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Showing papers on "Introduction to the mathematics of general relativity published in 2005"


Book
01 Jan 2005
TL;DR: In this article, the authors discuss the history of the Lorentz transformations in special relativity and quantum theory, including the trailblazers of Albert Keinstein and his followers.
Abstract: 1 Overview 2 The physics of coordinate transformations 3 The relativity principle and the fable of Albert Keinstein 4 The trailblazers 5 Einstein's principle-theory route to the Lorentz transformations 6 Variations on the Einstein theme 7 Unconventional voices on special relativity 8 What is special relativity? 9 The view from general relativity APPENDICES A Einstein on general covariance B Special relativity and quantum theory

345 citations



Journal ArticleDOI
TL;DR: In this paper, the unimodular theory of gravitation is compared with general relativity in the quadratic (Fierz-Pauli) regime, using a quite broad framework, and it is argued that quantum effects allow in principle to discriminate between both theories.
Abstract: The so called unimodular theory of gravitation is compared with general relativity in the quadratic (Fierz-Pauli) regime, using a quite broad framework, and it is argued that quantum effects allow in principle to discriminate between both theories.

133 citations


Journal ArticleDOI
TL;DR: Rindler as discussed by the authors explores in depth first special relativity, then general relativity, and finally relativistic cosmology, as well as the fundamental underlying ideas and principles that so successfully guided Einstein in his work.
Abstract: Wolfgang Rindler is known as a writer of exceptional clarity. This quality is evident in this book as it explores in depth first special relativity, then general relativity, and finally relativistic cosmology. He bases his writing in the fundamental underlying ideas and principles that so successfully guided Einstein in his work, clarifying their nature and implications in an illuminating way with many examples. The usual suspects are there: the relativity principle and equivalence principle, the abolishing of absolute space, invariance of the speed of light, analytic and geometric representations of the Lorentz transformation, its kinematic and dynamic consequences, relativistic optics, Minkowski spacetime, energy and momentum conservation, and the Compton effect. Particularly useful is the emphasis on the unity of the whole: for example (p 63) that the kinematic effect of length shortening must imply a corresponding detailed mechanical explanation of that shortening. The tensor formulation of Maxwell's equations leads to the transformation properties of the electromagnetic field and consequent elegant derivation of the field of an infinite straight current; in this case, relativity is important even for slowly moving charges because an ordinary current moves a very big charge (p 151). General relativity is systematically introduced in stages, starting with curved spaces and moving on through static and stationary spacetimes, geodesics, and tensor calculus to the field equations. A considerable strength of the book is the careful detailed examination of the local and global geometry of the major significant solutions of the equations: the Schwarzschild spacetime and its Kruskal extension, plane gravitational waves, de sitter and anti-de Sitter spacetimes, and Robertson-Walker cosmologies. The latter includes a clear presentation of the dust and radiation model dynamics for the variety of possible cases, a detailed examination of observational relations, and considered study of the properties of horizons. Linearized relativity is dealt with in depth leading to the standard weak field gravitational wave formulae and a study of their effects on test particles, together with a very useful discussion of the analogy between weak gravity and the electromagnetic field. Thus this is a straightforward detailed presentation of both special and general relativity theory and their applications. It has many examples and is well suited as a text on these topics, giving a clear relativists' view all the way through. It does not go into astrophysical or particle physics aspects, which is fine given its focus. Personally I would have liked a bit more emphasis on the geodesic deviation equation on the one hand, and on holonomy (which provides a link into particle gauge theories) on the other. But that is a matter of taste. This is an excellent book, which can be highly recommended. Just one quibble: what is the reason for the blurred picture of Einstein on the cover? The whole point of the book is its clarity: why the implication of this picture that it presents a blurred vision?

