Showing papers on "Mathematics of general relativity published in 1979"
TL;DR: In this article, it was shown that a static, spherically symmetric gravitational field is given by the Schwarzschild metric, and that in the weak-field approximation an antisymmetric field of zero mass and zero spin exists, besides gravitons.
Abstract: A gravitational theory is formulated on the Weitzenb\"ock space-time, characterized by the vanishing curvature tensor (absolute parallelism) and by the torsion tensor formed of four parallel vector fields. This theory is called new general relativity, since Einstein in 1928 first gave its original form. New general relativity has three parameters ${c}_{1}$, ${c}_{2}$, and $\ensuremath{\lambda}$, besides the Einstein constant $\ensuremath{\kappa}$. In this paper we choose ${c}_{1}=0={c}_{2}$, leaving open $\ensuremath{\lambda}$. We prove, among other things, that (i) a static, spherically symmetric gravitational field is given by the Schwarzschild metric, that (ii) in the weak-field approximation an antisymmetric field of zero mass and zero spin exists, besides gravitons, and that (iii) new general relativity agrees with all the experiments so far carried out.
1,075 citations
Book•
01 Jan 1979
TL;DR: In this article, the spacetime of general relativity and paths of particles are studied in the vicinity of a massive object, and the Chinese connection between the Chinese and general relativity is discussed.
Abstract: Introduction.- Vector and tensor fields.- The spacetime of general relativity and paths of particles.- Field equations and curvature.- Physics in the vicinity of a massive object.- Gravitational radiation.- Elements of cosmology.- Appendix A: Special relativity review.- Appendix B: The Chinese connection.- References.- Solutions.- Index.
211 citations
Proceedings Article•
01 Jan 1979
TL;DR: In this paper, the authors discuss some of the interrelationships between the initial value problem, the canonical formalism, linearization stability and the space of gravitational degrees of freedom.
Abstract: In this chapter we discuss some of the interrelationships between the initial value problem, the canonical formalism, linearization stability and the space of gravitational degrees of freedom. In the last decade, these topics have experienced a resurgence of interest as more advanced mathematical methods and viewpoints have begun to show the intimate relationships among these topics. At present, the literature regarding these areas of general relativity is a rapidly expanding body of knowledge. Our purpose here is to present the current state of affairs from our own point of view. We shall use geometric methods developed by the authors to establish various conn~ctions between the above-mentioned topics. The main tools we shall use to develop this material are nonlinear functional analysis, an adjoint formalism for Hamiltonian field theories, and infinite-dimensional symplectic geometry. As we shall see, these tools and the topics we shall consider are naturally related. For a more complete picture of the current state of affairs, the reader is urged to consult Choquet-Section 4.1 develops the Hamiltonian formalism for the dynamics of general relativity, usually called the ADM (Arnowitt, Deser and Misner) formalism. This is done using invariant concepts and the adjoint formalism developed by the authors. We show how to write the Einstein dynamical system explicitly in the compact form
102 citations
TL;DR: In this paper, the authors prove the existence and uniqueness of a zero-linear-momentum vector field and show the existence of a center-of-mass line which is a smooth timelike curve contained in a convex hull of the world tube of the body.
Abstract: In Dixon's theory of the dynamics of extended bodies in metric theories of gravity, a definition of a center-of-mass line is proposed. We prove the existence and uniqueness of a zero-linear-momentum vector field. Using this vector field we show the existence of a center-of-mass line which is a smooth timelike curve contained in a convex hull of the world-tube of the body.
65 citations
TL;DR: In this paper, a simple form and its physical interpretation are presented for a solution of the momentum constraint in the initial value problem of general relativity, and the physical interpretation is given.
Abstract: A simple form and its physical interpretation are presented for a solution of the momentum constraint in the initial-value problem of general relativity.
37 citations
TL;DR: In this paper, it was shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2form and the exterior derivative of related 2-forms are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group.
Abstract: It is shown that a space-time admitting a nonsingular 2-form satisfying the source-free Maxwell equations and a Lorentzian involution under which the 2-form and the exterior derivative of a related 2-form are skew invariant while the trace-free Ricci tensor and the covariant derivative of the involution itself are invariant possesses locally an invertible 2-parameter Abelian isometry group with nonsingular orbits
32 citations
TL;DR: In this article, it was shown that an arbitrarily close return to a previous initial state of the universe, such as predicted by the Poincare recurrence theorem, cannot occur in a closed universe governed by general relativity.
