Showing papers on "Mathematics of general relativity published in 1976"
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
Book•
30 Sep 1976
TL;DR: COSMOGRAPHY What the universe contains The cosmic distance hierarchy and the determination of galactic densities The red shift and the expansion of the universe PHYSICAL BASIS OF GENERAL RELATIVITY The need for relativistic ideas and a theory of gravitation DIFFICULTIES with Newtonian MECHANICS Gravity and inertial frames and absolute space Inadequacy of special relativity Mach's principle and gravitational waves Einstein's principle of equivalence.
Abstract: COSMOGRAPHY What the universe contains The cosmic distance hierarchy and the determination of galactic densities The red shift and the expansion of the universe PHYSICAL BASIS OF GENERAL RELATIVITY The need for relativistic ideas and a theory of gravitation DIFFICULTIES WITH NEWTONIAN MECHANICS Gravity and inertial frames and absolute space Inadequacy of special relativity Mach's principle and gravitational waves Einstein's principle of equivalence CURVED SPACETIME AND THE PHYSICAL MATHEMATICS OF GENERAL RELATIVITY Particle paths and the separation between events Geodesics Curved spaces Curvature and gravitation GENERAL RELATIVITY NEAR MASSIVE OBJECTS Spacetime near an isolated mass Around the world with clocks Precession of the perihelion of Mercury Deflection of light Radar echoes from planets Black holes COSMIC KINEMATICS Spacetime for the smoothed-out universe Red shifts and horizons Apparent luminosity Galactic densities and the darkness of the night sky Number counts COSMIC DYNAMICS Gravitation and the cosmic fluid Histories of model universes The steady state theory Cosmologies in which the strength of gravity varies IN THE BEGINNING Cosmic black-body radiation Condensation of galaxies Ylem APPENDIX A: Labeling astronomical objects APPENDIX B: Theorema egregium PROBLEMS SOLUTIONS TO ODD-NUMBERED PROBLEMS USEFUL NUMBERS BIBLIOGRAPHY INDEX
93 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 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.
24 citations
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 article, the Palatini variational principle is applied to the action integral of the Jordan-Brans-Dicke theory of gravitation and an affinity differing from the Christoffel symbols by an additional third-order tensor is obtained.
Abstract: The Palatini variational principle is applied to the action integral of the Jordan-Brans-Dicke theory of gravitation. An affinity differing from the Christoffel symbols by an additional third-order tensor is obtained. Field equations with the covariant derivatives and the Ricci tensor defined with respect to this affinity are found. Rewriting these equations using the covariant derivative and the Ricci tensor constructed from the Christoffel symbols yields equations physically equivalent to those of Jordan, Brans and Dicke.
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: In this paper, a relativistic version of deRham's theorem was used to find the holonomy group of a large class of space-times and showed that the concept of energy content needed above completely determines the energy content under suitable assumptions.
Abstract: We prove a relativistic version of deRham's theorem and use it to find the holonomy group of a large class of space-times. We also show that the concept of “energy content” needed above completely determinesg
ij
under suitable assumptions. Thus it brings us closer to a theorem that will express Mach's principle in general relativity.
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: In this article, a compact form of the dynamical equations of relativity is proposed, which clarifies the covariance of the equations under coordinate transformations of the space-time, and provides new insight into the infinite-dimensional symplectic geometry behind the dynamic equations, the decomposition of gravitational perturbations, and the space of gravitational degrees of freedom.
Abstract: A new compact form of the dynamical equations of relativity is proposed. The new form clarifies the covariance of the equations under coordinate transformations of the space-time. On a deeper level, we obtain new insight into the infinite-dimensional symplectic geometry behind the dynamical equations, the decompositions of gravitational perturbations, and the space of gravitational degrees of freedom. Prospects for these results in studying fields coupled to gravity and the quantization of gravity are outlined.
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: In this article, a necessary and sufficient criterion of inertia for the flat space-time theory of general frames of reference is presented, in terms of the vanishing of some typical components of the affine connection pertaining to curvilinear coordinate systems.
Abstract: A necessary and sufficient criterion of inertia is presented, for the flat space-time theory of general frames of reference, in terms of the vanishing of some typical components of the affine connection pertaining to curvilinear coordinate systems. The physical identification of inertial forces thus arises in the context of the special theory of relativity.
01 Sep 1976
TL;DR: In this paper, a region of space-time containing two null electromagnetic waves is considered, and a new class of exact solutions of the Einstein-Maxwell equations is given, describing two waves following shear-free null geodesics 'twisting' through each other with zero contraction, the twist of one wave being proportional to the field strength of the other.
Abstract: In the general theory of relativity electromagnetic waves interact nonlinearly through the field equations. A region of space-time containing two null electromagnetic waves is considered, and a new class of exact solutions of the Einstein-Maxwell equations is given. The solutions describe two waves following shear-free null geodesics 'twisting' through each other with zero contraction, the twist of one wave being proportional to the field strength of the other.
TL;DR: In this paper, the authors consider a Cartesian coordinate system in a pseudo-Euclidean space of events with the metric tensor gij (g~ = g22 = gaa = _ g44 = i, gij = 0 for i # j), d/dr = cui3 i is the symbol of the derivative with respect to the intrinsic time, c is the velocity of light in a vacuum.
Abstract: Here p is the fluid density, Pe = ep is the density of the free electric charge of the fluid, e is a constant, u i are the contravariant components of the imaginary dimensionless velocity vector of the individual points of the fluid, Fij = 3iAj -3jA. are the electromagnetic field tensor components, 3i = 3/Oxi is the s)nnbol of! the derivative with respect to the variables x i (i = I, 2, 3, 4) of the observed Cartesian coordinate system in a pseudo-Euclidean space of events with the metric tensor gij (g~ = g22 = gaa = _ g44 = i, gij = 0 for i # j), d/dr = cui3 i is the symbol of the derivative with respect to the intrinsic time, c is the velocity of light in a vacuum, S is the entropy, A m = Am(p, S) is a given function, KzJ = vpz] are the antisymmetric tensor components of the internal mechanical moment of the fluid, which are proportional to the volume density tensor components pij of the internal electromagnetic moment, v is a constant (the gyromagnetic ratio). The quantities p, u ~, ~lj in (i) are connected by the relationships (p is a constant)
TL;DR: In this article, a new series of solutions to the initial value constraints of general relativity is obtained by perturbing the trace of the extrinsic curvature, which is the same as the solution in this paper.
Abstract: This paper investigates a new series of solutions to the initial-value constraints of general relativity which is obtained by perturbing the trace of the extrinsic curvature. This analysis is important for two reasons. Firstly, it completes the proof that flat space is a local energy minimum. Secondly, it casts light on the extent to which this variable can be identified with the gauge degree of freedom in the initial-value problem.