120 citations


BookDOI
01 Nov 2005
TL;DR: From Newton to Einstein: Development of the concepts of Space, Time and Space-Time from Newton to E. Stachel (J Stachel) as mentioned in this paper, J Stachel's Universe: Gravitational Billiards, Dualities and Hidden Symmetries (H Nicolai) The Nature of Spacetime Singularities (A D Rendall) Black Holes -- An Introduction (P T Chru ciel) The Physical Basis of Black Hole Astrophysics (R H Price) Probing Space Time Through Numerical Simulations (P Laguna)
Abstract: From Newton to Einstein: Development of the Concepts of Space, Time and Space-Time from Newton to Einstein (J Stachel) Einstein's Universe: Gravitational Billiards, Dualities and Hidden Symmetries (H Nicolai) The Nature of Spacetime Singularities (A D Rendall) Black Holes -- An Introduction (P T Chru ciel) The Physical Basis of Black Hole Astrophysics (R H Price) Probing Space-Time Through Numerical Simulations (P Laguna) Understanding Our Universe: Current Status and Open Issues (T Padmanabhan) Was Einstein Right? Testing Relativity at the Centenary (C M Will) Receiving Gravitational Waves (P R Saulson) Relativity in the Global Positioning System (N Ashby) Beyond Einstein: Spacetime in Semiclassical Gravity (L H Ford) Space Time in String Theory (T Banks) Quantum Geometry and Its Ramifications (A Ashtekar) Loop Quantum Cosmology (M Bojowald) Consistent Discrete Space-Time (R Gambini & J Pullin) Causal Sets and the Deep Structure of Spacetime (F Dowker) The Twistor Approach to Space-Time Structures (R Penrose).

105 citations


Journal ArticleDOI
TL;DR: In this article, the authors set up the theory of elastic matter sources within the framework of general relativity in a self-contained manner, and applied the theory to static spherically symmetric configurations using a specific equation of state and consider models either having an elastic crust or core.
Abstract: The aim of this paper is twofold. First, we set up the theory of elastic matter sources within the framework of general relativity in a self-contained manner. The discussion is primarily based on the presentation of Carter and Quintana but also includes new methods and results as well as some modifications that in our opinion make the theory more modern and transparent. For instance, the equations of motion for the matter are shown to take a neat form when expressed in terms of the relativistic Hadamard elasticity tensor. Using this formulation, we obtain simple formulae for the speeds of elastic wave propagation along eigendirections of the pressure tensor. Secondly, we apply the theory to static spherically symmetric configurations using a specific equation of state and consider models either having an elastic crust or core.

88 citations


Journal ArticleDOI
TL;DR: The first systematic exposition of general relativity, submitted in March 1916 and published in May of that year, can be found in this article with the title "Outline (Entwurf) of a Generalized Theory of Relativity and of a Theory of Gravitation".
Abstract: Readers of this volume will notice that it contains only a few papers on general relativity. This is because most papers documenting the genesis and early development of general relativity were not published in Annalen der Physik.After Einstein took up his new prestigious position at the PrussianAcademy of Sciences in the spring of 1914, the Sitzungsberichte of the Berlin academy almost by default became the main outlet for his scientific production. Two of the more important papers on general relativity, however, did find their way into the pages of the Annalen [35,41].Although I shall discuss both papers in this essay, the main focus will be on [35], the first systematic exposition of general relativity, submitted in March 1916 and published in May of that year. Einstein’s first paper on a metric theory of gravity, co-authored with his mathematician friend Marcel Grossmann, was published as a separatum in early 1913 and was reprinted the following year in Zeitschrift fur Mathematik und Physik [50,51]. Their second (and last) joint paper on the theory also appeared in this journal [52]. Most of the formalism of general relativity as we know it today was already in place in this Einstein-Grossmann theory. Still missing were the generally-covariant Einstein field equations. As is clear from research notes on gravitation from the winter of 1912–1913 preserved in the so-called “Zurich Notebook,” Einstein had considered candidate field equations of broad if not general covariance, but had found all such candidates wanting on physical grounds. In the end he had settled on equations constructed specifically to be compatible with energy-momentum conservation and with Newtonian theory in the limit of weak static fields, even though it remained unclear whether these equations would be invariant under any non-linear transformations. In view of this uncertainty, Einstein and Grossmann chose a fairly modest title for their paper: “Outline (“Entwurf”) of a Generalized Theory of Relativity and of a Theory of Gravitation.” The Einstein-Grossmann theory and its fields equations are therefore also known as the “Entwurf” theory and the “Entwurf” field equations. Much of Einstein’s subsequent work on the “Entwurf” theory went into clarifying the covariance properties of its field equations. By the following year he had convinced himself of three things. First, generallycovariant field equations are physically inadmissible since they cannot determine the metric field uniquely. This was the upshot of the so-called “hole argument” (“Lochbetrachtung”) first published in an appendix to [51]. Second, the class of transformations leaving the “Entwurf” field equations invariant was as broad ∗ E-mail: janss011@tc.umn.edu 1 An annotated transcription of the gravitational portion of the “Zurich Notebook” is published as Doc.10 in [11]. For facsimile reproductions of these pages, a new transcription, and a running commentary, see [89]. 2 See Sect. 2 for further discussion of the hole argument.

