Showing papers on "Mathematics of general relativity published in 2004"

Symmetries and Curvature Structure in General Relativity

Abstract: Introduction topological spaces groups and linear algebra manifold theory transformation groups the Lorentz group general relativity theory space-time holonomy curvature structure in general relativity affine symmetries in space-time conformal symmetries in space-time curvature collineations sectional curvature structure.

Gravitational field equations on and off a 3-brane world

Abstract: The effective gravitational field equations on and off a 3-brane world possessing a mirror symmetry and embedded in a five-dimensional bulk spacetime with cosmological constant were derived by Shiromizu, Maeda and Sasaki (SMS) in the framework of the Gauss?Codazzi projective approach with the subsequent specialization to the Gaussian normal coordinates in the neighbourhood of the brane. However, the Gaussian normal coordinates imply a very special slicing of spacetime and clearly, the consistent analysis of the brane dynamics would benefit from complete freedom in the slicing of spacetime, pushing the layer surfaces in the fifth dimension at any rates of evolution and in arbitrary positions. We rederive the SMS effective gravitational field equations on a 3-brane and generalize the off-brane equations to the case where there is an arbitrary energy?momentum tensor in the bulk. We use a more general setting to allow for acceleration of the normals to the brane surface through the lapse function and the shift vector in the spirit of Arnowitt, Deser and Misner. We show that the gravitational influence of the bulk spacetime on the brane may be described by a traceless second-rank tensor , constructed from the 'electric' part of the bulk Riemann tensor. We also present the evolution equations for the tensor , as well as for the corresponding 'magnetic' part of the bulk curvature. These equations involve terms determined by both the nonvanishing acceleration of normals in the nongeodesic slicing of spacetime and the presence of other fields in the bulk.

The speed of gravity in general relativity and theoretical interpretation of the Jovian deflection experiment

Abstract: According to Einstein, the notions of geodesic, parallel transport (affine connection) and curvature of the spacetime manifold have a pure geometric origin and do not correlate with any electromagnetic concepts. At the same time, curvature is generated by matter which is not affiliated with the spacetime geometric concepts. For this reason, the fundamental constant c entering the geometric and matter sectors of the general theory of relativity has different conceptual meanings. Specifically, the letter c on the left-hand side of the Einstein equations (geometric sector) entering the Christoffel symbols and its time derivatives is the ultimate speed of gravity characterizing the upper limit on the speed of its propagation as well as the maximal rate of change of time derivatives of the metric tensor, that is gravitational field. The letter c on the right-hand side of the Einstein equations (matter sector) is the maximal speed of propagation of any other field rather than gravity. Einstein's general principle of relativity extends his principle of special relativity and equates the numerical value of the ultimate speed of gravity to that of the speed of light in the special theory of relativity but this general principle must be tested experimentally. To this end, we work out the speed of gravity parametrization of the Einstein equations (cg-parametrization) to keep track of the time-dependent effects associated with the geometric sector of general relativity and to separate them from the time-dependent effects of the matter sector. Parametrized post-Newtonian (PPN) approximation of the Einstein equations is derived in order to explain the gravitational physics of the Jovian deflection VLBI experiment conducted on 8 September 2002. The post-Newtonian series expansion in the cg-parametrized general relativity is with respect to a small parameter that is proportional to the ratio of the characteristic velocity of the bodies to the speed of propagation of the gravitational interaction cg. The Einstein equations are solved in terms of the Li?nard?Wiechert tensor potentials which are used for integrating the light-ray propagation equations. An exact analytic expression for the relativistic time delay in the propagation of a radio wave from a quasar to an observer is calculated under the assumption that the light-ray deflecting bodies move with constant velocities. A post-Newtonian expansion of the time delay proves that in general relativity the time delay is affected by the speed of gravity already to the first order in 1/cg beyond the leading (static) Shapiro term. We conclude that recent measurements of the propagation of the quasar's radio signal past Jupiter are directly sensitive to the time-dependent effect from the geometric sector of general relativity which is proportional to the speed of propagation of gravity cg but not the speed of light. It provides a first confirmative measurement of the fundamental speed c of the Einstein general principle of relativity for gravitational field. A comparative analysis of our formulation with the alternative interpretations of the experiment given by other authors is provided.

