Showing papers on "Introduction to the mathematics of general relativity published in 1992"
Book•
01 Jan 1992
TL;DR: In this paper, the authors discuss the applications of general relativity in astrophysics and cosmology but who would like to avoid mathematical complications, and provide an introduction to the subject that will enable students to consult more detailed treatises as well as current literature.
Abstract: This text is intended for students interested in the applications of general relativity in astrophysics and cosmology but who would like to avoid mathematical complications. It combines relativity, astrophysics and cosmology in a single volume. It provides an introduction to the subject that will enable students to consult more detailed treatises as well as the current literature. For prospective researchers in these fields, the book includes an appendix on differential forms, and an extensive list of references. The book is divided into three parts. The section on general relativity 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, Penrose processes and similar topics), and considers the energy-momentum tensor for various solutions. The section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects. The section on cosmology discusses various cosmological models, observational tests, and scenarios for the early universe.
60 citations
TL;DR: In this paper, a covariant formalism enabling the possibility of change of signature in classical General Relativity, when the geometry is that of a Robertson-Walker universe, is presented.
Abstract: This paper gives a covariant formalism enabling investigation of the possibility of change of signature in classical General Relativity, when the geometry is that of a Robertson-Walker universe. It is shown that such changes are compatible with the Einstein field equations, both in the case of a barotropic fluid and of a scalar field. A criterion is given for when such a change of signature should take place in the scalar field case. Some examples show the kind of resulting exact solutions of the field equations.
57 citations
TL;DR: In this article, it was shown that in many cases he was extremely curious about certain experimental results and that he could hardly wait for the moment when tests which he had suggested were actually done by skilled observers.
43 citations
TL;DR: In this paper, a new class of generally covariant gauge theories in four space-time dimensions is investigated, where field variables are taken to be a Lie algebra valued connection 1-form and a scalar density.
41 citations
TL;DR: In this article, a purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebras but not vice versa.
Abstract: A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the spacetime of a straight cosmic string with quasiregular singularity is constructed.
40 citations
Book•
30 Jan 1992
TL;DR: In this paper, a review of phenomena by the Einstein theory of gravitation gravitational waves great numbers, gravitation versus quantum phenomena cosmology the Einstein-Cartan theory, and the principle of equivalence geometric foundations of the general theory of relativity are discussed.
Abstract: Physical phenomena, models, theories Galilean spacetime in search of the ether predictions of the theory of relativity and their experimental verification Minkowski geometry the Lorentz group and the shape of bodies in motion particles and fields in special relativity theory spinors Newtonian theory of gravitation and the principle of equivalence geometric foundations of the general theory of relativity Einstein equations some aspects of the general relativity theory algebraic classification of gravitational fields review of phenomena by the Einstein theory of gravitation gravitational waves great numbers, gravitation versus quantum phenomena cosmology the Einstein-Cartan theory.
36 citations
TL;DR: In this paper, an exact solution of the gravitational and electromagnetic field equations with a charged rotating source in new general relativity was given, which has three parameters Q, h and a, and it gives a charged Kerr metric space-time.
Abstract: We give an exact solution of the gravitational and electromagnetic field equations with a charged rotating source in new general relativity. The solution has three parameters Q, h and a, and it gives a charged Kerr metric space-time. The parallel vector fields and the electromagnetic vector poten- " tial are axially symmetric. In this space-time, we cannot discriminate new general relativity "from general relativity, so far as scalar, the Dirac and the Yang-Mills fields and macroscopic bodies are used as probes. The space-time does not have singularities at all, although it has an "effective singularity". Two kinds of Reissner-Nordstrom metric solutions, one is our solution with h=O and the other is a solution given by Hayashi and Shirafuji, are physically equivalent with each other. Nevertheless, these are markedly different from each other with regard to the asymptotic behavior of the torsion tensor for r -+ 00 and the space-time singularities.
20 citations
TL;DR: The Lagrangian method to generate conserved currents in field theories starting from the so-called Poincare-Cartan form is reviewed in this article, with examples of application to general relativity.
Abstract: The general 'Lagrangian' method to generate conserved currents in field theories starting from the so-called Poincare-Cartan form is reviewed, with examples of application to general relativity.
17 citations
TL;DR: The current status of a nonperturbative canonical quantisation of general relativity is summarized in this article, where the authors present a survey of the current state of the canonical quantization.
Abstract: The current status of a programme for nonperturbative canonical quantisation of general relativity is briefly summarized.
13 citations
TL;DR: A brief review of theory and experimental work in general relativity can be found in this article, where the author goes from Galileo's work on gravity to Newton's work and on to Einstein's Equivalence Principle.
Abstract: The author gives a brief review of theory and experimental work in general relativity. He goes from Galileo's work on gravity to Newton's work and on to Einstein's Equivalence Principle.
12 citations
TL;DR: The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented in this article, which are the holonomy operator associated with special paths and the light cone cut function.
01 Jan 1992
TL;DR: The subject of classical field theory has received relatively little attention from both mathematicians and theoretical physicists as discussed by the authors, namely, from the perspective of Quantum Mechanics, namely, as subject of the elusive task of quantization.
Abstract: The subject of Classical Field Theory has received relatively little attention from both mathematicians and theoretical physicists. The former look at it primarily from the perspective of Quantum Mechanics, namely, as subject of the elusive task of quantization. Mathematicians, on the other hand, never quite took the underlying geometric structure of Classical Field Theory, the space-time, seriously. The geometers, though greatly emboldened by the success of Riemann’s visionary ideas in the formulation of General Relativity, have stayed away, with few notable exceptions, from the fundamental new twist given to them by Einstein who replaced the positive definite metric of Riemannian Geometry by a Lorentzian, or more appropriate, Einsteinian metric.
