Showing papers on "Mathematics of general relativity published in 2012"
TL;DR: This Letter addresses the implications of consistent nonlinear gravity-matter coupling and argues that such a completion of general relativity is viable from both an experimental and theoretical point of view through energy conditions, consistency, and singularity-avoidance perspectives.
Abstract: The coupling between matter and gravity in general relativity is given by a proportionality relation between the stress tensor and the geometry. This is an oriented assumption driven by the fact that both the stress tensor and the Einstein tensor are divergenceless. However, general relativity is in essence a nonlinear theory, so there is no obvious reason why the coupling to matter should be linear. On another hand, modified theories of gravity usually affect the vacuum dynamics, yet keep the coupling to matter linear. In this Letter, we address the implications of consistent nonlinear gravity-matter coupling. The Eddington-inspired Born-Infeld theory recently introduced by Banados and Ferreira provides an enlightening realization of such coupling modifications. We find that this theory coupled to a perfect fluid reduces to general relativity coupled to a nonlinearly modified perfect fluid, leading to an ambiguity between modified coupling and modified equation of state. We discuss observational consequences of this degeneracy and argue that such a completion of general relativity is viable from both an experimental and theoretical point of view through energy conditions, consistency, and singularity-avoidance perspectives. We use these results to discuss the impact of changing the coupling paradigm.
134 citations
TL;DR: In this article, it was shown that general relativity is invariant with respect to Weyl transformations in an arbitrary Weyl frame, and that WIST gravity theories are mathematically equivalent to Brans-Dicke theory when viewed in a particular Weyl framework.
Abstract: We show that the general theory of relativity can be formulated in the language of Weyl geometry. We develop the concept of Weyl frames and point out that the new mathematical formalism may lead to different pictures of the same gravitational phenomena. We show that in an arbitrary Weyl frame general relativity, which takes the form of a scalar–tensor gravitational theory, is invariant with respect to Weyl transformations. A key point in the development of the formalism is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in terms of Riemannian geometry we find that the theory has some similarities with Brans–Dicke gravitational theory. In this scenario, the gravitational field is not described by the metric tensor only, but by a combination of both the metric and a geometrical scalar field. We illustrate this point by examining how distinct geometrical and physical pictures of the same phenomena may arise in different frames. To give an example, we discuss the gravitational spectral shift as viewed in a general Weyl frame. We further explore the analogy of general relativity with scalar–tensor theories and show how a known Brans–Dicke vacuum solution may appear as a solution of general relativity theory when reinterpreted in a particular Weyl frame. Finally, we show that the so-called WIST gravity theories are mathematically equivalent to Brans–Dicke theory when viewed in a particular frame.
108 citations
TL;DR: In this article, the Bianchi identity of the new "Codazzi deviation tensor" with a geometric significance was shown to be equivalent to a Bianchi tensor with Riemann compatibility.
Abstract: Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation tensor" with a geometric significance. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mapping. Compatibility is extended to generalized curvature tensors with an application to Weyl's tensor and general relativity.
48 citations
01 Jan 2012
TL;DR: The geodesic principle can be recovered as a theorem in general relativity as mentioned in this paper, but it is not a consequence of Einstein's equation (or the conservation principle) alone, and other assumptions are needed to drive the theorems in question.
Abstract: It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein’s equation (via the conservation principle that is, itself, a consequence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic principle can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. My goal in this short note is to make this claim precise.
41 citations
TL;DR: Within the framework of projective geometry, this paper investigated kinematics and symmetry in (alpha, beta) spacetime one special types of Finsler spacetime The projectively flat spacetime with constant flag curvature is divided into four types and the corresponding invariant special relativity in the four types of spacetime contain two parameters, the speed of light and a geometrical parameter, which may relate to the new physical scale
Abstract: Within the framework of projective geometry, we investigate kinematics and symmetry in (alpha, beta) spacetime one special types of Finsler spacetime The projectively flat (alpha, beta) spacetime with constant flag curvature is divided into four types The symmetry in type A-Riemann spacetime with constant sectional curvature-is just the one in de Sitter special relativity The symmetry in type B-locally Minkowski spacetime-is just the one in very special relativity It is found that type C-Funk spacetime and type D-scaled Berwald's metric spacetime both possess the Lorentz group as its isometric group The geodesic equation, algebra and dispersion relation in the (alpha, beta) spacetime are given The corresponding invariant special relativity in the four types of (alpha, beta) spacetime contain two parameters, the speed of light and a geometrical parameter, which may relate to the new physical scale They all reduce to Einstein's special relativity while the geometrical parameter vanishes (C) 2012 Elsevier BV All rights reserved
31 citations
TL;DR: In this article, an alternative interpretation of the geodesic principle as a type of universality thesis analogous to the universality behavior exhibited in thermal systems during phase transitions was proposed. But it is difficult to apply the principle to massive bodies in a way that is coherent with Einstein's field equations.
