Showing papers on "Introduction to the mathematics of general relativity published in 1991"

Analogy between general relativity and electromagnetism for slowly moving particles in weak gravitational fields

Abstract: Starting from the equations of general relativity, equations similar to those of electromagnetic theory are derived. It is assumed that the particles are slowly moving (v≪c), and the gravitational field is sufficiently weak that nonlinear terms in Einstein’s field equations can be neglected. For static fields, the analogy to electrostatics and magnetostatics is very close. Results are compared with those of a previous derivation by Braginsky, Caves, and Thorne [Phys. Rev. D 15, 2047–2068 (1977)]. These results lead to very simple derivations of the Lense–Thirring precession [Phys. Z. 19, 156–163 (1918)] and the spin‐curvature force of Papepetrou [Proc. R. Soc. London Ser. A 209, 248–258 (1951)] and Pirani [Acta Phys. Pol. 15, 389–405 (1956)].

The lagrangian approach to conserved quantities in general relativity

Abstract: Publisher Summary This chapter reviews general ideas and results concerning the geometric “Lagrangian” approach to conserved currents in field theories and shows how these results apply to general relativity and to relativistic field theories. Although general relativity is a well-established theory of gravity interacting with external matter, there is still no general agreement on the definition of mass and of conserved quantities associated to the gravitational field itself. The chapter reviews the fundamentals of the theory of conserved quantities as it follows from the Poincare–Cartan form approach to the higher-order calculus of variations. It further analyzes the interaction between matter fields and gravitation, the first-order covariant formulation of general relativity, and the field theories on a fixed background.

A Kerr Metric Solution in New General Relativity

Abstract: We give an exact solution of the vacuum gravitational field equation in new general relativity. The solution gives the Kerr metric and the parallel vector fields are axially symmetric. A parameter h in the expression of the metric is related to the angular momentum of the rotating source, when the spin density S,f of the gravitational source satisfies the condition a~Si/=O. In the Kerr metric space· time, we cannot discriminate new general relativity from general relativity, so far as scalar, the Dirac and the Yang-Mills fields and macroscopic bodies ·are used as probes. The space-time given by the solution does not have singularities at all, although it has an "effective singularity". Two kinds of Schwarzschild metric solutions, one is our solution with h=O and the other is a solution given by Hayashi and Shirafuji, are physically equivalent with each other. Nevertheless, these are markedly different from each other with regard to the asymptotic behavior of the torsion tensor for r-->OO and the space-time singularities.

Quartic equations and classification of Riemann tensors in general relativity

Abstract: For space-times in general relativity, the Petrov classification of the Weyl conformai curvature and the Plebanski or Segre classification of the Ricci tensor each depend on the properties of the roots of quartic equations. The coefficients in these quartic equations are in general complicated functions of the space-time coordinates. We review the general theory of quartic equations, and discuss algorithms for determining the existence and values of multiple roots. We consider practical implementation of an algorithm and the consequent Petrov classification. Tests of programs embodying this algorithm, using the computer algebra system CLASSI based on SHEEP, are reported.

General Relativity and Fifth Force Experiments

Abstract: We calculate some general relativistic corrections to the Newtonian weak field limit of Einstein gravity, and discuss their possible relevance to considerations regarding the current set of fifth force experiments.

First-order field equations in general relativity

Abstract: The metric tensor of a Riemannian space-time is composed quadratically of timelike and spacelike tetrad fields. A system of first-order field equations for the tetrad fields is postulated, which determines the geometric structure of space-time as well as the second-order dynamics of the fields. The energy-momentum content of the fields automatically corresponds to the geometric structure of the space-time according to the Einstein field equations. The spin contribution of the fields is mutually compensated and therefore spin does not influence the space-time geometry. As a consequence, it is not necessary to include torsion into the general theory of relativity and nevertheless the equivalence principle can be used to consistently transfer energy-momentum tensors from flat to curved space.

Stationary Axially Symmetric Fields in General Relativity

Abstract: In addition to gravitational and electromagnetic fields, neutrino fields are objects of fundamental research in theoretical physics. The subject of this chapter is the nature of the interaction of these three types of material fields in fairly general situations.

The Relation between Complex Fluctuations of the Metric, the Cosmological Constant Problem and Electrodynamics in Einstein's Unified Field Theory

Abstract: The interpretation of Einstein's unified field theory with sources as a gravoelectrodynamics in a polarizable continuum is considered, and the influence exerted on the background stress energy momentum content of spacetime and on the polarizability by complex fluctuations of the metric field is investigated. It is shown that certain assumptions about the fluctuating metric field which comply with the requirement of local Lorentz invariance for freely falling observers simultaneously provide a possible answer to the question of the cosmological constant and ensure that the macroscopic behaviour of weak electromagnetic inductions and fields is just the one occurring in Einstein·Maxweli electrodynamics. We do not believe that Einstein's unified field theory!) is a complete theory, capable of accounting for the behaviour of charged matter through everywhere regular solutions of its field equations. We rather hold the opinion that the tensor quantities on the left-hand sides of those equations provide definitions of properties of charged matter in terms of geometric entities, as it occurs for uncharged matter in the general relativity of 1915, and that the conservation identities express laws imposed on the properties of matter by the non-Riemannian geometric structure of the theory. We introduce a real four-dimensional manifold with coordinates Xi and write 2 ) the definitions of material properties in terms of a nonsymmetric tensor density gik and of a nonsy~metric affinity Til as follows: