DOCSLIB.ORG

Introduction to 3+1 and BSSN Formalism

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to 3+1 and BSSN Formalism Introduction to 3+1 and BSSN formalism Yuichiro Sekiguchi (YITP) 1 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity Solving the Setting ‘realistic’ or constraint ‘physically motivated’ equations initial conditions Main loop GR-HD GR-MHD Solving Einstein’s equations GR-Rad(M)HD Solving gauge Solving source conditions filed equations Microphysics • EOS • weak processes Locating BH Extracting GWs (solving AH finder ) BH excision 2 APCTP International school on NR and GW July 28-August 3, 2011 Scope of this lecture Solving the Setting ‘realistic’ or constraint ‘physically motivated’ equations initial conditions Main loop GR-HD GR-MHD Solving Einstein’s equations GR-Rad(M)HD Solving gauge Solving source conditions filed equations Microphysics • EOS • weak processes Locating BH Extracting GWs (solving AH finder ) BH excision 3 APCTP International school on NR and GW July 28-August 3, 2011 The Goal The main goal that we are aiming at: “To Derive Einstein’s equations in BSSN formalism” 4 APCTP International school on NR and GW July 28-August 3, 2011 Notation and Convention The signature of the metric: ( - + + + ) (We will use the abstract index notation) e.g. Wald (1984); see also Penrose, R. and Rindler, W. Spinors and spacetime vol.1, Cambridge Univ. Press (1987) Geometrical unit c=G=1 symmetric and anti-symmetric notations 1 1 T T , T sgn( )T (a1...an ) a ()...a ( n ) [a1...an ] a ()...a ( n ) n! n! 1 1 T (T +T ), T (T -T ) (ab) 2 ab ba [ab] 2 ab ba 5 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity Solving the Setting ‘realistic’ or constraint ‘physically motivated’ equations initial conditions Main loop GR-HD GR-MHD Solving Einstein’s equations GR-Rad(M)HD Solving gauge Solving source conditions filed equations Microphysics • EOS • weak processes Locating BH Extracting GWs (solving AH finder ) BH excision 6 APCTP International school on NR and GW July 28-August 3, 2011 Solving Einstein’s equations on computers Einstein’s equations in full covariant form are a set of coupled partial differential equations The solution, metric gab, is not a dynamical object and represents the full geometry of the spacetime just as the metric of a two-sphere does To reveal the dynamical nature of Einstein’s equations, we must break the 4D covariance and exploit the special nature of time One method is 3+1 decomposition in which spacetime manifold and its geometry (graviational fields) are divided into a sequence of ‘instants’ of time Then, Einstein’s equations are posed as a Cauchy problem which can be solved numerically on computers 7 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of spacetime manifold Let us start to introduce foliation or slicing in the spacetime manifold M Foliation {S } of M is a family of slices (spacelike hypersurfaces) which do not intersect each other and fill the whole of M In a globally hyperbolic spacetime, each S is a Cauchy surface which is parameterized by a global time function, t , as St a at Foliation is characterized by the gradient one-form a at , [ab] 0 8 APCTP International school on NR and GW July 28-August 3, 2011 The lapse function The norm of a is related to a function called “lapse function”, a(xa) , as : ab ab 1 g ab g atbt - 2 a As we shall see later, the lapse function characterize the proper time between the slices Also let us introduce the normalized one-form : ab na -a a , g nanb -1 the negative sign is introduced so that the direction of n corresponds to the direction to which t increases na is the unit normal vector to S 9 APCTP International school on NR and GW July 28-August 3, 2011 The spatial metric of S : gab The spatial metric gab induced by