Showing papers on "Mathematics of general relativity published in 1999"
Journal Article•
TL;DR: In this paper, the authors focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $
abla g$ does not---it carries the ''gravitational force''.
161 citations
TL;DR: In this article, different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the ''first'' kind, corresponding to spacetimes admitting a homothetic vector.
Abstract: The different kinds of self-similarity in general relativity are discussed, with special emphasis on similarity of the `first' kind, corresponding to spacetimes admitting a homothetic vector. We then survey the various classes of self-similar solutions to Einstein's field equations and the different mathematical approaches used in studying them. We focus mainly on spatially homogenous and spherically symmetric self-similar solutions, emphasizing their possible roles as asymptotic states for more general models. Perfect fluid spherically symmetric similarity solutions have recently been completely classified, and we discuss various astrophysical and cosmological applications of such solutions. Finally, we consider more general types of self-similar models.
157 citations
TL;DR: The field equations of general relativity are applied to pressure-free spherically symmetrical systemsof particles in this paper, and the equations of motion are determined without the use of approximations and are compared with the Newtonian equations.
Abstract: The field equations of general relativity areapplied to pressure-free spherically symmetrical systemsof particles The equations of motion are determinedwithout the use of approximations and are compared with the Newtonian equations The total energyis found to be an important parameter, determining thegeometry of 3-space and the ratio of effectivegravitating to invariant mass The Doppler shift isdiscussed and is found to contain both the velocity shiftand the Einstein shift combined in a rather complexexpression
126 citations
Book•
01 Jan 1999TL;DR: In the early 20th century, Minkowski, Mathematicians and the Mathematical Theory of Relativity (MTL) were involved in the search for Gravitational Absorption in the early Twentieth Century.
Abstract: I Relativity in the Making- The Search for Gravitational Absorption in the Early Twentieth Century- Minkowski, Mathematicians, and the Mathematical Theory of Relativity- Heuristics and Mathematical Representation in Einstein's Search for a Gravitational Field Equation- Rotation as the Nemesis of Einstein's Entwurf Theory- II Relativity at Work- Einstein, Relativity and Gravitation Research in Vienna before 1938- Controversies in the History of the Radiation Reaction Problem in General Relativity- The Penrose-Hawking Singularity Theorems: History and Implications- III Relativity at Large- The Cosmological Woes of Newtonian Gravitation Theory- Genesis and Evolution of Weyl's Reflections on De Sitter's Universe- Milne, Bondi and the 'Second Way' to Cosmology- Steady-State Cosmology and General Relativity: Reconciliation or Conflict?- IV Relativity in Debate- Larmor versus General Relativity- Kretschmann's Analysis of Covariance and Relativity Principles- Point Coincidences and Pointer Coincidences: Einstein on the Invariant Content of Space-Time Theories- Contributors
111 citations
TL;DR: In this paper, the authors provide a clear and concise introduction to the theory of general relativity, suitable for final-year undergraduate mathematics or physics students, where the emphasis is on the geometric structure of spacetime rather than the traditional coordinate-dependent approach.
Abstract: Starting with the idea of an event and finishing with a description of the standard big-bang model of the Universe, this textbook provides a clear and concise introduction to the theory of general relativity, suitable for final-year undergraduate mathematics or physics students. Throughout, the emphasis is on the geometric structure of spacetime, rather than the traditional coordinate-dependent approach. Topics covered include flat spacetime (special relativity), Maxwell fields, the energy-momentum tensor, spacetime curvature and gravity, Schwarzschild and Kerr spacetimes, black holes and singularities, and cosmology. All physical assumptions are clearly spelled out and the necessary mathematics is developed along with the physics. Exercises are provided at the end of each chapter and key ideas are illustrated with worked examples. Solutions and hints to selected problems are provided at the end of the book. This textbook will enable the student to develop a sound understanding of the theory of general relativity.
60 citations
TL;DR: In this paper, it was shown that under particular circumstances a general relativistic spherically symmetric bounded distribution of matter could satisfy a non-local equation of state, which describes, at a given point, the components of the corresponding energy-momentum tensor not only as a function at that point, but as a functional throughout the enclosed configuration.