Posted Content•
TL;DR: A list of the major results achieved in the last six years in the program to construct quantum general relativity using the Ashtekar variables and the loop representation is given in this paper.
Abstract: I attempt to answer the question of the title by giving an annotated list of the major results achieved, over the last six years, in the program to construct quantum general relativity using the Ashtekar variables and the loop representation. A summary of the key open problems is also included. Also included are expositions of several new results including the construction of spatially diffeomorphism invariant observables constructed by coupling general relativity to matter fields.
01 Jan 1992
TL;DR: In this article, the authors review the present status of the null cone approach to numerical evolution being developed by the Pittsburgh group and demonstrate its effectiveness in revealing asymptotic physical properties of black hole formation in the gravitational collapse of a scalar field.
Abstract: We review the present status of the null cone approach to numerical evolution being developed by the Pittsburgh group. We describe the simplicity of the underlying algorithm as it applies to the global description of general relativistic spacetimes. We also demonstrate its effectiveness in revealing asymptotic physical properties of black hole formation in the gravitational collapse of a scalar field.
TL;DR: In this article, a comparative analysis of Einstein and Newton's spacetime theories is presented to show that spacetime is not introduced as an explanation of observable effects, but rather is defined through those effects in arguments like Newton's "water bucket" argument and Einstein's argument for special relativity.
Abstract: Einstein intended the general theory of relativity to be a generalization of the relativity of motion and, therefore, a radical departure from previous spacetime theories. It has since become clear, however, that this intention was not fulfilled. I try to explain Einstein's misunderstanding on this point as a misunderstanding of the role that spacetime plays in physics. According to Einstein, earlier spacetime theories introduced spacetime as the unobservable cause of observable relative motions and, in particular, as the cause of inertial effects of ‘absolute’ motion. I use a comparative analysis of Einstein and Newton to show that spacetime is not introduced as an explanation of observable effects, but rather is defined through those effects in arguments like Newton's ‘water bucket’ argument and Einstein's argument for special relativity. I then argue that to claim that a spacetime theory is true, or to claim that a spacetime structure is ‘real’, is not to claim that a theoretical object explai...
TL;DR: In this paper, exact solutions of Einstein's field equations for a conformally-invariant scalar field with trace-free energy-momentum tensor are presented for the Robertson-Walker models with K =+1, −1.
Abstract: Exact solutions of Einstein's field equations for a conformally-invariant scalar field with trace-free energy-momentum tensor is presented for the Robertson-Walker models withK=+1, −1. The physical properties of the solution are also studied.
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01 Jan 1992
TL;DR: In this article, the authors work out the relevant it-form-bit means to measure spacetime curvature and describe the essential new features of the knot description of gravity and the one index loop variable and the Einstein tensor.
Abstract: In this report the authors work out the relevant it-form-bit means to measure spacetime curvature. Also described are the essential new features of the knot description of gravity and the one index loop variable and the Einstein tensor. (LSP)
TL;DR: In this paper, the authors discuss the meaning of the canonical quantization of the fields in such reference systems, and discuss the significance of the singularity of the field fields in these reference systems.
Abstract: In general relativity the non-covariant ansatzAi=δ4i for the vectorpotentialAk gives the general solution of the Maxwell equations as four coordinate conditions which are the conditions of integrability of the Einstein equations. In the some sense the ansatzφ=X4 is a general solution of the scalar wave-equation in a reference system given by one coordinate-condition. We discuss the meaning of the canonical quantization of the fields in such reference systems.
01 Jan 1992
TL;DR: Chronometric-radial invariant (CRI) vectors of the electromagnetic field are introduced in the CRI formulation of the general theory of relativity proposed by Vladimirov.
Abstract: Chronometric-radial invariant (CRI) vectors of the electromagnetic field are introduced in the CRI formulation of the general theory of relativity proposed by Vladimirov. The general theory of relativity Maxwell equations for the electromagnetic field are transformed into equations for CRI quantities. It is postulated that the CRI components of vectors and tensors are the physically measurable quantities of the general theory of relativity.
TL;DR: In this article, three types of time derivatives of spatial geometrical quantities are considered in the framework of the general theory of relativity, and the form of equations of motion and their physical interpretation are determined by the type of time derivative that is employed.
Abstract: Three types of time derivatives of spatial geometrical quantities are considered in the framework of the general theory of relativity. The form of equations of motion (of spin precession and a geodesic) and their physical interpretation are determined by the type of time derivative that is employed.
TL;DR: In this paper, it was shown that in quantized general relativity one is led to Jordan-Fock type uncertainty relations implying the occurrence of cut-off lengths, and that these lengths represent limitations on the measurability of quantum effects of general relativity.
Abstract: It is demonstrated that in quantized general relativity one is led to Jordan-Fock type uncertainty relations implying the occurrence of cut-off lengths. We argue that these lengths (i) represent limitations on the measurability of quantum effects of general relativity and (ii) provide a cut-off length of quantum divergences.
Posted Content•
TL;DR: In this article, an exact solution of Einstein's field equations for a static spherically symmetric medium with a radially boost invariant energy-momentum tensor is presented.
Abstract: An exact solution of Einstein's field equations for a static spherically symmetric medium with a radially boost invariant energy-momentum tensor is presented. In the limit of an equation of state corresponding to a distribution of radially directed strings there is a $1/r$ correction to Newton's force law. At large distances and small accelerations this law coincides with the phenomenological force law invented by Milgrom in order to explain the flat rotation curves of galaxies without introducing dark matter. The present model explaines why the critical acceleration of Milgrom is of the same order of magnitude as the Hubble parameter.