Showing papers on "Introduction to the mathematics of general relativity published in 1976"
TL;DR: The string-model analog of general relativity is shown to be unphysically dependent on an embedding gauge as discussed by the authors, and an explicit example confirms that it is inequivalent to Einstein's theory.
Abstract: The string-model analog of general relativity is shown to be unphysically dependent on an embedding gauge. Moreover, an explicit example confirms that it is inequivalent to Einstein's theory. (AIP)
122 citations
TL;DR: In this article, a Lagrangian formalism in flat spacetime is used to derive the set of all possible energy-momentum and spin tensors compatible with the conservation laws.
121 citations
TL;DR: In this paper, an inhomogeneous GL (2,C) group of coordinate transformations, constrained to leave the tetrad form invariant, is constructed and used to simplify the equations and clarify the geometrical meaning of the parameters introduced during the integration process.
Abstract: Following Plebanski and Robinson, complex V4’s which admit a congruence of totally null surfaces are shown to have coordinates which, in pairs, have a spinor structure which generates the usual spinor structure of the 2‐forms over the space. This structure allows Einstein’s vacuum equations to fracture into three triples and a singlet, which allow for easy reduction of the entire set to one nonlinear partial differential equation needed for consistency. An inhomogeneous GL (2,C) group of coordinate transformations, constrained to leave the tetrad form invariant, is constructed and used to simplify the equations and clarify the geometrical meaning of the parameters introduced during the integration process.
66 citations
TL;DR: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory as discussed by the authors, and they are used for the classification of Ricci Tensor.
Abstract: Tangent space null rotations are used to give a straightforward classification of the Ricci tensor in general relativity theory.
64 citations
TL;DR: In this paper, a degeneracy distribution of a symplectic form Γ on a given 4-dimensional manifold is investigated and its connection with an action of the diffeomorphism group is established.
Abstract: A symplectic structure i.e. a symplectic form Γ on the set of all solutions of the Einstein equations on a given 4-dimensional manifold is defined. A degeneracy distribution of Γ is investigated and its connection with an action of the diffeomorphism group is established. A multiphase formulation of General Relativity is presented. A superphase space for General Relativity is proposed.
37 citations
TL;DR: In this paper, a general linearly connected spacetime with a metric (L4, g) is shown to be an appropriate geometrical framework for general relativistic field theory.
Abstract: In Part I** of this series we have introduced the new notion of hypermomentum Δijk as a dynamical quantity characterizing classical matter fields. In Part II, as a preparation for a general relativistic field theory, we look for a geometry of spacetime which will allow for the accomodation of hypermomentum into general relativity. A general linearly connected spacetime with a metric (L4, g) is shown to be the appropriate geometrical framework
33 citations
TL;DR: In this paper, a new representation of the simplest Tomimatsu-Sato solution of Einstein's vacuum field equations was devised, which allowed to dispose of the previously troublesome "directional singularities" through the introduction of an advanced (or retarded) time coordinate.
Abstract: We devise a new representation of the simplest Tomimatsu–Sato solution of Einstein’s vacuum field equations. This permits us to dispose of the previously troublesome ’’directional singularities’’ through the introduction of an advanced (or retarded) time coordinate. In the neighborhood of the locations in question the T–S space is shown to possess a Killing tensor of valence two, which allows us to solve the geodesic problem in this neighborhood completely. Finally, we present for future analysis a plausible toroidal model of the material source for the T–S solution.
27 citations
TL;DR: In this article, it was shown that a homothetic motion in the intrinsic metric can be transformed correctly under the spatial homothing motion if the extrinsic curvature is transformed correctly.
Abstract: Components of Killing’s equation are used to obtain constraints satisfied in a spacelike hypersurface by the intrinsic metric and extrinsic curvature in the presence of a spacetime conformal motion for a solution of Einstein’s equations. If the conformal motion is either a homothetic motion or a motion, it is shown that these Killing constraints are preserved by the Einstein evolution equations. It is then shown that the generator of the homothetic motion (homothetic Killing vector) can be constructed if the Killing constraints are satisfied by a set of initial data. It is shown that a homothetic motion in the intrinsic metric is a spacetime homothetic motion if the extrinsic curvature is transformed correctly under the spatial homothetic motion. Further restrictions on a proper conformal motion due to the fact that it is not identically a curvature collineation are obtained. Restrictions on the matter–stress–energy tensor are discussed. Examples are presented.
27 citations
TL;DR: In this paper, the higher order corrections to geometrical optics are studied in general relativity for an electromagnetic test wave, and an explicit expression is found for the average energy-momentum tensor which takes into account the first order corrections.
Abstract: The higher order corrections to geometrical optics are studied in general relativity for an electromagnetic test wave. An explicit expression is found for the average energy–momentum tensor which takes into account the first‐order corrections. Finally the first‐order corrections to the well‐known area‐intensity law of geometrical optics are derived.