Abstract: An arbitrarily close return to a previous initial state of the Universe, such as predicted by the Poincare recurrence theorem, cannot occur in a closed universe governed by general relativity The significance of this result for cosmology and thermodynamics is discussed
21 citations
TL;DR: In this paper, it was shown that the fast motion iteration method in General Relativity gives an asymptotic approximation to exact solutions of the reduced Einstein equations, where the error commited at each step of the iteration is derived.
Abstract: We show that the fast motion iteration method in General Relativity gives an asymptotic approximation to exact solutions of the reduced Einstein equations. Rigorous estimates of the error commited at each step of the iteration are derived.
21 citations
18 citations
TL;DR: In this paper, the field equations of general relativity have been solved to obtain a solution for a static charged fluid sphere, which is free from singularity and satisfies the necessary physical conditions.
Abstract: In the present paper the field equations of general relativity have been solved to obtain a solution for a static charged fluid sphere. This solution is free from singularity and satisfies the necessary physical conditions.
16 citations
01 Jan 1979
TL;DR: On the problem of singularity in the De Sitter model, the authors discusses the integral formulation of the Mach Principle in a Conformally Flat Space and Canonical Transformations and the Theory of Representations of Lie Groups.
Abstract: On the Problem of Singularity in the De Sitter Model.- On the Integral Formulation of the Mach Principle in a Conformally Flat Space.- Canonical Transformations and the Theory of Representations of Lie Groups.- The Newman-Penrose Method in the Theory of General Relativity.
TL;DR: In this article, a homogeneous, plane-symmetric, matter-free solution to a new theory of gravitation was presented, and the solution goes over into the Kasner metric of general relativity.
Abstract: We present a homogeneous, plane-symmetric, matter-free solution to a new theory of gravitation. In the limit of large t, the solution goes over into the plane-symmetric Kasner metric of general relativity.
01 Jan 1979
TL;DR: In this article, the role of the spacetime signature e is discussed and it is pointed out that e may be used as a perturbation parameter for general relativity in terms of surface deformations and its extension to supergravity.
Abstract: The formulation of general relativity in terms of surface deformations and its extension to supergravity are briefly reviewed The role of the spacetime signature e is discussed It is pointed out that e may be used as a perturbation parameter The “free” theory corresponds to e = 0, which is halfway between hyperbolic (e = −1) and Euclidean (e = +1) spacetime
TL;DR: In this paper, the authors define the concept of a nonvanishing Riemann tensor as a special case of the non-vanishing physical elements of the Christoffel symbols.
Abstract: Einstein's gravitational field equations in empty space outside a massive plane with infinite extension give a class of solutions describing a field with flat spacetime giving neutral, freely moving particles an acceleration. This points to the necessity of defining the concept “gravitational field” not simply by the nonvanishing of the Riemann curvature tensor, but by the nonvanishing of certain elements of the Christoffel symbols, called the physical elements, or the nonvanishing of the Riemann curvature tensor. The tidal component of a gravitational field is associated with a nonvanishing Riemann tensor, while the nontidal components are associated with nonvanishing physical elements of the Christoffel symbols. Spacetime in a nontidal gravitational field is flat. Such a field may be separated into a homogeneous and a rotational component. In order to exhibit the physical significance of these components in relation to their transformation properties, coordinate transformations inside a given reference frame are discussed. The mentioned solutions of Einstein's field equations lead to a metric identical to that obtained as a result of a transformation from an inertial frame to a uniformly accelerated frame. The validity of the strong principle of equivalence in extended regions for nontidal gravitational fields is made clear. An exact calculation of the weight of an extended body in a uniform gravitational field, from a global point of view, gives the result that its weight is independent of the position of the scale on the body.
TL;DR: In this article, an argument is given in favor of the conjecture that an isolated system in general relativity must have nonnegative total energy, which is the same as the one in this paper.
TL;DR: In this paper, the problem of locally imbedding a null hypersurface in a Riemannian manifold was studied and the generalized Gauss-Codazzi equations were derived.
