Showing papers on "Mathematics of general relativity published in 2009"

General Relativity and the Einstein Equations

Abstract: FOREWORD ACKNOWLEDGEMENTS 1. Lorentzian Geometry 2. Special Relativity 3. General Relativity and the Einstein Equations 4. Schwarzschild Space-time and Black Holes 5. Cosmology 6. Local Cauchy Problem 7. Constraints 8. Other Hyperbolic-Elliptic systems 9. Relativistic Fluids 10. Kinetic Theory 11. Progressive Waves 12. Global Hyperbolicity and Causality 13. Singularities 14. Stationary Space-times and Black Holes 15. Global Existence Theorems, Asymptotically Euclidean Data 16. Global existence theorems, cosmological case APPENDICES I. Sobolev Spaces II. Elliptic Systems III. Second Order Quasidiagonal Systems IV. General Hyperbolic Systems V. Cauchy Kovalevski and Fuchs theorems VI. Conformal Methods VII. Kaluza Klein Formulas

The Formation of Black Holes in General Relativity

Abstract: The subject of this work is the formation of black holes in pure general relativity, by the focusing of incoming gravitational waves. The theorems established in this monograph constitute the first foray into the long time dynamics of general relativity in the large, that is, when the initial data are no longer confined to a suitably small neighborhood of Minkowskian data. The theorems are general, no symmetry conditions on the initial data being imposed.

Structure and evolution of self-gravitating objects and the orthogonal splitting of the Riemann tensor

Abstract: The full set of equations governing the structure and the evolution of self-gravitating spherically symmetric dissipative fluids with anisotropic stresses is written down in terms of five scalar quantities obtained from the orthogonal splitting of the Riemann tensor, in the context of general relativity. It is shown that these scalars are directly related to fundamental properties of the fluid distribution, such as energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, and the active gravitational mass. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through these scalars. Some solutions are exhibited to illustrate this point.

The Cauchy Problem in General Relativity

Abstract: After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system, and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.

Extensions of the Stability Theorem of the Minkowski Space in General Relativity

Abstract: This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of $r$ and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A nontrivial solution of these equations is a curved spacetime with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced by Christodoulou and Klainerman. To generalize the original results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field $F$; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of $F$. In particular the Ricci curvature is a constant multiple of the stress-energy tensor for $F$. Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field.

General very special relativity in Finsler cosmology

Abstract: General very special relativity (GVSR) is the curved space-time of very special relativity (VSR) proposed by Cohen and Glashow. The geometry of general very special relativity possesses a line element of Finsler geometry introduced by Bogoslovsky. We calculate the Einstein field equations and derive a modified Friedmann-Robertson-Walker cosmology for an osculating Riemannian space. The Friedmann equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial-spurionic vector field introduced into the metric gives back an estimation of the energy evolution and inflation.

Spacetimes characterized by their scalar curvature invariants

Abstract: In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is $\mathcal{I}$-non-degenerate. This enables us to prove our main theorem that a spacetime metric is either $\mathcal{I}$-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of \emph{strong} and \emph{weak} non-degeneracy.

On quasi-Einstein spacetimes

Abstract: The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study quasi-Einstein spacetimes. Some basic geometric properties of such a spacetime are obtained. The applications of quasi-Einstein spacetimes in general relativity and cosmology are investigated. Finally, the existence of such spacetimes are ensured by several interesting examples.