Abstract: In this paper I critically review attempts to formulate and derive the geodesic principle, which claims that free massive bodies follow geodesic paths in general relativity theory. I argue that if the principle is (canonically) interpreted as a law of motion describing the actual evolution of gravitating bodies, then it is impossible to generically apply the law to massive bodies in a way that is coherent with Einstein's field equations. Rejecting the canonical interpretation, I propose an alternative interpretation of the geodesic principle as a type of universality thesis analogous to the universality behavior exhibited in thermal systems during phase transitions.
30 citations
TL;DR: In this paper, a decoupling of the geometrical spatial curvature term in the metric from the dynamical spatial curvatures in the Friedmann equation was investigated by fitting to a combination of HST, CMB, type Ia supernovae (SNIa), and baryon acoustic oscillation (BAO) data sets.
Abstract: Averaging in general relativity is a complicated operation, due to the general covariance of the theory and the nonlinearity of Einstein's equations. The latter of these ensures that smoothing spacetime over cosmological scales does not yield the same result as solving Einstein's equations with a smooth matter distribution, and that the smooth models we fit to observations need not be simply related to the actual geometry of spacetime. One specific consequence of this is a decoupling of the geometrical spatial curvature term in the metric from the dynamical spatial curvature in the Friedmann equation. Here we investigate the consequences of this decoupling by fitting to a combination of Hubble Space Telescope (HST), CMB, type Ia supernovae (SNIa), and baryon acoustic oscillation (BAO) data sets. We find that only the geometrical spatial curvature is tightly constrained and that our ability to constrain dark energy dynamics will be severely impaired until we gain a thorough understanding of the averaging problem in cosmology.
30 citations
TL;DR: In this article, the problem of finding modified gravitational dynamics is reduced to a clear mathematical task, in the sense that their solutions yield the actions governing the corresponding spacetime geometry, and the problem is reformulated as a system of linear partial differential equations.
Abstract: Only a severely restricted class of tensor fields can provide classical spacetime geometries, namely those that can carry matter field equations that are predictive, interpretable and quantizable. These three conditions on matter translate into three corresponding algebraic conditions on the underlying tensorial geometry, namely to be hyperbolic, time-orientable and energy-distinguishing. Lorentzian metrics, on which general relativity and the standard model of particle physics are built, present just the simplest tensorial spacetime geometry satisfying these conditions. The problem of finding gravitational dynamics—for the general tensorial spacetime geometries satisfying the above minimum requirements—is reformulated in this paper as a system of linear partial differential equations, in the sense that their solutions yield the actions governing the corresponding spacetime geometry. Thus the search for modified gravitational dynamics is reduced to a clear mathematical task.
25 citations
TL;DR: In this paper, a classification for the Weyl tensor in all four-dimensional manifolds, including all signatures and the complex case, in an unified and simple way is presented.
Abstract: It is well known that the classification of the Weyl tensor in Lorentzian manifolds of dimension four, the so called Petrov classification, was a great tool to the development of general relativity. Using the bivector approach it is shown in this article a classification for the Weyl tensor in all four-dimensional manifolds, including all signatures and the complex case, in an unified and simple way. The important Petrov classification then emerges just as a particular case in this scheme. The boost weight classification is also extended here to all signatures as well to complex manifolds. For the Weyl tensor in four dimensions it is established that this last approach produces a classification equivalent to the one generated by the bivector method.
20 citations
TL;DR: In this article, the authors recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime, and showed that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state.
Abstract: We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress?energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient ?Ai. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.
TL;DR: In this article, the authors discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment.
