scholarly journals Hyperbolic tetrad formulation of the Einstein equations for numerical relativity

2003 ◽  
Vol 67 (8) ◽  
Author(s):  
L. T. Buchman ◽  
J. M. Bardeen
Keyword(s):  
Numerical Relativity ◽  
Einstein Equations ◽  
Tetrad Formulation

Gravitation as a Composite Particle Effect in a Unified Spinor-Isospinor Preon Field Model I

1988 ◽  
Vol 43 (4) ◽  
pp. 345-359 ◽  
Author(s):  
H. Stumpf
Keyword(s):  
Field Model ◽  
Composite Particle ◽  
Low Energy ◽  
Einstein Equations ◽  
Energy Limit ◽  
Gauge Bosons ◽  
Mapping Procedure ◽  
Effective Interactions ◽  
Effective Dynamics ◽  
Tetrad Formulation

Abstract The model is defined by a selfregularizing nonlinear preon field equation, and all observable (elementary and non-elementary) particles are assumed to be bound (quantum) states of fermionic preon fields. Electroweak gauge bosons, leptons, quarks, gluons as preon composites and their effective dynamics etc. were studied in preceding papers. In this paper gravitons are introduced as four-preon composites and their effective interactions are discussed. This discussion is performed by the application of functional quantum theory to the model under consideration and subsequent evaluation of a weak mapping procedure, both introduced in preceding papers. In the low energy limit it is demonstrated that the effective graviton dynamics lead to the complete homogeneous Einstein equations in tetrad formulation.

scholarly journals Applied Mathematics of Space-time & Space+time: Problems in General Relativity and Cosmology

2021 ◽  
Author(s):  
◽  
Celine Cattoen
Keyword(s):  
Variational Formulation ◽  
Numerical Relativity ◽  
Cosmological Parameters ◽  
Spectral Element ◽  
Space Time ◽  
High Order ◽  
Einstein Equations ◽  
Element Discretization ◽  
Friedmann Equations ◽  
Hubble Flow

<p>Cosmography is the part of cosmology that proceeds by making minimal dynamic assumptions. That is, one does not assume the Friedmann equations (Einstein equations) unless and until absolutely necessary. On the other hand, cosmodynamics is the part of cosmology that relates the geometry to the density and pressure using the Friedmann equations. In both frameworks, we consider the amount of information and the nature of the constraints we can obtain from the Hubble flow in a FLRW universe. Indeed, the cosmological parameters contained in the Hubble relation between distance and redshift provide information on the behaviour of the universe (expansion, acceleration etc...). In the first framework, it is possible to concentrate more directly on the observational situation in a model-independent manner. We perform a number of inter-related cosmographic fits to supernova datasets, and pay particular attention to the extent to which the choice of distance scale and manner of representing the redshift scale affect the cosmological parameters. In the second framework, we use the class of w-parameter models which has become increasingly popular in the last decade. We explore the extent to which a constraint on the w-parameter leads to useful and non-trivial constraints on the Hubble flow in terms of cosmological parameters H(z), density p(z), density parameter O(z), distance scales d(z), and lookback time T(z). On another front, Numerical Relativity has experienced many breakthroughs since 2005, with full inspiral-merger-ringdown simulations now possible. One of the main goals is to provide very accurate templates of gravitational waves for ground-based and space-based interferometers. We explore the potential of a very recent and accurate numerical method, the Spectral Element Method (SEM), for Numerical Relativity, by treating a singular Schwarszchild black hole evolution as a test case. Spectral elements combine the theory of spectral and pseudo-spectral methods for high order polynomials and the variational formulation of finite elements and the associated geometric flexibility. We use the BSSN formulation of the Einstein equations with the method of the moving punctures. After applying the variational formulation to the BSSN system, we present several possible weak forms of this system and its spectral element discretization in space. We use a Runge-Kutta fourth order time discretization. The accuracy of high order methods can deteriorate in the presence of discontinuities or sharp gradients. We show that we can treat the element that contains the puncture with a filtering method to avoid artificial and spurious oscillations. These might form and propagate into the domain coming from discontinuous initial data from the BSSN system.</p>

Fundamentals of numerical relativity for gravitational wave sources

Science ◽  
2018 ◽  
Vol 361 (6400) ◽  
pp. 366-371 ◽  
Author(s):  
Bernd Brügmann
Keyword(s):  
General Relativity ◽  
Partial Differential Equations ◽  
Gravitational Wave ◽  
Tensor Field ◽  
Body Problem ◽  
Numerical Relativity ◽  
Gravitational Interaction ◽  
Einstein Equations ◽  
Two Body Problem ◽  
General Relativistic

Einstein’s theory of general relativity affords an enormously successful description of gravity. The theory encodes the gravitational interaction in the metric, a tensor field on spacetime that satisfies partial differential equations known as the Einstein equations. This review introduces some of the fundamental concepts of numerical relativity—solving the Einstein equations on the computer—in simple terms. As a primary example, we consider the solution of the general relativistic two-body problem, which features prominently in the new field of gravitational wave astronomy.

scholarly journals THE BSSN FORMULATION IS A PARTIALLY CONSTRAINED EVOLUTION SYSTEM

2010 ◽  
Vol 19 (02) ◽  
pp. 153-158
Author(s):  
ADRIAN P. GENTLE
Keyword(s):  
Numerical Relativity ◽  
Einstein Equations ◽  
Evolution System ◽  
Connection Coefficients ◽  
Constrained Evolution ◽  
Conformal Connection

It is shown that the BSSN formulation of the Einstein equations, which forms the basis of most simulations in numerical relativity, explicitly uses the momentum constraints to evolve the conformal connection coefficients.

