Research in Gravitational Physics:
Problems in Gravitational Radiation
General Relativity,
gravitational radiation, characteristic formulation, boundary conditions,
numerical methods, computational physics.
My research centers on the study and implementation of numerical algorithms for
time dependent, initial-boundary value problems. I apply these methods to the
numerical solution of Einstein's equations and the computation of gravitational
waves (both in vacuum and with matter sources), with particular emphasis in the
simulation of colliding black holes and black hole-neutron star systems. Both
areas are of great astrophysical interest, as these are the most likely sources
for the production of gravitational waves strong enough to be detected by the
next generation of gravitational interferometric
detectors, of which LIGO is the prime example. The following describes the most
recent projects.
We incorporate a massless scalar field
into a 3-dimensional code for the characteristic evolution of the gravitational
field. The extended 3-dimensional code for the Einstein--Klein--Gordon system
is calibrated to be second order convergent. It provides an accurate
calculation of the gravitational and scalar radiation at infinity. As an
application, we simulate the fully nonlinear evolution of an asymmetric scalar
pulse of ingoing radiation propagating toward an interior Schwarzschild black
hole and compute the backscattered scalar and gravitational outgoing radiation
patterns. The amplitudes of the scalar and gravitational outgoing radiation
modes exhibit the predicted power law scaling with respect to the amplitude of
the initial data. For the scattering of an axisymmetric
scalar field, the final ring down matches the complex frequency calculated perturbatively for the $\ell=2$ quasinormal
mode.
Follow this link for a published manuscript, Phys. Rev. D 71, 064028 (2005), gr-qc/0412066
In relation to the BSSN formulation of the Einstein equations, we write down
the boundary conditions that result from the vanishing of the projection of the
Einstein tensor normally to a timelike hypersurface. Furthermore, by setting up a well-posed
system of propagation equations for the constraints, we show explicitly that
there are three constraints that are incoming at the boundary surface and that
the boundary equations are linearly related to them. This indicates that such
boundary conditions play a role in enforcing the propagation of the constraints
in the region interior to the boundary. Additionally, we examine the related
problem for a strongly hyperbolic first-order reduction of the BSSN equations
and determine the characteristic fields that are prescribed by the three
boundary conditions, as well as those that are left arbitrary.
Follow this link for a published manuscript, Phys. Rev. D 70, 064008 (2004), gr-qc/0404070
We show how the use of the normal projection of the Einstein tensor as a set of boundary conditions relates to the propagation of the constraints, for two representations of the Einstein equations with vanishing shift vector: the ADM formulation, which is ill posed, and the Einstein-Christoffel formulation, which is symmetric hyperbolic. Essentially, the components of the normal projection of the Einstein tensor that act as non-trivial boundary conditions are linear combinations of the evolution equations with the constraints that are not preserved at the boundary, in both cases. In the process, the relationship of the normal projection of the Einstein tensor to the recently introduced ``constraint -preserving'' boundary conditions becomes apparent.
Follow this link for a published manuscript, Phys. Rev. D 69, 124020 (2004), gr-qc/0310064
We prescribe a choice of 18 variables in all that casts the equations of the fully nonlinear characteristic formulation of general relativity in first--order quasi-linear canonical form. At the analytical level, a formulation of this type allows us to make concrete statements about existence of solutions. In addition, it offers concrete advantages for numerical applications as it now becomes possible to incorporate advanced numerical techniques for first order systems, which had thus far not been applicable to the characteristic problem of the Einstein equations, as well as in providing a framework for a unified treatment of the vacuum and matter problems. This is of relevance to the accurate simulation of gravitational waves emitted in astrophysical scenarios such as stellar core collapse.
