Showing papers on "Introduction to the mathematics of general relativity published in 2006"

General Relativity: An Introduction for Physicists

Abstract: General Relativity: An Introduction for Physicists provides a clear mathematical introduction to Einstein's theory of general relativity. It presents a wide range of applications of the theory, concentrating on its physical consequences. After reviewing the basic concepts, the authors present a clear and intuitive discussion of the mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are then used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is then introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle. Written for advanced undergraduate and graduate students, this approachable textbook contains over 300 exercises to illuminate and extend the discussion in the text.

An Introduction to General Relativity and Cosmology

Abstract: 1. How the theory of relativity came into being (a brief historical sketch) Part I. Elements of Differential Geometry: 2. A short sketch of two-dimensional differential geometries 3. Tensors, tensor densities 4. Covariant derivatives 5. Parallel transport and geodesic lines 6. Curvature of a manifold: flat manifolds 7. Riemannian geometry 8. Symmetries of Rieman spaces, invariance of tensors 9. Methods to calculate the curvature quickly - Cartan forms and algebraic computer programs 10. The spatially homogeneous Bianchi-type spacetimes 11. The Petrov classification by the spinor method Part II. The Gravitation Theory: 12. The Einstein equations and the sources of a gravitational field 13. The Maxwell and Einstein-Maxwell equations and the Kaluza-Klein theory 14. Spherically symmetric gravitational field of isolated objects 15. Relativistic hydrodynamics and thermodynamics 16. Relativistic cosmology I: general geometry 17. Relativistic cosmology II: the Robertson-Walker geometry 18. Relativistic cosmology III: the Lemaitre-Tolman geometry 19. Relativistic cosmology IV: generalisations of L-T and related geometries 20. The Kerr solution 21. Subjects omitted in this book References.

The use of generalized functions and distributions in general relativity

Abstract: We review the extent to which one can use classical distribution theory in describing solutions of Einstein's equations. We show that there are a number of physically interesting cases which cannot be treated using distribution theory but require a more general concept. We describe a mathematical theory of nonlinear generalized functions based on Colombeau algebras and show how this may be applied in general relativity. We end by discussing the concept of singularity in general relativity and show that certain solutions with weak singularities may be regarded as distributional solutions of Einstein's equations.

The End Results of General Relativity Theory Via Just Energy Conservation and Quantum Mechanics

Abstract: Herein we present a whole new approach that leads to the end results of the general theory of relativity via just the law of conservation of energy (broadened to embody the mass and energy equivalence of the special theory of relativity) and quantum mechanics. We start with the following postulate. Postulate: The rest mass of an object bound to a celestial body amounts less than its rest mass measured in empty space, and this, as much as its binding energy vis-a-vis the gravitational field of concern. The decreased rest mass is further dilated by the Lorentz factor if the object in hand is in motion in the gravitational field of concern. The overall relativistic energy must be constant on a stationary trajectory. This yields the equation of motion driven by the celestial body of concern, via the relationship e α / √ 1 − r 0 2 / e 0 2 = constant, along with the definition α = GM / re 0 2 ; here M is the mass of the celestial body creating the gravitational field of concern; G is the universal gravitational constant, measured in empty space it comes into play in Newton's law of gravitation, which is assumed though to be valid for static masses only; r points to the location picked on the trajectory of the motion, the center of M being the origin of coordinates, as assessed by the distant observer; v 0 is the tangential velocity of the object at r; c 0 is the ceiling of the speed of light in empty space; v 0 and c 0 remain the same for both the local observer and the distant observer, just the same way as that framed by the special theory of relativity. The differentiation of the above relationship leads to − GM / r 2 (1 − v 0 2 / c 0 2 ) = v 0 dr dv0 or, via v 0 = dr / dt, − GM / r 2 (1 − v 0 2 / c 0 2 ) ṟ / r = d?? 0 / dt: ?? is the outward looking unit vector along r; the latter differential equation is the classical Newton's Equation of Motion, were v 0, negligible as compared to c 0; this equation is valid for any object, including a light photon. Taking into account the quantum mechanical stretching of lengths due to the rest mass decrease in the gravitational field, the above equation can be transformed into an equation written in terms of the proper lengths, yielding well the end results of the general theory of relativity, though through a completely different set up.

