Showing papers on "Introduction to the mathematics of general relativity published in 2003"

Spacetime and Geometry: An Introduction to General Relativity

Abstract: Spacetime and Geometry is an introductory textbook on general relativity, specifically aimed at students. Using a lucid style, Carroll first covers the foundations of the theory and mathematical formalism, providing an approachable introduction to what can often be an intimidating subject. Three major applications of general relativity are then discussed: black holes, perturbation theory and gravitational waves, and cosmology. Students will learn the origin of how spacetime curves (the Einstein equation) and how matter moves through it (the geodesic equation). They will learn what black holes really are, how gravitational waves are generated and detected, and the modern view of the expansion of the universe. A brief introduction to quantum field theory in curved spacetime is also included. A student familiar with this book will be ready to tackle research-level problems in gravitational physics.

General-covariant evolution formalism for numerical relativity

Abstract: A general-covariant extension of Einstein's field equations is considered with a view to numerical relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector ${Z}_{\ensuremath{\mu}}.$ Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition ${Z}_{\ensuremath{\mu}}=0.$ The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.

A Nonlocal metric formulation of MOND

Abstract: We study a class of nonlocal, but causal, covariant and conserved field equations for the metric. Although nonlocal, these equations do not seem to possess extra graviton solutions in weak field perturbation theory. Indeed, the equations reduce to those of general relativity when the Ricci scalar vanishes throughout spacetime. When a static matter source is present, we show how these equations can be adjusted to reproduce Milgrom's modified Newtonian dynamics in the weak field regime, while reducing to general relativity for strong fields. We compute the angular deflection of light in the weak field regime and demonstrate that it is the same as for general relativity, resulting in far too little lensing if no dark matter is present. We also study the field equations for a general Robertson–Walker geometry. An interesting feature of our equations is that they become conformally invariant in the MOND limit.

General relativity with a topological phase: An Action principle

Abstract: An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced and depending on the value it takes the theory has solutions that reduce it to 1) general relativity in Palatini form, 2) general relativity in the Ashtekar form, 3) $F\wedge F$ theory for SO(5) and 4) $BF$ theory for SO(5). This theory then makes it possible to describe explicitly the dynamics of phase transition between a topological phase and a gravitational phase where the theory has local degrees of freedom. We also find that a boundary between adymnamical and topological phase resembles an horizon.

Various features of quasiequilibrium sequences of binary neutron stars in general relativity

Abstract: Quasiequilibrium sequences of binary neutron stars are numerically calculated in the framework of the Isenberg-Wilson-Mathews (IWM) approximation of general relativity. The results are presented for both rotation states of synchronized spins and irrotational motion, the latter being considered as the realistic one for binary neutron stars just prior to merger. We assume a polytropic equation of state and compute several evolutionary sequences of binary systems composed of different-mass stars as well as identical-mass stars with adiabatic indices $\ensuremath{\gamma}=2.5,$ 2.25, 2, and 1.8. From our results, we propose as a conjecture that if the turning point of the binding energy (and total angular momentum) locating the innermost stable circular orbit is found in Newtonian gravity for some value of the adiabatic index ${\ensuremath{\gamma}}_{0},$ that of the ADM mass (and total angular momentum) should exist in the IWM approximation of general relativity for the same value of the adiabatic index.

Transformations of coordinates and Hamiltonian formalism in deformed special relativity

Abstract: We investigate the transformation laws of coordinates in generalizations of special relativity with two observer-independent scales. The request of covariance leads to simple formulas if one assumes noncanonical Poisson brackets, corresponding to noncommuting spacetime coordinates.

Detecting ill posed boundary conditions in general relativity

Abstract: A persistent challenge in numerical relativity is the correct specification of boundary conditions. In this work we consider a many-parameter family of symmetric hyperbolic initial-boundary value formulations for the linearized Einstein equations and analyze its well posedness using the Laplace–Fourier technique. By using this technique ill posed modes can be detected and thus a necessary condition for well posedness is provided. We focus on the following types of boundary conditions: (i) boundary conditions that have been shown to preserve the constraints, (ii) boundary conditions that result from setting the ingoing constraint characteristic fields to zero, and (iii) boundary conditions that result from considering the projection of Einstein’s equations along the normal to the boundary surface. While we show that in case (i) there are no ill posed modes, our analysis reveals that, unless the parameters in the formulation are chosen with care, there exist ill posed constraint violating modes in the remaining cases.

