Non-Metricity in Theories of Gravity and Einstein Equivalence Principle
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:0911.0334V2 [Gr-Qc] 4 Jul 2020
Classical Physics: Spacetime and Fields Nikodem Poplawski Department of Mathematics and Physics, University of New Haven, CT, USA Preface We present a self-contained introduction to the classical theory of spacetime and fields. This expo- sition is based on the most general principles: the principle of general covariance (relativity) and the principle of least action. The order of the exposition is: 1. Spacetime (principle of general covariance and 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, action for particles). In this order, a particle is a special case of a field existing in spacetime, and classical mechanics can be derived from field theory. I dedicate this book to my Parents: Bo_zennaPop lawska and Janusz Pop lawski. I am also grateful to Chris Cox for inspiring this book. The Laws of Physics are simple, beautiful, and universal. arXiv:0911.0334v2 [gr-qc] 4 Jul 2020 1 Contents 1 Spacetime 5 1.1 Principle of general covariance and tensors . 5 1.1.1 Vectors . 5 1.1.2 Tensors . 6 1.1.3 Densities . 7 1.1.4 Contraction . 7 1.1.5 Kronecker and Levi-Civita symbols . 8 1.1.6 Dual densities . 8 1.1.7 Covariant integrals . 9 1.1.8 Antisymmetric derivatives . 9 1.2 Affine connection . 10 1.2.1 Covariant differentiation of tensors . 10 1.2.2 Parallel transport . 11 1.2.3 Torsion tensor . 11 1.2.4 Covariant differentiation of densities . -
Testing General Relativity Through the Computation of Radiative Terms And
TESTING GENERAL RELATIVITY THROUGH THE COMPUTATION OF RADIATIVE TERMS AND WITHIN THE NEUTRON STAR STRONG-FIELD REGIME by Alexander Saffer A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics MONTANA STATE UNIVERSITY Bozeman, Montana April 2019 ©COPYRIGHT by Alexander Saffer 2019 All Rights Reserved ii DEDICATION Dedicated to my family, who provided me with their unconditional love and support. iii ACKNOWLEDGEMENTS I would like to acknowledge the hard work of my advisor Dr. Nicolás Yunes, whose dedication and compassion for physics and those in this field has taught me more than classes ever could. I would also like to thank the Dr. Kent Yagi and Dr. Hector Okada da Silva for their willingness to always help me no matter what I asked and for having the patience to put up with me whenever I struggled. In addition, I would like to thank Dr. Neil Cornish and the rest of the eXtreme Gravity Institute for providing an environment conducive to success. Lastly, I would like to thank my family and friends both within and outside of the MSU physics department who provided their support and guidance over these past 6 years, especially Meg. iv TABLE OF CONTENTS 1. INTRODUCTION ................................................................................................ 1 2. THE GRAVITATIONAL WAVESTRESS-ENERGY (PSEUDO)-TENSOR IN MODIFIED GRAVITY ...................................................................................12 Contribution of Authors and Co-Authors...............................................................12 -
Research Article Weyl-Invariant Extension of the Metric-Affine Gravity
CORE Metadata, citation and similar papers at core.ac.uk Provided by Open Access Repository Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 902396, 7 pages http://dx.doi.org/10.1155/2015/902396 Research Article Weyl-Invariant Extension of the Metric-Affine Gravity R. Vazirian,1 M. R. Tanhayi,2 and Z. A. Motahar3 1 Plasma Physics Research Center, Islamic Azad University, Science and Research Branch, Tehran 1477893855, Iran 2Department of Physics, Islamic Azad University, Central Tehran Branch, Tehran 8683114676, Iran 3Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia Correspondence should be addressed to M. R. Tanhayi; [email protected] Received 30 September 2014; Accepted 28 November 2014 Academic Editor: Anastasios Petkou Copyright © 2015 R. Vazirian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. Metric-affine geometry provides a nontrivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the spacetime (with nonvanishing torsion and nonmetricity). In this paper,westudythegenericformofactioninthisformalismandthenconstructtheWeyl-invariantversionofthistheory.Itisshown that, in Weitzenbock¨ space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally, the related field equations are obtained in the general case. 1. Introduction with the metric and torsion-free condition is relaxed; thus, in addition to Christoffel symbols, the affine connection would Extended theories of gravity have become a field of interest contain an antisymmetric part and nonmetric terms as well. -
Spacetime and Geometry: an Introduction to General Relativity Pdf, Epub, Ebook
