# Matthew Choptuik

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#### Supervision Enquiry

## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2019)

In the first part of this thesis, we solve the coupled Einstein-Vlasov system in spherical symmetry using direct numerical integration of the Vlasov equation in phase space. Focusing on the case of massless particles we study critical phenomena in the model, finding strong evidence for generic type I behaviour at the black hole threshold that parallels what has previously been observed in the massive sector. For differing families of initial data we find distinct critical solutions, so there is no universality of the critical configuration itself. However we find indications of at least a weak universality in the lifetime scaling exponent, which is yet to be understood. Additionally, we clarify the role that angular momentum plays in the critical behaviour in the massless case. The second part focuses on type II critical collapse. Using the critical collapse of a massless scalar field in spherical symmetry as a test case, we study a generalization of the BSSN formulation due to Brown that is suited for use with curvilinear coordinates. We adopt standard dynamical gauge choices, including 1+log slicing and a shift that is either zero or evolved by a Gamma-driver condition. With both choices of shift we are able to evolve sufficiently close to the black hole threshold to 1) unambiguously identify the discrete self-similarity of the critical solution, 2) determine an echoing exponent consistent with previous calculations, and 3) measure a mass scaling exponent, also in accord with prior computations. Our results can be viewed as an encouraging first step towards the use of hyperbolic formulations in more generic type II scenarios, including the as yet unresolved problem of critical collapse of axisymmetric gravitational waves. In the last part, we present simulations of nonlinear evolutions of pure gravity waves. We describe a new G-BSSN code in axial symmetry that is capable of evolving a pure vacuum content in a strong gravity regime for both Teukolsky and Brill initial data. We provide strong evidence for the accuracy of the numerical solver. Our results suggest that the G-BSSN is promising for type II critical phenomena studies.

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This thesis constitutes a numerical study concerning the dynamics of an inviscidfluid subject to Newtonian gravity. Type-II critical phenomena has been previously measured in gravitational collapse simulations of isothermal-gas-spheres in Newtonian gravity. Our first objective was to extend this work by applying the more general polytropic-gas equation-of-state to the spherically symmetric fluid. We showed that under generic conditions of critical collapse, the polytropic gas allows for scale-invariant solutions. These solutions display self-similarity of the first kind with non-linear scaling between the space and time variables. One of these solutions was identified as the critical solution in critical collapse simulations. Such solution was found to have a single unstable mode with a Lyapunov exponent whose value depends on the polytropic index (Γ) from the equation of state. We argued that this behavior constitutes evidence of type-II critical phenomena with a transition from type-II to type-I behavior occurring at Γ ≥ 6/5. Thus, the polytropic gas exhibits both types of critical behavior. These phenomena was investigated dynamically and also from perturbation analysis. In the second phase of the project we extended the hydrodynamic model to treataxi-symmetric gravitational collapse. This allowed us to study the effect of angular momentum on the critical solution. As previously predicted, infinitesimal initial rotation introduces a non-spherical, unstable axial mode into the dynamics. The measured scaling behavior of the specific angular momentum of the collapsed core agrees with the predicted growth rate (Lyapunov exponent) of the axial perturbation. This two-mode linear regime modifies the scaling laws via the introduction of universal functions that depend on the two-parameter family of initial data. The predicted universality of these functions was confirmed through careful measurements of the collapsed mass and its angular momentum near the collapse threshold. A two-parameter space survey reveals a universal behavior of the order-parameters, with no mass-gap even in the presence of finite initial rotation. The behavior changes slightly beyond some initial rotation threshold. The results then, can be interpreted as an intermediate convergence to a non-spherical self-similar critical solution with a single unstable mode.

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In this thesis we present the first numerical study of gravitational collapse in braneworld within the framework of the single brane model by Randall-Sundrum (RSII). We directly show that the evolutions of sufficiently strong initial data configurations result in black holes (BHs) with finite extension into the bulk. The extension changes from sphere to pancake (or cigar, as seen from a different perspective) as the size of BH increases. We find preliminary evidences that BHs of the same size generated from distinct initial data profiles are geometrically indistinguishable. As such, a no-hair theorem of BH (uniqueness of BH solution) is suggested to hold in the RSII spacetimes studied in this thesis—these spacetimes are axisymmetric without angular momentum and non-gravitational charges. In particular, the BHs we obtained as the results of the dynamical system, are consistent with the ones previously obtained from a static vacuum system by Figueras and Wiseman. We also obtained some results in closed form without numerical computation such as the equality of ADM mass of the brane with the total mass of the braneworld.The calculation within the braneworld requires advances in the formalism of numerical relativity (NR). The regularity problem in previous numerical calculations in axisymmetric (and spherically symmetric) spacetimes, is actually associated with neither coordinate systems nor the machine pre- cision. The numerical calculation is regular in any coordinates, provided the fundamental variables (used in numerical calculations) are regular, and their asymptotic behaviours at the vicinity of the axis (or origin) are compatible with the finite difference scheme. The generalized harmonic (GH) formalism and the BSSN formalism for general relativity are developed to make them suitable for calculations in non-Cartesian coordinates under non-flat background. A conformal function of the metric is included into the GH formalism to simulate the braneworld.