Numerical Relativity with the Conformal Field Equations

Abstract: I discuss the conformal approach to the numerical simulation of radiating isolated systems in general relativity The method is based on conformal compactification and a reformulation of the Einstein equations in terms of rescaled variables, the so-called “conformal field equations” developed by Friedrich These equations allow to include “infinity” on a finite grid, solving regular equations, whose solutions give rise to solutions of the Einstein equations of (vacuum) general relativity The conformal approach promises certain advantages, in particular with respect to the treatment of radiation extraction and boundary conditions I will discuss the essential features of the analytical approach to the problem, previous work on the problem— in particular a code for simulations in 3+1 dimensions, some new results, open problems and strategies for future work

Gravitational field of massive point particle in general relativity

Abstract: Using various gauges of the radial coordinate we give a description of the static spherically symmetric space-times with point singularity at the center and vacuum outside the singularity. We show that in general relativity (GR) there exist innitely many such solutions to the Einstein equations which are physically dieren t and only some of them describe the gravitational eld of a single massive point particle. In particular, we show that the widespread Hilbert’s form of Schwarzschild solution does not solve the Einstein equations with a massive point particle’s stress-energy tensor. Novel normal coordinates for the eld and a new physical class of gauges are proposed, in this way achieving a correct description of a point mass source in GR. We also introduce a gravitational mass defect of a point particle and determine the dependence of the solutions on this mass defect. In addition we give invariant characteristics of the physically and geometrically dieren t classes of spherically symmetric static space-times created by one point mass.

Introduction to relativity

Abstract: Special relativity is a cornerstone of the structure of all fundamental theories, and general relativity has blossomed from Einstein's original theory into a cutting-edge applied science. Applications of Einstein's field equations describe such phenomena as supermassive black holes at the center of galaxies, the spiraling paths of binary pulsars, gravitational lensing caused by massive compact halo objects (Macho's), and the possibility of detecting gravitational waves emitted in cataclysmic cosmic events. In Introduction to Relativity, physics teacher and researcher Bill McGlinn explains the fundamental concepts of Einstein's special and general theories of relativity. He describes the basic consequences of special relativity-length contraction and time dilation-and the enigma of the twin paradox, as well as the Doppler shift of light. Relativistic dynamics is contrasted to Newtonian dynamics, followed by a discussion of relativistic tensor fields, including those of the electromagnetic field and the energy-momentum density of fluids. After a study of Einstein's early attempt at incorporating the equivalence principle into physics, McGlinn presents the general theory of relativity, discussing the three classic tests of relativity: the deflection of light by a gravitational field; the precession of perihelia; and the gravitational redshift of light. He also discusses other important applications, such as the dynamics of orbiting gyroscopes, the properties of stellar interiors, and black holes. The book ends with a chapter on cosmology, which includes discussions of kinematics and dynamics of the famed Robertson-Walker metric, Hubble's constant, cosmological constant, and cosmic microwave background radiation. For anyone seeking a brief, clear overview of modern general relativity which emphasizes physics over mathematics, McGlinn's Introduction to Relativity is indispensable.

What is a mean gravitational field

Abstract: The equations of General Relativity are non-linear. This makes their averaging non-trivial. The notion of mean gravitational field is defined and it is proven that this field obeys the equations of General Relativity if the unaveraged field does. The workings of the averaging procedure on Maxwell’s field and on perfect fluids in curved space-times are also discussed. It is found that Maxwell’s equations are still verified by the averaged quantities but that the equation of state for other kinds of matter generally changes upon average. In particular, it is proven that the separation between matter and gravitational field is not scale-independent. The same result can be interpreted by introducing a stress-energy tensor for a mean-vacuum. Possible applications to cosmology are discussed. Finally, the work presented in this article also suggests that the signature of the metric might be scale-dependent too.

Nonlinear Connections and Nearly Autoparallel Maps in General Relativity

Abstract: We apply the method of moving anholonomic frames with associated nonlinear connections to the (pseudo) Riemannian space geometry and examine the conditions when locally anisotropic structures (Finsler like and more general ones) could be modeled in the general relativity theory and/or Einstein–Cartan–Weyl extensions [1]. New classes of solutions of the Einstein equations with generic local anisotropy are constructed. We formulate the theory of nearly autoparallel (na) maps generalizing the conformal transforms and formulate the Einstein gravity theory on na–backgrounds provided with a set of na–map invariant conditions and local conservation laws. There are illustrated some examples when vacuum Einstein fields are generated by Finsler like metrics and chains of na–maps.

Plane symmetric cosmological micro model in modified theory of Einstein’s general relativity

Abstract: In this paper, we have investigated an anisotropic homogeneous plane symmetric cosmological micro-model in the presence of massless scalar field in modified theory of Einstein's general relativity. Some interesting physical and geometrical aspects of the model together with singularity in the model are discussed. Further, it is shown that this theory is valid and leads to Ein­stein's theory as the coupling parameter λ →>• 0 in micro (i.e. quantum) level in general.