The extended relativity theory in clifford spaces

Abstract: An introduction to some of the most important features of the Extended Relativity theory in Clifford-spaces (C-spaces) is presented whose "point" coordinates are non-commuting Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes.... degrees of freedom associated with the collective particle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in target Ddimensional spacetime backgrounds. C-space Relativity naturally incorporates the ideas of an invariant length (Planck scale), maximal acceleration, non-commuting coordinates, supersymmetry, holography, higher derivative gravity with torsion and variable dimensions/signatures. It permits to study the dynamics of all (closed) p-branes, for all values of p, on a unified footing. It resolves the ordering ambiguities in QFT, the problem of time in Cosmology and admits superluminal propagation ( tachyons ) without violations of causality. A discussion of the maximalacceleration Relativity principle in phase-spaces follows and the study of the invariance group of symmetry transformations in phase-space allows to show why Planck areas are invariant under acceleration-boosts transformations . This invariance feature suggests that a maximal-string tension principle may be operating in Nature. We continue by pointing out how the relativity of signatures of the underlying n-dimensional spacetime results from taking different n-dimensional slices through C-space. The conformal group in spacetime emerges as a natural subgroup of the Clifford group and Relativity in C-spaces involves natural scale changes in the sizes of physical objects without the introduction of forces nor Weyl's gauge field of dilations. We finalize by constructing the generalization of Maxwell theory of Electrodynamics of point charges to a theory in C-spaces that involves extended charges coupled to antisymmetric tensor fields of arbitrary rank. In the concluding remarks we outline briefly the current promising research programs and their plausible connections with C-space Relativity.

Rotation in relativity and the propagation of light

Abstract: We compare and contrast the different points of view of rotation in general relativity, put forward by Mach, Thirring and Lense, and Goedel. Our analysis relies on two tools: (i) the Sagnac effect which allows us to measure rotations of a coordinate system or induced by the curvature of spacetime, and (ii) computer visualizations which bring out the alien features of the Goedel Universe. In order to keep the paper self-contained, we summarize in several appendices crucial ingredients of the mathematical tools used in general relativity. In this way, our lecture notes should be accessible to researchers familiar with the basic elements of tensor calculus and general relativity.

Spacetime and Fields

Abstract: We present a self-contained introduction to the classical theory of spacetime and fields. The order of the presentation is: 1. Spacetime (tensors, affine connection, curvature, metric, tetrad and spin connection, Lorentz group, spinors), 2. Fields (principle of least action, action for gravitational field, matter, symmetries and conservation laws, gravitational field equations, spinor fields, electromagnetic field).

Spin-spin interaction in general relativity and induced geometries with nontrivial topology

Abstract: We consider the dynamics of a self-gravitating spinor field and a self-gravitating rotating perfect fluid. It is shown that both these matter distributions can induce a vortex field described by the curl 4-vector of a tetrad: ω i = 1 � iklm e(a)ke (a) l;m ,w heree (a) k are components of the tetrad. The energy-momentum tensor Tik(ω) of this field has been found and shown to violate the strong and weak energy conditions which leads to possible formation of geometries with nontrivial topology like wormholes. The corresponding exact solutions to the equations of general relativity have been found. It is also shown that other vortex fields, e.g., the magnetic field, can also possess such properties.

The static extension problem in General Relativity

Abstract: We prove the existence of asymptotically flat solutions to the static vacuum Einstein equations with prescribed boundary data consisting of the induced metric and mean curvature on a 2-sphere, provided that the mean curvature is positive and has no critical points on regions of nonpositive Gaussian curvature. This gives a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.

Gauge fixing in the tensor model and emergence of local gauge symmetries

Abstract: The tensor model can be regarded as theory of dynamical fuzzy spaces, and gives a way to formulate gravity on fuzzy spaces. It has recently been shown that the low-lying fluctuations around the Gaussian background solutions in the tensor model agree correctly with the metric fluctuations on the flat spaces with general dimensions in the general relativity. This suggests that the local gauge symmetry (the symmetry of local translations) is also emergent around these solutions. To systematically study this possibility, I apply the BRS gauge fixing procedure to the tensor model. The ghost kinetic term is numerically analyzed, and it has been found that there exist some massless trajectories of ghost modes, which are clearly separated from the other higher ghost modes. Comparing with the corresponding BRS gauge fixing in the general relativity, these ghost modes forming the massless trajectories in the tensor model are shown to be identical to the reparametrization ghosts in the general relativity.

On Tensorial Concomitants and the Non-Existence of a Gravitational Stress-Energy Tensor

Abstract: Based on an analysis of what it may mean for one tensor to depend in the proper way on another, I prove that, under certain natural conditions, there can be no tensor whose interpretation could be that it represents gravitational stress-energy in general relativity. It follows that gravitational energy, such as it is in general relativity, is necessarily non-local. Along the way, I prove a result of some interest in own right about the structure of the associated jet bundles of the bundle of Lorentz metrics over spacetime.

