Showing papers on "Introduction to the mathematics of general relativity published in 1983"
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TL;DR: In this paper, a hyperbolic system from the 3+1 system of Einstein equations with zero shift was obtained by combining the evolution equations with the constraints, and the lapse was chosen by a suitable choice of the lapse, which in a natural way is connected to the space metric.
Abstract: By a suitable choice of the lapse, which in a natural way is connected to the space metric, we obtain a hyperbolic system from the 3+1 system of Einstein equations with zero shift; this is accomplished by combining the evolution equations with the constraints.
80 citations
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TL;DR: In this paper, the authors provide a fairly concise introduction to the basic mathematical concepts of the general theory of relativity and their applications, at a level suitable for postgraduate students, covering Riemannian geometry and Einstein's theory of gravitation, gravitational waves, the classification of exact solutions of the Einstein equations, black holes and cosmology.
Abstract: Hans Stephani 1982 Cambridge: Cambridge University Press xvi + 298 pp price £25 This textbook provides a fairly concise introduction to the basic mathematical concepts of the general theory of relativity and their applications, at a level suitable for postgraduate students. It covers Riemannian geometry and Einstein's theory of gravitation, gravitational waves, the classification of exact solutions of the Einstein equations, black holes and cosmology.
78 citations
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TL;DR: In this article, the authors analyzed the Newtonian limit of the initial value problem set on a family of null cones and found a simple relationship between the gravitational null data (i.e., the shear of the null cones) and Newtonian gravitational potential.
Abstract: For a general relativistic ideal fluid, we analyze the Newtonian limit of the initial value problem set on a family of null cones. The underlying Newtonian structure is described using Cartan’s elegant space–time version of Newtonian theory and a limiting process rigorously based upon the velocity of light approaching infinity. We find that the existence of a Newtonian limit imposes a strikingly simple relationship between the gravitational null data (i.e., the shear of the null cones) and the Newtonian gravitational potential. This result has immediate application to numerical evolution programs for calculating gravitational radiation and might serve as the basis for a post‐Newtonian approximation scheme.
65 citations
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TL;DR: For a general class of space-times, it was shown in this paper that the componentsR of the curvature tensor determine the metric components up to a constant conformal factor.
Abstract: It is shown that for a very general class of space-times, the componentsR
of the curvature tensor determine the metric components up to a constant conformal factor This general class contains most of those cases which are usually considered to be interesting from the point of view of Einstein's general relativity theory The connection between the above result and the existence of proper curvature collineations is given
61 citations
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TL;DR: In this article, the general solution for a symmetric second-order tensor of the Riemann tensor is given in terms of the curvature 2-form structure of a space-time manifold.
Abstract: The general solution for a symmetric second-order tensorX of the equationX
e(a
R
e
b
cd=0 whereR is the Riemann tensor of a space-time manifold, andX is obtained in terms of the curvature 2-form structure ofR by a straightforward geometrical technique, and agrees with that given by McIntosh and Halford using a different procedure. Two results of earlier authors are derived as simple corollaries of the general theorem.
54 citations
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TL;DR: In this article, the basic geometry of twistor theory is introduced as it arises both from Minkowski space-time and the more general curved Einstein models, and it is shown how this provides a CR-structure (this being, in essence, another of Poincare's poioneering concepts) in a natural way.
Abstract: Space-time views leading up to Einstein's general relativity are described in relation to some of Poincare's early ideas on the subject. The basic geometry of twistor theory is introduced as it arises both from Minkowski space-time and the more general curved Einstein models. It is shown how this provides a CR-structure (this being, in essence, another of Poincare's poioneering concepts) in a natural way. Nonrealizable CR-structure can arise, and an example is presented, due to C. D. Hill, G. A. J. Sparling and the author, of a complex manifold-with-boundary which cannot be extended as a complex manifold beyond its C/sup infinity/ boundary.
41 citations
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TL;DR: In this article, a new class of exact solutions of Einstein's equations is presented which contains two arbitrary functions of time and one additional parameter, which can be used for spacetime.
Abstract: Although spherically symmetric expanding (or contracting) stars are a very important class of objects for the application of general relativity theory, only very few perfect fluid solutions of Einstein's equations have been found so far. A new class of exact solutions is presented which contains two arbitrary functions of time and one additional parameter.