Dynamics of Pure Shape, Relativity, and the Problem of Time

Abstract: A new approach to the dynamics of the universe based on work by O Murchadha, Foster, Anderson and the author is presented. The only kinematics presupposed is the spatial geometry needed to define configuration spaces in purely relational terms. A new formulation of the relativity principle based on Poincare’s analysis of the problem of absolute and relative motion (Mach’s principle) is given. The entire dynamics is based on shape and nothing else. It leads to much stronger predictions than standard Newtonian theory. For the dynamics of Riemannian 3-geometries on which matter fields also evolve, implementation of the new relativity principle establishes unexpected links between special relativity, general relativity and the gauge principle. They all emerge together as a self-consistent complex from a unified and completely relational approach to dynamics. A connection between time and scale invariance is established. In particular, the representation of general relativity as evolution of the shape of space leads to a unique dynamical definition of simultaneity. This opens up the prospect of a solution of the problem of time in quantum gravity on the basis of a fundamental dynamical principle.

Hermann Weyl's Analysis of the “Problem of Space” and the Origin of Gauge Structures

Abstract: Hermann Weyl (1885–1955) was one of the early contributors to the mathematics of general relativity. This article argues that in 1929, for the formulation of a general relativistic framework of the Dirac equation, he both abolished and preserved in modified form the conceptual perspective that he had developed earlier in his “analysis of the problem of space.” The ideas of infinitesimal congruence from the early 1920s were aufgehoben (in all senses of the German word) in the general relativistic framework for the Dirac equation. He preserved the central idea of gauge as a “purely infinitesimal” aspect of (internal) symmetries in a group extension schema. With respect to methodology, however, Weyl gave up his earlier preferences for relatively a-priori arguments and tried to incorporate as much empiricism as he could. This signified a clearly expressed empirical turn for him. Moreover, in this step he emphasized that the mathematical objects used for the representation of matter structures stood at the center of the construction, rather than interaction fields which, in the early 1920s, he had considered as more or less derivable from geometrico-philosophical considerations.

Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold

Abstract: The smooth gravitational singularities of the differential spacetime manifold based General Relativity (GR) are viewed from the perspective of the background manifold independent and, in extenso, Calculus-free Abstract Differential Geometry (ADG). In particular, the inner Schwarzschild singularity is being `resolved' ADG-theoretically in two different ways. A plethora of important mathematical, physical and philosophical issues in current classical and quantum gravity research are addressed and tackled.

Histories approach to general relativity: II. invariance groups

Abstract: In this paper we show in detail how the histories description of general relativity carries representations of both the spacetime diffeomorphism group and the Dirac algebra of constraints. We show that the introduction of metric-dependent equivariant foliations leads to the crucial result that the canonical constraints are invariant under the action of spacetime diffeomorphisms. Furthermore, there exists a representation of the group of generalized spacetime mappings that are functionals of the 4-metric: this is a spacetime analogue of the group originally defined by Bergmann and Komar in the context of the canonical formulation of general relativity. Finally, we discuss the possible directions for the quantization of gravity in histories theory.

Special Relativity: From Einstein to Strings

Abstract: This book provides a thorough introduction to Einstein's special theory of relativity, suitable for anyone with a minimum of one year's university physics with calculus. It is divided into fundamental and advanced topics. The first section starts by recalling the Pythagorean rule and its relation to the geometry of space, then covers every aspect of special relativity, including the history. The second section covers the impact of relativity in quantum theory, with an introduction to relativistic quantum mechanics and quantum field theory. It also goes over the group theory of the Lorentz group, a simple introduction to supersymmetry, and ends with cutting-edge topics such as general relativity, the standard model of elementary particles and its extensions, superstring theory, and a survey of important unsolved problems. Each chapter comes with a set of exercises. The book is accompanied by a CD-ROM illustrating, through interactive animation, classic problems in relativity involving motion.

General relativity with torsion

Abstract: General Relativity with nonvanishing torsion has been investigated in the first order formalism of Poincare gauge field theory. In the presence of torsion, either side of the Einstein equation has the nonvanishing covariant divergence. This fact turned out to be self-consistent in the framework under consideration. By using Noether's procedure with the definition of the Lie derivative where the general coordinate transformation and the local Lorentz rotation are combined, the revised covariant divergence of the energy momentum is consistently obtained. Subsequently we have definitely derived the spin correction to the energy momentum tensor for the Dirac field and the Rarita-Schwinger field in the Einstein equation. We conjecture that the accelerated expansion of the universe possibly arises due to the spin correction in our framework.

Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics Vol 47)

Abstract: This book is concerned with mathematical topics related to general relativity. Chapters 1-6 are expositions of a number of parts of mathematics which are important for relativists. The main subjects covered are linear algebra, general topology, manifolds, Lie groups and, in particular detail, the Lorentz group. Chapters 7-13 analyse aspects of the geometrical structure of spacetimes. They focus on symmetries of various types and on properties of curvature. Subjects covered include the Petrov classification, holonomy groups, the relation between metric and curvature, affine vector fields, conformal symmetries, projective symmetries and curvature collineations. This part of the book is a treatise on the (mainly local) geometry of four-dimensional Lorentz manifolds, with attention to energy-momentum tensors of interest in general relativity. In the first six chapters the author has concentrated on giving definitions and statements of theorems, the proofs being left to the references that are quoted. He has clearly put much effort into producing a very smooth exposition which is easy to follow and has succeeded in giving us a readable and informative account of the mathematics covered. At the same time mathematical rigour is strictly observed. He also takes the time to carefully discuss many of the subtleties which arise. This part of the book has the character of a textbook suitable for students of general relativity but experienced researchers will also find it a useful reference and are likely to come across interesting facts they have not met elsewhere. The remaining chapters are more like a research monograph and are influenced by the author's own research interests. More proofs are included. Much of the material in this part will be of interest to a narrower audience of relativists than that of the first part. It should, however, be of interest to those working on exact solutions of the Einstein equations and related topics. It is also the case that most relativists are concerned with detailed properties of symmetries and curvature in some phases of their work and may find information which can help them in this book. Two examples of useful things to be found in this book for which it is difficult to find a comparable account elsewhere are the discussions of covering spaces and the subgroups of the Lorentz group. Section 3.10 treats covering spaces in a general topological context while section 4.14 shows how these ideas can be adapted to the smooth category. Section 6.4 lists all the connected subgroups of the Lorentz group and collects background information about these subgroups. I have no doubt that this book will be valuable for students taking courses in general relativity and for people preparing their PhD in the subject.