81 citations


Journal ArticleDOI
TL;DR: In this article, the energy-momentum tensor of the gravitational field is identified in the teleparallel equivalent of general relativity and the spatial components of this tensor yield a consistent definition of the gravity pressure.
Abstract: In the framework of the teleparallel equivalent of general relativity it is possible to establish the energy-momentum tensor of the gravitational field. This tensor has the following essential properties: (1) it is identified directly in Einstein's field equations; (2) it is conserved and traceless; (3) it yields expressions for the energy and momentum of the gravitational field; (4) is is free of second (and highest) derivatives of the field variables; (5) the gravitational and matter energy-momentum tensors take place in the field equations on the same footing; (6) it is unique. However it is not symmetric. We show that the spatial components of this tensor yield a consistent definition of the gravitational pressure.

75 citations


Book
07 Jul 2005

74 citations


BookDOI
01 Jan 2005
TL;DR: In this paper, Renn and Rynasiewicz discuss the importance of general Relativistic Cosmology in the development of Inflation in the early 20th century, and present their theory of General Relativity as a challenge to 19th century Optics of Moving Bodies.
Abstract: * Preface * Stachel: Fresnel's (Dragging) Coefficient as a Challenge to 19th Century Optics of Moving Bodies * Katzir: Poincare's Relativistic Theory of Gravitation * Renn: Standing on the Shoulders of a Dwarf: General Relativity-A Triumph of Einstein and Grossman's Erroneous Entwurf Theory * Renn: Before the Riemann Tensor: The Emergence of Einstein's Double Strategy * Norton: A Conjecture on Einstein, the Independent Reality of Spacetime Coordinate Systems and the Disaster of 1913 * Lehner: Einstein and the Principle of General Relativity, 1916-1921 * Kennefick: Einstein and the Problem of Motion: A Small Clue * Brading: A Note on General Relativity, Energy Conservation, and Noether's Theorems * Rynasiewicz: Weyl vs. Reichenbach on Lichtgeometrie * Gale: Dingle and de Sitter Against the Metaphysicians, or Two Ways to Keep Modern Cosmology Physical * Kragh: George Gamow and the 'Factual Approach' to Relativistic Cosmology * Sanchez-Ron: George McVittie, The Uncompromising Empiricist * Smeenk: False Vacuum: Early Universe Cosmology and the Development of Inflation * Majer and Sauer: Hilbert's 'World Equations' and His Vision of a Unified Science * Wunsch: Einstein, Kaluza and the Fifth Dimension * Goenner: Unified Field Theory: Early History and Interplay between Mathematics and Physics * Mattingly: Is Quantum Gravity Necessary? * Wazeck: Einstein in the Daily Press: A Glimpse into the Gehrcke Papers * Goldberg: Syracuse: 1949-1952 * Newman: A Biased and PersonalDescription of GR at Syracuse University, 1951-1961

60 citations


Journal ArticleDOI
TL;DR: In this paper, a simple formulation of the basic equation of general relativity in terms of the motion of freely falling test particles is given, and the consequences of this formulation are sketched.
Abstract: This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein’s equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of the consequences of this formulation and explain how it is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles, and websites suitable for the student of relativity.

