Showing papers on "Introduction to the mathematics of general relativity published in 1990"

An introduction to general relativity

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.

Lagrangian Interaction: An Introduction To Relativistic Symmetry In Electrodynamics And Gravitation

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

Superfluid 3He-B and gravity

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.

General relativity in Newtonian form

Abstract: It is shown how the use of coordinates where time is measured with clocks moving radially in a spherically symmetric gravitational field leads to general relativistic dynamical expressions that are exactly identical to corresponding expressions in Newtonian theory. The general formalism is developed for the case where the stress-energy tensor is that of a perfect fluid. Expressions like the Newtonian inverse square gravitational law, the Newtonian equation of continuity for fluid flow, Newtonian conservation of energy, etc., follow quite naturally from the fully-fledged exact general relativistic equations. Specific examples involving cosmology and gravitational collapse are given.

A static axisymmetric anisotropic fluid solution in general relativity

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.

Ashtekar's new variables and the vacuum constraint equations

Abstract: The constraint equations for vacuum general relativity, formulated in terms of the new Hamiltonian variables, are investigated A simple local coordinate presentation of the equations is used

The effects of general relativity on near-earth satellites

Abstract: Whether one uses a solar system barycentric frame or a geocentric frame when including the general theory of relativity in orbit determination for near-earth satellites, the results should be equivalent to some limiting accuracy. The purpose of this paper is to clarify the effects of relativity in each frame and to demonstrate their equivalence through the analysis of three years of laser tracking data taken on the Lageos satellite. It is demonstrated that the simpler formulation in the geocentric frame is adequate for the purpose of near-earth satellite orbit determination. A correction to the conventional barycentric equations of motion is shown to be required.

Einstein equations in numerical relativity

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.

Description of open cosmological models with matter by harmonic functions

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.

Non-static cylindrically symmetric fields in general projective relativity

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

Some geometry associated with the Einstein bundle

Abstract: If one could characterise LeBrun's Einstein bundle in terms of the intrinsic geometry of ambitwistor space, then one could say that one has a solution of the full Einstein equations in the same sense that the non-linear graviton construction solves the half-flat case. Some geometry associated with null geodesics is discussed which leads to a definition of the Einstein bundle in a way that requires locally only a knowledge of the tangent planes of the 2-quadrics representing spacetime points. In particular, it does not involve explicit consideration of the structure of the space of quadrics-i.e. the spacetime.

The maximum principle and relativity theory

Abstract: It will be shown in this paper that a very simple way of deducing the Schwarzschild's metric can be obtained only from postulate on light sphere "displacement". This is not an accidental fact to my mind. Possibly the postulate gives a mathematical model for describing the mechanism of interaction of a particle with gravitational field, and contains a physical foundation of the general relativity theory. More detailed information is available in [1].