Showing papers on "Introduction to the mathematics of general relativity published in 1982"
TL;DR: In this article, a concluding review exposition of the investigations aimed at the construction of a general cosmological solution of the Einstein equations with a singularity in time is presented, which is a continuation of the previous (1970) paper by the authors in this Journal.
Abstract: This paper is a concluding review exposition of the investigations aimed at the construction of a general cosmological solution of the Einstein equations with a singularity in time; thus it is a direct continuation of the previous (1970) paper by the authors in this Journal. A detailed description is given of the analysis which leads to the construction of such a solution, and of its properties.
707 citations
Book•
01 Jan 1982
TL;DR: In this paper, a spinor formulation of gravitation and a classification of the gravitational and gauge fields is proposed. But the classification is based on the spinor model and not on the geodesic geometry of curved space-time.
Abstract: The gravitational field the geometry of curved space-time the Einstein field equations gravitational fields of elementary mass systems properties of the gravitational field equations of motion in general relativity axisymmetric solutions of the Einstein field equations spinor formulation of gravitation and gauge fields classification of the gravitational and gauge fields gauge theory of gravitation and other fields extended bodies in general relativity.
362 citations
Book•
01 Jan 1982
TL;DR: The foundations of special relativity can be found in this paper, where the Minkowski map of spacetime rules for the manipulation of 4-tensors 4-velocity and 4-acceleration wave motion.
Abstract: Part 1 The foundations of special relativity: schematic account of the Michelson - Morley experiment inertial frames in special relativity Einstein's two axioms for special relativity coordinates - the relativity of time derivation of the Lorentz transformation properties of the Lorentz transformation. Part 2 Relativistic kinematics: length contraction the length contraction paradox time dilation the twin paradox velocity transformation. Part 3 Relativistic optics: the drag effect the Doppler effect aberration and the visual appearance of moving objects. Part 4 Spacetime: spacetime and 4-tensors the Minkowski map of spacetime rules for the manipulation of 4-tensors 4-velocity and 4-acceleration wave motion. Part 5 Relativistic particle mechanics: the conservation of 4-momentum the equivalence of mass and energy some 4-momentum identities relativistic billiards the centre of momentum frame threshold energies De Broglie waves photons the angular momentum 4-tensor 3-force and 4-force relativistic analytic mechanics. Part 6 Relativity and electromagnetism: the formal structure of Maxwell's theory transformation of "e" and "b" - the deal field potential and field of an arbitrarily moving charge field of a uniformly moving charge the electromagnetic energy tensor electromagnetic waves. Part 7 Relativistic mechanics of continua: preliminaries external and internal forces the augmented momentum and mass densities the equations of continuity and of motion the mechanical energy tensor perfect fluids and incoherent fluids integral conservation laws. Appendix - tensors for special relativity.
346 citations
09 Aug 1982
TL;DR: In this article, a structure of dynamical theories is proposed that implements Mach's ideas by being relational in its treatment of both motion and time, which is called intrinsic dynamics and by construction treats the evolution of the entire universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero.
Abstract: A structure of dynamical theories is proposed that implements Mach’s ideas by being relational in its treatment of both motion and time. The resulting general dynamics, which is called intrinsic dynamics and by construction treats the evolution of the entire Universe, is shown to admit as special cases Newtonian dynamics and Lorentz-invariant field theory provided the angular momentum of the Universe is zero in the frame in which its momentum is zero. The formal structure of Einstein’s general theory of relativity also fits the pattern of intrinsic dynamics and is Machian according to the criteria of this paper provided the so-called thin-sandwich conjecture is generically correct.
254 citations
TL;DR: The most general time-independent spherically symmetric (in the usual three space dimensions) solution to the five-dimensional vacuum Einstein equations is found, subject to the existence of a Killing vector in the fifth direction as mentioned in this paper.
Abstract: The most general time-independent spherically symmetric (in the usual three space dimensions) solution to the five-dimensional vacuum Einstein equations is found, subject to the existence of a Killing vector in the fifth direction. The significance of these solutions is discussed within the context of a previously proposed extension of the Kaluza-Klein model in which the universe, although (4+1)-dimensional, has evolved over cosmic times into an effectively (3+l)-dimensional one.