Abstract: We show that under particular circumstances a general relativistic spherically symmetric bounded distribution of matter could satisfy a non-local equation of state. This equation describes, at a given point, the components of the corresponding energy-momentum tensor not only as a function at that point, but as a functional throughout the enclosed configuration. We have found that these types of dynamic bounded matter configurations, with constant compactness or gravitational potentials at the surface, admit a conformal Killing vector field and fulfil the energy conditions for anisotropic imperfect fluids. We present several analytical and numerical models satisfying these equations of state which collapse as reasonable radiating anisotropic spheres in general relativity.
49 citations
Book•
19 Oct 1999
TL;DR: Local theory of space and time, and analysis on manifolds, from a global point of view, were studied in this article, where the singularity theorems were proved.
Abstract: Local theory of space and time.- Analysis on manifolds.- Space and time from a global point of view.- Pseudo-Riemannian manifolds.- General relativity.- Robertson-Walker cosmology.- Spherical symmetry.- Causality.- Singularity theorems.
46 citations
TL;DR: In this article, a relativistic, stationary, rigidly rotating disk was obtained using the full equations and the approximate approach suggested by Wilson and Mathews, and the Wilson-Mathews method has about the same accuracy as the first post-Newtonian approximation.
Abstract: Treating problems in full general relativity is highly complex and frequently approximate methods are employed to simplify the solution. We present comparative solutions of an infinitesimally thin relativistic, stationary, rigidly rotating disk obtained using the full equations and the approximate approach suggested by Wilson and Mathews. We find that the Wilson-Mathews method has about the same accuracy as the first post-Newtonian approximation.
29 citations
TL;DR: The Kaluza-Klein approach to general relativity of 4-dimensional spacetimes was introduced in this paper, where the metric coefficients can be decomposed into sets of fields, which transform as a tensor field, gauge fields, and scalar fields with respect to the infinite dimensional group of the diffeomorphisms of the 2-dimensional fibre space.
29 citations
TL;DR: In this article, a new numerical scheme was developed to obtain quasiequilibrium structures of binary neutron star systems and nonaxisymmetric compact stars as well as the space time around those systems in general relativity.
Abstract: We develop a new numerical scheme to obtain quasiequilibrium structures of binary neutron star systems and nonaxisymmetric compact stars as well as the space time around those systems in general relativity. Although, strictly speaking, there are no equilibrium states for binary configurations in general relativity, the time scale of changes in orbital motion due to gravitational wave radiation is long compared with the orbital period. Thus, we can assume that binary neutron star systems, and nonaxisymmetric systems in general are in ``quasiequilibrium'' states. Concerning the quasiequilibrium states of binary systems in general relativity, several investigations have been already carried out by assuming conformal flatness of the spatial part of the metric. However, the validity of the conformally flat treatment has not been fully analyzed except for axisymmetric configurations. Therefore, it is desirable to solve for the quasiequilibrium states by developing totally different methods from the conformally flat scheme. In this paper, we present a new numerical scheme to solve the Einstein equations for three-dimensional configurations directly, without assuming conformal flatness, although we use the simplified metric for the space time. This new formulation is an extension of the scheme which has been successfully applied for structures of axisymmetric rotating compact stars in general relativity. It is based on the integral representation of the Einstein equations, and takes into account the boundary conditions at infinity. We have checked our numerical scheme by computing equilibrium sequences of binary polytropic star systems in Newtonian gravity and those of axisymmetric polytropic stars in general relativity. We have applied this numerical code to binary star systems in general relativity and have succeeded in obtaining several equilibrium sequences of synchronously rotating binary polytropes with the polytropic indices $N=0.0,$ $0.5,$ and $1.0.$ It should be noted that our equilibrium sequences are not those of constant baryon mass star models because there is no unique choice of parameters to keep the baryon mass constant for our polytropic relation.
TL;DR: In this article, it was shown that the theorem of Duff on the existence and uniqueness of solutions to linear characteristic initial value problems holds in the case of linearized characteristic evolution in Bondi-Sachs coordinates in general relativity.
Abstract: We show that the theorem of Duff on the existence and uniqueness of solutions to linear characteristic initial-value problems holds in the case of linearized characteristic evolution in Bondi-Sachs coordinates in general relativity. This represents the characteristic equivalent to the manifest existence and uniqueness of the case of standard Cauchy problems. This extends Sachs' original work on the characteristic approach to the Einstein equations, by considering a null-timelike approach rather than a null-asymptotic one.
