Showing papers on "Mathematics of general relativity published in 1990"
Book•
01 May 1990
TL;DR: In this article, the background manifold structure and the curvature of the Weyl tensor are discussed. But the classification of the tensor tensor is not discussed. And it is not shown how to obtain axially symmetric solutions in space-time.
Abstract: Geometry and physics: an overview 1. The background manifold structure 2. Differentiation 3. The curvature 4. Space-time and tetrad formalism 5. Spinors and the classification of the Weyl tensor 6. Coupling between fields and geometry 7. Dynamics on curved manifolds 8. Geometry of congruences 9. Physical measurements in space-time 10. Spherically symmetric solutions 11. Axially symmetric solutions References Notation Index.
253 citations
Book•
01 Jan 1990TL;DR: The authors provided an introduction to general relativity for mathematics undergraduates or graduate physicists, focusing on an intuitive grasp of the subject and a calculational facility rather than on a rigorous mathematical exposition.
Abstract: This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.
99 citations
Book•
01 Jan 1990
TL;DR: Galilean Relativity And Newtonian Physics as mentioned in this paper is a generalization of the Lagrangian Interaction And Symmetry (LIM) of the Euler-Lagrange Equations of Motion (ELEM).
Abstract: Galilean Relativity And Newtonian Physics * Introduction * Space and Time * Euclidean Geometry * Tensor Fields in Euclidean Spacetime * Spacetime Kinematics and Galilean Relativity * Newtonian Mechanics * Newtonian Gravity Lagrangian Interaction And Symmetry * Euler-Lagrange Equations of Motion * Symmetries and Noethers Theorem * Non-relativistic Lagrangian Fields * Hamiltonian Dynamics Poincare Covariance And Einsteinian Physics * Special Relativity * Poincare Transformations * Tensor Fields in Minkowskian Spacetime * Einsteinian Mechanics Electrodynamic And Gravitational Fields * Relativistic Lagrangian Fields * Relativistic Scalar Gravity * Maxwells Equations * Electrodynamics * Gauge Principle * Relativistic Gravitation
42 citations
TL;DR: In this article, the cosmological term in 3He-B and corresponding modification of this term in the general relativity are discussed, where the modified cosmology term in general relativity equations contains both the general metric tensor and the equilibrium Minkowski metric tensors.
Abstract: The analogue of the gravitation field in superfluid 3He-B, where some combination of the order parameter components plays a part of the metric tensor, is considered. Broken relative spin-orbit symmetry and parity violation in 3He-B lead to the dirac equation for the fermionic quasiparticles in the presence of a Vielbein. The cosmological term in the 3He-B and corresponding modification of this term in the general relativity are discussed. The modified cosmological term in the general relativity equations contains both the general metric tensor and the equilibrium Minkowski metric tensor. This term becomes zero in equilibrium and therefore does not give the huge gravitational mass of the vacuum which is a plague to the conventional gravity theory due to vacuum fluctuations. The cosmological term nevertheless gives the mass for gravitons. Domain walls in gravity analogous to that in 3He-B are also discussed.
38 citations
TL;DR: In this article, interior field equations in general relativity are considered when spacetime is static and axisymmetric and the energy-momentum tensor represents an anisotropic fluid.
Abstract: Einstein's interior field equations in general relativity are considered when spacetime is static and axisymmetric and the energy-momentum tensor represents an anisotropic fluid. After imposing a set of simplifying assumptions a two-parameter solution is derived and its properties are discussed. The solution is found to be physically reasonable in a certain range of the parameters in which case the metric could represent a core of anisotropic matter.
6 citations
TL;DR: In this paper, the amount of arbitrariness with which one can describe a nontrivial special conformai mapping is determined, and necessary and sufficient conditions are found under which a space-timeV admits a special conformal mapping to space-Time ¯V in which the metric tensor of both V and ¯V satisfy the Einstein equations with the energy-momentum tensor having a geodesic velocity field.
Abstract: Some special conformal mappings of relativistic spaces are studied. The amount of arbitrariness with which one can describe a nontrivial special conformai mapping is determined. Necessary and sufficient conditions are found under which a space-timeV admits a special conformal mapping to space-time ¯V in which the metric tensor of bothV and ¯V satisfy the Einstein equations with the energy-momentum tensor of an ideal fluid having a geodesic velocity field.