gab onto S is defined by g g + n n ab ab a b g ab g acg bdg g ab + nanb cd Using this ‘induced’ metric, a tensor on M is decomposed into two parts: components tangent and normal to S The tangent-projection operator is defined as a a a b b + n nb a b a a The normal-projection operator is b -n na b - b Then, projection of a tensor into S is defined by a1...ar a1 ar b1 bs c1...cr T b ...b ... ... T d ...d 1 s c1 cr d1 ds 1 s It is easy to check c d gab a b gcd g ab 10 APCTP International school on NR and GW July 28-August 3, 2011 Covariant derivative associated with gab Covariant derivative acting on spatial tensors is defined by a1...ar a1...ar D T b1...bs T b1...bs e e f a1 ar b1 bs c1...cr ... ... T d ...d e c1 cr d1 ds f 1 s The covariant derivative must satisfy the following conditions It is a linear operator : (obviously holds from linearity of ∇) Torsion free : DaDb f = DbDa f , (easy to check by direct calculation) Compatible with the metric : Dc gab=0 , (easy to check also) Leibnitz’s rule holds : c b c b d c b d c Da (v wc ) a b (v wc ) a v d bwc + a wdc bv b d c c b d d c a v (d +d )bwc + a wd (c +c )bv c d b c d d c v Da wc + wd Dav + a (d v bwc + c wd bv ) c d c d d v Da wc + wd Dav for d v c wd 0 11 APCTP International school on NR and GW July 28-August 3, 2011 Intrinsic and extrinsic curvature for S The Riemann tensor for the slice S is defined by d (Da Db - DbDa ) vc vd R abc An other curvature tensor, the extrinsic curvature for S is defined by a Kab - (anb) , n : unit normal to S extrinsic curvature provides information on how much the normal direction changes and hence, how S is curved parallel the antisymmetric part vanish transported b n -K x due to Frobenius’s theorem : a ab a “For unit normal n to a slice a b n (x ) a b b n [ a b n c ] 0 ” n (x + x ) b a x 0 3(n n[abnc] ) - [bnc] b b S x + x 12 APCTP International school on NR and GW July 28-August 3, 2011 The other expressions of Kab First, note that c d c c d d anb ab cnd (a +a )(b +b )cnd anb - na ab - Kab - naab where ab is the acceleration of nb which is purely spatial b a b a b n ab n n anb n a (nbn ) 0 Because the extrinsic curvature is symmetric, we have 1 1 K - n - n a - L (g + n n ) - L g ab (a b) (a b) 2 n ab a b 2 n ab where L n is the Lie derivative with respect to n 1 Also, we simply have K - n - L g ab (a b) 2 n ab Thus the extrinsic curvature is related to the “velocity” of the spatial metric gab 13 APCTP International school on NR and GW July 28-August 3, 2011 3+1 decomposition of 4D Riemann tensor Geometry of a slice S is described by gab and Kab gab and Kab represent the “instantaneous” gravitational fields in S In order that the foliation {S} to “fits” the spacetime manifold, gab and Kab must satisfy certain conditions known as Gauss, Codazzi, and Ricci relations They are related to 3+1 decomposition g 4 of Einstein’s equations ab Rabcd These equations are obtained by K Rabcd ab taking the projections of the g ab 4D Riemann tensor 4 4 d Rabcd , Rabcdn , 4 b d Rabcdn n 14 APCTP International school on NR and GW July 28-August 3, 2011 Gauss relation: spatial projection to S Let us calculate the spatial Riemann tensor D D w ( w ) a b c a b c w + ( w )( d ) + ( w )( d ) a b c b d a c d c a b d d d abwc - KacKb w - Kabn d wc where we used (also note that n vanishes if n is uncontracted) d d d d d d a b na (bn ) + n (anb ) -nb (Ka + naa ) - n (Kab + naab ) Then we obtain the Gauss relation d 4 d d d (Da Db - Db Da )wc Rabc wd Rabc wd - KacKb wd + KbcKa wd 4 Rabcd Rabcd + KacKbd - KadKbc The contracted Gauss relations are 4 4 b d b Rac+ Rabcdn n Rac + KKac - KabKc 4 4 a b 2 ab R +2 Rabn n R + K - KabK 15 APCTP International school on NR and GW July 28-August 3, 2011 Codazzi relation : mixed projection to S and n Next, let us consider the “mixed” projection 4 d Rabc nd (abnc -banc ) where the right hand side is calculated as abnc a (-Kbc - nbac ) -Da Kbc- acanb -Da Kbc + ac Kab Then we obtain the Codazzi relation 4 d Rabc nd DbKac - Da Kbc The contracted Codazzi relation is 4 b b Rabn Da K - DbKa 16 APCTP International school on NR and GW July 28-August 3, 2011 Gauss and Codazzi relations Note that the Gauss and Codazzi relations depend only on the spatial metric gab, the extrinsic curvature Kab, and their spatial derivatives This implies that the Gauss-Codazzi relations represent integrability conditions that gab and Kab must satisfy for any slice to be embedded in the spacetime manifold The Gauss-Codazzi relations are directly associated with the constraint equations of Einstein’s equation 17 APCTP International school on NR and GW July 28-August 3, 2011 Ricci relation (1) Let us start from the following equation 4 b d b Rabcdn n n (abnc -banc ) b n [-a (Kbc + nbac ) + b (Kac + naac )] b b b [Kbcan + aac + n b Kac + acaa + nan bac ] [K (-K b - n ab ) + a + nb K + a a + n nb a ] bc a a a c b ac c a a b c b b - KbcKa + Daac + n b Kac + acaa The Lie derivative of Kac is L K (nb K + K nb + K nb ) nb K - K K b - K K b n ac b ac ab c cb a b ac ab c cb a Then we obtain the Ricci relation 4 b d b Rabcdn n Ln Kac + KabKc + Daac + acaa 18 APCTP International school on NR and GW July 28-August 3, 2011 Ricci relation (2) Note that the Lie derivative of Kab is purely spatial, as a a c a c a c a c n Ln Kab n n c Kab + n Kacbn + n Kbcan -Kaba + Kbca 0 Thus the Ricci relation is 4 b d b Rabcdn n Ln Kac + KabKc + Daac + acaa 19 APCTP International school on NR and GW July 28-August 3, 2011 Ricci relation (3) The acceleration ab is related to the lapse function a, as c c c c a n n 2n n -2n a -n ( a - a) b c b [c b] [c b] b c c b ncn ln a +
Load more

Recommended publications
  • 3+1 Formalism and Bases of Numerical Relativity
    3+1 Formalism and Bases of Numerical Relativity Lecture notes Eric´ Gourgoulhon Laboratoire Univers et Th´eories, UMR 8102 du C.N.R.S., Observatoire de Paris, Universit´eParis 7 arXiv:gr-qc/0703035v1 6 Mar 2007 F-92195 Meudon Cedex, France [email protected] 6 March 2007 2 Contents 1 Introduction 11 2 Geometry of hypersurfaces 15 2.1 Introduction.................................... 15 2.2 Frameworkandnotations . .... 15 2.2.1 Spacetimeandtensorfields . 15 2.2.2 Scalar products and metric duality . ...... 16 2.2.3 Curvaturetensor ............................... 18 2.3 Hypersurfaceembeddedinspacetime . ........ 19 2.3.1 Definition .................................... 19 2.3.2 Normalvector ................................. 21 2.3.3 Intrinsiccurvature . 22 2.3.4 Extrinsiccurvature. 23 2.3.5 Examples: surfaces embedded in the Euclidean space R3 .......... 24 2.4 Spacelikehypersurface . ...... 28 2.4.1 Theorthogonalprojector . 29 2.4.2 Relation between K and n ......................... 31 ∇ 2.4.3 Links between the and D connections. .. .. .. .. .. 32 ∇ 2.5 Gauss-Codazzirelations . ...... 34 2.5.1 Gaussrelation ................................. 34 2.5.2 Codazzirelation ............................... 36 3 Geometry of foliations 39 3.1 Introduction.................................... 39 3.2 Globally hyperbolic spacetimes and foliations . ............. 39 3.2.1 Globally hyperbolic spacetimes . ...... 39 3.2.2 Definition of a foliation . 40 3.3 Foliationkinematics .. .. .. .. .. .. .. .. ..... 41 3.3.1 Lapsefunction ................................. 41 3.3.2 Normal evolution vector . 42 3.3.3 Eulerianobservers ............................. 42 3.3.4 Gradients of n and m ............................. 44 3.3.5 Evolution of the 3-metric . 45 4 CONTENTS 3.3.6 Evolution of the orthogonal projector . ....... 46 3.4 Last part of the 3+1 decomposition of the Riemann tensor .
