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Computational Relativity
My main interest is the source modelling and computation of gravitational waves. So far, I have worked in perturbation theory and full numerical relativity. In the context of black hole perturbations, my work involves numerical studies of the evolution of gravitational perturbations using the Teukolsky formalism. On a separate path, it also includes the use of gauge-invariant Regge-Wheeler-Zerilli approach and its application to the problem of computing gravitational waves at a finite distance from the source. In the area of numerical relativity, I have worked with both, multi-block and adaptive mesh refinement techniques to study the binary black hole problem. Below, I expand on these and other topics that I find interesting.
Multi-block domains and binary black holes
Numerical relativity consists of solving Einstein field equations using computer algorithms. The solution consists in the evolution in time of a given initial data, defined in a finite region of a 3-dimensional spacelike hypersurface. One way of solving the equations consists in using a cartesian grid with adapting mesh refinement and finite differencing methods. This approach provides higher resolution near the black holes while using coarser grids far away from them. Another method consists in covering the computational domain with overlapping patches using pseudo-spectral methods to integrate Einstein's equations. These two programs have been carried out successfully to evolve the basic binary black hole spacetime. A third method exists which has not yet been widely exploited: the multi-block domain decomposition. The idea is to use non-overlapping blocks to cover the computational domain. Reasons to explore this idea are: a) Each block has a trivial topology. b) It is possible to cover domains with non-trivial topologies (such as a domain with excised regions). c) The distribution of multiple blocks can be tailored to the geometry of the problem. This is specially important for the binary black hole case, where spherical symmetry can be exploited in the vicinity of each black hole and far away from them. This feature implies that it is possible to keep the angular resolution fixed while only varying the radial one. With this possibility, one can increase the volume of the domain requiring only a linear increment (instead of cubic one) in the number of points needed for computation. Motivated by these ideas, the multi-block domains offer another venue of exploration for solving Einsteins equations. A report implementing this technique is currently in preparation; wherein we have implemented a multi-block domain (in collaboration with people at Cornell) suitable for evolving binary black holes using excision, the generalized harmonic formulation of Einstein's equations and high-order finite differences operators.
On a different path, also in collaboration with the Cornell group, I've started to use the turducken technique to evolve binary black holes with high spin. The turducken approach provides the functionality of excising the black hole interior by substituting the singularity with a smooth function. This enables the possibility of evolving non-puncture initial data (such as the one provided by the Cornell group) using the standard BSSN and adaptive mesh refinement machinery. As a matter of fact, results using this technique have already been produced for another simulation setup and I'm working jointly in the preparation of the publication.
Gravitational wave phenomenology
Over the past years, numerical relativity has steadily progressed to the point that it is now possible to go through the inspiral, merger and ring-down phases of a comparable mass binary black hole collision. However, 3-dimensional simulations are computationally expensive. This poses a challenge for the goal of creating a waveform template bank, because a large number of simulations is needed in order to cover the parameter space of such physical scenarios. On the other hand, post-Newtonian and perturbation theory do a good job describing the gravitational radiation of these systems in the inspiral and the ring-down (starting from the close-limit) phases, respectively. The idea is to make use of the proper tool in the proper regime to come up with hybrid methods to compute gravitational waveforms, in a computationally cheap fashion.
Other important aspect of phenomenological studies is their usefulness as a guide towards first-principles calculations. Notwithstanding all the insight coming from numerical treatment of Einstein equations, analytic solutions are not yet available. Such analytic results would be the fastest way of computing gravitational waveforms. But even more important, it would provide crucial insight into the theory of general relativity, enabling the possibility of making connections with related and even seemingly unrelated physical phenomena. A historic example of a somewhat analogous situation is the unification of electric and magnetic phenomena via Maxwell's equations, culminating in the description of light as an electromagnetic wave. Yet another similar situation is that of the Balmer series. Although the empiric formula was known since 1885, the true insight was provided by the solution of Schrodinger's equation for the hydrogen atom.
Relevant physical scenarios for General Relativity
General Relativity is necessary in the regimen of strong gravitational fields. Such situations are found in the vicinity of massive stellar bodies. Of special interest in physics is the description of phenomena occurring in the presence of strong gravitational fields, such as matter accreting onto black holes or neutron stars. The behavior of the matter that forms a neutron star and supernovae core collapse are some examples. These astrophysical situations require the description of matter as a fluid that moves under the influence of gravity and magnetic fields in curved spacetimes. Situations like those are important because gas and dust are present in stellar media, representing more realistic scenarios that can imprint additional features in the generated gravitational waves.
Computation with graphics cards
I am also interested in the application of the Graphics Processing Unit (GPU) to scientific computing. GPU's were originally designed for video game graphics rendering. However, recent developments have enabled people to program the GPU and use it as a general-purpose processor (with certain limitations such as the amount of memory available in the card itself). GPU's are an attractive option for scientific computing due to their low price and extremely high computing power. The task of exploring the parameter space for gravitational waveforms could benefit enormously from using this new technology. Here at Maryland, we have started to build a cluster of GPU's to explore its applications in numerical relativity.