Showing papers on "Introduction to the mathematics of general relativity published in 1968"
158 citations
TL;DR: In this article, a family of stationary cylindrically symmetric solutions of the Einstein-Maxwell equations corresponding to a charged dust distribution in rigid rotation is presented, where the Lorentz force vanishes everywhere and the ratio of the charge density and mass density may assume arbitrary value.
Abstract: The paper presents a family of stationary cylindrically symmetric solutions of the Einstein-Maxwell equations corresponding to a charged dust distribution in rigid rotation. The interesting feature of the solution is that the Lorentz force vanishes everywhere and the ratio of the charge density and mass density may assume arbitrary value. The solutions do not seem to have any classical analogue.
125 citations
TL;DR: In this paper, the field of a sphere of constant gravitational mass density is given for a sphere with constant mass density, where the field is characterized by a constant number of degrees of freedom.
Abstract: Formulae are given for the field of a sphere of constant gravitational mass density.
46 citations
TL;DR: In this article, the field equations for the coupled zero-mass scalar field and gravitational fields are solved in the axially symmetric static case, and it is found that the scalar fields obeys a flat-space Laplace's equation such that a large class of solutions exists.
Abstract: The field equations for the coupled zero-mass scalar field and gravitational fields are solved in the axially symmetric static case. It is found that the scalar field obeys a flat-space Laplace's equation such that a large class of solutions exists. The spherically symmetric limiting solution is shown to be a simplified version of the solution due to Jains et al.
46 citations
35 citations
TL;DR: In this article, the authors give a number of solutions corresponding to a stationary distribution of rotating charged dust with vanishing Lorentz force, and show that in all but one case there are closed time-like lines.
Abstract: In the present paper we give a number of solutions corresponding to a stationary distribution of rotating charged dust with vanishing Lorentz force. It is found that in all but one case there are closed time-like lines. Further, the paper shows the existence of some homogeneous stationary universes other than that of Godel and Einstein when, besides the matter field, there is an electromagnetic field as well.
28 citations
TL;DR: In this paper, it was shown that the symmetric tensor-antisymmetric tensors of the generalized metrical field are physically equivalent to a scalar-tensor formalism.
Abstract: The factorization of Einstein’s formalism into a pair of simultaneous quaternion field equations in general relativity entails the enlargement of the original formalism from 10 to 16 independent relations. By iteration, the quaternion field equations are shown to be physically equivalent to a symmetric-tensor-antisymmetric-tensor formalism. The symmetric-tensor part is in one-to-one correspondence with the 10 relations of Einstein’s original theory of gravitation. The remaining 6 (antisymmetric tensor) relations have no counterpart in the earlier gravitational theory. In addition to the generalization of the metrical field (and therefore the description of gravitational forces) that follows from the incorporation of the antisymmetric-tensor contribution, the covariant divergence of the latter formalism automatically leads to a system of field equations whose structure is in one-to-one correspondence with the Maxwell theory for electromagnetism. It is shown that Einstein’s original formalism entails 10, rather than 16 relations, because, in addition to its covariance under the group of general relativity (aconnected topological group) it is also covariant under time and space reflection transformations. The latter is an undue restriction since it is not required by the principle of general relativity alone. When these discrete symmetry elements are dropped, the (more general) quaternion formalism results. With the expression of the latter (which is not sensitive to the «handedness» of space or the direction of time) as thesum of two formalisms—one that is even and the other odd under space or time reflections—the original 10 relations of Einstein’s equationsplus the 6 relations that lead to the Maxwell field equations follow. Finally, with the application of the Schwarzschild conditions in the anti-symmetric-tensor part of the field equations, they reduce (in orderv/c) to a scalar-field formalism. With this approximation then, the full exploitation of the principle of general relativity—in terms of a symmetric-tensor-antisymmetric-tensor formalism—reduces to a scalar-tensor formalism, such as the one that has been studied by Brans and Dicke. It follows that to within the approximations that have been considered in this comparison (which would be applicable to a derivation of planetary motion) the present theory and a trulyscalar-tensor theory would be experimentally indistinguishable. In the derivation presented in this paper, however, the generalization of the metrical field to incorporate with the usual formalism an antisymmetric-tensor part (and therefore a scalar part in the application to planetary problems) follows in a natural way from the lowest-dimensional irreducible representations of the group of general relativity and no new fundamental constants need be introduced.