Follow this link for a published manuscript, Phys. Rev. D 68, 084013 (2003), gr-qc/0303104
In the 3+1 framework of the Einstein equations for the case of a vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value problem. Such conditions take the form of evolution equations along (as opposed to across) the boundary for certain components of the extrinsic curvature and for certain space derivatives of the three-metric. We argue that, in general, such boundary conditions do not follow necessarily from the evolution equations and the initial data, but need to be imposed on the boundary values of the fundamental variables. Using the Einstein-Christoffel formulation, which is strongly hyperbolic, we show how three of the boundary equations up to linear combinations should be used to prescribe the values of some incoming characteristic fields. Additionally, we show that the fourth one imposes conditions on some outgoing fields.
Follow this link for a published manuscript, Phys. Rev. D 68, 044014 (2003), gr-qc/0302071
In regards to the initial-boundary value problem of the Einstein equations, we argue that the projection of the Einstein equations along the normal to the boundary yields necessary and appropriate boundary conditions for a wide class of equivalent formulations. We explicitly show that this is so for the Einstein-Christoffel formulation of the Einstein equations in the case of spherical symmetry.
Follow this link for a published manuscript, Class. Quantum Grav 20 (11) 2379-2392, (2003) , gr-qc/0302032.
We present a method for computing the evolution of a spacetime containing a massive particle and a black hole. The essential idea is that the gravitational field is evolved using full numerical relativity, with the particle generating a non-zero source term in the Einstein equations. The matter fields are not evolved by hydrodynamic equations. Instead the particle is treated as a rigid body whose center follows a geodesic. The necessary theoretical framework is developed and then implemented in a computer code that uses the null-cone, or characteristic, formulation of numerical relativity. The performance of the code is illustrated in test runs, including a complete orbit (near r=9M) of a Schwarzschild black hole.
Follow this link for a published manuscript, Phys. Rev. D 68,
084015 (2003), gr-qc/0301060
We study the properties of the outgoing gravitational wave produced when a non-spinning black hole is excited by an ingoing gravitational wave. Simulations using a numerical code for solving Einstein's equations allow the study to be extended from the linearized approximation, where the system is treated as a perturbed Schwarzschild black hole, to the fully nonlinear regime. Several nonlinear features are found which bear importance to the data analysis of gravitational waves. When compared to the results obtained in the linearized approximation, we observe large phase shifts, a stronger than linear generation of gravitational wave output and considerable generation of radiation in polarization states which are not found in the linearized approximation. In terms of a spherical harmonic decomposition, the nonlinear properties of the harmonic amplitudes have simple scaling properties which offer an economical way to catalog the details of the waves produced in such black hole processes.
Follow this link for a published manuscript, Phys. Rev. D 68,
084014 (2003), gr-qc/0306098
We present a fully nonlinear calculation of the waveform of the gravitational radiation emitted in the fission of a vacuum white hole. At early times, the waveforms agree with close-approximation perturbative calculations but they reveal dramatic time and angular dependence in the nonlinear regime. The results pave the way for a subsequent computation of the radiation emitted after a binary black hole merger.
Follow this link for a published manuscript, Phys. Rev. D. 66,
064019 (2002), gr-qc/0205038
We use null hypersurface techniques in a new approach to calculate the retarded waveform from a binary black hole merger in the close approximation. The process of removing ingoing radiation from the system leads to two notable features in the shape of the close approximation waveform for a head-on collision of black holes: (i) an initial quasinormal ringup and (ii) weak sensitivity to the parameter controlling the collision velocity. Feature (ii) is unexpected and has the potential importance of enabling the design of an efficient template for extracting the gravitational wave signal from the noise.
Follow this link for a published manuscript, Phys. Rev. D. 65, 084034 (2002)
A partially first-order form of the characteristic formulation is introduced to control the accuracy in the computation of gravitational waveforms produced by highly distorted single black hole spacetimes. Our approach is to reduce the system of equations to first-order differential form on the angular derivatives, while retaining the proven radial and time integration schemes of the standard characteristic formulation. This results in significantly improved accuracy over the standard mixed-order approach in the extremely nonlinear post-merger regime of binary black hole collisions.
Follow this link for a published manuscript, Phys. Rev. D. 64, 024007 (2001)