(n + 1)-Dimensional Lorentzian Wormholes in an Expanding Cosmological Background

Abstract: We discuss (n+1)-dimensional dynamical wormholes in an evolving cosmological background with a throat expanding with time. These solutions are examined in the general relativity framework. A linear relation between diagonal elements of an anisotropic energy-momentum tensor is used to obtain the solutions. The energy-momentum tensor elements approach the vacuum case when we are far from the central object for one class of solutions. Finally, we discuss the energy-momentum tensor which supports this geometry, taking into account the energy conditions.

Tube dislocations in gravity

Abstract: We consider static massive thin cylindrical shells (tubes) as the sources in Einstein’s equations. They correspond to δ- and δ′-function-type energy-momentum tensors. The corresponding metric components are found explicitly. They are not continuous functions, in general, and lead to ambiguous curvature tensor components. Nevertheless all ambiguous terms in Einstein’s equations safely cancel. The interplay between elasticity theory, geometric theory of defects, and general relativity is analyzed. The elasticity theory provides a simple picture for defect creation and a new look on general relativity.

Outer Boundary Conditions in Numerical Relativity

Abstract: This thesis is concerned with outer boundary conditions in numerical relativity. In numerical simulations, the spatially infinite universe is typically modelled using a finite spatial domain, on the edge of which boundary conditions are imposed. These boundary conditions should mirror the unbounded physical domain as closely as possible. They should be transparent to outgoing gravitational radiation and should not introduce spurious incoming radiation via reflections of outgoing radiation off the boundary. The concepts of incoming and outgoing gravitational radiation are only well understood in certain specific charts and tetrads. The first half of this thesis investigates the relationship between these charts and tetrads and those used in numerical relativity. We begin by studying a previous calculation [134], in which quantities such as the Bondi mass and the news function were expressed in terms of the Newman-Penrose scalars in an axisymmetric spacetime. The calculation is generalized to spacetimes with no symmetries. The results above still require a specific choice of tetrad. By supposing that the region of spacetime far from an isolated gravitating source is in some sense Minkowskian, we demonstrate how to transform between the charts and tetrads used in theoretical studies of gravitational radiation and the charts and tetrads used in numerical relativity. This enables us to provide “numerical relativity recipes” in which the Weyl scalars, the Bondi mass and news function are expressed in terms of the metric variables in a numerical chart. The second half of this thesis addresses the problem of absorbing boundary conditions in numerical relativity. Using Hertz potentials, the far-field region of a spacetime can be expressed as a linear perturbation about Minkowski, Schwarzschild or Kerr backgrounds. The resulting field equations enable us to investigate the propagation of linearized gravitational radiation. On a Minkowski background, incoming and outgoing waves propagate independently. The presence of a curved background creates a “gravitational tail” whose behaviour near future null infinity we are able to estimate. This enables us to formulate absorbing boundary conditions for numerical relativity. Finally, we link the two threads mentioned above. The boundary conditions are expressed in terms of the metric variables in a numerical relativity chart.

The Fully Covariant Energy Momentum Stress Tensor For Gravity and the Einstein Equation in General Relativity

Abstract: We give a fully covariant energy momentum stress tensor for the gravitational field which is easily physically motivated, and which leads to a very general derivation of the Einstein equation for gravity. We do not need to assume any property of the surce matter fields' energy momentum stress tensor other than symmetry. We give a physical motivation for this choice using laser light pressure. As a consequence of our derivation, the energy momentum stress tensor for the total source matter and fields must be divergence free, when spacetime is 4 dimensional. Moreoverr, if the total source matter fields are assumed to be divergence free, then either spacetime is dimension 4 or the spacetime has constant scalar curvature.

