Showing papers on "Introduction to the mathematics of general relativity published in 2008"
Book•
01 Jan 2008
TL;DR: The 3+1 formalism has been studied extensively in the literature as discussed by the authors, where the authors present a brief review of general Relativity and its applications in 3+ 1 languages.
Abstract: 1. Brief Review of General Relativity 2. The 3+1 Formalism 3. Initial Data 4. Gauge Conditions 5. Hyperbolic Reductions of the Field Equations 6. Evolving Black Hole Spacetimes 7. Relativistic Hydrodynamics 8. Gravitational Wave Extraction 9. Numerical Methods 10. Examples of Numerical Spacetimes A. Total Mass and Momentum in General Relativity B. Spacetime Christoffel Symbols in 3+1 Language C. BSSNOK with Natural Conformal Rescaling D. Spin-weighted Spherical Harmonics References Index
564 citations
TL;DR: In this article, the authors show that the black-hole solutions of these theories are essentially indistinguishable from those of general relativity and conclude that a potential observational verification of the Kerr metric around an astrophysical black hole cannot, in and of itself, be used to distinguish between these theories.
Abstract: Considerable attention has recently focused on gravity theories obtained by extending general relativity with additional scalar, vector, or tensor degrees of freedom. In this Letter, we show that the black-hole solutions of these theories are essentially indistinguishable from those of general relativity. Thus, we conclude that a potential observational verification of the Kerr metric around an astrophysical black hole cannot, in and of itself, be used to distinguish between these theories. On the other hand, it remains true that detection of deviations from the Kerr metric will signify the need for a major change in our understanding of gravitational physics.
153 citations
01 Jan 2008
TL;DR: In this paper, the Lagrangian model was used to solve problems in the context of solving problems. But it was not suitable for solving general relativity problems, and it was shown that angular momentum, Part I (constant L) and Part II (general L) can not be used in general L.
Abstract: Preface 1. Strategies for solving problems 2. Statics 3. Using F=ma 4. Oscillations 5. Conservation of energy and momentum 6. The Lagrangian model 7. Central forces 8. Angular momentum, Part I (constant L) 9. Angular momentum, Part II (general L) 10. Accelerating frames of reference 11. Relativity (kinematics) 12. Relativity (dynamics) 13. 4-vectors 14. General relativity Appendices References Index.
145 citations
TL;DR: In this article, the authors extend the validity of Dain's angularmomentum inequality to maximal, asymptotically flat, initial data sets on a simply connected manifold with several invariant under a U(1) action and which admit a twist potential.
101 citations
TL;DR: In this article, the authors derived the field equations in vacuum and in the presence of perfect-fluid matter and discussed the related cosmological models for f(R)-gravity with geometric torsion.
Abstract: f(R)-gravity with geometric torsion (not related to any spin fluid) is considered in a cosmological context. We derive the field equations in vacuum and in the presence of perfect-fluid matter and discuss the related cosmological models. Torsion vanishes in vacuum for almost all arbitrary functions f(R) leading to standard general relativity. Only for f(R)=R2, torsion gives a contribution in the vacuum leading to accelerated behavior. When material sources are considered, we find that the torsion tensor is different from zero even with spinless material sources. This tensor is related to the logarithmic derivative of f' (R), which can be expressed also as a nonlinear function of the trace of the matter energy–momentum tensor Σμν. We show that the resulting equations for the metric can always be arranged to yield effective Einstein equations. When the homogeneous and isotropic cosmological models are considered, terms originated by torsion can lead to accelerated expansion. This means that, in f(R)-gravity, torsion can be a geometric source for acceleration.
59 citations
TL;DR: In this paper, the authors apply the energy-momentum tensor to calculate energy, momentum and angular momentum of two different tetrad fields and show that the energy associated with one of them is consistent, while the other one does not show this consistency.
