Results of Breakage Testing in MCF

Images correspond to the links in the table on page :    http://wwwrel.ph.utexas.edu/Members/deirdre/research/mcfbreaking.html

There are 3 grid sizes represented below, 5, 17, and 33.  For each grid size, the sparations 3.0, 3.1, and 3.2 have two plots one of curvature  versus theta and one of y versus z of the surface.  Both plots are slices of the 3-d space into 2-d along a common phi, utilizing the cylindrical symmetry of the 2 black hole  spaceime.  All plots are snapshots of the data, usually at the last time step.

Grid Size: 5 (Coarse)

Separation: 3.0

Separation 3.1 is not displayed at this grid size since the resolution is too poor for any indentation in the center to occur.  

Separation: 3.2

   This is beyond the critical separtion, therefore, no single apparent horizon exists to envelope both masses in this system.  The plot to the left is the y versus z surface plot showing the instability that arises when there is not a single minimal surface.

Grid Size: 17 (medium)

Separation: 3.0

Separation: 3.1

We see the expected "peanut" like indention in the shape. This separation should be near the critical separation beyond which there is no longer a single horizon.

Separation: 3.2

The critical separation is surpassed and the code contains instabilities as two surfaces are not necessary to obtain zero curvature as expected on the apparent horizons.  The instability grows starting from the center of the two masses.

Grid Size: 33 (Fine)

Separation: 3.0

Separation: 3.1

Separation: 3.2

  Once again the critical separation is surpassed and the instability begins at the center of the two masses.  The plot to the right is a snapshot of the curvature.  Spikes are beginning to grow at this time during the evolution but are obscured by the numbers at the top of the image.  Please look below for further investigations of the curvature instability.

Separation: 3.1

Plot of the time derivative of the curvature of the surface at a separation of 3.1, where no instabilities appeared in the plot of the surface (above).

Separation: 3.2

Plot of the time derivative of the curvature of the surface at a separation of 3.2 where instabilities did  appear in the plot of the surface (above).

The  plot below is the L-infinity norm is taken of the difference between two time levels in the curvature.  

  Note the increase late in the evolution corresponding to the growing instabilities depicted above.