28 citations
TL;DR: The Runge-Lenz vector, which is a constant of the motion in the nonrelativistic hydrogen atom, is shown to be a two-body quantity whose presence can be completely understood within the classical framwork of special relativity as mentioned in this paper.
14 citations
13 citations
TL;DR: In this article, the authors established two theorems for a spherically symmetric charged dust distribution: the first theorem states that the expansion (or contraction) of a spherical symmetric charge cannot be isotropic and the second one states that a static charge is unstable to perturbations, which do not destroy the spherical symmetry, and would eventually collapse.
Abstract: This note establishes two theorems for a spherically symmetric charged dust distribution. The first theorem states that the expansion (or contraction) of a spherically symmetric charged dust cannot be isotropic and the second theorem states that a static charge is unstable to perturbations, which do not destroy the spherical symmetry, and would eventually collapse.
TL;DR: In this article, the moments of the source in the Newtonian field with no spatial symmetries were investigated and it was shown that the stationary field contains twice the number of moments which appear in the static field, apart from the mass term.
Abstract: Stationary solutions of the Einstein field equations with no spatial symmetries are considered in order to discover the number of arbitrary constants, corresponding to the moments of the source in the Newtonian field, which appear in the solution. It is shown that the stationary field contains twice the number of moments which appear in the static field, apart from the mass term which appears singly. These additional moments are essentially angular momentum terms, which in Newtonian theory make no contribution to the field.
01 Jan 1968
TL;DR: In this paper, the uniqueness of general relativity is established by postulating the derivability of the equations of motion from the (gravitational) field equations, and it is also emphasized that even with regard to agreement with experiment (including cosmology) there exist, in my opinion, no compelling arguments against general relativity.
Abstract: The essential point of this note is to establish the uniqueness of general relativity by postulating the derivability of the equations of motion from the (gravitational) field equations. It is also emphasized that even with regard to agreement with experiment (including cosmology) there exist, in my opinion, no compelling arguments against general relativity.
01 Jan 1968
Abstract: The Rainich equations of the `already unified field theory' are studied in the case of non-static electromagnetic fields, and a solution is obtained for a space-time metric which admits a group G4 of automorphisms. There exists a divergence-free electromagnetic field for x4 > 0, except for x4 -> infinity. It is shown that the electromagnetic field vanishes for large values of time, and the solution for a completely empty flat space is then obtained.
TL;DR: In this paper, the central properties of the usual variational formulations of EINSTEIN's general theory of relativity are sketched and a more general variational formulation is introduced, which is more appropriate to the geometric foundations of the theory.
Abstract: The central properties of the usual variational formulations of EINSTEIN's general theory of relativity are sketched and a more general variational formulation is introduced, which is more appropriate to the geometric foundations of the theory. In particular, this formulation leads to the BIANCHI identities in their non-contracted form following in consequence of a general invariance property. This invariance property of the general variational principle relates to transformations (not, in general, coordinate transformations) that contain arbitrary fifth (or third) order tensor “generators”. These results can be interpreted as implying that the variational principle introduced here admits an “extended principle of general covariance” (i.e., a covariance principle more general than the usual principle relating to general coordinate covariance). Some of the formal implications of these results as well as their connection with general coordinate covariance are discussed briefly. In particular these results point to the existence of a fundamental transformation theory connecting all of the RIEMANNian spacetimes of general relativity.
TL;DR: In this article, it was shown that linear momentum is carried by the waves cyclically in such a way that the center of rotation is a fixed point outside the axis of symmetry of the rod.
Abstract: Gravitational waves emitted from a spinning rod - taken as a rigid axially symmetric distribution of matter of uniform small cross section - are studied within the framework of general relativity, with special interest on transport of linear momentum from the rotating source. It is found that linear momentum is carried by the waves cyclically in such a way that the centre of rotation is a fixed point outside the axis of symmetry of the rod. Two results for the rate of momentum flux, differing in numerical content, are derived from the solution of the linear approximation, one by the use of Synge's argument involving the energy-momentum tensor, and the other by means of the pseudo-tensor. The discrepancy between the results is explained.
TL;DR: In this article, a cylindrically symmetric distribution of stressless conducting matter with an axial magnetic field has been found, which can be matched with an outside pure magnetic field solution originally due to Bonnor.