Einstein-Cartan Theory

Abstract: The Einstein--Cartan Theory (ECT) of gravity is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsion to the density of intrinsic angular momentum This modification was put forward in 1922 by Elie Cartan, before the discovery of spin Cartan was influenced by the work of the Cosserat brothers (1909), who considered besides an (asymmetric) force stress tensor also a moments stress tensor in a suitably generalized continuous medium

Emergent General Relativity

Abstract: We review different approaches to quantum gravity in which spacetime is emerging. We discuss in some detail the proposals by G. Volovik and S. Lloyd and show how they differ in the way they treat time. We further propose an approach to quantum gravity in which the Einstein equations are derived rather then used. We call this approach Internal Relativity.

Another look at general covariance and the equivalence of reference frames

Abstract: In his general theory of relativity (GR) Einstein sought to generalize the special-relativistic equivalence of inertial frames to a principle according to which all frames of reference are equivalent. He claimed to have achieved this aim through the general covariance of the equations of GR. There is broad consensus among philosophers of relativity that Einstein was mistaken in this. That equations can be made to look the same in different frames certainly does not imply in general that such frames are physically equivalent. We shall argue, however, that Einstein's position is tenable. The equivalence of arbitrary frames in GR should not be equated with relativity of arbitrary motion, though. There certainly are observable differences between reference frames in GR (differences in the way particles move and fields evolve). The core of our defense of Einstein's position will be to argue that such differences should be seen as fact-like rather than law-like in GR. By contrast, in classical mechanics and in special relativity (SR) the differences between inertial systems and accelerated systems have a law-like status. The fact-like character of the differences between frames in GR justifies regarding them as equivalent in the same sense as inertial frames in SR.

Non-chaotic dynamics in general-relativistic and scalar–tensor cosmology

Abstract: In the context of scalar?tensor models of dark energy and inflation, the dynamics of vacuum scalar?tensor cosmology are analysed without specifying the coupling function or the scalar field potential. A conformal transformation to the Einstein frame is used and the dynamics of general relativity with a minimally coupled scalar field are derived for a generic potential. It is shown that the dynamics are non-chaotic, thus settling an existing debate.

The physical role of gravitational and gauge degrees of freedom in general relativity — I: Dynamical synchronization and generalized inertial effects

Abstract: This is the first of a couple of papers in which the peculiar capabilities of the Hamiltonian approach to general relativity are exploited to get both new results concerning specific technical issues, and new insights about old foundational problems of the theory. The first paper includes: (1) a critical analysis of the various concepts of symmetry related to the Einstein-Hilbert Lagrangian viewpoint on the one hand, and to the Hamiltonian viewpoint, on the other. This analysis leads, in particular, to a re-interpretation of active diffeomorphisms as passive and metric-dependent dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose the (not widely known)) connection of a subgroup of them to Hamiltonian gauge transformations on-shell; (2) a re-visitation of the canonical reduction of the ADM formulation of general relativity, with particular emphasis on the geometro-dynamical effects of the gauge-fixing procedure, which amounts to the definition of a global non-inertial, space-time laboratory. This analysis discloses the peculiar dynamical nature that the traditional definition of distant simultaneity and clock-synchronization assume in general relativity, as well as the gauge relatedness of the “conventions” which generalize the classical Einstein's convention. (3) a clarification of the physical role of Dirac and gauge variables, as their being related to tidal-like and generalized inertial effects, respectively. This clarification is mainly due to the fact that, unlike the standard formulations of the equivalence principle, the Hamiltonian formalism allows to define a generalized notion of “force” in general relativity in a natural way.

Averaging spherically symmetric spacetimes in general relativity

Abstract: We discuss the averaging problem in general relativity, using the form of the macroscopic gravity equations in the case of spherical symmetry in volume preserving coordinates. In particular, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. On cosmological scales, the correlation tensor in a Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background is found to be of the form of a spatial curvature. On astrophysical scales the correlation tensor can be interpreted as the sum of a spatial curvature and an anisotropic fluid. We briefly discuss the physical implications of these results.

Hyperboloidal data and evolution

Abstract: We discuss the hyperboloidal evolution problem in general relativity from a numerical perspective, and present some new results. Families of initial data which are the hyperboloidal analogue of Brill waves are constructed numerically, and a systematic search for apparent horizons is performed. Schwarzschild‐Kruskal spacetime is discussed as a first application of Friedrich’s general conformal field equations in spherical symmetry, and the Maxwell equations are discussed on a nontrivial background as a toy model for continuum instabilities.

Is general relativity `essentially understood'?