General relativity extended to non-Riemannian space-time geometry

Abstract: The gravitation equations of the general relativity, written for Riemannian space-time geometry, are extended to the case of arbitrary (non-Riemannian) space-time geometry. The obtained equations are written in terms of the world function in the coordinateless form. These equations determine directly the world function, (but not only the metric tensor). As a result the space-time geometry appears to be non-Rieamannian. Invariant form of the obtained equations admits one to exclude influence of the coordinate system on solutions of dynamic equations. Anybody, who trusts in the general relativity, is to accept the extended general relativity, because the extended theory does not use any new hypotheses. It corrects only inconsequences and restrictions of the conventional conception of general relativity. The extended general relativity predicts an induced antigravitation, which eliminates existence of black holes.

Covariant energy-momentum and an uncertainty principle for general relativity

Abstract: We introduce a naturally-defined totally invariant spacetime energy expression for general relativity incorporating the contribution from gravity. The extension links seamlessly to the action integral for the gravitational field. The demand that the general expression for arbitrary systems reduces to the Tolman integral in the case of stationary bounded distributions, leads to the matter-localized Ricci integral for energy-momentum in support of the energy localization hypothesis. The role of the observer is addressed and as an extension of the special relativistic case, the field of observers comoving with the matter is seen to compute the intrinsic global energy of a system. The new localized energy supports the Bonnor claim that the Szekeres collapsing dust solutions are energy-conserving. It is suggested that in the extreme of strong gravity, the Heisenberg Uncertainty Principle be generalized in terms of spacetime energy-momentum.

Extended Palatini action for general relativity

Abstract: In the Palatini action of general relativity, the connection and the metric are treated as independent dynamical variables. Instead of assuming a relation between these quantities, the desired relation between them is derived through the Euler-Lagrange equations of the Palatini action. In this manuscript we construct an extended Palatini action, where we do not assume any a priori relationship between the connection, the covariant metric tensor, and the contravariant metric tensor. Instead we treat these three quantities as independent dynamical variables. We show that this action reproduces the standard Einstein field equations depending on a single metric tensor. We further show that in vacuum and in the absence of cosmological constant this theory has an enhanced symmetry.

Emergent general relativity in the tensor models possessing Gaussian classical solutions

Abstract: This paper gives a summary of the author’s works concerning the emergent general relativity in a particular class of tensor models, which possess Gaussian classical solutions. In general, a classical solution in a tensor model may be physically regarded as a background space, and small fluctuations about the solution as emergent fields on the space. The numerical analyses of the tensor models possessing Gaussian classical background solutions have shown that the low-lying long-wavelength fluctuations around the backgrounds are in one-to-one correspondence with the geometric fluctuations on flat spaces in the general relativity. It has also been shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds can be identified with the local translation symmetry of the general relativity. Thus the tensor model provides an interesting model of simultaneous emergence of space, the general relativity, and its local gauge symmetry of translation.

Exact Space-Times in Einstein's General Relativity: Plebański–Demiański solutions

Abstract: A number of the previous chapters have described important black hole space-times that are of algebraic type D – specifically Chapters 8, 9, 11, 12 and 14. In fact these are all members of a larger family of solutions that can be expressed in a common form. The purpose of this chapter is to present this wider class, particularly showing the relation between these and related space-times, and to indicate their further generalisations. A general family of type D space-times with an aligned non-null electromagnetic field and a possibly non-zero cosmological constant can be represented by a metric that was given originally by Debever (1971), and in a more convenient form by Plebanski and Demianski (1976). These solutions are characterised by two related quartic functions, each of a single coordinate, whose coefficients are determined by seven arbitrary parameters which include Λ and both electric and magnetic charges. Together with cases that can be derived from it by certain transformations and limiting procedures, this gives the complete family of such solutions. Non-accelerating solutions of this class were obtained by Carter (1968b). For the vacuum case with no cosmological constant, they include all the particular solutions identified by Kinnersley (1969a). Metrics with an expanding repeated principal null congruence were analysed further by Debever (1969) and Weir and Kerr (1977), where the relations between the different forms of the line element can be deduced. They have also been studied by Debever and Kamran (1980) and Ishikawa and Miyashita (1982). The most general metric form which covers both expanding and non-expanding cases was given by Debever, Kamran and McLenaghan (1984) and Garcia D. (1984).

