Showing papers on "Introduction to the mathematics of general relativity published in 1967"
TL;DR: In this article, an orthonormal tetrad system and associated coordinate system is obtained, which may be used to locally describe any dust-filled space-time, and three classes of solution with σ = 0, ω ≠ 0 are studied.
Abstract: An orthonormal tetrad system and associated coordinate system is obtained, which may be used to locally describe any dust‐filled space‐time. This is used to study dust‐filled spaces in which there exist multiply transitive groups of motions; all such spaces are classified in detail. Spaces containing shear‐free dust are also considered; it is shown that σ = 0 ⇒ ωΘ = 0. Three classes of solution with σ = 0, ω ≠ 0 are studied. Several new solutions of the field equations are contained in these results.
405 citations
TL;DR: In this article, a number of theorems and definitions which are useful in the global analysis of relativistic world models are presented, and it is shown in particular that changes in the topology of spacelike sections can occur if and only if the model is acausal.
Abstract: A number of theorems and definitions which are useful in the global analysis of relativistic world models are presented. It is shown in particular that, under certain conditions, changes in the topology of spacelike sections can occur if and only if the model is acausal. Two new covering manifolds, embodying certain properties of the universal covering manifold, are defined, and their application to general relativity is discussed.
341 citations
223 citations
TL;DR: In this article, a four parametric group of isometries of type 5 in Bianchi's classification, which act on a three-dimensional negative definite subspace, are found.
Abstract: In a four‐dimensional Lorentzian manifold in which Einstein's gravitational field equations hold with the field produced by pressure free dust, a significant set of new solutions is found. Assuming that the manifold possesses a four parametric group of isometries of type 5 in Bianchi's classification, which act on a three‐dimensional negative definite subspace, all metrics are found. The solutions are unusual in that the geodesics which the particles follow are not orthogonal to the three‐dimensional negative‐definite subspace so that the space will not appear homogeneous to these observers. The isotropic expansion is nonzero for almost all these solutions.
54 citations
53 citations
TL;DR: In this paper, it is shown that a full exploitation of the principle of general relativity in the costruction of the metrical field equations implies that the fundamental variables should be quaternion fields rather than the metric tensor field of the conventional formulation.
Abstract: It is proposed that a full exploitation of the principle of general relativity in the costruction of the metrical field equations implies that the fundamental variables should be quaternion fields rather than the metric tensor field of the conventional formulation. Thus, the tensor property of Einstein's formalism is replaced here by a formalism that transforms as a quaternion—a vector field in co-ordinate space and a second-rank spinor field of the type η ⊗ η* in spinor space. The geometrical field variables of the Riemann space are derived in quaternion form. The principle of least action (with the Palatini technique) is then used to derive a pair of time-reversed quaternion field equations, from the (quaternionic form of ) Einstein's Lagrangian. It is then shown how the conventional tensor form of the Einstein formalism is recovered from a particular combination of the derived time-reversed quaternion equations.
35 citations
TL;DR: This article argued that Henri Poincare did not anticipate Einstein's special theory of relativity, but rather, was intent on an entirely different program, the Lorentz theory of electrons.
Abstract: Contrary to the opinion of some who have written on the history of relativity, the author argues that Henri Poincare did not anticipate Einstein's special theory of relativity; but rather, that Poincare was intent on an entirely different program—the perfection of the Lorentz theory of electrons. It is also suggested that Poincare was not consistent in his use of the so-called “conventional” point of view.
34 citations
01 Jan 1967
TL;DR: Complex vectorial formalism in general relativity, discussing Riemann connection related to representation of Lorentz group by three-dimensional linear space is discussed in this paper, where the authors propose a vectorial representation for general relativity.
Abstract: Complex vectorial formalism in general relativity, discussing Riemann connection related to representation of Lorentz group by three- dimensional linear space
34 citations
TL;DR: In this article, the authors investigated certain weakened vacuum field equations which have been suggested as alternatives to the vacuum field equation of General Relativity and showed that they each possess a solution which corresponds to an isolated mass at the origin which repels test particles.
Abstract: Certain weakened vacuum field equations which have been suggested as alternatives to the vacuum field equations of General Relativity are investigated. It is shown that they each possess a solution which corresponds to an isolated mass at the origin which repels test particles. In view of the contradiction with experiment this solution is an embarrassment to these proposed vacuum field equations.
24 citations
Book•
01 Jan 1967
TL;DR: In this paper, a class of static, axisymmetric, interior solutions of Einstein's field equations in general relativity is obtained, which can be interpreted as sources for Weyl and Levi-Civita's general exterior solution that is flat at infinity, by making the metric components and their derivatives continuous at the boundary of the matter.
Abstract: A class of static, axisymmetric, interior solutions of Einstein's field equations in general relativity is obtained. The solutions are physically reasonable, and can be interpreted as sources for Weyl and Levi-Civita's general exterior solution that is flat at infinity, by making the metric components and their derivatives continuous at the boundary of the matter. For a particular model, a correlation is exhibited between the structure of the source material and the exterior field, which resembles closely, but not exactly, that of the corresponding Newtonian model.
TL;DR: In this article, it was shown that the family of solutions of the Einstein field equations, determined by a solution of the Hamilton-Jacobi equations, is characterized by a complete commuting set of constants of the motion.