204 citations
Book•
30 Sep 1982
TL;DR: In this article, the theory of gravitation is introduced in an elementary way with those parts of the theory that are essential to the beginning student of general relativity, giving all the mathematics necessary to understand the theory.
Abstract: This book is an introduction to the theory of gravitation. It deals in an elementary way with those parts of the theory that are essential to the beginning student of general relativity, giving all the mathematics necessary to an understanding of the theory. Starting from the foundations of Riemannian geometry and the tensor calculus, the author formulates and works out the essential laws of physics in a Riemannian space. Next, the Einstein field equations are derived. All important applications of the theory are dealt with, including issues of current importance, in particular the Schwarzchild metric, gravitational waves, gravitational collapse, black holes and cosmological models. All the associated basic physical problems are fully discussed, but many results that draw heavily on mathematics are given without derivation. In the rather more demanding chapters on selected vector fields, groups of motion and the Petrov classification, methods are discussed which have proved to be especially fruitful in modern research.
139 citations
TL;DR: In this paper, a new Lagrangian theory of gravitation in which the metric and the arbitrary affine connection are regarded as independent field variables has been considered, making use of the pure geometrical objects only from the variational principle the empty field equations are derived.
Abstract: A new Lagrangian theory of gravitation in which the metric and the arbitrary affine connection are regarded as independent field variables has been considered. Making use of the pure geometrical objects only from the variational principle the empty field equations are derived. It is shown that the metric obeys the ordinary Einstein equations of general relativity. However, the covariant derivative of the metric tensor does not vanish, so that the vector's length is generally nonintegrable under the parallel displacement. The torsion trace vector turns out to be the natural dynamical variable, satisfying the Maxwell-like equations with tensor of homothetic curvature as the Maxwell tensor. The equations of motion are explored; they are shown to be identical to the motion of electric charge under the Lorentz force. The conservation laws are discussed.
67 citations
50 citations
TL;DR: In this paper, a variational principle was applied to the general relativity case, using a tetrad to express the spin density and the four-velocity of the fluid. And an energy-momentum tensor was defined for a spinning fluid.
Abstract: General relativity field equations are employed to examine a continuous medium with internal spin. A variational principle formerly applied in the special relativity case is extended to the general relativity case, using a tetrad to express the spin density and the four-velocity of the fluid. An energy-momentum tensor is subsequently defined for a spinning fluid. The equations of motion of the fluid are suggested to be useful in analytical studies of galaxies, for anisotropic Bianchi universes, and for turbulent eddies.
33 citations
TL;DR: In this paper, Heintzmann's generating method is used to build up a family of new exact solutions for each value of n, which are spherically symmetric and static with perfect fluid distributions satisfying a linear equation of state p = nρ and n∈(0, 1).
Abstract: New exact solutions to Einstein’s equations are given which are spherically symmetric and static with perfect fluid distributions satisfying a linear equation of state p = nρ and n∈(0,1]. Heintzmann’s generating method is then used to build up a family of new solutions for each value of n.
TL;DR: The authors showed that the non-conservative gravitational theories of the type considered by Rastall (1972), Smalley (1974), and Malin (1975) do have conservation laws and are formally equivalent to general relativity.
Abstract: Shows that the 'non-conservative' gravitational theories of the type considered by Rastall (1972), Smalley (1974) and Malin (1975) do have conservation laws and are formally equivalent to general relativity. The supposed 'difference' between these theories and general relativity is shown to lie entirely in the question of whether the stress-energy tensor of matter fields is conserved in special relativity (flat space-time). If one chooses to interpret these theories as non-conservative, then the coefficient lambda in these theories, which measures the degree to which stress-energy is not conserved, can be constrained to values mod lambda mod
TL;DR: In this paper, the physical ideas underlying general relativity theory are discussed and the basic mathematical techniques (tensor calculus, Riemann curvature) needed to describe them are developed.
Abstract: In this tutorial article the physical ideas underlying general relativity theory are discussed and the basic mathematical techniques (tensor calculus, Riemann curvature) needed to describe them are developed. The general relativity field equations are presented and are used in several applications including a discussion of black holes.