    [Show full text]
  • Graduate Student Research Day 2010 Florida Atlantic University
    Graduate Student Research Day 2010 Florida Atlantic University CHARLES E. SCHMIDT COLLEGE OF SCIENCE Improvements to Moving Puncture Initial Data and Analysis of Gravitational Waveforms using the BSSN Formalism on the BAM Code George Reifenberger Charles E. Schmidt College of Science, Florida Atlantic University Faculty Advisor: Dr. Wolfgang Tichy Numerical relativity has only recently been able to simulate the creation of gravitational wave signatures through techniques concerning both the natural complexities of general relativity and the computational requirements needed to obtain significant accuracies. These gravitational wave signatures will be used as templates for detection. The need to develop better results has divided the field into specific areas of code development, one of which involves improvement on the construction of binary black hole initial data. Gravitational waves are a prediction of general relativity, however none have currently been observed experimentally. The search for gravitational wave signatures from binary black hole mergers has required many sub-disciplines of general relativity to contribute to the effort; numerical relativity being one of these. Assuming speeds much smaller than the speed of light Post-Newtonian approximations are introduced into initial data schemes. Evolution of this initial data using the BSSN formalism has approximately satisfied constraint equations of General Relativity to produce probable gravitational wave signatures. Analysis of data convergence and behavior of physical quantities shows appropriate behavior of the simulation and allows for the wave signatures to be considered for template inclusion. High order convergence of our data validates the accuracy of our code, and the wave signatures produced can be used as an improved foundation for more physical models.
    [Show full text]
  • Status of Numerical Relativity: from My Personal Point of View Masaru Shibata (U
    Status of Numerical Relativity: From my personal point of view Masaru Shibata (U. Tokyo) 1 Introduction 2 General issues in numerical relativity 3 Current status of implementation 4 Some of our latest numerical results: NS-NS merger & Stellar core collapse 5 Summary & perspective 1: Introduction: Roles in NR A To predict gravitational waveforms: Two types of gravitational-wave detectors work now or soon. h Space Interferometer Ground Interferometer LISA LIGO/VIRGO/ Physics of GEO/TAMA SMBH BH-BH SMBH/SMBH NS-NS SN Frequency 0.1mHz 0.1Hz 10Hz 1kHz Templates (for compact binaries, core collapse, etc) should be prepared B To simulate Astrophysical Phenomena e.g. Central engine of GRBs = Stellar-mass black hole + disks (Probably) NS-NS merger BH-NS merger ? Best theoretical approach Stellar collapse of = Simulation in GR rapidly rotating star C To discover new phenomena in GR In the past 20 years, community has discovered e.g., 1: Critical phenomena (Choptuik) 2: Toroidal black hole (Shapiro-Teukolsky) 3: Naked singularity formation (Nakamura, S-T) GR phenomena to be simulated ASAP ・ NS-NS / BH-NS /BH-BH mergers (Promising GW sources/GRB) ・ Stellar collapse of massive star to a NS/BH (Promising GW sources/GRB) ・ Nonaxisymmetric dynamical instabilities of rotating NSs (Promising GW sources) ・ …. In general, 3D simulations are necessary 2 Issues: Necessary elements for GR simulations • Einstein’s evolution equations solver • GR Hydrodynamic equations solver • Appropriate gauge conditions (coordinate conditions) • Realistic initial conditions
    [Show full text]
  • Following the Trails of Light in Curved Spacetime
    Following the trails of light in curved spacetime by Darius Bunandar THESIS Presented in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Physics The University of Texas at Austin Austin, Texas May 2013 The Thesis Committee for Darius Bunandar certifies that this is the approved version of the following thesis: Following the trails of light in curved spacetime Committee: Richard A. Matzner, Supervisor Philip J. Morrison Greg O. Sitz © 2013 - D B A . Following the trails of light in curved spacetime Darius Bunandar The University of Texas at Austin, 2013 Supervisor: Richard A. Matzner Light has played an instrumental role in the initial development of the theory of relativity. In this thesis, we intend to explore other physical phe- nomena that can be explained by tracing the path that light takes in curved spacetime. We consider the null geodesic and eikonal equations that are equivalent descriptions of the propagation of light rays within the frame- work of numerical relativity. We find that they are suited for different phys- ical situations. The null geodesic equation is more suited for tracing the path of individual light rays. We solve this equation in order to visualize images of the sky that have been severely distorted by one or more black holes, the so-called gravitational lensing effect. We demonstrate that our procedure of solving the null geodesic equation is sufficiently robust to produce some of the world’s first images of the lensing effect from fully dynamical binary black hole coalescence. The second formulation of propagation of light that we explore is the eikonal surface equation.