Abstract: Solutions of the Einstein-Maxwell equations corresponding to cylindrically symmetric distributions of stressless conducting matter with an axial magnetic field have been found, which can be matched with an outside pure magnetic field solution originally due to Bonnor. It is also shown that Melvin's magnetic universe cannot be fitted with dust distributions in this way.
01 Jan 1968
TL;DR: In this article, it was shown that spherically symmetric electromagnetic fields cannot be embedded in a flat space of 5-dimensions, and the incompatibility of class one SPH-space-times with Rainich algebraic conditions is established.
Abstract: It is well known that a spherically symmetric space-time is, in general, of class two. A necessary and sufficient condition for a spherically symmetric space-time to be of class one has been obtained in terms of the Riemann curvature tensor. By means of a transformation property of s.s. space-time, three distinct cases are shown to exist. The incompatibility of class one spherically symmetric space-times with Rainich algebraic conditions is established in these three cases. It is concluded that spherically symmetric electromagnetic fields cannot be embedded in a flat space of 5-dimensions.
TL;DR: In this paper, the topological aspects of the Bel-Petrov classification of the curvature tensor are examined for compact orientable space-times in which the Einstein equations for the exterior case are satisfied.
Abstract: The topological aspects of the Bel-Petrov classification of the curvature tensor are examined for compact orientable space-times in which the Einstein equations for the exterior case are satisfied. It is shown that for such space-times of Bel Case III the metric tensor is singularity-free and that the Pontrjagin number identically vanishes. Bel Cases I and II are examined and conditions are given for which the metric is singularity-free and the Pontrjagin number vanishes. Applications to gravitional radiation in general relativity are discussed.
TL;DR: In this paper, the motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameter V/c, whereV denotes the velocity of the bodies andc is velocity of light.
Abstract: The motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameterV/c, whereV denotes the velocity of the bodies andc is the velocity of light. The following simplifying assumptions, consistent with a problem of astronomical interest, have been made: (1) the dimensions of the bodies are small compared with their mutual distance; (2) the bodies consist of matter in the fluid state with internal hydrostatic pressure and their oblateness is due to their own rotation; (3) there exist axial symmetry about the axis of rotation and symmetry with respect to the equatorial plane, the same symmetry properties apply to mass densities and stress tensors. The Fock-Papapetrou method was used to ascertain those terms in the equations of motion which are due to the rotation and to the oblateness of each component. Approximate solutions to the Poisson and wave equations were obtained to express the potential and retarded potential at large distances from the bodies generating them. The explicit evaluation of certain integrals has necessitated the use of the Laplace-Clairaut theory for the equibrium configuration of rotating bodies. The final expressions require the knowledge of the mass density as a function of the mean radius of the equipotential surfaces. As an interpretation of the results, the Lagrangian perturbation equations were employed to evaluate the secular motion of the nodal line for the relative orbit of the two components. The results constitute a generalization of Fock's work and furnish the contribution of the mass distribution to the rotation effect of general relativity.
TL;DR: In this article, the existence of a solution of the Einstein-Maxwell equations in general relativity where matter coexists with a cylindrically symmetric axial field is demonstrated.
Abstract: The paper demonstrates the existence of a solution of the Einstein-Maxwell equations in general relativity where matter coexists with a cylindrically symmetric axial field.
01 Jan 1968
TL;DR: In this article, it is shown that the momenta are constructed tensorially only with the two fundamental tensors of the hypersurface since Dirac's velocities are equated with the components of the second fundamental tensor.
Abstract: In the Hamiltonian formulation of relativistic theory of gravitation the gravitational field at a given time is defined on a spacelike hypersurface of Riemannian space-time by six gravitational potentials and as many conjugate momenta. The former characterize the metric of the hypersurface and it is shown that the momenta are constructed tensorially only with the two fundamental tensors of the hypersurface since Dirac's velocities are equated with the components of the second fundamental tensor. Invariants of the two fundamental tensors, geometrically interpretable, can then be introduced into the Hamiltonian. It is at once clear from the Gauss and Codazzi equations, relating the two fundamental tensors of a hypersurface that the four secondary constraints, which the conjugate dynamical variables satisfy, are equivalent to four Einstein gravitational equations.