Abstract: We apply the energy-momentum tensor to calculate energy, momentum and angular-momentum of two different tetrad fields. This tensor is coordinate independent of the gravitational field established in the Hamiltonian structure of the teleparallel equivalent of general relativity (TEGR). The spacetime of these tetrad fields is the charged dilaton. Our results show that the energy associated with one of these tetrad fields is consistent, while the other one does not show this consistency. Therefore, we use the regularized expression of the gravitational energy-momentum tensor of the TEGR. We investigate the energy within the external event horizon using the definition of the gravitational energy-momentum.
46 citations
TL;DR: In this paper, an extension of general relativity with two different metrics is proposed, and two types of fields, each of which moves according to one of the metrics and its connection, are considered.
Abstract: We propose an extension of general relativity with two different metrics. To each metric we define a Levi-Cevita connection and a curvature tensor. We then consider two types of fields, each of which moves according to one of the metrics and its connection. To obtain the field equations for the second metric we impose an exchange symmetry on the action. As a consequence of this ansatz, additional source terms for Einstein's field equations are generated. We discuss the properties of these additional fields, and consider the examples of the Schwarzschild solution, and the Friedmann-Robertson-Walker metric.
44 citations
TL;DR: In this article, an exact stationary axially symmetric solution to the four-dimensional Einstein equations with corotating pressureless perfect fluid sources is studied, and a particular solution with an approximately flat rotation curve is discussed.
Abstract: Exact stationary axially symmetric solutions to the four-dimensional Einstein equations with corotating pressureless perfect fluid sources are studied. A particular solution with an approximately flat rotation curve is discussed in some detail. We find that simple Newtonian arguments overestimate the amount of matter needed to explain such curves by more than 30%. The crucial insight gained by this model is that the Newtonian approximation breaks down in an extended rotating region, even though it is valid locally everywhere. No conflict with solar system tests arises.
35 citations
TL;DR: Gron and Hervik as mentioned in this paper presented a detailed description of the applications of the Einstein field equations to a universe with various matter contents, and present in a successful way the recent developments in this domain.
Abstract: The increasing prominence of general relativity in astrophysics and cosmology is reflected in the growing number of texts, particularly at the undergraduate level. A natural attitude before opening a new one is to ask i) what makes this different from those already published? And ii) does it follow the 'physics-first approach' as for instance the book by Hartle where the basic physical concepts are introduced first with as little formalism as possible, or does it follow the more traditional 'math-first approach' for which the mathematical formalism comes first and is then applied to phyics? As announced in the title, a distinctive feature of the book by Gron and Hervik is the space (almost half the book) devoted to cosmology and in particular to some of the most recent developments in this rapidly evolving field. It is also apparent that the authors have chosen, like the majority of current books on general relativity, the 'math-first approach'. The book is divided into six parts, each of them subdivided into chapters with part VI containing a few short technical appendices. The first part of the book briefly presents in chapter I the principles of relativity, Newtonian mechanics and the Newtonian theory of gravity. In chapter II, a short introduction to special relativity is given. It seems at first surprising that the four-dimensional structure of space-time is not more fully exploited so that the reader would gain familiarity early on with notions like 4-velocity, 4-momentum and the stress–energy tensor. This is in fact postponed to part II as an illustration of the mathematical formalism. The second part is devoted to those elements of differential geometry needed in this kind of course. The authors' presentation is somewhat similar to that of the books by Misner, Thorne and Wheeler and by Straumann (2nd edition). Vectors and forms are treated separately and the formalism of differential forms is introduced in detail. The various kinds of differentiation on forms and on vectors (exterior covariant and Lie derivatives) are presented, and emphasis is given to the Cartan formalism as it is later systematically used to derive the curvature tensor and for solutions of the Einstein field equations. One also finds the properties of hypersurfaces, such as the intrinsic and extrinsic curvatures and the Gauss–Codazzi relations. This makes this part of the book very useful and convenient since those important elements are gathered in one place. However the density