Relativity: An Introduction to Special and General Relativity, 3rd edn

Abstract: This new edition of Hans Stephani's book is a welcome addition to the many texts now available to the student wishing to study relativity. A new feature, as compared with earlier editions, is a substantial introductory section on special relativity amounting to almost a quarter of the total length of the book. This forms part I and covers all the usual topics one might expect, including particle mechanics, the formulae for aberration and the Doppler effect, tensors in Minkowski spacetime, Maxwell's equations and the energy-momentum tensor for the electromagnetic field, the equations of motion for charged point particles and their fields (with a brief discussion of radiation reaction and runaway solutions) and the energy-momentum tensor for a perfect fluid. But there is also a useful section on the algebraic classification of the electromagnetic field using the apparatus of null tetrads and self-dual bivectors. An unusual feature is a section on pole-dipole charged particles. General relativity is introduced and developed in parts II to VII of the book, which follow closely the text of the 1982 English edition. In part II, the reasons for regarding spacetime as a four-dimensional Riemannian manifold with a metric of Lorentz signature are clearly explained. There follows a concise treatment of the tensor calculus including the covariant derivative, parallel and Fermi-Walker transport, the Lie derivative and the properties of the Riemann tensor. Invariant forms for line, surface and volume integrals are discussed and Stokes' theorem is stated without proof. There is a final section on how physical laws within special relativity may be generalized to curved spacetime. Part III introduces the reader to the Einstein field equations. Schwarzschild's exterior solution and Birkoff's theorem are derived, as are the usual formulae for the perihelion advance for planetary orbits, the deflection of light rays, the gravitational red shift and the geodesic precession of a top. Gravitational lensing is briefly discussed. This part ends with a derivation of the Schwarzschild interior metric for a source with constant rest mass density. The linearized field equations and the question as to whether their solution for a time varying system gives reliable information about solutions to the full nonlinear equations are discussed in part IV. The Landau-Lifshitz pseudo energy-momentum tensor is used to obtain the standard formula for the outward flow of energy due to quadrupole radiation from a bounded system. While the limitations of this approach are emphasized by the author, particularly in regard to the question of whether gravitational waves lead to energy transfer, it is surprising that no mention is made of the important work by Bondi and others in the 1960s which used the nonlinear equations to show that such radiating systems must lose mass. The final sections contain standard treatments of plane wave solutions of both the linearized and nonlinear theory, and a discussion of the Cauchy problem for the vacuum equations. Part V contains a clear account of the Petrov classification of the Weyl tensor (using the same technique as that used earlier in the algebraic classification of the Maxwell field), and of Killing vectors (including a brief introduction to the Bianchi classification of groups of motion) and their relation to conservation laws. Gravitational collapse and black holes are discussed in part VI, which covers such topics as the Kruskal extension of the Schwarzschild metric, the critical mass of a star, the gravitational collapse of a sphere of dust and a brief discussion of the properties of the Kerr metric. Also included in this new edition is a short but useful section on the problem of quantization, of quantum field theory in a given curved spacetime and on the Hawking effect, and on the conformal structure of infinity leading to a definition of asymptotic flatness. In part VII on cosmology, after showing how the cosmological principle leads to the Robertson-Walker metric, the various Friedmann models are derived and discussed. The final sections discuss a Bianchi type I model and the Godel universe. For its size (at about four hundred pages) the book covers a great deal of material, and it does this very clearly and well. There are some misprints, but almost all of these are of an obvious kind and easily corrected. At the end of most sections there are some well chosen exercises (perhaps there could have been more of these) and useful suggestions for further reading. Students who work through the book carefully, completing the details for those calculations and proofs where these are outlined in the text, will learn a great deal about relativity and be well placed for further study in the subject. While it is probably too advanced in its scope for the average final year (UK) undergraduate course, I would recommend it as a reference book for further reading at this level or as a text for study by a beginning graduate student.

A New Strong-Field Effect in Scalar Tensor Gravity: Spontaneous Violation of the Energy Conditions

Abstract: A decade ago, it was shown that a wide class of scalar-tensor theories can pass very restrictive weak-field tests of gravity and yet exhibit nonperturbative strong-field deviations away from general relativity. This phenomenon, called "spontaneous scalarization," causes the (Einstein frame) scalar field inside a neutron star to rapidly become inhomogeneous once the star's mass increases above some critical value. For a star whose mass is below the threshold, the field is instead nearly uniform (a state that minimizes the star's energy) and the configuration is similar to the general relativity one. Here we show that the spontaneous scalarization phenomenon is linked to another strong-field effect: a spontaneous violation of the weak energy condition.

Cheeger-Gromov Theory and Applications to General Relativity

Abstract: This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M, and some recent applications of this theory to general relativity. The basic point of view of convergence/degeneration described here originates in the work of Gromov, cf. [31]—[33], with important prior work of Cheeger [16], leading to the joint work of [18].

Physical time and physical space in general relativity

Abstract: This paper comments on the physical meaning of the line element in general relativity. We emphasize that, generally speaking, physical spatial and temporal coordinates (those with direct metrical significance) exist only in the immediate neighborhood of a given observer, and that the physical coordinates in different reference frames are related by Lorentz transformations (as in special relativity) even though those frames are accelerating or exist in strong gravitational fields.

Relativity in introductory physics

Abstract: A century after its formulation by Einstein, it is time to incorporate special relativity early in the physics curriculum. The approach advocated here employs a simple algebraic extension of vector formalism that generates Minkowski spacetime, displays covariant symmetries, and enables calculations of boosts and spatial rotations without matrices or tensors. The approach is part of a comprehensive geometric algebra with applications in many areas of physics, but only an intuitive subset is needed at the introductory level. The approach and some of its extensions are given here and illustrated with insights into the geometry of spacetime. PACS Nos.: 03.30.+p, 01.40.Gm, 03.50.De, 02.10.Hh

Anisotropic Fluid Distribution in Bimetric Theory of Relativity

Abstract: In this paper we have presented a procedure to obtain exact analytical solutions of field equations for spherically symmetric self-gravitating distribution of anisotropic matter in bimetric theory of gravitation. The solution agrees with the Einstein's general relativity for a physical system compared to the size of universe such as the solar system.