NUMERICAL RELATIVITY

1994 ◽  
Vol 05 (02) ◽  
pp. 181-187 ◽  
Author(s):  
Edward Seidel ◽  
Wai-Mo Suen
Keyword(s):  
Initial Conditions ◽  
Numerical Relativity ◽  
Numerical Study ◽  
Dimensional Space ◽  
Numerical Data ◽  
Einstein Equations ◽  
3 Dimensional ◽  
Mathematical Properties ◽  
Dimensional Spacetime ◽  
Intrinsic Complexity

The present status of numerical relativity is reviewed. There are five closely interconnected aspects of numerical relativity: (1) Formulation. The general covariant Einstein equations are reformulated in a way suitable for numerical study by separating the 4-dimensional spacetime into a 3-dimensional space evolving in time. (2) Techniques. A set of tools is developed for determining gauge choices, setting boundary and initial conditions, handling spacetime singularities, etc. As required by the special physical and mathematical properties of general relativity, such techniques are indispensable for the numerical evolutions of spacetime. (3) Coding. The optimal use of parallel processing is crucial for many problems in numerical relativity, due to the intrinsic complexity of the theory. (4) Visualization. Numerical relativity is about the evolutions of 3-dimensional geometric structures. There are special demands on visualization. (5) Interpretation and Understanding. The integration of numerical data in relativity into a consistent physical picture is complicated by gauge and coordinate degrees of freedoms and other difficulties. We give a brief overview of the progress made in these areas.

NEW METHOD FOR CHARACTERISTIC EVOLUTIONS IN NUMERICAL RELATIVITY

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
ZHOUJIAN CAO
Keyword(s):  
Cauchy Problem ◽  
Finite Difference ◽  
Differential Equations ◽  
Numerical Relativity ◽  
Einstein Equations ◽  
New Method ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Special Difference

The main task of numerical relativity is to solve Einstein equations with the aid of supercomputer. There are two main schemes to write Einstein equations explicitly as differential equations. One is based on 3 + 1 decomposition and reduces the Einstein equations to a Cauchy problem. The another takes the advantage of the characteristic property of Einstein equations and reduces them to a set of ordinary differential equations. The latter scheme is called characteristic formalism which is free of constraint equations in contrast to the corresponding Cauchy problem. Till now there is only one well developed code (PITT code) for characteristic formalism. In PITT code, special finite difference algorithm is adopted for the numerical calculation. And it is this special difference algorithm that restricts the numerical accuracy order to second-order. In addition, this special difference algorithm makes the popular Runge–Kutta method used in Cauchy problem not available. In this paper, we modify the equations for characteristic formalism. Based on our new set of equations, we can use usual finite difference method as done in usual Cauchy evolution. And Runge–Kutta method can also be adopted naturally. We develop a set of code in the framework of AMSS-NCKU code based on our new method and some numerical tests are done.

scholarly journals Applied Mathematics of Space-time & Space+time: Problems in General Relativity and Cosmology

2021 ◽  
Author(s):  
◽  
Celine Cattoen
Keyword(s):  
Variational Formulation ◽  
Numerical Relativity ◽  
Cosmological Parameters ◽  
Spectral Element ◽  
Space Time ◽  
High Order ◽  
Einstein Equations ◽  
Element Discretization ◽  
Friedmann Equations ◽  
Hubble Flow

<p>Cosmography is the part of cosmology that proceeds by making minimal dynamic assumptions. That is, one does not assume the Friedmann equations (Einstein equations) unless and until absolutely necessary. On the other hand, cosmodynamics is the part of cosmology that relates the geometry to the density and pressure using the Friedmann equations. In both frameworks, we consider the amount of information and the nature of the constraints we can obtain from the Hubble flow in a FLRW universe. Indeed, the cosmological parameters contained in the Hubble relation between distance and redshift provide information on the behaviour of the universe (expansion, acceleration etc...). In the first framework, it is possible to concentrate more directly on the observational situation in a model-independent manner. We perform a number of inter-related cosmographic fits to supernova datasets, and pay particular attention to the extent to which the choice of distance scale and manner of representing the redshift scale affect the cosmological parameters. In the second framework, we use the class of w-parameter models which has become increasingly popular in the last decade. We explore the extent to which a constraint on the w-parameter leads to useful and non-trivial constraints on the Hubble flow in terms of cosmological parameters H(z), density p(z), density parameter O(z), distance scales d(z), and lookback time T(z). On another front, Numerical Relativity has experienced many breakthroughs since 2005, with full inspiral-merger-ringdown simulations now possible. One of the main goals is to provide very accurate templates of gravitational waves for ground-based and space-based interferometers. We explore the potential of a very recent and accurate numerical method, the Spectral Element Method (SEM), for Numerical Relativity, by treating a singular Schwarszchild black hole evolution as a test case. Spectral elements combine the theory of spectral and pseudo-spectral methods for high order polynomials and the variational formulation of finite elements and the associated geometric flexibility. We use the BSSN formulation of the Einstein equations with the method of the moving punctures. After applying the variational formulation to the BSSN system, we present several possible weak forms of this system and its spectral element discretization in space. We use a Runge-Kutta fourth order time discretization. The accuracy of high order methods can deteriorate in the presence of discontinuities or sharp gradients. We show that we can treat the element that contains the puncture with a filtering method to avoid artificial and spurious oscillations. These might form and propagate into the domain coming from discontinuous initial data from the BSSN system.</p>

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