Posted Content•
TL;DR: In this article, a model based on (3+1) dimensional general relativity in Riemannian spacetime is presented, which is explicit and calculable in details for a physical application, a computation formula for spacetime volume density correlations is presented.
Abstract: A surface theoretic view of non-perturbative quantum gravity as "spin-foams" was proposed by Baez. A possibility of constructing such a model was studied some time ago based on (2+1) dimensional general relativity as a reformulation of the Ponzano-Regge model in Riemannian spacetime. In the present work, a model based on (3+1) dimensional general relativity in Riemannian spacetime is presented. The construction is explicit and calculable in details. For a physical application, a computation formula for spacetime volume density correlations is presented. Remarks for further investigations are made.
TL;DR: In this paper, all homogeneous Lorentzian structures on the oscillator group equipped with a bi-invariant metric and the associated reductive pairs are obtained, and some of them are solutions of the Einstein-Yang-Mills equations.
Abstract: All of the homogeneous Lorentzian structures on the oscillator group equipped with a bi-invariant Lorentzian metric, and then the associated reductive pairs, are obtained. Some of them are solutions of the Einstein–Yang–Mills equations.
TL;DR: In this article, the Segre classification of the energy-momentum tensors associated with a fluid with anisotropic pressure and heat flux and a perfect fluid with electromagnetic field is presented.
Abstract: We present, in detail, the Segre classification of the energy-momentum tensors associated with: (a) a fluid with anisotropic pressure and heat flux, and (b) a perfect fluid with an electromagnetic field. These are the most widely used energy-momentum tensors in general relativity. We discuss algebraic consistency and how it may be used to simplify the process of obtaining exact solutions to Einstein's field equations. Further, we shed considerable light on the meaning of syzygies amongst the Riemann invariants.
TL;DR: In this article, it was shown that a spacetime admitting Lie-group actions may be disjointly decomposed into a closed subset with no interior plus a dense finite union of open sets in each of which the character and dimension of the group orbits as well as the Petrov type are constant.
Abstract: Following a recent work in which it is shown that a spacetime admitting Lie-group actions may be disjointly decomposed into a a closed subset with no interior plus a dense finite union of open sets in each of which the character and dimension of the group orbits as well as the Petrov type are constant, the aim of this work is to include the Segre types of the Ricci tensor (and hence of the Einstein tensor) into the decomposition. We also show how this type of decomposition can be carried out for any type of property of the spacetime depending on the existence of a continuous endomorphism.
Posted Content•
TL;DR: In this paper, a self-contained introduction to holomorphic ideas in general relativity is given, focusing on complex manifolds, spinor and twistor methods, and heaven spaces.
Abstract: An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven spaces.
01 Jan 1999
TL;DR: Wave maps from a pseudo-Riemannian manifold of hyperbolic (Lorentzian) signature (V, g) of scalar functions on (V and g) are the generalization of the usual wave equations for scalar function on V, g as discussed by the authors.
Abstract: Wave maps from a pseudo-Riemannian manifold of hyperbolic (Lorentzian) signature (V, g) into a pseudo-Riemannian manifold are the generalization of the usual wave equations for scalar functions on (V, g) They are the counterpart in hyperbolic signature of the harmonic mappings between properly Riemannian manifolds The first wave maps to be considered in physics were the σ-models, eg, the mapping from the Minkowski spacetime into the 3-sphere which models the classical dynamics of 4-meson fields linked by the relation
$$ \sum\limits_{a = 1}^4 {|{f_a}} {|^2} = 1$$
TL;DR: In this paper, it was shown that under somewhat different conditions on the curvature, the b-boundary will be non-Hausdorff, and illustrate the degeneracy by applying the conditions to some well known exact solutions of general relativity.
Abstract: The b-boundary construction by B. Schmidt is a general way of providing a boundary to a manifold with connection [12]. It has been shown to have undesirable topological properties however. C. J. S. Clarke gave a result showing that for space-times, non-Hausdorffness is to be expected in general [3], but the argument contains some errors. We show that under somewhat different conditions on the curvature, the b-boundary will be non-Hausdorff, and illustrate the degeneracy by applying the conditions to some well known exact solutions of general relativity.
Posted Content•
TL;DR: In this article, a new method to generate rotating solutions of the Einstein-Maxwell equations from static solutions is presented, and several examples of its application are discussed, as well as its general properties.