TL;DR: In this paper, an analytical formalism is developed to deal with the occurrence of jump discontinuities in the gmu nu or their derivatives across a hypersurface Sigma, and it is shown that the equations of relativity remain meaningful at Sigma, even when Sigma does not inherit a unique intrinsic geometry, so that the gm nu are discontinuous across Sigma in natural coordinates.
Abstract: An analytical formalism is developed to deal with the occurrence of jump discontinuities in the gmu nu or their derivatives across a hypersurface Sigma . It is shown that the equations of relativity remain meaningful at Sigma , even when Sigma does not inherit a unique intrinsic geometry, so that the gmu nu are discontinuous across Sigma in natural coordinates. The spherically symmetric surface layer at the Schwarzschild-Minkowski junction is used to illustrate these techniques, and to establish rigorously the existence of C0 solutions of the Einstein equations and the conservation equations. The possible validity of relativity at the microscopic level is examined, and it is concluded that, if relativity is valid at the microscopic level, then it is likely that the gmu nu are not globally continuously differentiable.
TL;DR: In this paper, the authors discuss the meaning and prove the accordance of general relativity, wave mechanics, and the quantization of Einstein's gravitation equations themselves, and they show that classical and quantum gravitation have the same physical meaning according to limitations of measurements given by Einstein's strong principle of equivalence and the Heisenberg uncertainties for the mechanics of test bodies.
Abstract: We discuss the meaning and prove the accordance of general relativity, wave mechanics, and the quantization of Einstein's gravitation equations themselves. Firstly, we have the problem of the influence of gravitational fields on the de Broglie waves, which influence is in accordance with Eeinstein's weak principle of equivalence and the limitation of measurements given by Heisenberg's uncertainty relations. Secondly, the quantization of the gravitational fields is a “quantization of geometry.” However, classical and quantum gravitation have the same physical meaning according to limitations of measurements given by Einstein's strong principle of equivalence and the Heisenberg uncertainties for the mechanics of test bodies.
TL;DR: In this paper, the authors analyzed Einstein's introductory comments to "Zur Elektrodynamik Bewegter Korper" and concluded that length and time are relative quantities.
Abstract: 1 Electrodynamics: 1890-1905.- 2 Einstein's Philosophic Viewpoint in 1905.- 3 Analysis of Einstein's Introductory Comments to "Zur Elektrodynamik Bewegter Korper".- 4 Simultaneity and Time.- 5 Length and Time Are Relative Quantities.- 6 The Relativistic Transformations.- 7 The Relativity of Length and Time.- 8 The Theorem of Addition of Velocities.- 9 The Relativity of the Electric and Magnetic Fields.- 10 Doppler's Principle and Stellar Aberration.- 11 Light Quanta, Radiation and Relativity.- 12 On the Electrodynamics of Moving Bodies.- Epilogue.
TL;DR: An interior solution of the field equations of general relativity has been obtained for a static and spherically symmetric charged body as discussed by the authors, which is joined smoothly to the Reissner-Nordstrom solution at the surface of the body.
TL;DR: In this article, the authors consider the singularity problem in deriving the metric external to a static spherically symmetric body of mass qn, which is the Schwarzschild curvature singularity.
Abstract: In recent years much work has been devoted to the development of the (i singularity theorems ~ of Hawking and Penrose in order to have a bet ter understanding of the properties of space-times in Einstein 's general theory of re la t iv i ty (1). In spite, however, of the significant advances made in the subject, many impor tant problems of the classical theory of singularities remain still unsolved. The results obtained t i l l now, support the feeling tha t physically realistic solutions to the Einstein 's equations that are singulari ty free can only be obtained either by modifying the field equations or by looking for some missing physics in the applied general relat ivi ty. To be more concrete, let us consider the singulari ty which occurs in deriving the metric external to a static spherically symmetr ic body of mass qn, tha t is the Schwarzschild curvature singularity. This is a part icularly simple situation so, assuming the val id i ty of Einstein 's equations of general relativity, we can t ry to look for a minimal physical ansatz which can lead to singularity-free solutions. The most obvious suggestion might be to explore which physical proper ty one could at t r ibute to the body, besides its mass m, in order to get a well-behaved line element. Otherwise stated, between the many parameter familes of asymptot ical ly flat solutions which generalize che Schwarzschild case, one should succeed in choosing a suitable set of parameter to which correspond regular solutions. In stat ionary nonspherieal situations, however, i t is known tha t the Kerr-Newman solutions, where the parameters are the mass m, the electric charge q, the magneticmonopole charge p and a rotation parameter a, still display horizons, naked singularities and other unwanted phenomena. The simplest generalization of the Schwarzsehild problem, in which to the mass m is associated a scalar charge a, does not seem to have been adequately t reated in the li terature.