Journal ArticleDOI
TL;DR: The field equations of general relativity are derived from a limit to force or to power in nature as mentioned in this paper, and the limits have the value of c4/4G and c5/ 4G.
Abstract: The field equations of general relativity are shown to derive from a limit to force or to power in nature. The limits have the value of c4/4G and c5/4G. The proof makes use of a result of Jacobson. All known experimental data are consistent with the limits. Applied to the universe, the limits predict its darkness at night and the observed scale factor. Other experimental tests of the limits are proposed. The main counterarguments and paradoxes are discussed, such as the transformation under boosts, the force felt at a black hole horizon, the mountain problem, and the contrast to scalar–tensor theories of gravitation. The resolution of the paradoxes also clarifies why the maximum force and the maximum power have remained hidden for so long. The derivation of the field equations shows that the maximum force or power plays the same role for general relativity as the maximum speed plays for special relativity.

Book
01 Jan 2005
TL;DR: Tensors, Relativity, and Cosmology, Second Edition as discussed by the authors combines tensors, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject that is followed by detailed mathematical derivations.
Abstract: Tensors, Relativity, and Cosmology, Second Edition, combines relativity, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject that is followed by detailed mathematical derivations. The book includes a section on general relativity that gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes and Penrose processes), and considers the energy-momentum tensor for various solutions. In addition, a section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects, with a final section on cosmology discussing cosmological models, observational tests, and scenarios for the early universe.

Journal ArticleDOI
TL;DR: In this paper, the authors derived a new expression for the metric in the near zone of an isolated matter system in post-Newtonian approximations of general relativity, which is formally valid up to any order, and cast in the form of a particular solution of the wave equation, plus a specific homogeneous solution which ensures the asymptotic matching to the multipolar expansion of the gravitational field in the exterior of the system.
Abstract: In the continuation of a preceding work, we derive a new expression for the metric in the near zone of an isolated matter system in post-Newtonian approximations of general relativity. The post-Newtonian metric, a solution of the field equations in harmonic coordinates, is formally valid up to any order, and is cast in the form of a particular solution of the wave equation, plus a specific homogeneous solution which ensures the asymptotic matching to the multipolar expansion of the gravitational field in the exterior of the system. The new form provides some insights on the structure of the post-Newtonian expansion in general relativity and the gravitational radiation reaction terms therein.

Journal ArticleDOI
TL;DR: In this article, the authors used the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and Moller to compute the energy-momentum densities for two exact solutions of Einstein field equations.
Abstract: This paper is aimed to elaborate the problem of energy–momentum in general relativity. In this connection, we use the prescriptions of Einstein, Landau–Lifshitz, Papapetrou and Moller to compute the energy–momentum densities for two exact solutions of Einstein field equations. The space–times under consideration are the nonnull Einstein–Maxwell solutions and the singularity-free cosmological model. The electromagnetic generalization of the Godel solution and the Godel metric become special cases of the nonnull Einstein–Maxwell solutions. It turns out that these prescriptions do not provide consistent results for any of these space–times. These inconsistent results verify the well-known proposal that the idea of localization does not follow the lines of pseudotensorial construction but instead follows from the energy–momentum tensor itself. These differences can also be understood with the help of the Hamiltonian approach.

Journal ArticleDOI
TL;DR: In this article, the Riemann-Einstein tensor of general relativity as well as chiral and super-symmetry are utilized to develop various extended versions of the standard model of high energy physics.
Abstract: The Riemann–Einstein tensor of general relativity as well as chiral and super-symmetry are utilized to develop various extended versions of the standard model of high energy physics. Based on these models, it is possible to predict that few new elementary particles conjectured to be the Higgs are likely to be found experimentally at an energy scale which is just above that of the electroweak. Connections to the massless states of different super-string theories as well as super-gravity are also discussed.