Abstract: It is shown that the family of solutions of the Einstein field equations, determined by a solution of the Hamilton-Jacobi equations, is characterized by a complete commuting set of constants of the motion. It is then pointed out that the quantization program based on the utilization of the transition from the Hamilton-Jacobi equations to the corresponding Schr\"odinger equations yields a quantum thoery of gravitation which is "frozen" (that is, it contains no dynamics), thus suffering from the same malady as does the canonical quantization program.
01 Apr 1967
TL;DR: In this paper, the authors generalize a previous result of the author about the impossibility of constructing a physically acceptable solution of the Einstein-Maxwell equations corresponding to an infinite charged cylinder.
Abstract: The present paper generalizes a previous result of the author about the impossibility of constructing a physically acceptable solution of the Einstein-Maxwell equations corresponding to an infinite charged cylinder.
01 Jan 1967
TL;DR: Relativity theory and astrophysics - Conference, Ithaca, July-August 1965, Volume 1, Relativity and cosmology as discussed by the authors, Theodorakopoulos et al.
Abstract: Relativity theory and astrophysics - Conference, Ithaca, July-August 1965, Volume 1, Relativity and cosmology
TL;DR: In this paper, the Fresnel equation is derived in general relativity using the classical method applied by Levi-Civita in the study of a nonrelativistic theory of electromagnetic induction.
Abstract: The Fresnel equation is derived in general relativity using the classical method applied by Levi‐Civita in the study of a nonrelativistic theory of electromagnetic induction. For the description of the anisotropic medium the theory proposed by Quan is adopted. The study of the Cauchy problem is also presented and the convergence of results assures us that the equation proposed is the good one.
TL;DR: In this paper, the authors investigated the relation between Einstein's equations and electromagnetism and found that the strongly nonlinear lattice transmits low-frequency perturbations only if the perturbation field satisfies a set of field equations which become practically identical with Maxwell's equations.
Abstract: Einstein's equations for infinitesimal gravitational fields are investigated from the standpoint of replacing Einstein's flat Minkowskian background space by a strongly agitated Riemannian lattice space, with the added demand that the perturbation field be of very low frequency compared with the lattice frequency. It is found that the strongly nonlinear lattice transmits low‐frequency perturbations only if the perturbation field satisfies a set of field equations which become practically identical with Maxwell's equations. While this result demonstrates the close relation between Einstein equations and electromagnetism, it also opens a new door toward the deeper understanding of the quantum phenomena.
Journal Article•
TL;DR: In this paper, a method for writing the Einstein equations in integral form by means of a co-variant Green's function which is a two-point tensor is proposed, which includes a natural way of eliminating the difficulties associated with the nonlinearity of the equations.
Abstract: A method is proposed for writing the Einstein equations in integral form by means of a co variant Green's function which is a two-point tensor. This approach includes a natural way of eliminating the difficulties associated with the nonlinearity of the Einstein equations. With a definite choice of the equation for the Green's function the integral form of the equations is a covariant way of writing Mach's principle, which is now equivalent to the requirement that the Einstein equations be valid not in differential form but in integral form. A number of ex amples are considered. In particular it is shown that the Friedmann model of a homogeneous and isotropic universe is incompatible with Mach's principle, because of the behavior of the metric near a singular point which is characteristic of this model.
TL;DR: Chandrasekhar et al. as mentioned in this paper considered the post-Newtonian effects of general relativity on the Maclaurin and Jacobian sequences of the Roche model.
Abstract: As is well known, in the classical investigations of uniformly rotating bodies, two models have played conspicuous roles: the homogeneous model (leading to the sequences of Maclaurin and Jacobi) and the Roche model (consisting of a tenuous envelope surrounding a highly concentrated central mass). In Papers II and III of this series (Chandrasekhar 1967a and b) the post-Newtonian effects of general relativity on the Maclaurin and the Jacobian sequences have been considered; and in this paper the same effects on the Roche model will be considered.
TL;DR: The phenomenological-metric method is applied to four tests of general relativity as discussed by the authors, and the difficulty in attaching meaning to anr co-ordinate obstructs the process of determining deviations from flatness by this method.
Abstract: The phenomenological-metric method is applied to four tests of general relativity The difficulty in attaching meaning to anr co-ordinate obstructs the process of determining deviations from flatness by this method An attempt to alleviate this difficulty by using idealized models of flat and curved space-time does simplify the partial evaluation of the metric of the curved space-time for first-order effects, but the flaw in the definition of ther co-ordinate persists unless the curved space-time is actually flat in the region of interest
01 Aug 1967
TL;DR: General relativistic implication testing within solar system using artificial satellites and space probes using space probes is described in this article, where the authors present the results of the general relativism implication testing.
Abstract: General relativistic implication testing within solar system using artificial satellites and space probes
01 Jan 1967
TL;DR: In this article, it was shown that a spherical massive star becomes unstable long before it has contracted to the stage at which nuclear reactions become important, using a detailed stability analysis on the spherically symmetric relativistic equations and calculating the relaxation oscillations from a variational principle.
Abstract: THE suggestion by Hoyle and Fowler1 that stars with masses of 106–1010 M\cirċ may provide the energy for radio sources, and the subsequent discovery of quasars, has stimulated considerable interest in the structure of very massive stars2. Iben3, using a binding-energy argument, showed that within the framework of general relativity a spherical massive star becomes unstable long before it has contracted to the stage at which nuclear reactions become important. A similar conclusion was obtained by Chandrasekhar4, using a detailed stability analysis on the spherically symmetric relativistic equations and calculating the relaxation oscillations from a variational principle. Similar results have been obtained by Fowler5 using a virial theorem approach.