    [Show full text]
  • Aspectos De Relatividade Numérica Campos Escalares E Estrelas De Nêutrons
    UNIVERSIDADE DE SÃO PAULO INSTITUTO DE FÍSICA Aspectos de Relatividade Numérica Campos escalares e estrelas de nêutrons Leonardo Rosa Werneck Orientador: Prof. Dr. Elcio Abdalla Uma tese apresentada ao Instituto de Física da Universidade de São Paulo como parte dos requisitos necessários para obter o título de doutor em Física. Banca examinadora: Prof. Dr. Elcio Abdalla (IF-USP) – Presidente da banca Prof. Dr. Arnaldo Gammal (IF-USP) Prof. Dr. Daniel A. Turolla Vanzela (IFSC-USP) Prof. Dr. Alberto V. Saa (IFGW-UNICAMP) Profa. Dra. Cecilia Bertoni Chirenti (UFABC/UMD/NASA) São Paulo 2020 FICHA CATALOGRÁFICA Preparada pelo Serviço de Biblioteca e Informação do Instituto de Física da Universidade de São Paulo Werneck, Leonardo Rosa Aspectos de relatividade numérica: campos escalares e estrelas de nêutrons / Aspects of Numerical Relativity: scalar fields and neutron stars. São Paulo, 2020. Tese (Doutorado) − Universidade de São Paulo. Instituto de Física. Depto. Física Geral. Orientador: Prof. Dr. Elcio Abdalla Área de Concentração: Relatividade e Gravitação Unitermos: 1. Relatividade numérica; 2. Campo escalar; 3. Fenômeno crítico; 4. Estrelas de nêutrons; 5. Equações diferenciais parciais. USP/IF/SBI-057/2020 UNIVERSITY OF SÃO PAULO INSTITUTE OF PHYSICS Aspects of Numerical Relativity Scalar fields and neutron stars Leonardo Rosa Werneck Advisor: Prof. Dr. Elcio Abdalla A thesis submitted to the Institute of Physics of the University of São Paulo in partial fulfillment of the requirements for the title of Doctor of Philosophy in Physics. Examination committee: Prof. Dr. Elcio Abdalla (IF-USP) – Committee president Prof. Dr. Arnaldo Gammal (IF-USP) Prof. Dr. Daniel A. Turolla Vanzela (IFSC-USP) Prof.
    [Show full text]
  • Trumpet Initial Data for Highly Boosted Black Holes and High Energy Binaries
    TRUMPET INITIAL DATA FOR HIGHLY BOOSTED BLACK HOLES AND HIGH ENERGY BINARIES Kyle Patrick Slinker A dissertation submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Physics and Astronomy. Chapel Hill 2017 Approved by: Charles R. Evans J. Christopher Clemens Louise A. Dolan Joaqu´ınE. Drut Fabian Heitsch Reyco Henning c 2017 Kyle Patrick Slinker ALL RIGHTS RESERVED ii ABSTRACT Kyle Patrick Slinker: Trumpet Initial Data for Highly Boosted Black Holes and High Energy Binaries (Under the direction of Charles R. Evans) Initial data for a single boosted black hole is constructed that analytically contains no initial transient (junk) gravitational radiation and is adapted to the moving punctures gauge conditions. The properties of this data are investigated in detail. It is found to be generally superior to canonical Bowen-York data and, when implemented numerically in simulations, yields orders of magnitude less junk gravitational radiation content and more accurate black hole velocities. This allows for modeling of black holes that are boosted faster than previously possible. An approximate superposition of the data is used to demonstrate how a binary black hole system can be constructed to retain the advantages found for the single black hole. Extensions to black holes with spin are considered. iii TABLE OF CONTENTS LIST OF TABLES ............................................. vii LIST OF FIGURES ............................................ viii LIST OF ABBREVIATIONS AND SYMBOLS ........................... xii 1 Introduction ............................................... 1 1.1 Testing General Relativity Through Gravitational Wave Astronomy . 1 1.2 Numerical Relativity . 2 1.3 Project Goals .