of exposition in this part might appear a bit steep to a reader without some previous knowledge of differential geometry. Part III deals with Einstein's field equations, and their applications to gravitational waves and black holes. The field equations are derived from a variational principle, the geometrical part (Einstein tensor) from the Einstein–Hilbert action, and the matter part (stress–energy tensor) from a generic action integral for matter. Various examples of stress–energy tensors and in particular, for fluids, are considered, and several are used later in cosmology (for instance quintessence and Lorentz invariant vacuum energy). A short chapter on the linear approximation and gravitational waves then follows and it is good to see a section on gravito-electromagnetism. This part ends with a chapter devoted to black holes which is perhaps the weakest part of the book as it is quite sketchy. However this is to be expected in a book with an emphasis on cosmology, and such topics are extensively described in other books. The rest of the book (parts IV and V) is essentially concerned with cosmology. The authors give a detailed description of the applications of the Einstein field equations to a universe with various matter contents, and present in a successful way the recent developments in this domain. The first chapter of part IV describes the standard homogeneous and isotropic cosmological model. It is followed by an interesting chapter dealing with universes composed of vacuum energy. There one finds, after the description of the Einstein static universe and the de Sitter solution, sections on inflation, on the Friedman–Lemaitre model and on models with quintessence and dark energy. This chapter ends with sections on cosmic density perturbations, temperature fluctations in the cosmic microwave background and on the history of our universe. With an additional chapter on anisotropic and homogeneous universes, part IV appears to be a very good and complete introduction to the basic and classical (i.e. non-quantum) elements of cosmology. In part V some advanced tools, such as Lie groups and the Lagrangian and Hamiltonian formalism are introduced and applied to cosmology. Also part V contains a chapter on the extrinsic curvature formalism for surface layers and its application to the recently introduced braneworld models. Finally it is a pleasant surprise to find an introduction to the Kaluza–Klein theory as the last chapter of part V. This book by Gron and Hervik certainly has its place in any good library. It covers most of the classical aspects of the theory of general relativity. The authors have made the effort to discuss many observational aspects and to illustrate the different chapters with many problems. One might regret that the authors' style is generally rather terse and not enough space is always reserved for explanation of physical concepts and for motivations of the theory (for instance, why curvature is so fundamental). This book would be most appropriate for graduate students and I will definitely recommend it as a reference textbook as well as a useful complement to other textbooks on general relativity.
TL;DR: In this article, it was shown that the Dirac operator for the four-dimensional nonholonomic distribution can be extended to functions defined on a manifold M ≥ 4 × S ≥ 1, where S ≥ S is the circle.
Abstract: The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M
4 × ℝ1. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, −, −, −) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M
4 × S
1, where S
1 is the circle. For such functions, electric charges are topologically quantized.
TL;DR: In this paper, a new method of imposing gauge conditions that preserve hyperbolicity for a much wider class of conditions, including as special cases many of the standard ones used in numerical relativity: e.g., K freezing, Gamma freezing, Bona-Masso slicing, conformal Gamma drivers, etc.
Abstract: The generalized harmonic representation of Einstein's equations is manifestly hyperbolic for a large class of gauge conditions. Unfortunately most of the useful gauges developed over the past several decades by the numerical relativity community are incompatible with the hyperbolicity of the equations in this form. This paper presents a new method of imposing gauge conditions that preserves hyperbolicity for a much wider class of conditions, including as special cases many of the standard ones used in numerical relativity: e.g., K freezing, Gamma freezing, Bona-Masso slicing, conformal Gamma drivers, etc. Analytical and numerical results are presented which test the stability and the effectiveness of this new gauge-driver evolution system.
TL;DR: In this article, all 2-covariant tensors naturally constructed from a semiriemannian metric g which are divergence-free and have weight greater than 2 were shown to lead to the field equation of General Relativity.
TL;DR: In this paper, the authors describe a theory of gravity in (1 + 1) dimensions that can be thought of as a toy model of general relativity, and derive the theory from fundamental physical principles using two different methods.
