# Numerical relativity coordinate system displayed

In a picture or video of a numerical relativity simulation, such as a neutron star merger into a black hole, how do they set up their coordinate system? Lets take the point in a video corresponding to x=10km, y=20km, z=30km, t=1ms. Spacetime itself is distorted, in a very complex way, so how do you make sense of these numbers?

Website to find some nice videos: http://numrel.aei.mpg.de/images

Just to clarify: There are simulations in which space-time is fixed to the a well defined metric (e.g. Kerr black hole accretion disk MHD simulation with no disk self-gravity). But for true numerical relativity, in which the shape of space-time itself has to be simulated, there is no "clean" metric.

• Do you have an example picture or video? – tpg2114 Jul 22 '13 at 1:04
• tpg2114: Added a link just now. – Kevin Kostlan Jul 22 '13 at 13:55

There is a huge variance in how these coordinates are set up, and very often the coordinate systems are chosen for computational convenience (having more data points in place where the metric varies a lot, and fewer far from, say, your colliding black holes), in addition to more physical choices. Once you have run the simulation and have found a solution, however, you can apply math and create any coordinate system you wish for visualizations.

• SO there is no convention. Omitting "minor" details is one of the many ways journal articles distort their results. – Kevin Kostlan Jul 23 '13 at 22:30
• @KevinKostlan: journal articles most definitely will not omit details like this. Web visualizaitons, sure. But not journal articles. – Jerry Schirmer Jul 23 '13 at 23:53

While I do not work on GRHD/GRMHD simulations, I am fairly certain that the numbers you would see in a simulation (displayed as an image or a video, or even the initial conditions file) would be the laboratory reference frame.

• The question is: how do you define the lab reference frame? In special rel it is trivial, but this is general rel! Riemann normal coords are one way I can think of, but they are prone to "caustics". – Kevin Kostlan Jul 22 '13 at 13:52
• @KevinKostlan: see my answer. The reference frame is arbitrary. For Binary black hole collisions, the choice is usually something like the background Minkowski spacetime of the binary system in Schild coordinates filtered through a fisheye transformation, but there are many possible choices. – Jerry Schirmer Jul 22 '13 at 13:58
• @Jerry, I will mark you if no one else can give a more concrete answer. Isn't there any convention that people use? Or a small set of conventions? – Kevin Kostlan Jul 22 '13 at 22:39

In the program David Madore has used to simulate Kerr black holes, the description mentions

Note: this program does not handle correctly the region where two horizons cross (where neither ingoing nor outgoing Kerr coordinates are adequate). This will result in some “snow” around such regions. This is a numerical artefact.

Thus, the program is using ingoing and outgoing Kerr coordinates internally for the computations, but that might not be the best solution. See also some notes on what all the different coordinate systems mean.

For visualization, a free falling camera is used.

• This is a simulation in which we have a known geometry. However, my question adresses cases in which space time itself does not have a simple metric (i.e. colliding black holes). – Kevin Kostlan Jul 22 '13 at 22:39