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This thesis presents results from numerical studies of the dynamics of three classical nonlinear fieldtheories, each of which possesses stable, localized solutions called solitons. We focus on two of thetheories, known as Skyrme models, which have had application in various areas of physics. Thethird, which treats a complex scalar field, is principally viewed as a model problem to developsolution techniques. In all cases, time dependent, nonlinear partial differential equations in severalspatial dimensions are solved computationally. Simulation of high energy collisions of the solitons is of particular interest. The solitons of the complex scalar field theory are known as Q-balls and carry a charge Q. Weinvestigate the scattering of these objects, reproducing previous findings for collisions where the Q-balls have charge of the same sign. We obtain new results for interactions where the Q-ballshave opposite charge, and for scattering of Q-balls against potential obstructions. The chief contributions of this thesis come from simulations performed within the context of a Skyrme model in two spatial dimensions. This is a multi-scalar theory with solitons known as baby skyrmions. We concentrate on the rich phenomenology seen in high-energy scattering of pairs of these objects, each of which has a topological charge that can be either positive or negative. We extend the understanding of the role of different parameters of the model in governing the outcome of head-on and off-axis collisions. The study of instabilities seen in previous simulations of Skyrme models is of central interest. Our results confirm that the governing partial differential equations become of mixed hyperbolic-elliptic type for interactions at sufficiently high-energy, as originally suggested by Crutchfield and Bell. We present strong evidence for the loss of energy conservation and smoothness of the dynamical fields in these instances. This supports the conclusion that the initial value problem at hand becomes ill-posed, so that the observed instabilities result from the nature of the equations themselves, and are not numerical artifacts. Finally, we present preliminary results for the typical phenomenology seen in head-on scattering of solitons in the Skyrme model in three spatial dimensions.

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In this thesis, we present numerical studies of models for the accretion of fluids and magnetofluids onto rotating black holes. Specifically, we study three main scenarios, two of which treat accretion of an unmagnetized perfect fluid characterized by an internal energy sufficiently large that the rest-mass energy of the fluid can be ignored. We call this the ultrarelativistic limit, and use it to investigate accretion flows which are either axisymmetric or restricted to a thin disk. For the third scenario, we adopt the equations of ideal magnetohydrodyamics and consider axisymmetric solutions. In all cases, the black hole is assumed to be moving with fixed velocity through a fluid which has constant pressure and density at large distances. Because all of the simulated flows are highly nonlinear and supersonic, we use modern computational techniques capable of accurately dealing with extreme solution features such as shocks.In the axisymmetric ultrarelativistic case, we show that the accretion is described by steady-state solutions characterized by well-defined accretion rates which we compute, and are in reasonable agreement with previously reported results by Font and collaborators [1,2,3]. However, in contrast to this earlier work with moderate energy densities, where the computed solutions always had tail shocks, we find parameter settings for which the time-independent solutions contain bow shocks. For the ultrarelativistic thin-disk models, we find steady-state configurations with specific accretion rates and observe that the flows simultaneously develop both a tail shock and a bow shock. For the case of axisymmetric accretion using a magnetohydrodynamic perfect fluid, we align the magnetic field with the axis of symmetry. Preliminary results suggest that the resulting flows remain time-dependent at late times, although we cannot conclusively rule out the existence of steady-state solutions. Moreover, the flow morphology is different in the magnetic case: additional features are apparent that include an evacuated region near the symmetry axis and close to the black hole.

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This thesis describes a numerical study of binary boson stars within the context of an approximation to general relativity. Boson stars, which are static, gravitationally bound configurations of a massive complex scalar field, can be made gravitationally compact. Astrophysically, the study of gravitationally compact binaries---in which each constituent is either a neutron star or a black hole---and especially the merger of the constituents that generically results from gravitational wave emission, continues to be of great interest. Such mergers are among the most energetic phenomena thought to occur in our universe. They typically emit copious amounts of gravitational radiation, and are thus excellent candidates for early detection by current and future gravitational wave experiments.The approximation we adopt places certain restrictions on the dynamical variables of general relativity (conformal flatness of the 3-metric), and on the time-slicing of thespacetime (maximal slicing), and has been previously used in the simulation of neutron stars mergers. The resulting modeling problem requires the solution of a coupled nonlinear system of 4 hyperbolic, and 5 elliptic partial differential equations (PDEs) in three space dimensions and time. We approximately solve this system as an initial-boundary value problem, using finite difference techniques and well known, computationally efficient numerical algorithms such as the multigrid method in the case of the elliptic equations. Careful attention is paid to the issue of code validation, and a key part of the thesis is the demonstration that, as the basic scale of finite difference discretization is reduced, our numerical code generates results that converge to a solution of the continuum system of PDEs as desired.The thesis concludes with a discussion of results from some initial explorations of the orbital dynamics of boson star binaries. In particular, we describe calculations in which motion of such a binary is followed for more than two orbital periods, which is a significant advance over previous studies. We also present results from computations in which the boson stars merge, and where there is evidence for black hole formation.

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##### Master's Student Supervision (2010 - 2018)

We study numerically the gravitational collapse of two massless scalar fields in spherical symmetry using an approach which incorporates some of the effects of angular momentum. Each field is characterized by a distinct angular momentum parameter, l, with l = 0, 1, 2, ..., and has distinct equations of motion. We focus on black critical behaviour, in which we look for solutions, and the properties thereof, that represent the onset of black hole formation in parametrized families of initial data. Although many different critical solutions have been discovered since the early 1990's, almost all work has involved a single matter source, and by coupling more than one type of matter to the gravitational field, new and interesting questions arise.In particular, we consider the issue of the relative stability of the critical solutions in our model, in which we look for signs of instability of one field in the presence of the critical solution associated with the other. Our main result is that fields corresponding to larger values of l appear to be unstable in the context of lower-l critical solutions. This implies that in a two-field universe, the field with larger l would always dominate at precise criticality.

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