SPACETIME AND GEOMETRY: AN INTRODUCTION TO GENERAL RELATIVITY PDF, EPUB, EBOOK Sean M. Carroll,John E. Neely,Richard R. Kibbe | 513 pages | 28 Sep 2003 | Pearson Education (US) | 9780805387322 | English | New Jersey, United States Spacetime and Geometry: An Introduction to General Relativity PDF Book Likewise an explorer from region IV could have a brief look at region I before perishing. Mathematics Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor. Thorne, John Archibald Wheeler Gravitation. When using coordinate transformations as described above, the new coordinate system will often appear to have oblique axes compared to the old system. Indeed, they find some remarkable new regions of spacetime! In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress—energy tensor representing matter, for instance. Views Read Edit View history. Modern cosmological models are a bit more complicated, but retain those features. I was misled even lied to by JG on math. Bitte versuchen Sie es erneut. Considering the number of things I did not know it was a good idea to postpone reading this section until now. Tuesday, 25 February Einstein's equation. So I decided to test the flatness idea in two dimensions Principles of Cosmology and Gravitation. It was supposed to be impossible to travel between regions I and IV of the Kruskal diagram and here Carroll -
Relativistic Spacetime Structure
Relativistic Spacetime Structure Samuel C. Fletcher∗ Department of Philosophy University of Minnesota, Twin Cities & Munich Center for Mathematical Philosophy Ludwig Maximilian University of Munich August 12, 2019 Abstract I survey from a modern perspective what spacetime structure there is according to the general theory of relativity, and what of it determines what else. I describe in some detail both the “standard” and various alternative answers to these questions. Besides bringing many underexplored topics to the attention of philosophers of physics and of science, metaphysicians of science, and foundationally minded physicists, I also aim to cast other, more familiar ones in a new light. 1 Introduction and Scope In the broadest sense, spacetime structure consists in the totality of relations between events and processes described in a spacetime theory, including distance, duration, motion, and (more gener- ally) change. A spacetime theory can attribute more or less such structure, and some parts of that structure may determine other parts. The nature of these structures and their relations of determi- nation bear on the interpretation of the theory—what the world would be like if the theory were true (North, 2009). For example, the structures of spacetime might be taken as its ontological or conceptual posits, and the determination relations might indicate which of these structures is more fundamental (North, 2018). Different perspectives on these questions might also reveal structural similarities with other spacetime theories, providing the resources to articulate how the picture of the world that that theory provides is different (if at all) from what came before, and might be different from what is yet to come.1 ∗Juliusz Doboszewski, Laurenz Hudetz, Eleanor Knox, J. -
Solutions of Conformal Gravity with Dynamical Mass Generation in the Solar System
Solutions of Conformal Gravity with Dynamical Mass Generation in the Solar System A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy in Physics in the University of Canterbury by Joshua N. Wood University of Canterbury 2000 11 PHYSICAl.. SCIENCES LIBRARY Qc (1~ , W87() Abstract An investigation is made into the viability of the fourth-order conformal theory of gravity with dynamical mass generation. This is done by considering the analytical behaviour of the equations of motion and using this as a guide to producing numerical solutions to these equations. A review of some criticisms of the fourth-order gravity theory is included. Numerical solutions of the equations of motion are produced for the domain interior to the source and the exterior region within the solar system, with a variety of source conditions and under different formulations. These are analysed with consideration of reasonable physical and observational requirements, based on the well established solar system gravitational effects. The possibility of extra gravitational effects to explain anomalies in current gravitational theory is investigated. These effects include the possible anomalous acceleration of the Pioneer spacecraft, and could be extended to cover galaxy-scale phenomena such as galactic rotation curves. Conclusions are then drawn about what formulations and parameter sets are viable for good representation of the physics. The effects of the dynamical mass generation on observed phenomena are discussed, with reference to the potential resolution of gravitational anomalies. 1 4 zaU1 iv Contents 1 Introduction 1 1.1 Problems for General Relativity 2 1.2 Conformal Gravity Theories . -
Coframe Geometry, Gravity and Electromagnetism