First-order quasilinear canonical representation of the characteristic formulation of the Einstein equations

Abstract: We prescribe a choice of 18 variables in all that casts the equations of the fully nonlinear characteristic formulation of general relativity in first--order quasi-linear canonical form. At the analytical level, a formulation of this type allows us to make concrete statements about existence of solutions. In addition, it offers concrete advantages for numerical applications as it now becomes possible to incorporate advanced numerical techniques for first order systems, which had thus far not been applicable to the characteristic problem of the Einstein equations, as well as in providing a framework for a unified treatment of the vacuum and matter problems. This is of relevance to the accurate simulation of gravitational waves emitted in astrophysical scenarios such as stellar core collapse.

The Pioneer anomalous acceleration: a measurement of the cosmological constant at the scale of the solar system

Abstract: An anomalous constant acceleration of (8.7 ± 1.3) × 10 −8 cm.s −2 directed toward the Sun has been discovered by Anderson et al. in the motion of the Pioneer 10/11 and Galileo spacecrafts. In parallel, the WMAP results have definitively established the existence of a cosmological constant � = 1/L 2 , and therefore of an invariant cosmic length-scale LU = 2.85 ± 0.25 Mpc. We show that the existence of this invariant scale definitively implements Mach’s principle in Einstein’s theory of general relativity. Then we demonstrate, in the framework of an exact cosmological solution of Einstein’s field equations which is valid both locally and globally, that the definition of inertial systems ultimately depends on this length-scale. As a consequence, usual local coordinates are not inertial, so that the motion of a free body is expected to contain an additional constant acceleration aP = c 2 / p 3 LU = (5.9±0.5)×10 −8 cm.s −2 . Such an effect represents a major contribution to the Pioneer acceleration. The recent definitive proof of the existence, in Einstein’s general relativity equations, of a cosmological constant term � = 0.73 ± 0.05 [1] (or of an equivalent contribution coming e.g. from vacuum energy) can be considered as a corner stone in the history of cosmology. We shall in this paper investigate one of its possible consequences: namely, its very existence allows the full theory of general relativity to come under Mach’s principle, as was initially required by Einstein in its construction.

Ellipsoidal shapes in general relativity: general definitions and an application

Abstract: A generalization of the notion of ellipsoids to curved Riemannian spaces is given and the possibility of using it in describing the shapes of rotating bodies in general relativity is examined. As an illustrative example, stationary, axisymmetric perfect-fluid spacetimes with a so-called confocal inside ellipsoidal symmetry are investigated in detail under the assumption that the 4-velocity of the fluid is parallel to a timelike Killing vector field. A class of perfect-fluid metrics representing interior NUT-spacetimes is obtained along with a vacuum solution with a non-zero cosmological constant.

On the Appeal to a Pre-Established Harmony Between Pure Mathematics and Relativity Physics

Abstract: Soon after its appearance in 1905, the Einsteinian relativity with its relativistically admissible 3-velocities was recognized by Vladimir Varicak in 1908 as the realization in physics of the hyperbolic geometry of Bolyai and Lobachevski At the same time, however, during the years 1907–1909 Minkowski reformulated the Einsteinian relativity in terms of a space of 4-velocities that now bears his name As a result, the special theory of relativity that we find in the mainstream literature is not the one originally formulated by Einstein but, rather, the one reformulated by Minkowski Thus, in particular, one of the most powerful ideas of Einstein in 1905, the Einstein addition of relativistically admissible 3-velocities that need not be parallel, is unheard of in most texts on relativity physics Following our recently published book, Beyond the Einstein Addition, Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1], the aim of this article is to employ the principle of pre-established harmony between mathematics and physics to demonstrate that the original Einsteinian relativity, as opposed to the Minkowskian relativity, is the legitimate formulation of special relativity whose time has returned

Maximum force and minimum distance: physics in limit statements

Abstract: The newly discovered principle of maximum force makes it possible to summarize special relativity, quantum theory\se, and general relativity in one fundamental limit principle each. The three principles fully contain the three theories and are fully equivalent to their standard formulations. In particular, using a result by Jacobson based on the Raychaudhuri equation, it is shown that a maximum force implies the field equations of general relativity. The maximum force in nature is thus equivalent to the full theory of general relativity. Taken together, the three fundamental principles imply a bound for every physical observable, from acceleration to size. The new, precise limit values differ from the usual Planck values by numerical prefactors of order unity. They are given here for the first time. Among others, a maximum force and thus a minimum length imply that the non-continuity of space-time is an inevitable result of the unification of quantum theory and relativity.