Abstract: We briefly discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely, the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment. Furthermore, in his analysis of the foundations of Newtonian mechanics, Mach provided an operational definition for inertial mass and pointed out that time and space are conceptually distinct from their operational definitions by means of masses. Mach recognized that this circumstance is due to the lack of any a priori connection between the inertial mass of a body and its Newtonian state in space and time. One possible way to improve upon this situation in classical physics is to associate mass with an extra dimension. Indeed, Einstein's theory of gravitation can be locally embedded in a Ricci-flat 5D manifold such that the 4D energy-momentum tensor appears to originate from the existence of the extra dimension. An outline of such a 5D Machian extension of Einstein's general relativity is presented.
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TL;DR: In this article, a local version of Shape Dynamics that is equivalent to General Relativity is constructed, in the sense that the algebras of Dirac observables weakly coincide.
Abstract: In this conceptual paper we construct a local version of Shape Dynamics that is equivalent to General Relativity in the sense that the algebras of Dirac observables weakly coincide. This allows us to identify Shape Dynamics observables with General Relativity observables, whose observables can now be interpreted as particular representative functions of observables of a conformal theory at fixed York time. An application of the observable equivalence of General Relativity and Shape Dynamics is to define the quantization of General Relativity through quantizing Shape Dynamics and using observable equivalence. We investigate this proposal explicitly for gravity in 2+1 dimensions.
TL;DR: In this article, an approximate coordinate transformation to an accelerated frame was proposed, which turns out to be closely related to Rindler coordinates, widely used in modern general relativity, leading him directly to interpret gravitation in terms of spacetime curvature.
Abstract: On his way to general relativity, Einstein used the equivalence principle to formulate a theory of the static gravitational field. In this context he introduced an approximate coordinate transformation to an accelerated frame which turns out to be closely related to Rindler coordinates, widely used in modern general relativity. This work, published in the Annalen, led him directly to interpret gravitation in terms of spacetime curvature.
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TL;DR: In this article, the influence of quantum corrections consisting of quadratic curvature invariants on the Einstein-Hilbert action is considered and exact vacuum solutions of these curvature singularities are studied in arbitrary dimension.
Abstract: In the first part of this thesis, Kerr-Schild metrics and extended Kerr-Schild metrics are analyzed in the context of higher dimensional general relativity Employing the higher dimensional generalizations of the Newman-Penrose formalism and the algebraic classification of spacetimes based on the existence and multiplicity of Weyl aligned null directions, we establish various geometrical properties of the Kerr-Schild congruences, determine compatible Weyl types and in the expanding case discuss the presence of curvature singularities We also present known exact solutions admitting these Kerr-Schild forms and construct some new ones using the Brinkmann warp product In the second part, the influence of quantum corrections consisting of quadratic curvature invariants on the Einstein-Hilbert action is considered and exact vacuum solutions of these quadratic gravities are studied in arbitrary dimension We investigate classes of Einstein spacetimes and spacetimes with a null radiation term in the Ricci tensor satisfying the vacuum field equations of quadratic gravity and provide examples of these metrics
TL;DR: In this article, the second Bianchi identity can be recast as an evolution equation for the Riemann curvatures for a vacuum static spherically symmetric spacetime.
Abstract: The second Bianchi identity can be recast as an evolution equation for the Riemann curvatures. Here we will report on such a system for a vacuum static spherically symmetric spacetime. This is the rst of two papers. In the following paper we will extend the ideas developed here to general vacuum spacetimes. In this paper we will demonstrate our ideas on a Schwarzschild spacetime and give detailed numerical results. For suitable choices of lapse function we nd that the system gives excellent results with long term stability.
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TL;DR: In this article, the equivalence of General Relativity and Shape Dynamics can be extended to a theory, that respects the BRST-symmetries of general Relativity as well as the ones of an extended version of Shape Dynamics.
Abstract: We show that the equivalence of General Relativity and Shape Dynamics can be extended to a theory, that respects the BRST-symmetries of General Relativity as well as the ones of an extended version of Shape Dynamics. This version of Shape Dynamics implements local spatial conformal transformations as well as a local an abstract analogue of special conformal transformations. Standard effective field theory arguments suggest that the definition of a gravity theory should implement this duality between General Relativity and Shape Dynamics, thus the name “Doubly General Relativity.” We briefly discuss several consequences: bulk/bulk- duality in classical gravity, experimental falsification of Doubly General Relativity and possible implications for the renormalization of quantum gravity in the effective field theory framework.