IARD2012 IOP Publishing Journal of Physics: Conference Series 437 (2013) 012003 doi:10.1088/1742-6596/437/1/012003 Coframe geometry, gravity and electromagnetism Yakov Itin Institute of Mathematics, Hebrew University of Jerusalem and Jerusalem College of Technology, Jerusalem 91904, Israel E-mail: [email protected] Abstract. The extensions of GR for description of fermions on a curved space, for supergravity, and for the loop quantum gravity ordinary use a set of 16 independent variables instead of 10 components of metric. These variables can be assembled in a coframe field, i.e., a set of four linearly independent 1-forms. In this presentation we review a geometrical structure based on the coframe field. We construct a complete class of the coframe connections which are linear in the first order derivatives of the coframe field on an n dimensional manifolds with and without a metric. The subclasses of the torsion-free, metric-compatible and flat connections are derived. We also study the behavior of the geometrical structures under local transformations of the coframe. The remarkable fact is an existence of a subclass of connections which are invariant when the infinitesimal transformations satisfy the Maxwell-like system of equations. 1. Introduction. Why do we have to go beyond Riemannian geometry? General relativity (GR) is, probably, the best of the known theories of gravity. From mathematical and aesthetic points of view, it can be used as a standard of what a physical theory has to be. Einstein's theory is in a very good agreement with all up-to-date observation data. -
On the Interpretation of the Einstein-Cartan Formalism
33 On the Interpretation of the Einstein-Cartan Formalism Jobn Staebell 33.1 Introduction Hehl and collaborators [I] have suggested the need to generalize Riemannian ge ometry to a metric-affine geometry, by admitting torsion and nonmetricity of the connection field. They assume that this geometry represents the microstructure of space-time, with Riemannian geometry emerging as some sort of macroscopic average over the metric-affine microstructure. They thereby generalize the ear lier approach to the Einstein-Cartan formalism of Hehl et al. [2] based on a metric connection with torsion. A particularly clear statement of this point of view is found in Hehl, von der Heyde, and Kerlick [3]: "We claim that the [Einstein-Cartan] field equations ... are, at a classical level, the correct microscopic gravitational field equations. Einstein's field equation ought to be considered a macroscopic phenomenological equation oflimited validity, obtained by averaging [the Einstein-Cartan field equations]" (p. 1067). Adamowicz takes an alternate approach [4], asserting that "the relation between the Einstein-Cartan theory and general relativity is similar to that between the Maxwell theory of continuous media and the classical microscopic electrodynam ics" (p. 1203). However, he only develops the idea of treating the spin density that enters the Einstein-Cartan theory as the macroscopic average of microscopic angu lar momenta in the linear approximation, and does not make explicit the relation he suggests by developing a formal analogy between quantities in macroscopic electrodynamics and in the Einstein-Cartan theory. In this paper, I shall develop such an analogy with macroscopic electrodynamics in detail for the exact, nonlinear version of the Einstein-Cartan theory. -
A Note on Parallel Transportation in Symmetric Teleparallel Geometry
A novel approach to autoparallels for the theories of symmetric teleparallel gravity Caglar Pala1∗ and Muzaffer Adak2† , 1 2Department of Physics, Faculty of Arts and Sciences, Pamukkale University, 20017 Denizli, Turkey 1Department of Physics, Faculty of Science, Erciyes University, 38030 Kayseri, Turkey Abstract Although the autoparallel curves and the geodesics coincide in the Riemannian geometry in which only the curvature is nonzero among the nonmetricity, the torsion and the curvature, they define different curves in the non-Riemannian ones. We give a novel approach to autoparallel curves and geodesics for theories of the symmetric teleparallel gravity written in the coincident gauge. PACS numbers: 04.50.Kd, 11.15.Kc, 02.40.Yy Keywords: Non-Riemannian geometry, Geodesic, Autoparallel curve arXiv:1102.1878v2 [physics.gen-ph] 8 Sep 2021 ∗[email protected] †[email protected] 1 Introduction There have been two main ingredients for the theories of gravity since Isaac Newton; field equations and trajectory equations. The field equations are cast for describing the dynamics of a source and the trajectory ones are the for determining path followed by a spinless massive point test particle. For the Newton’s theory of gravity they are written respectively as ∇2φ =4πGρ (1) d2~r + ∇~ φ =0 (2) dt2 where G is the Newton’s constant of gravity, ~r is the three-dimensional Euclidean position vector of the test particle, t is the time disjoint from the Euclidean space, ρ is the volume density of mass of the source, ∇~ is the gradient operator in the three-dimensional Euclidean space and φ is the gravitational field from a source at the position of ~r. -
Professor John W. Norbury
GENERAL RELATIVITY & COSMOLOGY for Undergraduates Professor John W. Norbury Physics Department University of Wisconsin-Milwaukee P.O. Box 413 Milwaukee, WI 53201 1997 Contents 1 NEWTONIAN COSMOLOGY 5 1.1 Introduction ............................ 5 1.2 Equation of State ......................... 5 1.2.1 Matter ........................... 6 1.2.2 Radiation ......................... 6 1.3 Velocity and Acceleration Equations .............. 7 1.4 Cosmological Constant ...................... 9 1.4.1 Einstein Static Universe ................. 11 2 APPLICATIONS 13 2.1 Conservation laws ........................ 13 2.2 Age of the Universe ....................... 14 2.3 Ination .............................. 15 2.4 Quantum Cosmology ....................... 16 2.4.1 Derivation of the Schrodinger equation ......... 16 2.4.2 Wheeler-DeWitt equation ................ 17 2.5 Summary ............................. 18 2.6 Problems ............................. 19 2.7 Answers .............................. 20 2.8 Solutions ............................. 21 3 TENSORS 23 3.1 Contravariant and Covariant Vectors .............. 23 3.2 Higher Rank Tensors ....................... 26 3.3 Review of Cartesian Tensors ................... 27 3.4 Metric Tensor ........................... 28 3.4.1 Special Relativity ..................... 30 3.5 Christoel Symbols ........................ 31 1 2 CONTENTS 3.6 Christoel Symbols and Metric Tensor ............. 36 3.7 Riemann Curvature Tensor ................... 38 3.8 Summary ............................. 39 3.9 Problems ............................ -
Scalar-Tensor Black Holes Embedded in an Expanding Universe
Review Scalar-Tensor Black Holes Embedded in an Expanding Universe Daria Tretyakova 1,* ID and Boris Latosh 2 ID 1 Department Theoretical Physics, Ural Federal University, Ekaterinburg 620075, Russia 2 Department of Physics & Astronomy, University of Sussex, Brighton BN1 9RH, UK; [email protected] * Correspondence: [email protected] Received: 22 November 2017; Accepted: 27 December 2017; Published: 6 February 2018 Abstract: In this review, we focus our attention on scalar-tensor gravity models and their empirical verification in terms of black hole and wormhole physics. We focus on black holes, embedded in an expanding universe, describing both cosmological and astrophysical scales. We show that in scalar-tensor gravity it is quite common that the local geometry is isolated from the cosmological expansion, so that it does not backreact on the black hole metric. We try to extract common features of scalar-tensor black holes in an expanding universe and point out the issues that are not fully investigated. Keywords: scalar-tensor gravity; black holes; cosmology; modified gravity; Brans–Dicke; Horndeski; de Sitter space 1. Introduction General Relativity (GR) is nowadays the most successful theory of gravity providing the best fit for all the observed gravitational phenomena. For a century, GR was only tested in the weak field regime [1–3] until the recent detection of gravitational waves [4–6] that provided a completely new way to study strong-field gravity [7–10] via optical and gravitational channels [11,12]. Furthermore, constantly increasing resolution of satellite and terrestrial observational facilities gives one hope to resolve the SgrA* in the near future [13,14]. -
Arxiv:1512.05453V1 [Gr-Qc] 17 Dec 2015 the Black Hole Spacetime
Extremal Black Holes in Dynamical Chern-Simons Gravity Robert McNees,1 Leo C. Stein,2 and Nicolás Yunes3 1Loyola University Chicago, Department of Physics, Chicago, IL 60660, USA. 2TAPIR, Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA 3Department of Physics, Montana State University, Bozeman, MT 59717, USA. (Dated: December 18, 2015) Rapidly rotating black hole solutions in theories beyond general relativity play a key role in experimental gravity, as they allow us to compute observables in extreme spacetimes that deviate from the predictions of general relativity (GR). Such solutions are often difficult to find in beyond- GR theories due to the inclusion of additional fields that couple to the metric non-linearly and non-minimally. In this paper, we consider rotating black hole solutions in one such theory, dynamical Chern-Simons gravity, where the Einstein-Hilbert action is modified by the introduction of a dynamical scalar field that couples to the metric through the Pontryagin density. We treat dynamical Chern- Simons gravity as an effective field theory and thus work in the decoupling limit, where corrections are treated as small perturbations from general relativity. We perturb about the maximally-rotating Kerr solution, the so-called extremal limit, and develop mathematical insight into the analysis techniques needed to construct solutions for generic spin. First we find closed-form, analytic expressions for the extremal scalar field, and then determine the trace of the metric perturbation, giving both in terms of Legendre decompositions. Retaining only the first three and four modes in the Legendre representation of the scalar field and the trace respectively suffices to ensure a fidelity of over 99% relative to full numerical solutions.