Constrained evolution in numerical relativity

Abstract: vi

An unification of general theory of relativity with Dirac's large number hypothesis

Abstract: Taking a hint from Dirac's large number hypothesis, we note the existence of cosmologically combined conservation laws that work to cosmologically long time. We thus modify Einstein's theory of general relativity with fixed gravitation constant $G$ to a theory for varying $G$, with a tensor term arising naturally from the derivatives of $G$ in place of the cosmological constant term usually introduced \textit{ad hoc}. The modified theory, when applied to cosmology, is consistent with Dirac's large number hypothesis, and gives a theoretical Hubble's relation not contradicting the observational data. For phenomena of duration and distance short compared with that of the universe, our theory reduces to Einstein's theory with $G$ being constant outside the gravitating matter, and thus also passes the crucial tests of Einstein's theory.

On Metric and Matter in Unconnected, Connected, and Metrically Connected Manifolds

Abstract: From Einstein's point of view, his General Relativity Theory had strengths as well as failings For him, its shortcoming mainly was that it did not unify gravitation and electromagnetism and did not provide solutions to field equations which can be interpreted as particle models with discrete mass and charge spectra, As a consequence, General Relativity did (and does) not solve the quantum problem, either Einstein tried to get rid of the shortcomings without losing the achievements of General Relativity Theory Stimulated by papers of Weyl (Sitzungsber Preuss Akad Wiss (1918) 465) and Eddington (Proc R Soc Hond 99 (1921) 194), from 1923 onward, he believed that, to reach this goal, one has to transit to space–times which possess more comprehensive geometrical structures than the Riemann space–time This was the beginning of a decade's lasting search for a unitary field theory We describe this exciting part of the history of physics, discuss achievements and failures of this development, and ask how these early attempts of a unified theory strike us today Taking into account the fact that the Equivalence Principle only speaks for a geometrization of gravitation, we consider an alternative way to give those non-Riemannian structures which were introduced by the unitary field approach a physical meaning, namely the meaning of a generalized gravitational field This is interesting since there are arguments in favor of such a generalization of General Relativity Theory, eg, the problems the latter theory meets with if one tries to quantize it and to unify gravitation with other interactions

A Novel View of Spacetime Permitting Faster‐Than‐Light Travel

Abstract: Recent discoveries across many disciplines of physics have supported a driving need for a “new” science to explain the apparent relationship between phenomenon at cosmological scales and those at the quantum, subatomic level while still supporting the classical mechanics of motion, electromagnetism and relativity A novel view of both the spacetime continuum and the universe is postulated that not only connects these fields of interest, but proposes a method to travel at superluminal speeds by examining the underlying equations of special relativity The governing mathematics of special relativity describe a symmetrical continuum that supports not just one, but three, independent spacetimes each with a unique set of physical laws founded on the speed of light, c These spacetimes are the subluminal (where v/c 1) comprising a ‘tri‐space’ universe Relativistic symmetry illustrates that there can be up to three velocities (one for each spacetime) for a given absolute energy state The similar characteristics of mass and energy in each spacetime may permit faster‐than‐light (FTL) travel through a quantum transformation/exchange of energy and mass (at the quark level or beyond) between the subluminal and superluminal realms Based on the suggested characteristics of superluminal spacetime, the ‘trans‐space’ method of FTL travel would allow a particle to traverse sublight space by traveling through the superlight continuum without incurring the penalties of special relativity or causal relations In addition, the spacetime construct and superluminal realm of the ‘tri‐space’ universe may offer a different perspective than the current ideologies that could better represent physical phenomena including universal expansion, the zero‐point field, dark matter, and the source of inertia

General Relativity as a Genuine Connection Theory

Abstract: The Palatini formulation is used to develop a genuine connection theory for general relativity, in which the gravitational field is represented by a Lorentz-valued spin connection. The existence of a tetrad field, given by the Fock-Ivanenko covariant derivative of the tangent-space coordinates, implies a coupling between the spin connection and the coordinate vector-field, which turns out to be the responsible for the onset of curvature. This connection-coordinate coupling can thus be considered as the very foundation of the gravitational interaction. The peculiar form of the tetrad field is shown to reduce both Bianchi identities of general relativity to a single one, which brings this theory closer to the gauge theories describing the other fundamental interactions of Nature. Some further properties of this approach are also examined.

