Showing papers on "Introduction to the 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
137 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, it is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects.
Abstract: A simple example is given to show that the gauge equivalence classes of physical states in Chern-Simons theory are not in one-to-one correspondence with those of Einstein gravity in three spacetime dimensions. The two theories are therefore not equivalent. It is shown that including singular metrics into general relativity has more, and in fact a quite counter-intuitive, impact on the theory than one naively expects.
94 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
TL;DR: In this paper, the pseudo-local gravitoelectromagnetic stress-energy tensor for an arbitrary gravitational field within the framework of general relativity was studied and it was shown that there exists a current of gravitational energy around a rotating mass.
Abstract: We study the pseudo-local gravitoelectromagnetic stress-energy tensor for an arbitrary gravitational field within the framework of general relativity. It is shown that there exists a current of gravitational energy around a rotating mass. This gravitational analogue of the Poynting flux is evaluated for certain classes of observers in the Kerr field.
43 citations
TL;DR: The results have provided endless fascination and puzzlement to the general public, and have had an enormous impact on our conceptual framework for understanding nature as discussed by the authors, which has been the most profound conceptual advance in 20th century physics.
Abstract: Except for quantum mechanics—a more than modest exception—relativity has been the most profound conceptual advance in 20th century physics. Both in developing special and general relativity, Albert Einstein’s hallmark was to anchor his theory on a few simple but profound principles. The results have provided endless fascination and puzzlement to the general public, and have had an enormous impact on our conceptual framework for understanding nature.
39 citations
TL;DR: Cactus as mentioned in this paper is a 3D multi-purpose parallel code for general relativistic astrophysics, which can be used to simulate colliding black holes and neutron stars, as well as accelerate the complete set of Einstein's equations for the first time.
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.
TL;DR: In this article, it was shown that in general, finite perturbations in the gravitational field travel no faster than light, and that it is impossible to construct a warp drive as considered by Alcubierre (1994 The warp drive: hyper-fast travel within general relativity Class. 11 L73-7) in the absence of exotic matter.
Abstract: Some standard results on the initial value problem of general relativity in matter are reviewed. These results are applied first to show that in a well defined sense, finite perturbations in the gravitational field travel no faster than light, and second to show that it is impossible to construct a warp drive as considered by Alcubierre (1994 The warp drive: hyper-fast travel within general relativity Class. Quantum Grav. 11 L73-7) in the absence of exotic matter.
Book•
17 Aug 1999
TL;DR: The Relativity Principle and its applications in Newtonian physics can be found in this article, where it is used to describe the four-momentum conservation using invariant intervals and space-time diagrams.
Abstract: Preliminaries. The Relativity Principle, and its Applications in Newtonian Physics. Einstein's Relativity Principle. KINEMATICS. Lorentz Transformations. Invariant Intervals and Space-Time Diagrams. Proper Time and Nonuniform Motion. Four-Vectors. Four-Acceleration. MOMENTUM AND ENERGY. Particle Dynamics: Momentum and Energy. Natural Units, and the Prevalence of MeV. Systems of Particles: Four-Momentum Conservation using Invariants. WAVES. Plane Waves. Light Waves in Empty Space: Aberration and Doppler Effect. Appendices. Problems. Index.
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.
Posted Content•
TL;DR: In this article, a re-examination of the definition of a true gravitational energy tensor has been carried out, due to its unquestionable logical soundness and to the unique manner of propagation for gravitational energy that it entails.
Abstract: Foreword. While most textbooks of general relativity and research articles discuss at length the relative merits of the pseudo tensors proposed by Einstein and by other authors for representing the energy of the gravitational field, Levi Civita's definition of a true gravitational energy tensor has succumbed to the authority of Einstein and is nearly forgotten. It seems however worthy of a careful re-examination, due to its unquestionable logical soundness and to the unique manner of propagation for gravitational energy that it entails.
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.
TL;DR: In this paper, the Runge-Lenz vector was used to derive the results for precession of particle orbits and bending of null rays in the case of spherically symmetric Schwarzschild geometry.