Abstract: We describe a theory of gravity in (1 + 1) dimensions that can be thought of as a toy model of general relativity. The theory should be a useful pedagogical tool, because it is mathematically much simpler than general relativity but shares much of the same conceptual structure; in particular, it gives a simple illustration of how gravity arises from spacetime curvature. We derive the theory from fundamental physical principles using two different methods, one based on extrapolating from Newtonian gravity and one based on the equivalence principle, and present several exact solutions.
01 Jan 2008
TL;DR: A number of conceptual anomalies occurring in the Standard exposition of Einstein's Theory of Relativity are discussed in this article, including the fact that Einstein's field equations for the so-called static vacuum configuration, violate his Principle of Equivalence, and therefore are therefore erroneous.
Abstract: There are a number of conceptual anomalies occurring in the Standard exposition of Einstein’s Theory of Relativity. These anomalies relate to issues in both mathematics and in physics and penetrate to the very heart of Einstein’s theory. This paper reveals and amplifies a few such anomalies, including the fact that Einstein’s field equations for the so-called static vacuum configuration, , violates his Principle of Equivalence, and is therefore erroneous. This has a direct bearing on the usual concept of conservation of energy for the gravitational field and the conventional formulation for localisation of energy using Einstein’s pseudo-tensor. Misconceptions as to the relationship between Minkowski spacetime and Special Relativity are also discussed, along with their relationships to the pseudo-Riemannian metric manifold of Einstein’s gravitational field, and their fundamental geometric structures pertaining to spherical symmetry. In a series of papers [1‐17] I have previously provided mathematical demonstrations of the invalidity of the concept of the black hole and also of the expansion of the Universe with a Big Bang cosmology. In those papers I took on face value the fundamental line-elements from which these physical concepts have allegedly been derived by the Standard Model relativists, and demonstrated in purely mathematical terms that they are inconsistent with the geometrical structure of those line-elements, and are therefore false. I do not reiterate those demonstrations herein, referring the reader to the relevant papers for the details, and instead consider, in the main, various conceptual matters underlying the structure of Einstein’s Theory of Relativity, and show that there are some very serious anomalies in the usual exposition, which render much of what has been claimed for General Relativity to be false. 2 Misconception: that Ricci 0 fully describes the gravitational field Setting imposes upon an observer in the alleged gravitational field, a consideration of the perceived source of the field in terms of its centre of mass, and so is not a physically meaningful condition. In other words, the notion of gravitational collapse to a point-mass is not justified: it is ill-posed. A centre of mass is not a physical object, only a mathematical artifice. This same artifice occurs in Newton’s theory as well, and in Newton’s theory it is not a physical object either, and nobody, quite rightly, considers it a physical object in Newton’s universe. Oddly, the centre of mass is taken, by unconscious assumption or blind conviction, to be a real object in Einstein’s theory. Gravitational collapse is a conceptual anomaly in General Relativity that has no basis in the physical world or in General Relativity. It is built upon a false idea as a result of not realising that imposes
TL;DR: In this article, a tensor model around its classical background of two-, three-and four-dimensional fuzzy flat tori is studied and it is shown that the properties of low-lying low-momentum modes are in clear agreement with general relativity.
Abstract: Tensor models can be regarded as theories of dynamical fuzzy spaces, and provide background independent theories of space. Their classical solutions correspond to classical background spaces, and the small fluctuations around them can be regarded as fluctuations of fields on them. In this paper, I numerically study a tensor model around its classical backgrounds of two-, three- and four-dimensional fuzzy flat tori and show that the properties of low-lying low-momentum modes are in clear agreement with general relativity. Numerical analysis also suggests that the lowest-order effective action is composed of curvature square terms, which is consistent with general relativity in view of the form of the considered action of the tensor model.
TL;DR: In this paper, the authors examine the claim of Babak and Grishchuk to have solved the problem of localizing the energy and momentum of the gravitational field and conclude that this object has no physical significance.