Use Of Approximate Symmetry Methods To Define Energy Of Gravitational Waves

Abstract: Not AvailableIn this thesis approximate Lie symmetry methods for differential equations are used to investigate the problem of energy in general relativity and in particular in gravitational waves.For this purpose second-order approximate symmetries of the system of geodesic equations for the Reissner-Nordstrom (RN) spacetime are studied. It is shown that in the second-order approximation, energy must be rescaled for the RN spacetime.Then the approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime are investigated.Taking the Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional.This algebra is related to the 15 dimensional algebra of conformal isometries of the Minkowski spacetime.First introducing spin angular momentum per unit mass as a small parameter first-order approximate symmetries of the Kerr spacetime as a first perturbation of the Schwarzschild spacetime are considered. We then investigate the second-order approximate symmetries of the Kerr spacetime as a second perturbation of the Minkowski spacetime.Next, second-order approximate symmetries of the system of geodesic equations for the charged-Kerr spacetime are investigated. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr spacetime has to be rescaled. Since gravitational wave spacetimes are time-varying vacuum solutions of Einstein's field equations, there is no unambiguous means to define their energy content. Here a definition, using slightly broken Noether symmetries is proposed.A problem is noted with the use of the proposal for plane-fronted gravitational waves.To attain a better understanding of the implications of this proposal we also use an artificially constructed time-varying non-vacuum plane symmetric metric and evaluate its Weyl and stress-energy tensors so as to obtain the gravitational and matter components separately and compare them with the energy content obtained by our proposal.The procedure is also used for cylindrical gravitational wave solutions.The usefulness of the definition is demonstrated by the fact that it leads to a result on whether gravitational waves suffer self-damping.

The group aspect in the physical interpretation of general relativity theory

Abstract: When, at the end of the year 1915, both Einstein and Hilbert arrived at what were named the field equations of general relativity, both of them thought that their fundamental achievement entailed, inter alia, the realisation of a theory of gravitation whose underlying group was the group of general coordinate transformations. This group theoretical property was believed by Einstein to be a relevant one from a physical standpoint, because the general coordinates allowed to introduce reference frames not limited to the inertial reference frames that can be associated with the Minkowski coordinate systems, whose transformation group was perceived to be restricted to the Poincar\'e group. Two years later, however, Kretschmann published a paper in which the physical relevance of the group theoretical achievement in the general relativity of 1915 was denied. For Kretschmann, since any theory, whatever its physical content, can be rewritten in a generally covariant form, the group of general coordinate transformations is physically irrelevant. This is not the case, however, for the group of the infinitesimal motions that bring the metric field in itself, namely, for the Killing group. This group is physically characteristic of any given spacetime theory, since it accounts for the local invariance properties of the considered manifold, i.e., for its ``relativity postulate''. In the present chapter it is shown what are the consequences for the physical interpretation of some of these solutions whose relativity postulate is of intermediate content, when Kretschmann's standpoint is consistently adhered to.

Poincare-Snyder Relativity with Quantization

Abstract: Based on a linear realization formulation of a quantum relativity -- the proposed relativity for quantum `space-time', we introduce the Poincar\'e-Snyder relativity and Snyder relativity as relativities in between the latter and the well known Galilean and Einstein cases. We discuss how the Poincar\'e-Snyder relativity may provide a stronger framework for the description of the usual (Einstein) relativistic quantum mechanics and beyond. In particular, we discuss a geometric quantization picture through the U(1) central extension of the relativity group, which had been establish to work well for the Galilean case but not for the Einstein case. We discuss similarities and differences between our Poincar\'e-Snyder picture with a still not fully understood $\sigma$ variable as the `evolution' parameter and some use of an invariant time or the proper time parameter in some earlier formulations with very similar mathematical structure. The study is a first step towards the investigation of physics of the $\sigma$ variable at the Poincar\'e-Snyder setting, plausible leading to experimental signatures to be studied.