Abstract: I present a new method to generate rotating solutions of the Einstein-Maxwell equations from static solutions, give several examples of its application, and discuss its general properties.
TL;DR: In this paper, a new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time.
Abstract: A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold.
01 Jan 1999
TL;DR: In this paper, a new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as physicM space-time.
Abstract: A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory based on a five-dimensional manifold as the physicM space-time. The velocity space of moving particles in the model remains four-dimensionM as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold.
TL;DR: In this article, the authors developed the general theory of relativity in a formalism with extended causality that describes physical interaction through discrete, transversal and localized point-like fields.
Abstract: We develop the general theory of relativity in a formalism with extended causality that describes physical interaction through discrete, transversal and localized pointlike fields. The essence of this approach is of working with fields defined with support on straight lines and not on hypersurfaces as usual. The general relativity homogeneous field equations are then solved for a finite, singularity-free, point-like field that we associate with a `classical graviton'. The standard Einstein continuous formalism is retrieved by means of an averaging process, and its continuous solutions are determined by the chosen imposed symmetry. The Schwarzschild metric is obtained by imposing spherical symmetry on the averaged field.
Posted Content•
TL;DR: In this article, the authors discuss extreme alternative (the Gauss's and the Coulomb's) ways of obtaining and interpreting the averaged fields, how they depend on the kind of measurements involved, and how they fit with the experimental data.
Abstract: In a previous work General Relativity has been presented as a microscopic theory of finite and discrete point-like fields that we associate to a classical description of gravitons. The standard macroscopic continuous field is retrieved as an average-valued field through an integration over these gravitons. Here we discuss extreme alternative (the Gauss's and the Coulomb's) ways of obtaining and interpreting the averaged fields, how they depend on the kind of measurements involved, and how do they fit with the experimental data. The field measurements in the classical tests of general relativity correspond to the Coulomb's mode whereas the determination of the overall spacetime curvature in a cosmological scale is clearly a Gauss's mode. As a natural consequence there is no missing mass and, therefore, no such a need of dark mass as the value predicted by General Relativity, in the context of the Gauss's mode, agrees with the observed one.
TL;DR: In this paper, a linear approximate theory of parametrized gravitation in the spirit of the flat spacetime approach to general relativity was developed, where the Hamiltonian constraint for τ is removed from the Euler-Lagrange equations.
Abstract: In a framework describing manifestly covariant relativistic evolution using a scalar time τ, consistency demands that τ-dependent fields be used. In recent work by the authors, general features of a classical parametrized theory of gravitation, paralleling general relativity where possible, were outlined. The existence of a preferred “time” coordinate τ changes the theory significantly. In particular, the Hamiltonian constraint for τ is removed From the Euler-Lagrange equations. Instead of the 5-dimensional stress-energy tensor, a tensor comprised of 4-momentum density mid flux density only serves as the source. Building on that foundation, in this paper we develop a linear approximate theory of parametrized gravitation in the spirit of the flat spacetime approach to general relativity. Using a modified form of Kraichnan's flat spacetime derivation of general relativity, we extend the linear theory to a family of nonlinear theories in which the flat metric and the gravitational field coalesce into a single effective curved metric.
Book•
01 Jan 1999
TL;DR: The immortal Bel Robinson tensor, S. Deser relativistic quasiparticles, a covariant approach, R. Hakim and H.D. Sivak successive approximations to the regular order reduction of singular equations, J.M. Herrero and J.A. Mars as mentioned in this paper, E. Ruiz alternative statistical analysis of CMB maps, F.L. Sanz cosmological parameters, N. Dadhich radial conformal motions, A.
Abstract: The immortal Bel Robinson tensor, S. Deser relativistic quasiparticles, a covariant approach, R. Hakim and H.D. Sivak successive approximations to the regular order reduction of singular equations, J.M. Aguirregabiria et al a universal law of gravitational deformation for general relativity, B. Coll an attempt to derive Dirac's equation for an electron in an external electromagnetic field from a classical dynamical system, E. Ruiz alternative statistical analysis of CMB maps, J.L. Sanz cosmological parameters, F. Atrio-Barandela electrogravity duality in general relativity, N. Dadhich radial conformal motions, A. Herrero and J.A. Morales divergence-free vector fields in stationary and axially symmetric vacuum spacetimes, M. Mars. (Part contents)