Abstract: This paper is concerned with the problem of locally imbedding a null hypersurface in a Riemannian manifold. More precisely, on a one‐parameter family of null hypersurfaces, rigged by an arbitrary null vector field, in a four‐dimensional space–time manifold, a particular symmetric affine connection is used to derive the corresponding generalized Gauss–Codazzi equations. In addition, expressions are obtained for the projections of the Ricci tensor, which are relevant to the characteristic initial‐value problem of general relativity.
TL;DR: In this article, it was shown that a static electric field can produce a force on a moving current element, and the solutions are based on the cardinal equations of dynamics of special relativity, both in the microscopical and the macroscopical points of view.
Abstract: Solutions to some apparent paradoxes in the special theory of relativity are given. In particular it is shown that a static electric fieldE
y
can produce a forceF
x
on a moving current element. The solutions are based on the cardinal equations of dynamics of special relativity. A general treatment of such equations is also given both in the microscopical and the macroscopical points of view.
TL;DR: In this article, the restrictions imposed on the energy-momentum tensor in general relativity when the space-time admits certain types of symmetries were discussed, in particular for a G 4 of motions acting on null or non-null hypersurfaces.
TL;DR: In this paper, an anisotropic interior solution for the Kerr metric is constructed and examined and is found to be free of all singularities, and the principal value of the energy tensor corresponding to positive matter density is positive everywhere and is much greater than the other three principal values.
Abstract: The Kerr metric, which is the simplest example of the Tomimatsu-Sato family of solutions of Einstein's vacuum field equations, is written in Lewis' extension of the canonical co-ordinates of Weyl for stationary axially symmetric fields. An anisotropic interior solution for the Kerr metric is constructed and examined and is found to be free of all singularities. The principal value of the energy tensor corresponding to positive matter density is positive everywhere and is much greater than the other three principal values.
TL;DR: In this article, the well-known algebraic classification of the electromagnetic tensor field is used to provide the space-time manifold of general relativity with the latest technique of differential geometry.
Abstract: The well-known algebraic classification of the electromagnetic tensor field is used to provide the space-time manifold of general relativity with the latest technique of differential geometry.
Journal Article•
TL;DR: In this article, Modugno et al. developed a systematic and geometrical analysis of shock structures in a Riemannian manifold and defined globally the jump, the infinitesimal jump and the covariant derivative jump of a tensor.
Abstract: A systematic and geometrical analysis of shock structures in a Riemannian manifold is developed. The jump, the infinitesimal jump and the covariant derivative jump of a tensor are defined globally. By means of derivation laws induced on the shock hypersurface, physically significant operators are defined. As physical applications, the charged fluid electromagnetic and gravitational interacting fields are considered. INTRODUCTION Several authors have developed the shock waves from different points of view, under both mathematical and physical aspects. In General Relativity shock waves assume a peculiar theoretical role. In fact they constitute one of the few strictly covariant signals occurring in the space-time manifolds, where the usual way, to describe waves (as plane waves, Fourier series, etc.) are globally meaningless. Of course shock may be considered as a mathematical abstraction that approximates more realistic physical phenomena. A very large bibliography on shock waves in General Relativity is quoted in [9]. Annales de l’Institut Henri Poincaré Section A Vol. XXX, 0020-23 39/1979/27/$ 4.00/ @ Gauthier-Villars. 28 M. MODUGNO AND G. STEFANI We refer chiefly to Lichnerowicz’s researches [1], [2], [3], which range over this topic and employ refined mathematical techniques. We believe that a deep understanding of shock waves in General Relativity requires an adequate geometrical analysis. In fact Hadamard’s formulas have not a tensorial character and their application to the complex entities occurring in General Relativity leads to results that could seem involved, if the geometrical structures utilized are not emphasized. Our purpose is, following Lichnerowicz’s approach, to develop a systematic geometric theory of tensor jumps in a Riemannian manifold and to apply it to General Relativity. We get a global theory, expressed by an intrinsical language, adequate for a geometrical point of view. Care is devoted to distinguish the role played by different structures, as the differential structure, the metric, the connection, etc. The case which requires distributional techniques will be treated in a subsequent work. We consider a Coo manifold M and an embedded hypersurface E (1), first we define the jump [t] of a tensors t across E. By means of Lie derivatives we define the higher jumps Ekt, which involve only the