Numerical studies of constraints and gravitational wave extraction in general relativity

Abstract: Title of Dissertation: Numerical studies of constraints and gravitational wave extraction in general relativity David Robert Fiske, Doctor of Philosophy, 2004 Dissertation directed by: Professor Charles W. Misner Department of Physics Within classical physics, general relativity is the theory of gravity. Its equations are non-linear partial differential equations for which relatively few closed form solutions are known. Because of the growing observational need for solutions representing gravitational waves from astrophysically plausible sources, a subfield of general relativity, numerical relativity, has a emerged with the goal of generating numerical solutions to the Einstein equations. This dissertation focuses on two fundamental problems in modern numerical relativity: (1) Creating a theoretical treatment of the constraints in the presence of constraint-violating numerical errors, and (2) Designing and implementing an algorithm to compute the spherical harmonic decomposition of radiation quantities for comparison with observation. On the issue of the constraints, I present a novel and generic procedure for incorporating the constraints into the equations of motion of the theory in a way designed to make the constraint hypersurface an attractor of the evolution. In principle, the prescription generates non-linear corrections for the Einstein equations. The dissertation presents numerical evidence that the correction terms do work in the case of two formulations of the Maxwell equations and two formulations of the linearized Einstein equations. On the issue of radiation extraction, I provide the first in-depth analysis of a novel algorithm, due originally to Misner, for computing spherical harmonic components on a cubic grid. I compute explicitly how the truncation error in the algorithm depends on its various parameters, and I also provide a detailed analysis showing how to implement the method on grids in which explicit symmetries are enforced via boundary conditions. Finally, I verify these error estimates and symmetry arguments with a numerical study using a solution of the linearized Einstein equations known as a Teukolsky wave. The algorithm performs well and the estimates prove true both in simulations run on a uniform grid and in simulations that make use of fixed mesh refinement techniques. Numerical studies of constraints and gravitational wave extraction in general relativity

The Field theoretical formulation of general relativity and gravity with non-zero masses of gravitons

Abstract: It is a review paper General relativity (GR) is presented in the field theoretical form, where gravitational field (metric perturbations) together with other physical fields are propagated in an auxiliary arbitrary curved background spacetime Conserved currents are constructed and expressed through divergences of antisymmetrical tensor densities (superpotentials) This permits to connect local properties of perturbations with the quasi-local nature of the conserved quantities in GR The problem of the non-localization of energy in GR is presented in exact mathematical expressions A modification of GR developed recently by Babak and Grishchuk on the basis of the field formulation of GR is described Their theory includes massive of spin-2 and spin-0 gravitons All its local weak-field predictions are in agreement with experimental data The exact equations of the massive theory eliminate the black hole event horizons and give an oscillator behavior for the homogeneous isotropic universe

A Possible Modification of Einstein's Theory of General Relativity

Abstract: This article suggests a new metric theory of gravitation, in which metric field is determined not only by matter and nongravitational field but also by vector graviton field, and in principle there is no need to introduce the Einstein's tensor. In order to satisfy automatically the geodesic postulate, an additional coordinate condition is needed. For the spherically symmetric static field, it leads us to quite different conclusions from those of Einstein's general relativity in the interior region of the surface of infinite redshift. Accurate to the first order of , it obtains the same results about the four experimental tests of general relativity.

Kinematic self-similar solutions in general relativity

Abstract: The gravitational interaction is scale-free in both Newtonian gravity and general theory of relativity The concept of self-similarity arises from this nature Self-similar solutions reproduce themselves as the scale changes This property results in great simplification of the governing partial differential equations In addition, some self-similar solutions can describe the asymptotic behaviors of more general solutions Newtonian gravity contains only one dimensional constant, the gravitational constant, while the general relativity contains another dimensional constant, the speed of light, besides the gravitational constant Due to this crucial difference, incomplete similarity can be more interesting in general relativity than in Newtonian gravity Kinematic self-similarity has been defined and studied as an example of incomplete similarity in general relativity, in an effort to pursue a wider application of self-similarity in general relativity We review the mathematical and physical aspects of kinematic self-similar solutions in general relativity