sequence of wire frame snapshots shows the refinement structure at t=0: phi_0.jpeg , phi_1.jpeg , phi_2.jpeg , phi_3.jpeg .
Run time of about 1 week on 32 nodes of UBC's vn4 cluster
Maximum total memory usage ~ 2.5GB
~ 650,000 time steps taken on all levels combined

Black hole - boson star collision in axisymmetry

Case 1: M_bs/M_bh ~ 0.75, R_bs/R_bh ~ 12.5, black hole initially just outside boson star moving inwards with v~0.1

The magnitude of the complex field bhbs_phi4_L2.mpg (1.5MB) from the collision of a boson star with a black hole. The intial data for the boson star is only an approximation to the ground state configuration, hence there is some oscillation in the star which, interestingly, appears to persist after there is no longer an identifiable boson star (this still needs to be convergence tested though)

The black hole is formed via scalar field collapse, using a "boosted" profile such that its initial velocity is approximately 0.1. The initial mass of the black hole is roughly 0.012 (which is used to define the unit of time "m" in the simulation), and that of the boson star is roughly 0.009 (its radius, containing 95% of its mass, is approximately 0.3 ~ 25m). By the end of the simulation the black hole has essentially consumed the entire boson star. This is a "high-res" simulation, that took about 72 hours on 32 nodes of westgrid's glacier cluster, maximum total memory usage of roughly 320MB, a base grid of 128x64 points, up to 8 additional levels of 2:1 refinement, and roughly 1/2 million time steps on the finest level.

Case 2: M_bs/M_bh ~ 3.0, R_bs/R_bh ~ 50, black hole initially at rest at the center of the boson star

Scalar field "low" res. : bhbs_c_phi4_L0.mpg (1.2MB)

Plot of black hole mass as a function of time: M.eps

Case 3: M_bs/M_bh ~ 3.0, R_bs/R_bh ~ 50, black hole initially at rest just outside the boson star

Scalar field "low" res. : bhbs_r_phi4_L0.mpg (1.8MB)

Scalar field "medium" res. : bhbs_r_phi4_L1.mpg (0.8MB)

Plot of black hole mass as a function of time: M.eps

Plot of coordinate position of black hole as a function of time: d.eps

These plots suggest that the low resolution data should probably not be trusted after around 200M.

note: the medium and high resolution simulations are still running.

last updated: Jan. 2005