    [Show full text]
  • Mathematics of General Relativity - Wikipedia, the Free Encyclopedia Page 1 of 11
    Mathematics of general relativity - Wikipedia, the free encyclopedia Page 1 of 11 Mathematics of general relativity From Wikipedia, the free encyclopedia The mathematics of general relativity refers to various mathematical structures and General relativity techniques that are used in studying and formulating Albert Einstein's theory of general Introduction relativity. The main tools used in this geometrical theory of gravitation are tensor fields Mathematical formulation defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity. Resources Fundamental concepts Note: General relativity articles using tensors will use the abstract index Special relativity notation . Equivalence principle World line · Riemannian Contents geometry Phenomena 1 Why tensors? 2 Spacetime as a manifold Kepler problem · Lenses · 2.1 Local versus global structure Waves 3 Tensors in GR Frame-dragging · Geodetic 3.1 Symmetric and antisymmetric tensors effect 3.2 The metric tensor Event horizon · Singularity 3.3 Invariants Black hole 3.4 Tensor classifications Equations 4 Tensor fields in GR 5 Tensorial derivatives Linearized Gravity 5.1 Affine connections Post-Newtonian formalism 5.2 The covariant derivative Einstein field equations 5.3 The Lie derivative Friedmann equations 6 The Riemann curvature tensor ADM formalism 7 The energy-momentum tensor BSSN formalism 7.1 Energy conservation Advanced theories 8 The Einstein field equations 9 The geodesic equations Kaluza–Klein
    [Show full text]
  • Ray Tracing and Hubble Diagrams in Post-Newtonian Cosmology
    Ray tracing and Hubble diagrams in post-Newtonian cosmology Viraj A A Sanghai,a Pierre Fleury,b Timothy Cliftona aSchool of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS, United Kingdom bD´epartment de Physique Th´eorique, Universit´ede Gen`eve, 24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland E-mail: [email protected], pierre.fl[email protected], [email protected] Abstract. On small scales the observable Universe is highly inhomogeneous, with galaxies and clusters forming a complex web of voids and filaments. The optical properties of such configurations can be quite different from the perfectly smooth Friedmann-Lema^ıtre-Robertson- Walker (FLRW) solutions that are frequently used in cosmology, and must be well understood if we are to make precise inferences about fundamental physics from cosmological observations. We investigate this problem by calculating redshifts and luminosity distances within a class of cosmological models that are constructed explicitly in order to allow for large density contrasts on small scales. Our study of optics is then achieved by propagating one hundred thousand null geodesics through such space-times, with matter arranged in either compact opaque objects or diffuse transparent haloes. We find that in the absence of opaque objects, the mean of our ray tracing results faithfully reproduces the expectations from FLRW cosmology. When opaque objects with sizes similar to those of galactic bulges are introduced, however, we find that the mean of distance measures can be shifted up from FLRW predictions by as much as 10%.
    [Show full text]
  • Introduction to 3+1 and BSSN Formalism
    Introduction to 3+1 and BSSN formalism Yuichiro Sekiguchi (YITP) 1 APCTP International school on NR and GW July 28-August 3, 2011 Overview of numerical relativity Solving the Setting „realistic‟ or constraint „physically motivated‟ equations initial conditions Main loop GR-HD GR-MHD Solving Einstein‟s equations GR-Rad(M)HD Solving gauge Solving source conditions filed equations Microphysics • EOS • weak processes Locating BH Extracting GWs (solving AH finder ) BH excision 2 APCTP International school on NR and GW July 28-August 3, 2011 Scope of this lecture Solving the Setting „realistic‟ or constraint „physically motivated‟ equations initial conditions Main loop GR-HD GR-MHD Solving Einstein‟s equations GR-Rad(M)HD Solving gauge Solving source conditions filed equations Microphysics • EOS • weak processes Locating BH Extracting GWs (solving AH finder ) BH excision 3 APCTP International school on NR and GW July 28-August 3, 2011 The Goal The main goal that we are aiming at: “To Derive Einstein‟s equations in BSSN formalism” 4 APCTP International school on NR and GW July 28-August 3, 2011 Notation and Convention The signature of the metric: ( - + + + ) (We will use the abstract index notation) e.g. Wald (1984); see also Penrose, R. and Rindler, W. Spinors and spacetime vol.1, Cambridge Univ. Press (1987) Geometrical unit c=G=1 symmetric and anti-symmetric notations 1 1 T T , T sgn( )T (a1...an ) a (1)...a ( n ) [a1...an ] a (1)...a ( n ) n! n! 1 1 T (T +T ), T (T -T ) (ab) 2 ab ba [ab] 2 ab ba 5 APCTP International school
    [Show full text]
  • Ray Tracing and Hubble Diagrams in Post-Newtonian Cosmology
    Ray tracing and Hubble diagrams in post-Newtonian cosmology Viraj A A Sanghai,a Pierre Fleury,b Timothy Cliftona aSchool of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS, United Kingdom bD´epartment de Physique Th´eorique, Universit´ede Gen`eve, 24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland E-mail: [email protected], pierre.fl[email protected], [email protected] Abstract. On small scales the observable Universe is highly inhomogeneous, with galaxies and clusters forming a complex web of voids and filaments. The optical properties of such configurations can be quite different from the perfectly smooth Friedmann-Lema^ıtre-Robertson- Walker (FLRW) solutions that are frequently used in cosmology, and must be well understood if we are to make precise inferences about fundamental physics from cosmological observations. We investigate this problem by calculating redshifts and luminosity distances within a class of cosmological models that are constructed explicitly in order to allow for large density contrasts on small scales. Our study of optics is then achieved by propagating one hundred thousand null geodesics through such space-times, with matter arranged in either compact opaque objects or diffuse transparent haloes. We find that in the absence of opaque objects, the mean of our ray tracing results faithfully reproduces the expectations from FLRW cosmology. When opaque objects with sizes similar to those of galactic bulges are introduced, however, we find that the mean of distance measures can be shifted up from FLRW predictions by as much as 10%.