Abstract: We examine the claim of Babak and Grishchuk [S. V. Babak and L. P. Grishchuk, Phys. Rev. D 61, 024038 (1999)] to have solved the problem of localizing the energy and momentum of the gravitational field. After summarizing Grishchuk's flat-space formulation of gravity, we demonstrate its equivalence to general relativity at the level of the action. Two important transformations are described (diffeomorphisms applied to all fields, and diffeomorphisms applied to the flat-space metric alone) and we argue that both should be considered gauge transformations: they alter the mathematical representation of a physical system, but not the system itself. By examining the transformation properties of the Babak-Grishchuk gravitational energy-momentum tensor under these gauge transformations (infinitesimal and finite) we conclude that this object has no physical significance.
TL;DR: In this article, a method to find the exact solutions of the Einstein's field equations by using which they construct time-periodic solutions was developed, and some new physical phenomena, such as the time periodic event horizon, were found.
Abstract: In this paper, we develop a new method to find the exact solutions of the Einstein's field equations by using which we construct time-periodic solutions. The singularities of the time-periodic solutions are investigated and some new physical phenomena, such as the time-periodic event horizon, are found. The applications of these solutions in modern cosmology and general relativity are expected.
Book•
01 Dec 2008
TL;DR: The Tully-Fisher law is obtained directly from the theory, and thus it is found that there is no necessity to assume the existence of dark matter in the halo of galaxies, nor in galaxy clusters as mentioned in this paper.
Abstract: This book describes Carmeli's cosmological general and special relativity theory, along with Einstein's general and special relativity. These theories are discussed in the context of Moshe Carmeli's original research, in which velocity is introduced as an additional independent dimension. Four- and five-dimensional spaces are considered, and the five-dimensional braneworld theory is presented. The Tully–Fisher law is obtained directly from the theory, and thus it is found that there is no necessity to assume the existence of dark matter in the halo of galaxies, nor in galaxy clusters.The book gives the derivation of the Lorentz transformation, which is used in both Einstein's special relativity and Carmeli's cosmological special relativity theory. The text also provides the mathematical theory of curved spacetime geometry, which is necessary to describe both Einstein's general relativity and Carmeli's cosmological general relativity. A comparison between the dynamical and kinematic aspects of the expansion of the universe is made. Comparison is also made between the Friedmann–Robertson–Walker theory and the Carmeli theory. And neither is it necessary to assume the existence of dark matter to correctly describe the expansion of the cosmos.
TL;DR: In this paper, the field equations for perfect fluid coupled with massless scalar field are solved with two conditions p=ρ and R=AS n for Kantowski-Sachs space time in general theory of relativity.
Abstract: The field equations for perfect fluid coupled with massless scalar field are solved with two conditions p=ρ and R=AS n for Kantowski-Sachs space time in general theory of relativity. Various physical and geometrical properties of the model have also been discussed.
TL;DR: In this article, the authors study the evolution of homogeneous and isotropic, flat cosmological models within the general scalar-tensor theory of gravity with arbitrary coupling function and potential and scrutinize its limit to general relativity.
Abstract: We study the evolution of homogeneous and isotropic, flat cosmological models within the general scalar-tensor theory of gravity with arbitrary coupling function and potential and scrutinize its limit to general relativity. Using the methods of dynamical systems for the decoupled equation of the Jordan frame scalar field we find the fixed points of flows in two cases: potential domination and matter domination. We present the conditions on the mathematical form of the coupling function and potential which determine the nature of the fixed points (attractor or other). There are two types of fixed points, both are characterized by cosmological evolution mimicking general relativity, but only one of the types is compatible with the Solar System PPN constraints.
TL;DR: In this article, the generation method developed by Newman et al. which creates from spherically symmetric solutions of the field equations by a complex coordinate transformation axisymmetric solutions is extended to five-dimensional relativity.
Abstract: The generation method developed by Newman et al. which creates from spherically symmetric solutions of the field equations by a complex coordinate transformation axisymmetric solutions is extended to five-dimensional relativity.