Journal ArticleDOI
TL;DR: Considering the Moller energy definition in both Einstein's theory of general relativity and tele-parallel theory of gravity, this article found the energy of the universe based on viscous Kasner-type metrics.
Abstract: Considering the Moller energy definition in both Einstein's theory of general relativity and tele-parallel theory of gravity, we find the energy of the universe based on viscous Kasner-type metrics. The energy distribution which includes both the matter and gravitational field is found to be zero in both of these different gravitation theories and this result agrees with previous works of Cooperstock and Israelit et al., Banerjee–Sen, Vargas who investigated the problem of the energy in Friedmann–Robertson–Walker universe in Einstein's theory of general relativity and Aydogdu–Salti who considered the same problem in tele-parallel gravity. In all of these works, they found that the energy of the Friedmann–Robertson–Walker spacetime is zero. Our result is the same as that obtained in the studies of Salti and Havare. They used the viscous Kasner-type metric and found the total energy and momentum by using Bergmann–Thomson energy–momentum formulation in both general relativity and tele-parallel gravity. The result that the total energy and momentum components of the universe is zero supports the viewpoints of Albrow and Tryon.


Journal ArticleDOI
TL;DR: In this article, the authors give plausibility arguments for deriving the number of elementary particles of the standard model from general relativity and maximally symmetric spaces, and show that this is possible.
Abstract: The essay gives plausibility arguments for deriving the number of elementary particles of the standard model from general relativity and maximally symmetric spaces.

Journal ArticleDOI
TL;DR: In this paper, the cosmological constant, a term in the most general form of the Einstein field equations which causes free floating objects to accelerate apart, cannot be causally explained except by reference to spacetime itself.
Abstract: I offer a novel argument for spacetime substantivalism: We should take the spacetime of general relativity to be a substance because of its active role in gravitational causation. As a clear example of this causal behavior I offer the cosmological constant, a term in the most general form of the Einstein field equations which causes free floating objects to accelerate apart. This acceleration cannot, I claim, be causally explained except by reference to spacetime itself.

Journal ArticleDOI
TL;DR: In this article, the authors derived an integral equation satisfied by the curvature of a vacuum solution to the Yang-Mills field equations of general relativity, which they used for the proof of global existence for Yang-mills fields, propagating in curved, 4D, globally hyperbolic, background spacetimes.
Abstract: A key step in the proof of global existence for Yang-Mills fields, propagating in curved, 4-dimensional, globally hyperbolic, background spacetimes, was the derivation and reduction of an integral equation satisfied by the curvature of an arbitrary solution to the Yang-Mills field equations. This article presents the corresponding derivation of an integral equation satisfied by the curvature of a vacuum solution to the Einstein field equations of general relativity. The resultant formula expresses the curvature at a point in terms of a ‘direct’ integral over the past light cone from that point, a so-called ‘tail’ integral over the interior of that cone and two additional integrals over a ball in the initial data hypersurface and over its boundary. The tail contribution and the integral over the ball in the initial data surface result from the breakdown of Huygens’ principle for waves propagating in a general curved, 4-dimensional spacetime. By an application of Stokes’ theorem and some integration by parts lemmas, however, one can re-express these ‘Huygens-violating’ contributions purely in terms of integrals over the cone itself and over the 2-dimensional intersection of that cone with the initial data surface. Furthermore, by exploiting a generalization of the parallel propagation, or Cronstrom, gauge condition used in the Yang-Mills arguments, one can explicitly express the frame fields and connection one-forms in terms of curvature. While global existence is certainly false for general relativity one anticipates that the resulting integral equation may prove useful in analyzing the propagation, focusing and (sometimes) blow up of curvature during the course of Einsteinian evolution and thereby shed light on the natural alternative conjecture to global existence, namely Penrose’s cosmic censorship conjecture.