4 citations
TL;DR: In this article, a given spacetime theoryT is characterized as the theory of a certainspecies of structure in the sense of Bourbaki [1], and it is then possible to clarify in a rigorous way the concepts ofpassive and active covariance of T under the action of the manifold mapping groupG M.
Abstract: In this paper a given spacetime theoryT is characterized as the theory of a certainspecies of structure in the sense of Bourbaki [1]. It is then possible to clarify in a rigorous way the concepts ofpassive andactive covariance ofT under the action of the manifold mapping groupG M . For eachT, we define also aninvariance groupG I T and, in general,G I T ≠G M . This group is defined once we realize that, for eachτ ∈ModT, each explicit geometrical object defining the structure can be classified as absolute or dynamical [2]. All spacetime theories possess alsoimplicit geometrical objects that do not appear explicitly in the structure. These implicit objects are not absolute nor dynamical. Among them there are thereference frame fields, i.e., “timelike” vector fieldsX ∈TU,\(U \subseteq M\)M, whereM is a manifold which is part ofST, a substructure for eachτ ∈ModT, called spacetime. We give a physically motivated definition of equivalent reference frames and introduce the concept of theequivalence group of a class of reference frames of kind X according to T, G X T. We define thatT admits aweak principle of relativity (WPR) only ifG X T ≠ identity for someX. IfG X T =G I T for someX, we say thatT admits a strong principle of relativity (PR).
4 citations
2 citations
01 Dec 1990
TL;DR: In this article, the authors briefly review how to deal with those equations in relativistic astrophysics and cosmology and introduce two examples-the Centrella and Wilson's cosmology, and the Shapiro and Teukolsky's stellar cluster.
Abstract: Rapid progress in modern computer industries now enables us to solve the Einstein equations numerically. In the first part of this paper we briefly review how to deal with those equations in relativistic astrophysics and cosmology. In the second part we introduce two examples-the Centrella and Wilson's cosmology and the Shapiro and Teukolsky's relativistic stellar cluster.
2 citations
TL;DR: In this paper, an approach to a description of conformally flat open models in general relativity with the energy-momentum tensor of an ideal fluid for an equation of state of the type p=β · e.
Abstract: An approach is considered to a description of conformally flat open models in general relativity with the energy-momentum tensor of an ideal fluid for an equation of state of the type p=β · e. In the approximation of rational β, in order to describe such models by harmonic functions it is necessary to increase the dimensionality of the spacelike hypersurface N of 4-dimensional spacetime. A similarity is pointed out between N and N·β and the orbital and magnetic numbers in quantum mechanics. It is shown that a new variable can be introduced such that the problem is reduced to the equivalent problem of planetary model of a hydrogen-like atom.
1 citations
TL;DR: In this paper, the structure of the Riemannian spacetime of new general relativity was studied as probed by spin-1/2 particles and photons, and the static, isotropic spacetime was described by the parallel vector fields.
Abstract: New general relativity predicts a static, isotropic spacetime q\1ite different from the Schwarz schild spacetime of general relativity, unless one of the two unknown parameters, c, is exactly zero. In this paper we study its structure as a Riemannian spacetime, paying special attentions to motions of test particles theliein. Recently Kawai and Tomal} have discussed singularities of the static, isotropic spacetime in new general relativity, a gravitational theory based on absolute (or tele-) parallelism. 2 ),*) Here we study structures of the spacetime as probed by spin-1/2 particles and photons. The static, isotropic spacetime of new general relativity is described by the parallel vector fields b={bkl'(x)} with
TL;DR: In this paper, exact solutions for the vacuum field equations of General Projective Relativity of Arcidiacono (1986) are obtained for the non-static Einstein-Rosen metric (Einstein and Rosen, 1937).
Abstract: Exact solutions for the vacuum field equations of General Projective Relativity of Arcidiacono (1986) are obtained for the non-static Einstein-Rosen metric (Einstein and Rosen, 1937) The nature of singularities in these solutions is discussed
01 Jan 1990
TL;DR: In this article, general conditions are formulated within the framework of general relativity for "suspension" of rotating particles in static gravitational fields having the form of systems of differential equations for the components of the spin tensor.
Abstract: General conditions are formulated within the framework of general relativity for “suspension” of rotating particles in static gravitational fields having the form of systems of differential equations for the components of the spin tensor. In the special case of suspension of particles in gravitational fields created by nonrotating objects, the spin components are a superposition of quantities which vary harmonically over time. As an example the effect of suspension of a spin particle within a gravitating liquid sphere is considered.