Formulation Of General Relativity Research Articles

Test of the gravitomagnetic field via laser-ranged satellites

We describe a new experiment to measure the gravitomagnetic field of the Earth. This field, a consequence of the general relativistic formulation of Mach's principle (WEM—Wheeler-Einstein-Mach principle), has never been detected. The idea is to measure the Lense-Thirring precession of the nodal lines of two laser-ranged satellites with supplementary inclinations. In this way it is possible to separate the relativistic nodal precession from the classical nodal precession due to the multipole moments of the Earth.

Measurement of the Lense-Thirring drag on high-altitude, laser-ranged artificial satellites.

We describe a new method of measuring the Lense-Thirring relativistic nodal drag using LAGEOS together with another high-altitude, laser-ranged, similar satellite with appropriately chosen orbital parameters. We propose, for this purpose, that a future satellite such as LAGEOS II have an inclination supplementary to that of LAGEOS. The experiment proposed here would provide a method for experimental verification of the general relativistic formulation of Mach's principle and measurement of the gravitomagnetic field.

Generator algebra of the asymptotic Poincar� group in the general theory of relativity

This paper obtains the Poisson brackets of the generators of the Hamiltonian formalism for general relativity with allowance for surface terms of aritrary form. For Minkowski space, there exists the asymptotic Poincare group, which is the semi-direct product of the Poincare group and an infinite subgroup for which the algebra of generators with surface terms closes. A criterion invariant with respect to the choice of the coordinate system on the hypersurfaces is obtained for realization of the Poincare group in asymptotically flat space-time. The ''background'' flat metric on the hypersurfaces and Poincare group that preserve it are determined nonuniquely; however, the numerical values of the generators do not depend on the freedom of this choice on solutions of the constraint equations. For an asymptotically Galilean metric, the widely used boundary cnoditins are determined more accurately. A prescription is given for application of the Arnowitt-Deser-Misner decomposition in the case of a slowly decreasing contribution from coordinate and time transformations.

Poincaré gauge theory, gravitation, and transformation of matter fields

AbstractIn the generalized Kaluza‐Klein theories a (4 + N)‐dimensional Riemannian space is used as a configuration space to unify gravitation with a nonabelien gauge symmetry characterized by a gauge group G of order N. The original metric space V4+N, however, is broken down in these theories to a local product structure M4 × VN (M4 being Minkowski space‐time) leading thereby to a description in terms of a fiber bundle over space‐time V4 with structural group G. In this paper we base the discussion from the very beginning on a bundle structure as the underlying geometric stratum and treat the quantum mechanical aspect of matter by representing it in the form of a generalized wave function, ϕ(x, X), defined as a section on a fiber bundle associated to a principal G‐bundle over space time having a homogeneous space of the group G as fiber. To make contact with gravitation we consider various soldered bundles possessing the Poincaré group, G = ISO(3, 1), as structural or gauge group and define Poincaré gauge fields as generalized matter fields on them. After reviewing the formulation of general relativity as a gauge theory of the Lorentz group we turn to the affine case and present a geometric formulation of a Poincaré gauge theory based on a Riemann‐Cartan space‐time U4 with axial vector torsion. Two sets of field equations are discussed relating the material source quantities to the geometric fields. Besides Einstein's equations coupling the metric to the classical distribution of energy and momentum of matter a second set of nonlinear field equations is set up relating a bilinear current in the Poincaré gauge fields to the axial vector torsion field of the underlying space‐time geometry. Finally, a breaking of the Poincaré symmetry is introduced in a manner involving the generalized matter field ϕ(x, X) freezing thereby the translational gauge degrees of freedom, however, leaving the Lorentz gauge symmetry unaffected.

Algebraic programming of Hamiltonian formalism in general relativity: Application to inhomogeneous space-times

We describe the structure and the use of a program written in the algebraic programming languagereduce 2, giving the super-Hamiltonian and supermomenta constraints, as well as Hamilton's canonical equations in terms of the canonical variables, for vacuum relativistic space-times. The program uses as input the components of the spatial metric tensor and of the corresponding canonically conjugate momenta in a coordinate or in a spatial Cartan basis. The results of the application of the program to a series of inhomogeneous (cosmological as well as noncosmological) space-times are given: in particular, the constraints, the Dirac Hamiltonian and the canonical equations are explicitly written for axisymmetric space-times, constituting the starting point for the study of the dynamics and of the canonical quantization of these configurations.