39 citations
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TL;DR: In this paper, it was shown that if the unit tangent vector to any curve of the congruence is everywhere orthogonal to the 4-velocity field u a of a self-gravitating fluid, then observers comoving with the fluid can be employed along a curve of congruences if and only if the curves are material curves in the fluid.
Abstract: The theory of spacelike congruences in general relativity is briefly reviewed and the physical interpretation of the rotation tensorR a b , the expansion E, and the shear tensorS a b , of the curves is discussed. It is proved that if the unit tangent vector to any curve of the congruence is everywhere orthogonal to the 4‐velocity field u a of a self‐gravitating fluid, then observers comoving with the fluid can be employed along a curve of the congruence if and only if the curves are material curves in the fluid. A congruence of vortex lines is studied in detail. Starting from the Ricci identity for u a and using Einstein’s equations, general expressions in terms of the kinematic quantities and fluid variables are derived for R a b , C, and S a b for a vortex congruence. It is found that E and S a b depend explicitly on the gravitational field only through the magnetic part of the Weyl tensor, and R a b only through a term proportional to the total energy flux q a derived from Einstein’s equations. With the aid of Maxwell’sequations, properties of congruences of magnetic field lines, electric field lines, and a certain combination of vortex and magnetic field lines are determined. For a congruence of magnetic field lines in an electrically conducting fluid and assuming the magnetohydrodynamic approximation of vanishing electric field, it is proved that for a comoving observer, R a b =0 if and only if the conduction current in the fluid is orthogonal to the magnetic field. The propagation equations for R a b , E, and S a b along a curve of a spacelike congruence are considered. These equations are developed in full for the special case of a congruence of material curves in a fluid. The divergence, or constraint, equation for the rotation vector is also derived. Where appropriate, corresponding results in Newtonian gravitation theory are given for comparison.
38 citations
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TL;DR: In this article, a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity is presented, where the equations to be integrated have regular coefficients, but the nonlinearity leads to the occurrence of singularities after a finite evolution time.
Abstract: This is the second of a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity. Although the equations to be integrated have regular coefficients, the nonlinearity leads to the occurrence of singularities after a finite evolution time. In this paper we first discuss some novel techniques for integrating the equations right up to the singularities. The second half of the paper presents as examples the numerical evolution of the Schwarzschild and certain colliding plane wave space-times.
33 citations
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TL;DR: In this paper, a set of junction conditions is stated in terms of the Newman-Penrose variables (tetrad vectors and spin coefficients) and it is shown that these conditions are equivalent to those of Darmois and Lichnerowicz.
Abstract: A set of junction conditions is stated in terms of the Newman-Penrose variables (tetrad vectors and spin coefficients). It is shown that these conditions are equivalent to those of Darmois and Lichnerowicz. As an example we study the matching of the Schwarzschild metric with an axially and reflection-symmetric metric. For this particular example we study the propagation of the Killing vectors and show how the propagation is conditioned by the fulfillment of the junction conditions.
29 citations
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TL;DR: The energy-momentum problem for the gravitational field in general relativity has been a stumbling block for theoreticians for more than six decades as discussed by the authors, but none of them has led to sucoess.
Abstract: The energy-momentum problem for the gravitational field in general relativity has been a stumbling block for theoreticians for more than six decades. During this time different authors have proposed in the scientific literature numerous approaches to the problem, but none of them has led to sucoess. Extensive analysis of the energy-momentum problem in general relativity shows [8, 9, 11–17] that in Einstein's theory there is in principle no solution to this problem, since it does not contain conservation laws for the matter and gravitational field taken together.
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TL;DR: The proper framework for testing Rastall's theory and its generalizations is in the case of non-negligible (i.e. discernible) gravitational effects such as gravity gradients.
Abstract: The proper framework for testing Rastall's theory and its generalizations is in the case of non-negligible (i.e. discernible) gravitational effects such as gravity gradients. These theories have conserved integral four-momentum and angular momentum. The Nordtvedt effect then provides limits on the parameters which arise as the result of the non-zero divergence of the energy-momentum tensor.
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TL;DR: In this article, it was shown that a consistent gauging of the Poincar\'e group is capable of including Einstein's general relativity, provided the nontrivial part of the vierbein is taken as the fundamental gravitational field.
Abstract: It is shown that a consistent gauging of the Poincar\'e group is capable of including Einstein's general relativity. This statement holds for matter particles of arbitrary spin, provided the nontrivial part of the vierbein is taken as the fundamental gravitational field, thus giving rise to a known modification of the original theory. Since the gauge approach implies that gravitation is an ordinary field theory over flat space, the standard prescriptions for calculating the asymmetric momentum tensor of both matter and gravitation are available. Applying Belinfante's flat-space symmetrization procedure to the latter, we prove that the symmetrization of the asymmetric matter tensor just gives the dynamically defined symmetric matter tensor, whereas the symmetrization of the asymmetric gravitational momentum tensor leads to another version of the field equations that reveals a deep analogy to the equations of electrodynamics. Furthermore a method is developed that admits an unambiguous calculation of gauge-fixing conditions from a given gauge-breaking term. Besides the harmonic gauge, which can be reproduced by means of this method, new gauge-fixing conditions for local translations and local Lorentz transformations are obtained. These gauge-fixing techniques, as well as the symmetrization procedure, may equally be generalized to the case of nonvanishing torsion.
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TL;DR: In this paper, the problem of zero-mass scalar fields coupled to the gravitational field in the static plane-symmetric case is completely solved for a traceless energy-momentum tensor.
Abstract: The problem of zero-mass scalar fields coupled to the gravitational field in the static plane-symmetric case is completely solved for a traceless energy-momentum tensor.
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TL;DR: In this paper, the field equations for coupled gravitational and zero mass scalar fields in the presence of a point charge are obtained with the aid of a static spherically symmetric conformally flat metric.
Abstract: Field equations for coupled gravitational and zero mass scalar fields in the presence of a point charge are obtained with the aid of a static spherically symmetric conformally flat metric. A closed from exact solution of the field equations is presented which may be considered as describing the field of a charged particle at the origin surrounded by the scalar meson field in a flat space-time.
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TL;DR: In this paper, a connection between torsion and a macroscopical quantity, the tensor Ω ij = 2∇[j V i ], was made.
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TL;DR: The Hermitian theory of Relativity as mentioned in this paper starts form the Einstein Lagrangian with the Lagrangians with the conditions of Hermiticity, and then the field masses MA are given by the Hermitians.
Abstract: Die Ausdehnung der Riemannschen Metrik der Allgemeinen Relativitatstheorie ins Komplexe bedeutet die Ersetzung der Symmetrie-Bedingungen
fur den metrischen Tensor, Affinitat und Ricci-Tensor durch die Hermiteschen Bedingungen
Mit diesen Bedingungen fuhrt das Einstein-Hilbert-Hamilton-Prinzip
zu einer erweiterten Gravitationstheorie (Einstein), die im Sinne der EIH-Approximation neben der Newton-Einsteinschen Gravodynamik, auch die Chromodynamik der Elementarteilchenphysik enthalt.
Die von den Einstein-Schrodingerschen Feldgleichungen der Hermiteschen Relativitatstheorie implizierte Wechselwirkung zwischen Gravo- und Chromodynamik erzwingt das „Confinement”. Ohne dieses Confinement wurde das Gravitationspotential divergieren, d.h., es konnte keine — nach Masgabe der Einstein-Schrodingerschen Feldgleichungen — Riemannsche Raum-Zeit-Metrik gik = aik geben.
Hermitian Relativity, Chromodynamics and Confinement
The extension of the Riemannian metrics of General Relativity to the complex domain, i.e. the substitution of the symmetry conditions
for the fundamental tensor gik, the affinity lik and the Ricci curvature Rik by the conditions of Hermiticity
gives a „Generalized Theory of Gravity” (Einstein) which describes the Newton-Einstein gravodynamics combined with the chromodynamics of quarks.
The Hermitian Theory of Relativity starts form the Einstein Lagrangian with
and the Einstein-Schrodinger field equations result:
In the Einstein-Infeld-Hoffmann approximation (EIH) of General Relativity between point-like particles forces
are given by the conditions of integrability (generalized Bianchi-identics).