Journal ArticleDOI
TL;DR: In this paper, it is shown that the Noether stress energy tensor is equivalent to the gravitational tensor for general matter fields under the influence of gravity, and the full equivalence is established for matter fields that do not couple to the metric derivatives.
Abstract: It is dealt with the question, under which circumstances the canonical Noether stress-energy tensor is equivalent to the gravitational (Hilbert) tensor for general matter fields under the influence of gravity. In the framework of general relativity, the full equivalence is established for matter fields that do not couple to the metric derivatives. Spinor fields are included into our analysis by reformulating general relativity in terms of tetrad fields, and the case of Poincare gauge theory, with an additional, independent Lorentz connection, is also investigated. Special attention is given to the flat limit, focusing on the expressions for the matter field energy (Hamiltonian). The Dirac-Maxwell system is investigated in detail, with special care given to the separation of free (kinetic) and interaction (or potential) energy. Moreover, the stress-energy tensor of the gravitational field itself is briefly discussed.

Journal ArticleDOI
TL;DR: In this paper, Moller's energy-momentum complex was used to explicitly evaluate the energy and momentum density distributions associated with the three-dimensional magnetic solution to the Einstein-Maxwell equations.
Abstract: We use Moller's energy–momentum complex in order to explicitly evaluate the energy and momentum density distributions associated with the three-dimensional magnetic solution to the Einstein–Maxwell equations. The magnetic spacetime under consideration is a one-parametric solution describing the distribution of a radial magnetic field in a three-dimensional AdS background, and representing the superposition of the magnetic field with a 2+1 Einstein static gravitational field.

Journal ArticleDOI
TL;DR: In this paper, the authors place the relativistic velocity addition in the foundations of both special relativity and its underlying hyperbolic geometry, enabling them to present special relativity in full three space dimensions rather than the usual one-dimensional space.
Abstract: We add physical appeal to Einstein velocity addition law of relativistically admissible velocities, thereby gaining new analogies with classical mechanics and invoking new insights into the special theory of relativity. We place Einstein velocity addition in the foundations of both special relativity and its underlying hyperbolic geometry, enabling us to present special relativity in full three space dimensions rather than the usual one-dimensional space, using three-geometry instead of four-geometry. Doing so we uncover unexpected analogies with classical results, enabling readers to understand the modern and unfamiliar in terms of the classical and familiar. In particular, we show that while the relativistic mass does not mesh up with the four-geometry, it meshes extraordinarily well with the three-geometry, providing unexpected insights that are not easy to come by, by other means.

Book ChapterDOI
01 Jan 2005
TL;DR: A comprehensive overview of the final version of the general theory of relativity can be found in this paper, which includes a self-contained exposition of the elements of tensor calculus that are needed for the theory.
Abstract: Publisher Summary This chapter discusses Albert Einstein's reviews paper on general relativity theory. This paper was the first comprehensive overview of the final version of Einstein's general theory of relativity after several expositions of preliminary versions and latest revisions of the theory in November 1915. It includes a self-contained exposition of the elements of tensor calculus that are needed for the theory. It presented a conceptual analysis of the notions of space and time, with a critical reassessment of the meaning of simultaneity at its core. Its most salient features are length contraction and time dilation in a system that is in uniform relative motion to an observer with a speed comparable to that of light. Einstein concluded that generally covariant field equations cannot uniquely determine the physical processes in a gravitational field. Consequently, there had to be restriction of the admissible coordinate systems to what he began to call “adapted coordinates.” Einstein tried to encourage experimental efforts aimed at testing the two main predictions of the theory. A confirmation of the gravitational red shift was difficult to determine because of the many competing effects that result in a shifting or broadening of solar or stellar spectral lines.

Journal ArticleDOI
TL;DR: In this article, a first-order purely frame-formulation of general relativity is obtained in the gauge natural bundle framework, where a new space is introduced and a first order purely frameformulation is obtained.
Abstract: In the gauge natural bundle framework, a new space is introduced and a first-order purely frame-formulation of general relativity is obtained.