Modification of the Palatini variational principle in general relativity

A new classical action formulation of general relativity is presented. The aim is to develop a variational principle which derives both Riemannian geometry and Einstein's field equations in the general case of a gravitational field interacting with matter. The method is applied to the Hilbert action functional with a variable matter-gravity coupling. Surprisingly, the resulting theory is not general relativity with a variable G; instead, the method yields general relativity with a constant G and a cosmological constant. It is shown that the variational principle has a physical content different from that of the usual Hilbert and Palatini action formulations. A possible application of the new variational principle to the path-integral quantization of gravity is discussed.

Perfect fluid sources for spatially homogeneous spacetimes

Spatially homogeneous perfect fluid spacetimes are studied from a point of view which emphasizes the spatial geometry and the action of that subgroup of the spatial gauge group of the three-plus-one formulation of general relativity which is compatible with the spatial homogeneity. The specializations of the dynamics which correspond to the existence of additional spacetime symmetries are classified. An unconstrained set of gravitational and fluid variables is obtained by elimination of the gravitational constraints using an approach which obtains the gravitational evolution equations from a suitably modified Lagrangian/Hamiltonian formalism. A slightly different choice of variables is then described which allows one to take full advantage of the spatial gauge group and of the 1-parameter group of scale transformations of the unit of length.

Structure and stability of rotating fluid disks around massive objects. II. General relativistic formulation

In this paper we have considered the structure of a thick perfect fluid disk of constant density rotating around a Schwarzschild black hole and its stability under axisymmetric perturbation. The inner edge of such disk cannot lie within 4m. The critical γc for neutral stability is found to be much less than 4/3 indicating that the disks are generally stable

The Hilbert space of quantum gravity

In a recent paper one of us (MC) suggested a new approach to the quantization of the gravitational field—a presentation of gravity as the Hilbert space of the quantized system. This approach is developed and extended in the present paper. States in the Hilbert space are defined in terms of the representation of the Weyl tensor in the Newman-Penrose formulation of general relativity. The determination of first-order states in perturbation theory is shown to reduce to a solution of a set of equations for the coefficients of the expansion of one complex function in the matrix elements of the irreducible representations of the groupSU2. For exact (nonperturbative) solutions it is shown that in the stationary case each solution of the field equations corresponds to a unique state in the Hilbert space. When gravitational radiation is present, an additional parameter is introduced. The classification of states in the Hilbert space is given.

Evolving dust shells

The reduction of the Arnowitt-Deser-Misner canonical formulation of general relativity developed in the first paper of this series is applied to the full time-evolution problem for spherically symmetric charged dust shells. Detailed pictures of shell evolution are produced. Among other things, it is found that under certain well-defined circumstances the asymptotically flat spacelike hypersurfaces of constant time ''pinch off'' and become completely closed, the closure point being a locally naked singularity.

Computational techniques for the Arnowitt-Deser-Misner formalism

In order to apply the Arnowitt-Deser-Misner canonical formulation of general relativity to the study of systems involving material sources, a particular set of coordinate conditions and a stipulated initial absence of gravitational radiation are applied to the reduction on the initial hypersurface of the constraint equations, determining equations for the lapse and shift functions, and evolution equations to forms that are sufficiently tractable for practical computations.

Boundary terms in the action for the Regge calculus

The boundary terms in the action for Regge's formulation of general relativity on a simplicial net are derived and compared with the boundary terms in continuum general relativity.

General relativistic analysis of structure and stability of charged fluid disks around compact objects

In this paper we give a detailed general relativistic formulation of the study of structure and stability of charged fluid disks around compact objects like black holes neglecting the self-gravitation of the disk itself. Having presented the general equations for equilibrium as well as for perturbations we solve explicitly the cases of rigidly and differentially rotating thin disks, with constant charge density and zero pressure, confined to the equatorial plane of the black hole. By using normal mode analysis we have analysed the stability of such disks under purely radial perturbations and find that the disks are generally stable.