Furthermore, solutions for the symmetrical part
of the equations exist only if there is “confinement” for the “charges” ΣQA = 0. Then the field masses MA are given by
with the distance L between the charges QA.
The equations of motions with distance-independent forces ∼QAQB and the confinement ΣQA = 0 are a consequence of the Einstein-Schrodinger equations for Hermitian Relativity. The forte Einstein-Straus equations (with Rik = 0) do not give “charges” QA and therefore the Newtonian forces only.
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TL;DR: In this paper, the equations of motion in general relativity using a fast motion approach are worked out explicitly to second order to derive the derivation from first principles with the assumptions made, and the application of these equations to small-angle scattering is given.
Abstract: The equations of motion in general relativity using a 'fast motion approach' are worked out explicitly to second order. The derivation from first principles with the assumptions made is given. The application of these equations to small-angle scattering will be given in a later paper.
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TL;DR: Using equations of motion for a charged particle acted on by the new electromagnetic fields allowed in six-dimensional relativity, it was shown how much a particle moving in the field of a central charge can have a space trajectory which the standard four-dimensional theory does not allow as mentioned in this paper.
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TL;DR: In this paper, a new classical action formulation of general relativity is presented, which derives both Riemannian geometry and Einstein's field equations in the general case of a gravitational field interacting with matter.
Abstract: A new classical action formulation of general relativity is presented. The aim is to develop a variational principle which derives both Riemannian geometry and Einstein's field equations in the general case of a gravitational field interacting with matter. The method is applied to the Hilbert action functional with a variable matter-gravity coupling. Surprisingly, the resulting theory is not general relativity with a variable G; instead, the method yields general relativity with a constant G and a cosmological constant. It is shown that the variational principle has a physical content different from that of the usual Hilbert and Palatini action formulations. A possible application of the new variational principle to the path-integral quantization of gravity is discussed.
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TL;DR: In this paper, the canonical treatment of general relativity in a vierbein formulation was discussed, and the primary constraints were derived from an algebra the same as that of the generators of local Poincare transformations.
Abstract: Discusses the canonical treatment of general relativity in a vierbein formulation. The authors derive the primary constraints, and find that they satisfy an algebra the same as that of the generators of local Poincare transformations, namely the Poincare algebra.
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TL;DR: In this article, a theory which makes it possible to classify the points of Riemannian space is used, together with the corresponding associated reference systems for the gravitational field in the classical general theory of relativity, to obtain the expression for the field energy density in the form of a four-dimensional scalar (not a pseudoscalar) and energy-impulse field tensor as an energy-imperceptible tensor of second rank.
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TL;DR: In this article, the relative motion of test bodies in general relativity attracts a considerable interest and various sides of this problem have been studied in a series of papers, including the geometrical aspects of the problem, the exact equation of the test body in an external field was written, and for all essential geometric quantities, in particular, for the connection coefficients up to the fourth order, expansions in powers of normal coordinates were obtained.
Abstract: Studying the relative motion of test bodies in general relativity attracts a considerable interest. Various sides of this problem have been studied in a series of papers. In [i], the exponential mapping in the neighborhood of reference trajectory was used in order to obtain and study the exact solutions for the relative motion of geodesic lines, as well as their approximations up to and including the third order in the deviation vector and its derivatives. In [2], the geometrical aspects of the problem were studied, the exact equation of the relative motion of test bodies in an external field was written, and for all essential geometric quantities, in particular, for the connection coefficients up to the fourth order, expansions in powers of normal coordinates were obtained. In the series of papers [3], the asymptotic methods of the theory of nonlinear oscillations were applied to study nonlinear effects in the relative motion of test bodies.
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TL;DR: In this paper, the fundamental result of Lanczos is used in a new type of quadratic variational principle whose field equations are the Einstein field equations together with the Yang-Mills type equations for the Riemann curvature.
Abstract: The fundamental result of Lanczos is used in a new type of quadratic variational principle whose field equations are the Einstein field equations together with the Yang-Mills type equations for the Riemann curvature. Additionally, a spin-2 theory of gravity for the special case of the Einstein vacuum is discussed.