Multigrid Ulrich Trottenberg Pdf Creator

Multigrid ulrich trottenberg pdf creator

Refworks Account Login. UBC Theses and Dissertations. Featured Collection. 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 the spacetime 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. Arnowitt, S. Deser and C.

Thank you very much for all your support and wise guidance over the years this project took place.

Second, I would like to thank all the members of my graduate committee: William G. Unruh, Jeremy S. Heyl and Ingrid Stairs for all their input on this project and for asking all those hard questions that made me think deeper about physics in general and general relativity in particular.

Also I would like to thank Bill Unruh for fostering interesting discussions taking place in the gravity group meeting. I enjoyed being part of it and certainly will miss it a lot. I would like to thank Gerhard Huisken and the Albert Einstein Institute for their hospitality and support during the time that I spent there.

Multigrid ulrich trottenberg pdf creator

It was great being a member of the numerical relativity group at UBC. I learned a lot over the years with the innumerable interactions among different members of the group. My thanks goes to Roland Stevenson as well for sharing his programming expertise. Thank you guys for all your friendship. I would like to thank my friends and colleagues from UBC for all their friendship and for making my stay at UBC so vibrant. Special thanks for my friend Sanaz Vafaei for all her support during the writing of this manuscript.

I am deeply indebted to several of friends in Brazil that have actively encouraged me to pursue my dreams.

I am grateful to my wife, Janine Kurtz, for all her dedication, love, companionship and patience during my final steps to conclude this project. Thanks for helping me to get through this difficult time.

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I wish to thank my entire family for providing me a loving and nurturing environment. Without xi their support and sacrifices during my education, I would not have made it so far. I would like to dedicate this thesis to my parents, Elson R. Mundim and Selmara C.

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Mundim, for all their love, support and encouragement during my entire life. Binary systems composed of gravitationally compact objects 1, such as two black holes, two neutron stars or one neutron star and one black hole, are among the most promising sources of these waves.

Multigrid ulrich trottenberg pdf creator

An international network of laser interfer- ometer detectors e. However, even for such strong sources, the terrestrial signal strength from a typical event not likely in our own galaxy, or even in our local group of galaxies, on the time scale of years is expected to be much less than the inherent noise in the instruments.

Faced with this situation, the most promising technique to extract the signals from the noise is matched filtering [5], which involves comparison of the measured signal against a known precomputed family of waveforms. Construction of such template waveforms is thus an urgent problem, and it requires accurate theoretical modelling of the process of compact binary inspiral.

Due to the complexity and nonlinearity of the Einstein field equations for the relativistic grav- itational field—to which one must add the governing equations for any matter fields which are involved such as a perfect fluid in the case of neutron stars —accurate modelling of the late phases of binary inspiral and merger requires a numerical approach.

Now, as we will discuss in more detail shortly Sec.

This is especially true for the case of the inspiral and collision of two black holes, which although a formidable problem, is simplified relative to its neutron star counterpart by the fact that it does not involve matter fields.

For neutron stars, the need to solve the equations of general relativistic hydrodynamics in concert with the Einstein equations leads to a host of other difficulties, including the need to deal with shocks, turbulence and uncertainties concerning the equation of state.

Moreover, the parameter space describing the generic collision of two compact object has many dimensions, so that even with improved sim- ulation techniques, identifying and extracting the key physics from the calculations will present a huge computational challenge for years to come. Here we should emphasize that calculations of interacting binaries must be done in three spatial dimensions and time, so that even a single calculation that is adequately resolved requires the use of high performance computing facilities.

We have already noted that the plunge and merger phase of neutron stars in inspiral is characterized by a strong and dynamical gravitational field. At least heuristically, the dynamics is dominated by the bulk motion of the two stars, so that localized features in the matter—such as individual shocks, or small-scale turbulence—should have relatively little impact on the overall dynamics, or on the gravitational radiation which is produced.

We thus look for a matter model which can describe gravitationally compact objects, but where the equations of motion for the matter are easier to treat computationally than those for a relativistic fluid.


The studies in this thesis involve precisely such a matter model. Specifically, we adopt a single, massive, complex scalar field as a matter source.

Multigrid ulrich trottenberg pdf creator

The model admits localized, time-independent, gravitationally-bound configurations known as boson stars, which, through an appropriate choice of parameters, can be made compact. We can also set up initial data representing two stars, with subsequent evolution describing a variety of different kinds of encounters.

The equation of motion for the scalar field is simply the general relativistic wave equation, or Klein-Gordon equation, whose numerical solution using finite difference techniques is quite straightforward.

Multigrid ulrich trottenberg pdf creator

Although it is certainly possible to simulate boson stars in a fully general relativistic setting [6, 7, 8, 9, 10], we opt to study them within the context of a relatively simple approximation to general relativity 3which has previously been used to study strong-gravity effects in several scenarios of astrophysical interest, including the interaction of neutron stars. More specifically, and as will be discussed in detail in Chap. Although the literature has traditionally used the acronym CFC for the resulting approximation to Einsteinian gravity, we prefer to use CFA, for conformally flat approximation.

This stresses the fact that we are dealing with an approximation to general relativity. Also, we emphasize that although neither acronym makes explicit reference to the maximal slicing condition, that choice of time coordinatization has always been an essential ingredient of the approximation, and is so here.