Abstract: The content of Einstein’s theory of gravitation is encoded in the properties of the solutions to his field equations. There has been obtained a wealth of information about these solutions in the ninety years the theory has been around. It led to the prediction and the observation of physical phenomena which confirm the important role of general relativity in physics. The understanding of the domain of highly dynamical, strong field configurations is, however, still quite limited. The gravitational wave experiments are likely to provide soon observational data on phenomena which are not accessible by other means. Further theoretical progress will require, however, new methods for the analysis and the numerical calculation of the solutions to Einstein’s field equations on large scales and under general assumptions. We discuss some of the problems involved, describe the status of the field and recent results, and point out some open problems.

Towards a wave-extraction method for numerical relativity. III. Analytical examples for the Beetle-Burko radiation scalar

Abstract: Beetle and Burko recently introduced a background-independent scalar curvature invariant for general relativity that carries information about the gravitational radiation in generic spacetimes, in cases where such radiation is incontrovertibly defined. In this paper we adopt a formalism that only uses spatial data as they are used in numerical relativity and compute the Beetle-Burko radiation scalar for a number of analytical examples, specifically linearized Einstein-Rosen cylindrical waves, linearized quadrupole waves, the Kerr spacetime, Bowen-York initial data, and the Kasner spacetime. These examples illustrate how the Beetle-Burko radiation scalar can be used to examine the gravitational wave content of numerically generated spacetimes, and how it may provide a useful diagnostic for initial data sets.

General relativity research trends

Abstract: CONTENTS: Preface The Chrono-Geometrical Structure of Special and General Relativity: Towards a Background-Independent Description of the Gravitational Field and Elementary Particles Leibniz-Mach Foundations for GR and Fundamental Physics Kinematic Self-Similar Solutions in General Relativity The Dynamic Space of General Relativity in Second Atomisation Reinstating Schwarzschilds Original Manifold and its Singularity Index

Gravitational amplitudes in black-hole evaporation: the effect of non-commutative geometry

Abstract: Recent work in the literature has studied the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat spacetime and weak radiation at a very late time. The relevant quantum amplitudes have been evaluated for bosonic and fermionic fields, showing that no information is lost in collapse to a black hole. On the other hand, recent developments in non-commutative geometry have shown that, in general relativity, the effects of non-commutativity can be taken into account by keeping the standard form of the Einstein tensor on the left-hand side of the field equations and introducing a modified energy–momentum tensor as a source on the right-hand side. The present paper, relying on the recently obtained non-commutativity effect on a static, spherically symmetric metric, considers from a new perspective the quantum amplitudes in black hole evaporation. The general relativity analysis of spin-2 amplitudes is shown to be modified by a multiplicative factor F depending on a constant non-commutativity parameter and on the upper limit R of the radial coordinate. Limiting forms of F are derived which are compatible with the adiabatic approximation here exploited. Approximate formulae for the particle emission rate are also obtained within this framework.

Simple Derivation of Minimum Length, Minimum Dipole Moment and Lack of Space–Time Continuity

Abstract: The principle of maximum power makes it possible to summarize special relativity, quantum theory and general relativity in one fundamental limit principle each. Special relativity contains an upper limit to speed; following Bohr, quantum theory is based on a lower limit to action; recently, a maximum power given by c 5/4G was shown to be equivalent to the full field equations of general relativity. Taken together, these three fundamental principles imply a limit value 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. Among others, minimum length and time intervals appear. The limits imply that elementary particles are not point-like and suggest a lower limit on electric dipole values. The minimum intervals also imply that the non-continuity of space–time is an inevitable result of the unification of quantum theory and relativity, independently of the approach used.

Redshifts and Killing vectors

Abstract: Current approaches to physics stress the importance of conservation laws due to spacetime and internal symmetries. In special and general relativity the generators of these symmetries are known as Killing vectors. We use them for the rigorous determination of gravitational and cosmological redshifts.

Polynomial form of the Hilbert–Einstein action

Abstract: Configuration space of general relativity is extended by inclusion of the determinant of the metric as a new independent variable. As the consequence the Hilbert–Einstein action takes a polynomial form.

Finding and using exact solutions of the Einstein equations

Abstract: The evolution of the methods used to find solutions of Einstein’s field equations during the last 100 years is described. Early papers used assumptions on the coordinate forms of the metrics. Since the 1950s more invariant methods have been deployed in most new papers. The uses to which the solutions found have been put are discussed, and it is shown that they have played an important role in the development of many aspects, both mathematical and physical, of general relativity.

Physics of quantum relativity through a linear realization

Abstract: The idea of quantum relativity as a generalized, or rather deformed, version of Einstein (special) relativity has been taking shape in recent years. Following the perspective of deformations, while staying within the framework of Lie algebra, we implement explicitly a simple linear realization of the relativity symmetry, and explore systematically the resulting physical interpretations. Some suggestions we make may sound radical, but are arguably natural within the context of our formulation. Our work may provide a new perspective on the subject matter, complementary to the previous approach(es), and may lead to a better understanding of the physics.