manifold structure : so we get a first generalization of Hadamard’s formulas (which are local and hold for functions). As a particular case, we consider the jump of Riemannian metric To describe the jump of the Riemannian connection we get a veritable tensor [r]k, which is directly expressed by means of The jump of the covariant derivative is obtained by means of Ekg and [r]k : this is a second generalization of Hadamard’s formulas (which operate only on functions by partial derivatives). In this way we get a global expression of the jump [R] of the curvature tensor. Particular interest have several derivation laws, induced on E, when the latter is singular, which replace the induced connection (that cannot be defined, for the tangent space of M does not split into the tangent space to E and into its orthogonal one). Some of these maps, as div\" and divl, intervene in the physical conservation laws. In physical applications we analyse the charged fluid, electromagnetic and gravitational field, as an example. We get compact formulas, that resemble Lichnerowicz’s results. In particular we get the ((shock conditions », the « conservation conditions )) and an intrinsical definition of the shock energy tensor. (1) Lichnerowicz considers a manifold, to get the physical significant part of gravitational potentials. But, for our purposes it seems more simple to assume M COO deferring to consideration on the Cauchy problem the statement about the physically significant part of Ekg (namely Annales de l’Institut Henri Poincaré Section A 29 SHOCK WAVES IN GENERAL RELATIVITY 1. THE BASIC ASSUMPTIONS Let M be a ceo manifold without boundary with dimension n 2, connected, paracompact, oriented and endowed with a pseudo-Riemannian metric g at least of class C°. We are mainly concerned with the case in which n = 4 and g is Lorentztype, for obvious physical reasons. But we don’t need such a requirement, as our results are more general. Moreover, let j : E 2014~ M be a Coo embedded orientable submanifold of M without boundary and with dimensions 11 1 (L:t: are the two orientations). E will be the support of the shock waves. In General Relativity the physical fields satisfy equations which impose shock conditions for X. The most important among them is that E is « singular », i. e. the induced metric j*g is degenerate. Thus we are led to make a study of the geometry of E which holds in the singular case too. Let us introduce some notations : is the subspace of tensors, p times contravariant and g times covariant of M, restricted to E (we say that such tensors are oh E) ; is the subspace of tensors, p times contravariant, of M, that are tangent to E; is the subspace of tensors, q times covariant, of M, generated by 1-forms orthogonal to E (by duality) ; is the subspace of tensors, p times contravariant, of M, generated by vectors orthogonal to E (by the metric). If L is singular, is the subspace, of tensors, p times contravariant, of E, generated by vectors orthogonal to E (by the metric). The symbols « /1: o, « \" o, » may be combined, with obvious meaning. For simplicity, we write also T for and T* for The spaces of sections, for each one of the preceding spaces, is denoted replacing « T » by « ~ o. Furthermore, the spaces of antisymmetric tensors are denoted by A and those of their sections by SZ. The class of differentiability of tensor fields is denoted by an upper suffix on and Q. If necessary, the labels ~, t and t will denote the contravariant, covariant and mixed form (induced by the metric) of a tensor t. 2. THE ORIENTATION OF ~ For the orientability of E there exists an « Orthogonal form ))
01 Jan 1979
TL;DR: The structure, goals and current status of supergravity are reviewed from a space-time, rather than superspace, point of view in this article, where its relations to general relativity are emphasized.
Abstract: The structure, goals and current status of supergravity are reviewed from a space-time, rather than superspace, point of view. Its relations to general relativity are emphasized.
TL;DR: For the case of stationary cylindrically symmetric dust distribution in nonrigid rotation, singularities invariably occur in contrast to the singularity-free solution obtained by Maitra (1966) in the corresponding case in general relativity as mentioned in this paper.
Abstract: Solutions of the field equations of Brans–Dicke theory for the case of stationary cylindrically symmetric dust distribution in nonrigid rotation are studied. It is seen that singularities invariably occur in contrast to the singularity‐free solution obtained by Maitra (1966) in the corresponding case in general relativity.
01 Jan 1979
TL;DR: Recently, recent discoveries suggest that now, sixty-two years after the birth of general relativity, the general solution to one physically significant problem, the spinning mass problem, may be close at hand as mentioned in this paper.
Abstract: Recent discoveries suggest that now, sixty-two years after the birth of general relativity, the general solution to one physically significant problem, the spinning mass problem, may be close at hand.
TL;DR: In this article, a symmetric energy-momentum tensor derived by Taub from a general variational principle was shown to predict the Pauli effect, which is a force term additional to that given by Minkowski.
Journal Article•