TL;DR: The discrepancy in General Relativity between its general statements and the vacuum singular solutions of Einstein's equations is analyzed in this article, where the discrepancy between the general statements of Relativity and the solutions of the singular solution of the equations is investigated.
Abstract: The discrepancy in General Relativity between its general statements and the vacuum singular solutions of Einstein’s equations is analyzed.
TL;DR: In this paper, the authors evaluated the energy-momentum distributions of Yilmaz-Rosen metric in general relativity (GR) and teleparallel gravity (TG) using the energy momentum definitions of the Einstein, Bergmann-Thomson, Landau-Lifshitz and Moller.
Abstract: In this paper, using the energy momentum definitions of the Einstein, Bergmann-Thomson, Landau-Lifshitz and Moller in general relativity (GR) and teleparallel gravity (TG), we have evaluated the energy-momentum distributions of Yilmaz-Rosen metric. We have obtained that these different energy-momentum definitions give different results in GR and TG. Furthermore these results are same in different gravitation theories and we get that both general relativity and teleparallel gravity are equivalent theories for Einstein, Bergmann-Thomson and Landau-Lifshitz prescriptions. Also, while the Moller energy definitions are same and zero but the momentum prescriptions are disagree in GR and TG.
Posted Content•
TL;DR: In this article, the authors give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer.
Abstract: We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivation for this choice using light pressure. Mathematics Subject Classification (2000) : 83C05, 83C40, 83C99.
TL;DR: In this paper, the elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is analyzed and principal directions associated with this tensor are compared with eigendirections of the material metric.
Abstract: The elasticity difference tensor, used in [1] to describe elasticity properties of a continuous medium filling a space-time, is here analysed. Principal directions associated with this tensor are compared with eigendirections of the material metric. Examples concerning spherically symmetric and axially symmetric space-times are then presented.
01 Jan 2008
TL;DR: In this article, the origin of inertia is explained without violating Einstein's two postulates that form the basis for Special Relativity, and the relativistic momentum becomes a property of curved spacetime during acceleration, and Newton's second law of motion is derived from a line element in General Relativity.
Abstract: A proposed theory explains the origin of Inertia without violating Einstein’s two postulates that form the basis for Special Relativity. The new model agrees with observational aspects of Special Relativity and is compatible with General Relativity. The relativistic momentum becomes a property of curved spacetime during acceleration, and Newton’s second law of motion is derived from a line-element in General Relativity. The new model unambiguously resolves the Twin Paradox, since aging always progresses at the same pace, and it admits an absolute temporal reference.
TL;DR: In this paper, the fundamental properties of the generalized Einstein's Lagrangian density for a gravitational system, the theoretical foundations of the modified Einstein's field equations and the Lorentz and Levi-Civita's conservation laws are systematically studied.
Abstract: Through discussions on the fundamental properties of the generalized Einstein’s Lagrangian density for a gravitational system, the theoretical foundations of the modified Einstein’s field equations and the Lorentz and Levi-Civita’s conservation laws are systematically studied The theory of cosmology founded on them is discussed in detail and some new properties and new effects of the cosmos are deduced; these new properties and new effects could be tested via future experiments and observations
TL;DR: In this paper, it was shown that the Reissner-nordstrom solution to the Einstein field equations tells us that charge, like mass, has a unique space-time signature.
Abstract: Charge, like mass in Newtonian mechanics, is an irreducible element of electromagnetic theory that must be introduced ab initio. Its origin is not properly a part of the theory. Fields are then defined in terms of forces on either masses—in the case of Newtonian mechanics, or charges in the case of electromagnetism. General Relativity changed our way of thinking about the gravitational field by replacing the concept of a force field with the curvature of space-time. Mass, however, remained an irreducible element. It is shown here that the Reissner-Nordstrom solution to the Einstein field equations tells us that charge, like mass, has a unique space-time signature.