Doubly Special Relativity: A New Relativity or Not?

Abstract: Double Special Relativity theories are the relativistic theories in which the transformations between inertial observers are characterized by two observer-independent scales of the light speed and the Planck length. We study two main examples of these theories and want to show that these theories are not the new theories of relativity, but only are re-descriptions of Einstein's special relativity in the non-conventional coordinates.

Cosmological Relativity: The Special and General Theories for the Structure of the Universe

Abstract: Cosmological Special Relativity Elements of General Relativity Cosmological General Relativity Cosmological General Relativity in Five Dimensions.

Modified GR and Helium Nucleosynthesis

Abstract: We show that a previously proposed cosmological model based on general relativity with non vanishing divergence for the energy-momentum tensor is consistent with the observed values for the nucleosynthesis of helium for some values of the arbitrary parameter $\alpha$ presented in this model. Further more values of $\alpha$ can be accommodated if we adopt the Randall-Sundrum single brane model.

Alternative proposal to modified newtonian dynamics

Abstract: From a study of conserved quantities of the so-called Modified Newtonian Dynamics (MOND) we propose an alternative to this theory. We show that this proposal is consistent with the Tully-Fisher law, has conserved quantities whose Newtonian limit are the energy and angular momentum, and can be useful to explain cosmic acceleration. The dynamics obtained suggests that, when acceleration is very small, time depends on acceleration. This result is analogous to that of special relativity where time depends on velocity. PACS numbers: 95.35.+d, 45.20.Dd Nowadays there are various observational results in astrophysics whose explanation represents a challenge for theoretical physics. One of those problems is to explain the rotation curves of the galaxies. Observations indicate a relationship V 4 ∝ M for the speed V of the distant stars in a galaxy of mass M. However, as the only force acting on those stars is gravity and their trajectories are circles, Newtonian dynamics indicates that the relationship to hold is V 2 = GM/r, where r is the distance from the star to the center of the galaxy. To account for the difference, some authors assume the existence of a sort of matter that does not radiate: the so-called dark matter. There are, however, other proposals which assume modifications to the gravitational field or to the laws of dynamics. By considering the behavior of the speed of the distant stars, M. Milgrom proposed a modification to Newton’s second law as [1] m� (z) d 2 x i dt 2 = F

Experimental Concepts for Generating Negative Energy in the Laboratory

Abstract: Implementation of faster‐than‐light (FTL) interstellar travel via traversable wormholes, warp drives, or other spacetime modification schemes generally requires the engineering of spacetime into very specialized local geometries The analysis of these via Einstein’s General Theory of Relativity (GTR) field equations plus the resultant equations of state demonstrate that such geometries require the use of “exotic” matter in order to induce the requisite FTL spacetime modification Exotic matter is generally defined by GTR physics to be matter that possesses (renormalized) negative energy density, and this is a very misunderstood and misapplied term by the non‐GTR community We clear up this misconception by defining what negative energy is, where it can be found in nature, and we also review the experimental concepts that have been proposed to generate negative energy in the laboratory

The Einstein Postulates: 1905-2005 A Critical Review of the Evidence

Abstract: The Einstein postulates assert an invariance of the propagation speed of light in vacuum for any observer, and which amounts to a presumed absence of any preferred frame. The postulates appear to be directly linked to relativistic effects which emerge from Einstein's Special Theory of Relativity, which is based upon the concept of a flat spacetime ontology, and which then lead to the General Theory of Relativity with its curved spacetime model for gravity. While the relativistic effects are well established experimentally it is now known that numerous experiments, beginning with the Michelson-Morley experiment of 1887, have always shown that the postulates themselves are false, namely that there is a detectable local preferred frame of reference. This critique briefly reviews the experimental evidence regarding the failure of the postulates, and the implications for our understanding of fundamental physics, and in particular for our understanding of gravity. A new theory of gravity is seen to be necessary, and this results in an explanation of the `dark matter' effect entailing the discovery that the fine structure constant is a 2nd gravitational constant.

The physical meaning of the “boost-rotation symmetric” solutions within the general interpretation of Einstein's theory of gravitation

Abstract: The answer to the question, what physical meaning should be attributed to the so-called boost-rotation symmetric exact solutions to the field equations of general relativity, is provided within the general interpretation scheme for the “theories of relativity,” based on group theoretical arguments, and set forth by Erich Kretschmann already in the year 1917.