Re-formulating the Einstein equations for stable numerical simulations -F ormulation Problem in Numerical Relativity -

Abstract: We review recent efforts to re-formulate the Einstein equations for fully relativistic numerical simulations. The so-called numerical relativity (computational simulations in general relativity) is a promising research field matching with ongoing astrophysical observations such as gravitational wave astronomy. Many trials for longterm stable and accurate simulations of binary compact objects have revealed that mathematically equivalent sets of evolution equations show different numerical stability in free evolution schemes. In this article, we first review the efforts of the community, categorizing them into the following three directions: (1) modifications of the standard Arnowitt-Deser-Misner equations initiated by the Kyoto group, (2) rewriting of the evolution equations in hyperbolic form, and (3) construction of an “asymptotically constrained” system. We next introduce our idea for explaining these evolution behaviors in a unified way using eigenvalue analysis of the constraint propagation equations. The modifications of (or adjustments to) the evolution equations change the character of constraint propagation, and several particular adjustments using constraints are expected to diminish the constraint-violating modes. We propose several new adjusted evolution equations, and include some numerical demonstrations. We conclude by discussing some directions for future research.

The most intensive fluctuation in chaotic time series and relativity principle

Abstract: Relativity principle in mechanics and principle of invariant speed of light lead to Einstein theory. The exponent of porder momentum, derived from a piece of multi-scale chaotic time series, varies with the order p and cannot exceeds a maximum, so there exists the principle of scale relativity. Its special case is the same one as Lorenz transformation from Einstein theory. 2002 Elsevier Science Ltd. All rights reserved.

A Qualitative Analysis of the Attractor Mechanism of General relativity

Abstract: In this work we present a phase-space analysis of the cosmological relaxation of generalized gravity theories where the gravitational constant G varies towards general relativity. We assess the interplay of the two main mechanisms that yield general relativity as the cosmological attractor: (i) the vanishing of the coupling function α(φ), and (ii) the existence of a minimum of the scalar field potential. The latter mechanism is shown to supersede the first. We classify the fixed points associated to different types of potentials and discuss the late time self-similar solutions that arise when the scalar field potential exhibits an asymptotic exponential behaviour from the viewpoint of the relaxation mechanism.

Hyperbolicity of the Kidder-Scheel-Teukolsky formulation of Einstein's equations coupled to a modified Bona-Masso slicing condition

Abstract: We show that the Kidder-Scheel-Teukolsky family of hyperbolic formulations of the 3+1 evolution equations of general relativity remains hyperbolic when coupled to a recently proposed modified version of the Bona-Masso slicing condition.

Applications of Symmetry to General Relativity

Abstract: The equations of general relativity are highly nonlinear partial differential equations and require special techniques to solve exactly. Symmetry considerations have sometimes helped to find solutions. Three examples will be considered here. The first one considers a scaling symmetry in the equations for perfect fluid with a spherical symmetry, which is of historical importance in understanding neutron stars and black holes. The second involves symmetries in the Ernst equation for stationary axially symmetric fields, which are related to the methods worked out in the 1970s for finding solutions to that equation. The third example is an ongoing investigation into critical gravitational collapse, a topic of current interest, using a symmetry of the equations and other analytic techniques.

Completeness theorems in general relativity

Abstract: The global existence problem in general relativity does not reduce toa global existence theorem for a solution of the Einstein equations withsome choice of time coordinate. The physical problem is the existenceof spacetimes for an infinite proper time. But proper time depends on theobserver, i.e. on the timelike line along which it is observed. The singularitytheorems of general relativity consider as a singularity the timelike or nullgeodesic incompleteness of a spacetime and are established under variousgeometric hypotheses.On the other hand, existence theorems for a solution of generic Cauchyproblems for the Einstein equations with an infinite proper time of existencefor a family of observers have been obtained but their timelike or null

Remarks on the Total Angular Momentum in General Relativity

Abstract: We verify that the total angular momentum 3-vector defined by the author [X. Zhang, Commun. Math.Phys. 206 (1999) 137] is equal to (0, 0, ma) forany time slice in both the Kerr and the Kerr-Newman spacetimes.

The Principle of Physical Identity of Units of Measure in Einstein's Special Relativity

Abstract: Einstein's principle of special relativity (SR) implies the existence of physical processes giving identical lengths and times within all inertial frames. This theory of measurement of length and duration, that occupies an essential place in Einstein's SR, can only be wholly justified in quantum theory. Einstein's principle of identity of units is not necessary to establish a complete theory of SR. Indeed Poincare established in 1905 another SR, without Einstein's measurement theory, based on the principle of relativity (group structure of Lorentz's transformations, LT) and on the Lorentz principle of real contraction of units of lengths. Each SR is based on its own system of two principles. Poincare's classical SR supposes a specific use of LT and a specific definition of units not only of space but also of time. In particular the contrast between 1905 Poincare's relativistic use of classical astronomical clocks and Einstein's 1905 relativistic use of identical quantum atoms clocks meets questions which have a certain importance today in physics. In the next few years indeed several experiments will measure the relativistic effects with cold-atoms clocks in space. So the clear separation of the "standard mixture" SR into its two components (Einstein's SR and Poincare's SR) may help to solve the delicate problems which still persist at the interface between quantum theory and general relativity.