Picture of a 2 black hole coordinate system.
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Crazy Coordinates...
At left we see an applet that generates grids for the collision of two black holes of unequal size. The ``throat'' of each black hole is illustrated with a grey circle. The radial coordinate depicted here is designed to be zero on the surface of each circle and on the line connecting them.

Each black hole is parameterized by a number Mu which characterizes the radius of the black hole (smaller with larger Mu) and distance from the x-axis (larger with larger Mu to a maximum of 1.0).


The java program is freely available.

Generating Coordinate Systems for Black Hole Spacetimes


The collision of black holes represents one of the most outstanding problems in current research involving the general theory of relativity. The extremely strong gravitational fields in the vicinity of black holes makes it very difficult to solve the problem of colliding black holes. As a result, scientists at various acedemic institutions are now taking advantage of the rapid advances in supercomputing technologies and focusing on a large scale computational effort to simulate the collision process on modern parallel computing platforms. However, the availability of powerful supercomputers does not simplify the problem much. Many difficult computational and physical issues remain to be worked out, applied and tested. In particular, at the very heart of the theory of relativity is the arbitrariness of spacetime coordinates. Of course, relativity theory is built around the notion that there is no preferred observer nor coordinate system, But from a practical and numerical point of view, certain coordinate systems may simplify the calculations, or perhaps make them more stable and accurate given constraints on the computational resources.

It is with this in mind that we have developed a numerical program to generate discrete and orthogonal grids that are designed to conform to the geometry of two black holes with arbitrary masses and separations. The specific problem we consider here is the axisymmetric or head-on collision of two black holes. Due to the axisymmetric nature of the interaction, it is only necessary, computationally, to consider the problem to be two-dimensional (in other words, any cross-section of this spacetime through the z-axis will look the same). In designing the coordinate systems, it is necessary to account for the nature of the physical systems. First, each black hole is characterized by a 2-sphere, called the throat, which provides a surface across which a particular type of boundary condition must be imposed in order to avoid the physical singularity hidden within the black hole throat. Second, as the black holes collide, they will emit gravitational radiation that will come off the system predominantly as quadrupole waves, and the final merged state of the two holes will evolve to the spherically symmetric single black hole (the Schwarzschild solution). Hence it is natural to evolve this system with asymptotically spherical coordinates to simplify the calculations of the gravity waves, and to better resolve the spherical component of the merged state. Third, it is important to extend the coordinate systems to cover very large distances from the black holes, and resolve the gravity waves which emanate from the system at the speed of light. In short, the coordinate systems meeting these objectives should have the properties of conforming to the spherical throats of the two black holes and become spherical (as measured from a point centered on the origin midway between the two holes) at asymptotic infinity. In addition, they must be orthogonal (the coordinate lines must be perpendicular) to simplify the computations.

We have developed a program to generate two different classes of coordinates that meet the above criterion. It is topologically not possible to conform the coordinates smoothly from two disconnected spherical domains and also be spherical at large distances. Hence coordinate singularities (i.e. ``saddle points'' in which the coordinates turn by 90 degrees) are unavoidable. The two grid classes we have developed are characterized by the placement and the number of singularities. The grid generation procedure is based on specifying one of the two (in two-dimensions) coordinates analytically, then solving essentially the equations enforcing the orthogonality condition to determine the second coordinate.

The first coordinate system has a coordinate (saddle-type) singularity where each of the throats meet the axis closest to the origin. A singularity is generated at these points by requiring that a radial-like coordinate be zero on both of the throats and the entire section of the axis connecting the throats. Lines of constant ``radius'' will then transform from ``peanut''-like surfaces near the throats to radial circles at infinity. The two throats and the axis between the throats in this case make up the constant zero radial line. The axis above (below) the top (bottom) hole is the angular coordinate value of zero (180 degrees). In the equal mass case, the equator corresponds to the line angle 90 degrees.

The second coordinate system has a saddle point at the origin midway between the two black holes. A singularity is generated at the origin by requiring that a radial-like coordinate be zero only on the two throats, and asymptotes to the usual radial coordinate at infinity. These coordinates are analogous to the field and equipotential lines of two charged metallic cylinders located at the centers of the two throats.


Resources:
  1. Source Code: Lines.java
  2. A zip file Lines.zip containing the compiled java code.