Book
01 Jan 2005
TL;DR: In this article, the authors present a survey of the most relevant aspects of numerical relativity, including boundary conditions, singularity-avoidant gauge conditions and hyperbolic evolution formalisms.
Abstract: We became involved with numerical relativity under very different circumstances. For one of us (C.B.) it dates back to about 1987, when the current Laser-Interferometer Gravitational Wave Observatories were just promising proposals. It was during a visit to Paris, at the Institut Henri Poincar´e, where some colleagues were pushing the VIRGO proposal with such a contagious enthusiasm that I actually decided to reorient my career. The goal was to be ready, armed with a reliable numerical code, when the first detection data would arrive. Allowing for my experience with the 3+1 formalism at that time, I started working on singularity-avoidant gauge conditions. Soon, I became interested in hyperbolic evolution formalisms. When trying to get some practical applications, I turned to numerical algorithms (a really big step for a theoretically oriented guy) and black hole initial data. More recently, I became interested in boundary conditions and, closing the circle, again in gauge conditions. The problem is that a reliable code needs all these ingredients to be working fine at the same time. It is like an orchestra, where strings, woodwinds, brass and percussion must play together in a harmonic way: a violin virtuoso, no matter how good, cannot play Vivaldi’s Four Seasons by himself. During that time, I have had many Ph.D. students. The most recent one is the other of us (C.P.). All of them started with some specific topic, but they needed a basic knowledge of all the remaining ones: you cannot work on the saxophone part unless you know what the bass is supposed to play at the same time. This is where this book can be of a great help. Imagine a beginning graduate student armed only with a home PC. Imagine that the objective is to build a working numerical code for simple black-hole applications. This book should first provide him or her with a basic insight into the most relevant aspects of numerical relativity. But this is not enough; the book should also provide reliable and compatible choices for every component: evolution system, gauge, initial and boundary conditions, even for numerical algorithms. This pragmatic orientation may cause this book to be seen as biased. But the idea was not to produce a compendium of the excellent work that has been made in numerical relativity during these years. The idea is rather to present a well-founded and convenient way for a beginner to get into the field. He or she will Quickly discover everything else. The structure of the book reflects the peculiarities of numerical relativity research: • It is strongly rooted in theory. Einstein’s relativity is a general-covariant theory. This means that we are building at the same time the solution and the coordinate system, a unique fact among physical theories. This point is stressed in the first chapter, which could be omitted by more experienced readers. • It turns the theory upside down. General covariance implies that no specific coordinate is more special than the others, at least not a priori. But this is at odds with the way humans and computers usually model things: as functions (of space) that evolve in time. The second chapter is devoted to the evolution (or 3+1) formalism, which reconciles general relativity with our everyday perception of reality, in which time plays such a distinct role. • It is a fertile domain, even from the theoretical point of view. The structure of Einstein’s equations allows many ways of building well-posed evolution formalisms. Chapter 3 is devoted to those which are of first order in time but second order in space. Chapter 4 is devoted instead to those which are of first order both in time and in space. In both cases, suitable numerical algorithms are provided, although the most advanced ones apply mainly to the fully first order case. • It is challenging. The last sections of Chaps. 5 and 6 contain frontedge developments on constraint-preserving boundary conditions and gauge pathologies, respectively. These are very active research topics, where new developments will soon improve on the ones presented here. The prudent reader is encouraged to look for updates of these front-edge areas in the current scientific literature.

Book ChapterDOI
01 Jan 2005

Book ChapterDOI
01 Jan 2005
TL;DR: The authors conjecture that a second hitherto unrecognized error also defeated Einstein's efforts, which was the assumption that weak, static gravitational fields must be spatially flat and a corresponding assumption about general covariant field equations.
Abstract: Two fundamental errors led Einstein to reject generally covariant gravitational field equations for over two years as he was developing his general theory of relativity. The first is well known in the literature. It was the presumption that weak, static gravitational fields must be spatially flat and a corresponding assumption about his weak field equations. I conjecture that a second hitherto unrecognized error also defeated Einstein's efforts. The same error, months later, allowed the hole argument to convince Einstein that all generally covariant gravitational field equations would be physically uninteresting.