My primary research focus is the field of numerical relativity, specifically critical phenomena, black hole astrophysics and gravitational wave generation. Ultimately, my goal is to simulate and understand the details of:
- black hole formation from electromagnetically interacting matter;
- accretion onto black holes; and,
- gravitational waves generated in collisions between black holes and compact astrophysical objects.

My doctoral dissertation research concerns the numerical study of black hole formation from charged, massive scalar fields in spherical symmetry, and the axially symmetric accretion of such matter onto rotating black holes and jet formation. In the gravitational collapse to a black hole, and in accretion onto black holes, astrophysically realistic matter will likely be magnetohydrodynamic (MHD)---fluid-like, and ionized. Only in very few highly symmetric cases has the fully general relativistic problem (where matter sources dynamically influence the curvature of spacetime) been completely studied. Even then, computational techniques are required. Shock formation in fluid systems further complicates the details. Numerical simulation of general relativistic MHD systems is in its infancy, and insight gained through the study of MHD analogues like the charged, massive scalar field is required for progress.

Black Hole Critical Phenomena

One focus of my thesis is investigating spherically symmetric black hole formation. The threshold solution between black hole formation and dispersal exhibits unique behavior. Originally studied for the uncharged, massless scalar field, Choptuik found the threshold solution exhibits discrete self-similarity (exponential rescaling in space and time reveals an identical replica) and universality (all threshold initial data evolve to the same solution). Infinitesimal black holes can be formed as the threshold is approached. However, while my research shows similar results with sufficiently small charge and mass parameters, if the field is very massive (and possibly highly charged), threshold solutions are no longer discretely self-similar or universal. The solution in this case is different---it is now a periodically oscillating star of charged, massive scalar material. Furthermore, there is now a theoretical minimum mass for formed black holes. These results will be important to the analysis of less symmetric configurations. While similar solutions have been found for other sources, of the charged models studied, the charged, massive scalar field most closely resembles MHD matter.

- Methods

Investigation uses computational techniques---analytic analysis can suggest details, but solving Einstein's equations of general relativity requires numerical computation in all but the simplest of cases. I apply finite difference methods (discrete analogue of differential methods), and use an adaptive mesh refinement algorithm (to dynamically add resolution where necessary and remove resolution when not required).

- Equations and Results

Equations in Postscript
Equations in PDF

Results in Postscript
Results in PDF

The critical solution for large scalar field mass parameter is periodic---an oscillating perturbed boson star with nonzero electric charge. (Click on image above for an MPEG animation of the function 2m/r.)

Black Hole Accretion

In black hole accretion, astronomical evidence shows that a portion of inspiralling plasma and generated electromagnetic radiation collimate into bipolar jets from the black hole. Only modest progress has been made in MHD systems, and again the details are not well understood. In order to distinguish between matter and gravitational effects---essential to discussion of the fully general relativistic system---I include the axially symmetric study of charged, massive scalar fields (i.e., the Maxwell-Klein-Gordon system) accreting onto rotating black holes. In this case, I treat the scalar field as test matter (it evolves about the black hole, but does not itself contribute to the curvature of spacetime) and observe its dynamics.

- Methods

Investigation again uses the computational techniques of finite difference methods and adaptive mesh refinement. In addition, the size of the computational domain in axial symmetry requires the use of parallel computation in order to obtain the results within a reasonable period of human time.

- Results

My results show that both the mass and charge coupling parameters play significant roles in structure formation and energy collimation. When the mass parameter lies with in a specific range, the scalar field persists in a region near the black hole, marking the onset of structure formation. Outside of this range, the scalar field either plunges into the back hole or mostly scatters away. But when inside this range, an additional charge coupling leads to amplification of the energy collimated along the black hole rotation axis. In addition, the dynamics display strong dynamo-like behavior in the accretion-disk plane as the axial current oscillates between the clockwise and anti-clockwise directions. This indicates how both electromagnetic and non-electromagnetic properties of the matter dramatically affect its evolution in a rotating black hole spacetime.

Energy density showing oblique scattering of the massless scalar field off an extremally rotating black hole. The angular momentum per unit mass has a small effect on the dynamics. Note that the color gradient is scaled logarithmically. (Click on image above for an MPEG animation of the energy density.)

Energy density showing oblique scattering of the massive scalar field off an extremally rotating black hole. The nonzero scalar field mass has a significant effect on the dynamics. Note that the color gradient is scaled logarithmically. (Click on image above for an MPEG animation of the energy density.)

Energy density showing oblique scattering of the scalar off an extremally rotating black hole. The mass parameter is very large so the bulk of the field spreads at a slower rate. Eventually the field is entirely consumed by the black hole. Again, the color gradient is scaled logarithmically. (Click on image above for an MPEG animation of the energy density.)

Energy density showing collimation of a charged massive Maxwell-Klein-Gordon field along the axis of an extremally rotating black hole. The color gradient is scaled logarithmically. Calculations were performed in parallel across 64 processors. (Click on image above for an MPEG animation of the energy density.)

Axial current density showing dynamo-like behavior of a charged massless Maxwell-Klein-Gordon field on an extremally rotating black hole spacetime. Now, the color gradient is scaled linearly. Calculations were performed in parallel across 64 processors. (Click on image above for an MPEG animation of the axial current density.)

Dynamics of the angular momentum density for the charged massless Maxwell-Klein-Gordon field on an extremally rotating black hole spacetime. Again, the color gradient is scaled linearly. Calculations were performed in parallel across 64 processors. (Click on image above for an MPEG animation of the angular momentum density.)