TL;DR: In this paper, the axis of symmetry is at the same time the direction of the magnetic field, and of the aligned spins in axially symmetric cosmological models of the Einstein-Cartan theory.
Abstract: In axially-symmetric cosmological models of the Einstein-Cartan theory (which may be briefly called ‘general relativity plus spin’), the axis of symmetry is at the same time the direction of the magnetic field, and of the aligned spins. The general set of relevant equations is given. Some exact solutions of this set constitute quasi-Euclidean and semiclosed cosmologies with a uniform magnetic field and aligned spinning matter. In contrast to the situation in the framework of general relativity, one may obtain non-singular solutions. Such a behaviour of the solutions of the Einstein-Cartan theory is rendered possible by the specific spin-spin repulsive interaction which is inherent in the theory.
TL;DR: In this paper, the invariants associated with the Weyl conform tensor of the δ=2 Tomimatsu-Sato solution of Einstein's field equations are described.
Abstract: Extremely simple expressions are presented for the hitherto uncalculated invariants associated with the Weyl conform tensor of the δ=2 Tomimatsu–Sato solution of Einstein’s field equations.
TL;DR: Time-dependent solutions of Yang's pure-space equations have been shown to violate Birkhoff's theorem as discussed by the authors, and they are shown to be unphysical and require the explicit presence of a source in vacuum.
Abstract: Recently I argued that Yang's pure-space equations must be supplemented by restrictions on the class of allowable space-times. I now consider time-dependent solutions of the pure-space equations and prove the violation of Birkhoff's theorem. Time-dependent spherically symmetric solutions are displayed, as well as solutions representing plane gravitational waves. The suggestion is made that pure spaces are unphysical and Yang's theory requires the explicit presence of a source in vacuum, in contrast to general relativity. (AIP)
TL;DR: The difference of opinion between these authors stems basically from the fact that a rigorous calculation of this force must be carried out using the general theory of relativity, and as mentioned in this paper gave here such a calculation.
Abstract: THE gravitational analogue of the magnetic force has been discussed1,2 within the framework of special relativity. The difference of opinion between these authors stems basically from the fact that a rigorous calculation of this force must be carried out using the general theory of relativity, and we give here such a calculation.
TL;DR: In this paper, the equilibrium configurations for static spherically symmetric self-gravitating perfect fluids in general relativity were considered and the new solutions were generalized Einstein static universes which have a Killing horizon similar to the one we are familiar with from the vacuum Schwarzschild solution.
Abstract: We consider the equilibrium configurations for static spherically symmetric self-gravitating perfect fluids in general relativity. The fluid obeys an equation of state which is the extreme limit allowed by Hawking's strong energy condition. The new solutions we have found are generalized Einstein static universes which have a Killing horizon similar to the one we are familiar with from the vacuum Schwarzschild solution.
TL;DR: In this paper, the neutrino field equations are given in Newman-Penrose formalism, and an exact solution of the Einstein-neutrino equations is obtained which describes the collision and subsequent interaction of two neutrinos fields.
Abstract: The neutrino field equations are given in Newman-Penrose formalism, and an exact solution of the Einstein-neutrino equations is obtained which describes the collision and subsequent interaction of two neutrino fields The gravitational interaction of the two fields is found to be completely different from that between two similar electromagnetic fields
TL;DR: In this paper, a technique for constructing physically meaningful initial data in the integration of Einstein's equations, and a method for characterization and analysis of the spacelike mass, momentum, angular momemtum, and multipole moments of gravitational fields are presented.
Abstract: Recent investigations of the initial-value problem of general relativity have shown that the initial-value constraints can be formulated in all cases as a system of elliptic equations with well-defined physical and mathematical properties. The solutions of these equations can be regarded as generalized gravitational potentials. These potentials are interrelated and depend on their sources quasilinearly. They are particularly useful in analyzing asymptotically flat solutions of Einstein's equations. We have found from these results (1) a technique for constructing physically meaningful initial data in the integration of Einstein's equations, and (2) a method for characterization and analysis of the spacelike mass, momentum, angular momemtum, and multipole moments of gravitational fields.
TL;DR: In this article, a number of Robertson-Walker-type solutions for certain cases, namely, for noncharged massless scalar meson fields, viscous fluids, Hookean elastic mediums, and Kelvin-Voigt viscoelastic systems, are presented.
Abstract: Robertson-Walker solutions are important in general relativity as universe solutions. This paper contains a number of Robertson-Walker-type solutions for certain cases, namely, for noncharged massless scalar meson fields, viscous fluids, Hookean elastic mediums, and Kelvin-Voigt viscoelastic systems.
TL;DR: Yang's new integral theory for gauge fields associated with the group $\mathrm{GL}(n)$ is discussed for the particular case when the field variables are written as operators.