    [Show full text]
  • Binary Neutron Star Mergers
    Living Rev. Relativity, 15, (2012), 8 LIVINGREVIEWS http://www.livingreviews.org/lrr-2012-8 in relativity Binary Neutron Star Mergers Joshua A. Faber Center for Computational Relativity and Gravitation and School of Mathematical Sciences Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester NY 14623, USA email: [email protected] http://ccrg.rit.edu/people/faber Frederic A. Rasio Center for Interdisciplinary Exploration and Research in Astrophysics, and Department of Physics & Astronomy, Northwestern University, 2145 Sheridan Road, Evanston IL 60208, USA email: [email protected] http://ciera.northwestern.edu/rasio/ Accepted on 22 May 2012 Published on 4 July 2012 Abstract We review the current status of studies of the coalescence of binary neutron star systems. We begin with a discussion of the formation channels of merging binaries and we discuss the most recent theoretical predictions for merger rates. Next, we turn to the quasi-equilibrium formalisms that are used to study binaries prior to the merger phase and to generate initial data for fully dynamical simulations. The quasi-equilibrium approximation has played a key role in developing our understanding of the physics of binary coalescence and, in particular, of the orbital instability processes that can drive binaries to merger at the end of their life- times. We then turn to the numerical techniques used in dynamical simulations, including relativistic formalisms, (magneto-)hydrodynamics, gravitational-wave extraction techniques, and nuclear microphysics treatments. This is followed by a summary of the simulations per- formed across the field to date, including the most recent results from both fully relativistic and microphysically detailed simulations.
    [Show full text]
  • Gravitational Waves from Merging Compact Binaries
    Gravitational waves from merging compact binaries The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Hughes, Scott A. “Gravitational Waves from Merging Compact Binaries.” Annu. Rev. Astro. Astrophys. 47.1 (2011): 107-157. As Published http://dx.doi.org/10.1146/annurev-astro-082708-101711 Publisher Annual Reviews Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/60688 Terms of Use Attribution-Noncommercial-Share Alike 3.0 Unported Detailed Terms http://creativecommons.org/licenses/by-nc-sa/3.0/ Compact binary gravitational waves 1 Gravitational waves from merging compact binaries Scott A. Hughes Department of Physics and MIT Kavli Institute, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 Key Words gravitational waves, compact objects, relativistic binaries Abstract Largely motivated by the development of highly sensitive gravitational-wave detec- tors, our understanding of merging compact binaries and the gravitational waves they generate has improved dramatically in recent years. Breakthroughs in numerical relativity now allow us to model the coalescence of two black holes with no approximations or simplifications. There has also been outstanding progress in our analytical understanding of binaries. We review these developments, examining merging binaries using black hole perturbation theory, post-Newtonian expansions, and direct numerical integration of the field equations. We summarize these ap- arXiv:0903.4877v3 [astro-ph.CO] 18 Apr 2009 proaches and what they have taught us about gravitational waves from compact binaries. We place these results in the context of gravitational-wave generating systems, analyzing the impact gravitational wave emission has on their sources, as well as what we can learn about them from direct gravitational-wave measurements.
    [Show full text]