TL;DR: In this paper, the OPERA data were examined in the framework of special relativity with de Sitter spacetime symmetry (dS-SR) and it was shown that the ICARUS anomaly is in agreement with the prediction of dS- SR with R≃1.95×1012 l.y.
Abstract: We explore the recent OPERA experiment of superluminal neutrinos in the framework of special relativity with de Sitter spacetime symmetry (dS-SR). According to Einstein, a photon is treated as a massless particle in the framework of special relativity. In special relativity (SR) we have the universal parameter c, the photon velocity cphoton and the phase velocity of a light wave in vacuum cwave = λν. Due to the null experiments of Michelson–Morley we have c = cwave. The parameter cphoton is determined by the Noether charges corresponding to the spacetime symmetries of SR. In Einstein's special relativity (E-SR) we have c = cphoton. In dS-SR, i.e. the special relativity with SO(4, 1) de Sitter spacetime symmetry, we have cphoton > c. In this paper, the OPERA datum are examined in the framework of dS-SR. We show that OPERA anomaly is in agreement with the prediction of dS-SR with R≃1.95×1012 l.y. Based on the p-E relation of dS-SR, we also prove that the Cohen and Glashow's argument of possible superluminal neutrino's Cherenkov-like radiation is forbidden. We conclude that OPERA and ICARUS results are consistent and they are explained in the dS-SR framework.
TL;DR: In this article, a complete set of equations, in the form of an Einstein-Bianchi system, was presented to describe the evolution of generic smooth lattices in spacetime. But this work is restricted to general 3+1 vacuum spacetimes.
Abstract: We will present a complete set of equations, in the form of an EinsteinBianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg-lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalisation of our previous paper [1] on the Einstein-Bianchi system for the Schwarzschild spacetime to general 3+1 vacuum spacetimes.
07 Apr 2012
TL;DR: In this paper, the authors describe infinitesimal symmetries of geometrical structures induced naturally on the phase space of the classical spacetime by a metric field and an electromagnetic field.
Abstract: We describe infinitesimal symmetries of geometrical structures induced naturally on the phase space of the classical spacetime by a metric field and an electromagnetic field.
06 Nov 2012
TL;DR: In this article, a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of Cosmological Special Relation is discussed.
Abstract: Geometric algebra and Clifford algebra are important tools to describe and analyze the physics of the world we live in. Although there is enormous empirical evidence that we are living in four dimensional spacetime, mathematical worlds of higher dimensions can be used to present the physical laws of our world in an aesthetical and didactical more appealing way. In physics and mathematics education we are therefore confronted with the question how these high dimensional spaces should be taught. But as an immediate confrontation of students with high dimensional compactified spacetimes would expect too much from them at the beginning of their university studies, it seems reasonable to approach the mathematics and physics of higher dimensions step by step. The first step naturally is the step from four dimensional spacetime of special relativity to a five dimensional spacetime world. As a toy model for this artificial world cosmological special relativity, invented by Moshe Carmeli, can be used. This five dimensional non-compactified approach describes a spacetime which consists not only of one time dimension and three space dimensions. In addition velocity is regarded as a fifth dimension. This model very probably will not represent physics correctly. But it can be used to discuss and analyze the consequences of an additional dimension in a clear and simple way. Unfortunately Carmeli has formulated cosmological special relativity in standard vector notation. Therefore a translation of cosmological special relativity into the mathematical language of Grassmann and Clifford (Geometric algebra) is given and the physics of cosmological special relativity is discussed.
Dissertation•
01 Jan 2012
TL;DR: A study of shear-free perfect fluids in general relativity (rotating and/or expanding) and a characterisation of the orthogonal Bianchi class A perfect fluids as geodesic, non-rotating fluids with vanishing divergence of the electric and magnetic part of the Weyl tensor are presented in this paper.
Abstract: A study of shear-free perfect fluids in general relativity (rotating and/or expanding) and a characterisation of the orthogonal Bianchi class A perfect fluids as geodesic, non-rotating fluids with vanishing divergence of the electric and magnetic part of the Weyl tensor.
TL;DR: In this article, the Einstein tensor is shown to appear naturally from the Bianchi identities, thus emphasizing the pure geometrical nature of the left hand side of the equations of general relativity.
