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# Einstein-Vlasov numerics

This page provides additional content to the paper "A numerical stability analysis for the Einstein-Vlasov system" (available at https://arxiv.org/abs/2009.08163) by Günther, Körner, Lebeda, Pötzl, Rein, Straub, and Weber. In particular, we present movies of the behaviors on the particle level in the different situations. All animations can be found on our Youtube Channel "Mathematik VI Universität Bayreuth".

All the following movies result from fixing a set of numerical particles and tracking their position and energy over time. We want to emphasize that we only track about one in a thousand numerical particles and only plot one in 200 time steps, i.e., our simulations are way more accurate than indicated by these movies.
As in the paper, we only present the results for the King model, but point out that the movies are qualitatively similar for other ansatz functions. In addition, we restrict ourselves to Schwarzschild coordinates here.

We start by considering the case of a stable steady state, i.e., y_0<y_{max}. We fix the steady state corresponding to y_0=0.1 and a rather strong dynamically accessible perturbation of \gamma>0 in this section, in order to see the occuring effects clearly enough.
The first movie shows the position of the fixed set of numerical particles in (r,w)-coordinates over time. The "structures" within the shown particles at the start of the movie solely originate from the selection of numerical particles we track. We observe that these initial structures disperse over time.
In addition, the existence of particles whose radii become significantly larger than the initial outer radius of the support is due to the choice of a rather strong perturbation.

The second movie illustrates the evolution of the fixed set of numerical particles in (r,E)-coordinates, where E is the particle energy. In this movie, the oscillating behavior of the solution can be seen at the lowest occurring energy.

## COLLAPSE

In this section we consider the case of the unstable steady state corresponding to y_0=0.6>y_{max} perturbed by a dynamically accessible perturbation of \gamma<0.
The first movie in this section shows the evolution of the positions of the fixed set of numerical particles in (r,w)-coordinates over time. We observe that the "collapse" happens rather quickly and that it affects all the numerical particles (however, recall that we will never be able to actually “see” the formation of a trapped surface in Schwarzschild coordinates, since these coordinates do not exist on both sides of the event horizon).

In the second movie, we chose a larger w-scale compared to the first one and marked the Schwarzschild radius, i.e., two times the (time-independent) ADM mass of the solution. We observe that the radius of the support of the solution indeed tends to the Schwarzschild radius.

The third movie illustrates the "collapse" in (r,E)-coordinates.

## HETEROCLINIC ORBITS

In this last section we consider the case of the unstable steady state corresponding to y_0=0.6>y_{max} perturbed by a dynamically accessible perturbation of \gamma>0.
The first movie in this section shows  the evolution of the positions of the fixed set of numerical particles in (r,w)-coordinates over time. As described in the paper,  the particles initially spread in space. While some particles keep moving away from the spatial origin manifesting the initial elevation, a considerable amount returns after some time, completing the first oscillation. This behavior of particles getting expelled and returning repeats several times until the system eventually oscillates around a seemingly unchanging configuration.

Compared to the first case of a stable steady state, there exist particles which keep moving away from the radial origin with seemingly constant radial velocity. This can be seen in the following movie, where we enlarged the r-scale.

The last movie illustrates the above effects in  (r,E)-coordinates and corresponds to Figure 11 in the paper. Here, we observe a portion of low energy particles that seems to form some dense structure which is independent of the less dense particles with higher energy.

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