Much more than documents.

The CFA is based on the heuristic assumption that the dynamical degrees of freedom of the gravitational field, i. The CFA effectively eliminates the two dynamical degrees of freedom present in general relativity, but still allows for investigation of the same kinds of phenomena ob- served in the full general relativistic case.

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These include the description of compact objects, the dynamics of their interaction, and black hole formation. It is also worth mentioning that it is still possible to study gravitational wave generation within this approximation via a perturbative mul- tipole expansion of the metric components. Briefly, the incorporation of radiation effects, although far from a trivial matter, can be realized through the introduction of a radiation reaction potential in the equations of motion for the matter model.

We also note that for spherically symmetric sys- tems the CFA is not an approximation, but can always be adopted through an appropriate choice of coordinates.

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Furthermore, use of the approximation for axially symmetric problems has indi- cated that the results obtained mimic those of general relativity quite accurately [11, 12, 13, 14]. There is thus considerable motivation to perform additional studies of strong-field gravity using this approach. In this prior work, Wilson and 4his collaborators presented evidence that, for a realistic neutron-star equation of state, general relativistic effects might cause the stars to individually collapse to black holes prior to merging.

Furthermore, they observed that at least for some set of initial orbital parameters, strong-field effects caused the last innermost stable circular orbit, or ISCO, to occur at a larger separation distance, and thus at lower frequency, than was previously estimated by post-Newtonian methods. This result had significance for the possible detection of gravitational waves, since it placed the frequency of radiation from coalescence closer to the maximum sensitivity range of current laser- interferometric detectors.

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However, the Wilson-Mathews compression effect was unexpected and controversial, and raised questions concerning the validity of the CFA. Subsequently, Flanagan [24] identified an inconsistency in the derivation of some of the equations of motion used in this study, and suggested that use of the correct equations would reduce the crushing effect. A revised version of simulations was published shortly thereafter [25]: a key claim resulting from this work was that the crushing effect was still present, although the magnitude of the observed effect was reduced relative to the previous calculations.

Further comparisons between a fully relativistic code and its CFA counterpart in the context of head-on collisions of neutron stars showed the presence of this effect and lent more credibility to the earlier calculations [26]. However, the most recent simulations [27] aimed at studying the possible crushing phenomenon actually indicate a decompressing effect on the neutron stars.

For a more complete review of the history of this controversy, as well as a possible explanation for the neutron star crushing effect, the reader should refer to the work by Favata [28] and references therein.

An ultimate goal of the work started in this thesis is to determine to what extent the CFA is a good approximation for the modelling of general compact binaries. From this point of view, it is particularly interesting to study the CFA within the context of a simpler matter model than that previously adopted, and this provides an important motivation for our use of boson stars. Favata [28] posited a mechanism that would tend to 5compress the neutron stars given a particular set of conditions.

Does that analysis apply to the binary boson star as well? Can we shed some light on the nature of the radiative degrees of freedom in the full theory from a detailed study of the differences between results obtained using the CFA and the full Einstein equations?

Can we compare the results to those obtained from other techniques, including fully general relativistic calculations? Do the results obtained match those seen in collisions of fluid stars, or can they at least be compared qualitatively?

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In order to answer these and related questions, the Einstein equations and ultimately the equations that result from adopting the CFA have to be cast in a form suitable for numerical computation. What follows is a brief description of our modelling procedure; more details will be provided in subsequent chapters. Specifically, the spacetime metric components are grouped into 4 kinematical components which encode the coordinate freedom of the theory the lapse function and three components of the shift vector , and 6 dynamical ones the components of a 3-dimensional metric induced in a constant- time hypersurface.

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Now, for any specific calculation, the coordinate system must be completely fixed by giving prescriptions for the lapse and shift. In particular, the time coordinate is fixed by a choice of the lapse function, and, as already mentioned, in this work we follow previous studies using the CFA and adopt so-called maximal slicing a very commonly adopted slicing condition in numerical relativity that was originally proposed by Lichnerowicz [33].

A key property of maximal slicing is that its use inhibits the focusing of the world lines of observers that move orthogonally to the hypersurfaces: such focusing can result in the development of coordinate singularities, and is also associated with the formation of physical singularities.

Another important point is that this coordinate choice generally results in a well-posed elliptic equation for the lapse function.

Multigrid ulrich trottenberg pdf creator

This reduces the number of independent components of the spatial metric from 6 to 1, and the single, non-trivial function that then defines the spatial metric is called the conformal factor.

Given the maximal slicing condition, it transpires that conformal flatness implies that all non-trivial components of the 4-metric are governed by elliptic equations.

Within the CFA, the dynamics of the gravitational field is completely determined by the dynam- ics of the matter source s , which in our case is a complex scalar field that satisfies the Klein-Gordon equation.

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We treat this system as an initial-boundary value problem and the thesis focuses on the development, testing and preliminary use of a code for its numerical solution. Our computational approach is based on finite differencing [34], and we use approximations that are second-order accurate in the grid spacing. A numerical evolution of our model starts with the specification of initial conditions for the complex scalar field.

Multigrid ulrich trottenberg pdf creator

This sets initial values representing a binary system.