TL;DR: A number of exact solutions of Einstein's equations are obtained in this paper, which describe the collisions between one scalar plane wave and another scalar, neutrino, electromagnetic or gravitational plane wave.
Abstract: A number of exact solutions of Einstein's equations are obtained, which describe the collisions between one scalar plane wave and one scalar, neutrino, electromagnetic or gravitational plane wave.
TL;DR: In this paper, the undressed extrinsic curvature of a general compact momentum source is presented in terms of multipole moment integrals, and the momentum constraint of the initial value problem of general relativity is addressed.
Abstract: The momentum constraint of the initial-value problem of general relativity is addressed. The undressed extrinsic curvature of a general compact momentum source is presented in terms of multipole moment integrals.
TL;DR: In this article, a solution for the equations governing the space-time transformation in six-dimensional special relativity is found, and the energy required to turn the time vector of a particle is calculated.
Abstract: A solution is found for the equations governing the space-time transformation in six-dimensional special relativity. The energy required to turn the time vector of a particle is calculated.
TL;DR: The solution of the conformastat vacuum problem in general relativity is presented in this article, which consists of three axially symmetric Levi-Civita metrics, disproving the alleged spherical symmetry of the space-times.
TL;DR: In this article, a unification scheme for the Einstein and Maxwell theories is developed within a Riemann-Cartan geometry, where the electromagnetic field is introduced at the level of the connection and it is, in fact, coded in the torsion.
TL;DR: In this paper, the dominant energy condition in general relativity theory is examined in connection with the types of energy-momentum tensors it permits, and the condition that the energy momentum tensor be "stable" in obeying it is defined in terms of a suitable topology.
Abstract: The dominant energy condition in general relativity theory, which says that every observer measures a nonnegative local energy density and a nonspacelike local energy flow, is examined in connection with the types of energy-momentum tensor it permits. The condition that the energy-momentum tensor be “stable” in obeying the dominant energy condition is then defined in terms of a suitable topology on the set of energy-momentum tensors on space-time and the consequences are evaluated and discussed.
TL;DR: In this paper, the unusual correspondence of a closed cosmological model and the field equations of the Brans-Dicke theory with general relativity is discussed, and the meaning of the correspondence is also discussed briefly from the Machian point of view.
Abstract: The unusual correspondence of a closed cosmological model and the field equations of the Brans-Dicke theory with general relativity is discussed. This cosmological model is correspondent to the Einstein universe and the Brans-Dicke scalar field behaves like the cosmological term in the field equations when the coupling parameterω is very small. The meaning of the correspondence is also discussed briefly from the Machian point of view.
TL;DR: In this article, different models for the symmetric, time-symmetric charged two-body problem of general relativity were constructed by solving the source-free initial-value equations on two-and three-sheeted manifolds.
Abstract: It is shown that different models for the symmetric, time-symmetric charged two-body problem of general relativity lead to different quantitative predictions. The models are constructed by solving the source-free initial-value equations on two- and three-sheeted manifolds. The tidal accelerations at the midpoint of the two-body system are shown to be different for these models after they are normalized for mass, charge, and electric dipole moment. This discrepancy vanishes with the charge, but the second derivatives of the Ricci curvature tensor still differ in the limit of vanishing charge.
TL;DR: In this article, the spin propagation equation at first order is checked with the known result on the precession, which is obtained by means of slow motion approximation of the result of our result.
Abstract: Within the framework of the previous paper, we complete the set of equations of motion by including the spin propagation equation at first order. We check this equation with the known result on the precession, which is obtained by means of slow motion approximation of our result. A new scheme of expanding equations of motion is also introduced. It will be useful to undertake higher-order calculations.