Topological Charges in Gauge Theories

AbstractTopological and geometric aspects of gauge theories are examined. The geometry of the fiber‐bundle formulation of gauge theories is discussed and compared with the formalism of general relativity. The basic role played by the parallel displacement operator of this geometry is examined. With this operator a gauge independent characterization of various topological singularities and non‐singular soliton configurations is carried out.

Even parity junction conditions for perturbations on most general spherically symmetric space–times

A new highly efficient and versatile general relativistic perturbational formalism for general matter occupied spherically symmetric space–times is developed. The perturbations are geometrical objects on the two dimensional totally geodesic submanifold spanned by the radial and time coordinates. The geometrical objects are ’’gauge invariant’’ scalars, vectors, and tensors which are independent of infinitesimal coordinate transformations on the background space–time. This article gives the even parity gauge invariant perturbation objects for arbitrary background scalars, vectors, and symmetric tensors on a spherically symmetric space–time. In particular, metric, matter, first and second fundamental forms, as well as vacuum-matter interface gauge invariant perturbations for a collapsing star are given. In addition four even parity continuity conditions across discontinuous timelike hypersurfaces are given. Two are conditions on the metric gauge invariants, one is a condition on the perturbation away from the spherical contour of the interface, and the fourth couples that contour perturbation to the metric gauge invariants.

De Sitter gauge invariance and the geometry of the Einstein-Cartan theory

A formulation of general relativity as a gauge theory of the de Sitter group SO(3,2) is used to analyse the geometrical structure of the Einstein-Cartan theory. The SO(3,2) symmetry must be spontaneously broken to the Lorentz group in order to reproduce the usual four-dimensional geometry of gravity. Special emphasis is placed upon the role of the Goldstone field of the symmetry breaking mechanism and also that of the original SO(3,2) gauge fields. The latter are not directly identified with the gravitational vierbein and spin connection, but instead generate a kind of parallel transport known as development which is the necessary construction to interpret the effects of space-time torsion and curvature.

General relativity as a generalized Hamiltonian system

A rigorous setting for Dirac's generalized Hamiltonian dynamics in an infinite number of dimensions is presented. It is shown that the dynamical formulation of general relativity fits into this scheme.

Quantum gravity on a lattice

A formulation of quantum gravity with a gauge-invariant cutoff is given by transcribing MacDowell and Mansouri's formulation of general relativity onto a Wilson lattice. The weak coupling and strong coupling expansions are developed. The strong coupling expansion reveals a typical strong coupling phase with short range order, confinement (of spin) and asymptotic freedom. If the lattice theory has a second-order phase transition, then the continuum theory may be well-defined.

Conservation laws in the tetrad formulation of the general theory of relativity

The covariant consequences of a weak conservation law in the tetrad formulation of general relativity that do not contain noncovariant complexes of energy-momentum or external and internal spin momenta are considered. The relationship between a group of arbitrary tetrad Lorentz transformations and a generally covariant definition of the spin angular momentum of nongravitational matter is outlined.

A non-singular quantum Universe

CLASSICAL relativistic singularities remain one of the most puzzling problems of contemporary physics. Although no consensus of opinion has been attained on their status, a great deal of work has been devoted to the search for physical mechanisms capable of eliminating the cosmological singularities, whose existence follows from the theorems of Hawking and Penrose1,2. The alternative view that the initial singularity of big-bang models is essential for the coming into being of the Universe, has also been advocated3,4. Some suggestions of cosmological singularity avoidance have recently come to light, in the ‘classical’ quantum regime (from 10−43–10−23s) (refs 5–8) or even without resorting to quantum effects9. The quantum gravitational period (t < 10−43 s) remains potentially very interesting for searching for singularity avoidance mechanisms. An interesting approach is that of quantum cosmology, based on the hamiltonian formulation of general relativity (Dirac's10 or Arnowitt, Deser and Misner's11 (ADM) method). This approach was initiated by De Witt12 and then followed by Misner13 and many others (see refs 14,15) and is used here to derive a non-singular quantum cosmological model.