Abstract: The standard General Relativity results for precession of particle orbits and for bending of null rays are derived as special cases of perturbation of a quantity that is conserved in Newtonian physics, the Runge–Lenz vector. First, this method is applied to give a derivation of these General Relativity effects for the case of the spherically symmetric Schwarzschild geometry. Then the lowest order correction due to an angular momentum of the central body is considered. The results obtained are well known, but the method used is rather more efficient than that found in the standard texts, and it provides a good occasion to use the Runge–Lenz vector beyond its standard applications in Newtonian physics.
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, the inspiral of a binary system of compact objects due to gravitational radiation is investigated using the toy model of two infinitely long lines of mass moving in a fixed circular orbit.
Abstract: The inspiral of a binary system of compact objects due to gravitational radiation is investigated using the toy model of two infinitely long lines of mass moving in a fixed circular orbit. The two Killing fields in the toy model are used, according to a formalism introduced by Geroch, to describe the geometry entirely in terms of a set of tensor fields on the two-manifold of Killing vector orbits. Geroch's derivation of the Einstein equations in this formalism is streamlined and generalized. The explicit Einstein equations for the toy model spacetime are derived in terms of the degrees of freedom which remain after a particular choice of gauge.
Posted Content•
TL;DR: The significance of past-pointing four-vectors and negative energies in general relativity is discussed in this article, where it is shown that the description of the interaction of past pointing and future pointing matter requires two metric tensors for self-consistency.
Abstract: The significance of past-pointing four-vectors and negative energies in general relativity is discussed. The sign of the energy is not absolute, but relative to the four-velocity of the observer, and every particle/observer always measures its own mass as positive. It is shown that the description of the interaction of past-pointing and future-pointing matter in general relativity requires two metric tensors for self-consistency. This aspect of general relativity might account for the observations that led to the proposal of "dark energy" and non-baryonic "dark matter".
TL;DR: In this article, a formulation of the Einstein equations is presented that could yield advantages in the study of collisions of binary compact objects during regimes between linear-nonlinear transitions, and the key idea behind this formulation is a separation of the dynamical variables into (i) a fixed conformal 3-geometry, (ii) a conformal factor possessing nonlinear dynamics and (iii) transverse-traceless perturbations of the conformal geometry.
Abstract: A formulation of Einstein equations is presented that could yield advantages in the study of collisions of binary compact objects during regimes between linear-nonlinear transitions. The key idea behind this formulation is a separation of the dynamical variables into (i) a fixed conformal 3-geometry, (ii) a conformal factor possessing nonlinear dynamics and (iii) transverse-traceless perturbations of the conformal 3-geometry.
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
$$ \sum\limits_{a = 1}^4 {|{f_a}} {|^2} = 1$$
Book•
01 Jan 1999
TL;DR: In this article, the authors provide a survey of differential geometry with zero Ricci curvature, rigidity and compactness of Einstein metrics, general relativity, stability of Minkowski space-time, and more.
Abstract: This is the sixth volume in a series providing surveys of differential geometry. It addresses: Einstein manifolds with zero Ricci curvature; rigidity and compactness of Einstein metrics; general relativity; the stability of Minkowski space-time; and more.
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.
Posted Content•
TL;DR: In this article, the behavior of model clocks and measuring rods can be derived directly from the field equations of General Relativity using the Einstein-Infeld-Hoffmann (EIH) approiximation procedure.
Abstract: The nexus between the gravitational field and the space-time metric was an essential element in Einstein's development of General Relativity and led him to his discovery of the field equations for the gravitational field/metric. I will argue here that the metric is in fact an inessential element of this theory and can be dispensed with entirely. Its sole function in the theory was to describe the space-time measurements made by ideal clocks and rods. However, the behavior of model clocks and measuring rods can be derived directly from the field equations of General Relativity using the Einstein-Infeld-Hoffmann (EIH) approiximation procedure. Therefore one does not need to introduce these ideal clocks and rods and hence has no need of a metric.