Abstract: In the context of the theory of fiber bundles and connections, mainly restricted to the frame and associated bundles of a Riemannian or pseudo-Riemannian differentiable manifold, we present the global and local versions of the concepts of covariant derivative, parallel transport, geodesics, metric compatible connections with (Riemann-Cartan) or without (Levi-Civita) torsion, and curvature and torsion with their geometric interpretations. The Einstein tensor is shown to appear naturally from the Bianchi identities, thus emphasizing the pure geometrical nature of “the left hand side” of the equations of general relativity.
Dissertation•
01 Feb 2012
TL;DR: Conboye et al. as mentioned in this paper proposed Axial Symmetry and Transverse Trace-Free Tensors in Numerical Relativity, a transverse trace-free tensors in numerical relativity.
Abstract: Axial symmetry and transverse trace-free tensors in numerical relativity Author(s) Conboye, Rory P. A. Publication date 2012-02 Original citation Conboye, Rory P.A. 2012. Axial Symmetry and Transverse Trace-Free Tensors in Numerical Relativity. PhD Thesis, University College Cork. Type of publication Doctoral thesis Rights © 2012, Rory Patrick Albert Conboye http://creativecommons.org/licenses/by-nc-nd/3.0/
01 Jan 2012
TL;DR: In this paper, the curving of spacetime in the general theory of gravity is discussed and a new aspect of physics brought about by Einstein's Relativity is discussed. But not a word was mentioned about gravity and it is with the inclusion of gravity that the theory takes on a new and exciting complexion.
Abstract: In the preceding chapters, we witnessed the new aspects of physics brought about by Einstein’s Relativity. All of this arose because: a) there is a speed limit in nature, \(c\), the maximum speed at which influences can be propagated and b) because all inertial observers are physically equivalent, they must all agree on this speed limit. However, not a word was mentioned about gravity and it is with the inclusion of gravity that Einstein’s Relativity takes on a whole new and exciting complexion. Einstein’s Relativity without gravity is called “Special Relativity” to distinguish it from Relativity with gravity which is called “General Relativity”. In brief, General Relativity is Einstein’s theory of gravity. In what follows, we will delve into General Relativity, showing how it is the curving of spacetime in the general theory that replaces the old Newtonian idea of gravity being just another force.
TL;DR: In this paper, an axiomatic approach is presented in a coherently structured, deductive manner, which leads to a system of Lorentz-covariant electromagnetic field equations in (N + 1 ) -spacetime, including the corresponding LSTM transformations of the spacetime coordinates, the field values and the source terms.
TL;DR: In this article, the system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered and the general solution to a problem representing two classes of line elements with arbitrary functions g00 and g22 is obtained.
Abstract: The system of the spherical-symmetric vacuum equations of the General Relativity Theory is considered. The general solution to a problem representing two classes of line elements with arbitrary functions g00 and g22 is obtained. The properties of the found solutions are analyzed.
TL;DR: This paper deals with the cosmological models for the static spherically symmetric spacetime for perfect fluid with anisotropic stress energy tensor in general relativity by introducing the generating functions 𝑔(𝑟) and 𝓂(𝓚) and also discussing their physical and geometric properties.
Abstract: This paper deals with the cosmological models for the static spherically symmetric spacetime for perfect fluid with anisotropic stress energy tensor in general relativity by introducing the generating functions 𝑔(𝑟) and 𝑤(𝑟) and also discussing their physical and geometric properties.
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TL;DR: In this article, the authors examined the conceptual characteristics of the (3+1)- decomposition of the Einstein field equations, and the preeminent role of geodesic motions was emphasized.
Abstract: The computations of numerical relativity make use of (3+1)- decompositions of Einstein field equations. We examine the conceptual characteristics of this method; instances of compact-star binaries are considered. The preeminent role of the geodesic motions is emphasized.
16 May 2012
TL;DR: In this article, the authors give complementary characterizations of the relativity of spacetime locality that affects certain Planck-scale-deformed phase-space constructions, and summarize some of the results they obtained.
Abstract: We summarize some of the results we obtained in arXiv:1006.2126 (Physical Review Letters 106, 071301), arXiv:1102.4637 (Physics Letters B 700, 150-156) and in arXiv:1107.3334, giving complementary characterizations of the relativity of spacetime locality that affects certain Planck-scale-deformed phase-space constructions.