Abstract: Yang's new integral theory for gauge fields associated with the group $\mathrm{GL}(n)$ is discussed for the particular case when $\mathrm{GL}(n)$ is reduced to $\mathrm{SL}(2, C)$. It is pointed out that, in this particular case, the theory gives the vacuum Einstein equations, but not the full equations. A modification giving nonvacuum equations consistent with the Einstein theory is pointed out. The field variables are then written as operators.
TL;DR: In this paper, the equations for two-killing-vector solutions of Einstein's equations are looked at from the point of view of the Lagrangian formalism, and a convenient method for dealing with Lagrangians quadratic in the velocities is outlined.
Abstract: The equations for two‐Killing‐vector solutions of Einstein’s equations are looked at from the point of view of the Lagrangian formalism. The original Lagrangian, written in terms of the metric, will undergo a Legendre transformation leading to the Ernst Lagrangian and from there by the same procedure to others. So one gets a sequence of Lagrangians, and by performing an invariance transformation at each step one obtains new solutions of Einstein’s equations. A convenient method for dealing with Lagrangians quadratic in the velocities is outlined.
Abstract: We calculate the Lagrangian up to the fourth order (2-post-Newtonian approximation) of an isolated system ofN particles under their mutual gravitational attractions, which forms the post-Newtonian theory of Einstein, Infeld and Hoffman for the motion.
TL;DR: In this article, a tensors of contravariant rank two which are divergence-free on one index, concomitants of a spinor field σiAX′ together with its first two partial derivatives, and scalars under spin transformations are constructed.
Abstract: All tensors of contravariant rank two which are divergence‐free on one index, concomitants of a spinor field σiAX′ together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates.
TL;DR: The relationship between the singularities in the solutions of the field equations and soliton type is analyzed in this paper, where it is shown that the Yang-Mills gauge theory is of Einstein type.
Abstract: A field theory is said to be of “Einstein type” if it has the property that the field equations imply the equations of motion. It is known that general relativity is of Einstein type, it is demonstrated here that the Yang-Mills gauge theory is of Einstein type. The relationship between the singularities in the solutions of the field equations and soliton type is analyzed.
TL;DR: In this paper, the authors present a new method of calculating multipole moments which, they hope, is complementary to Geroch's method and containing all the results obtained by the above authors.
Abstract: Multipole moments in general relativity are defined as coefficients of a multipole expan sion of appropriate potentials, as they are so in Newton's theory of gravitation. The essential point is the introduction of Fock's harmonic coordinate system in which the potentials are expanded in inverse powers of the distance from the source. First several moments are obtained for the Kerr, Tomimatsu-Sato and a class of the Weyl solutions of the Einstein equation, and then are inferred all moments for the Kerr and Weyl solutions. § l. Introduction The problem of obtaining multipole moments of a solution of the Einstein equation is the problem of interpreting the solution in terms of its Newtonian limit. In particular the knowledge of multipole moments serves to infer a possible source distribution which produces the gravitational field in question. Since the interior solutions which may be considered to describe the interior metric of the source of the Weyl,n Kerr') or Tomimatsu-Sato (T-S) 3l gravitational field have not been discovered at this stage, it is desirable to have a systematic method of obtain ing multipole moments of these fields. One of the methods was developed by Geroch using conformal Killing vec tors,<)· 5) and by means of this method Hansen was able to obtain multipole moments of the Kerr solution.6l Although it is not so easy to find the necessary conformal factor, once it is found, Geroch's method allows us to calculate all moments in principle. There are also other methods of finding multipole moments, although they are in many cases not adequate to find higher moments. For example, Voor hees obtained the quadrupole moment of a W eyl solution ;n Hernandez obtained all moments of the Kerr solutions ;8l Tomimatsu and Sa to obtained the quadrupole moments of their solutions. 3l The purpose of this paper is to present a new method of calculating multipole moments which, we hope, is complementary to Geroch's method and containing all the results obtained by the above authors. The idea is to introduce Fock's harmonic coordinate system 9l in which appropriate potentials are expanded in inverse powers of the distance from the source. This expansion will be hoped to be of the form of a multipole expansion in the Newtonian limit. Of cource, there is, a priori, no assurance that such a expansion will be just of the form of a multipole expansion. Then the main purpose of this paper is to show that this is the case
01 Sep 1976
TL;DR: In this article, the existence of a formal identity between Einstein's and Ernst's stationary axisymmetric gravitational field equations and the Einstein-Maxwell and the Ernst equations for the electrostatic and magnetostatic cases was shown.
Abstract: We show the existence of a formal identity between Einstein’s and Ernst’s stationary axisymmetric gravitational field equations and the Einstein–Maxwell and the Ernst equations for the electrostatic and magnetostatic axisymmetric cases. Our equations are invariant under very simple internal symmetry groups, and one of them appears to be new. We also obtain a method for associating two stationary axisymmetric vacuum solutions with every electrostatic known.