Gauge-Invariant Localization of Infinitely Many Gravitational
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The Levi-Civita Tensor Noncovariance and Curvature in the Pseudotensors
The Levi-Civita tensor noncovariance and curvature in the pseudotensors space A. L. Koshkarov∗ University of Petrozavodsk, Physics department, Russia Abstract It is shown that conventional ”covariant” derivative of the Levi-Civita tensor Eαβµν;ξ is not really covariant. Adding compensative terms, it is possible to make it covariant and to be equal to zero. Then one can be introduced a curvature in the pseudotensors space. There appears a curvature tensor which is dissimilar to ordinary one by covariant term including the Levi-Civita density derivatives hence to be equal to zero. This term is a little bit similar to Weylean one in the Weyl curvature tensor. There has been attempted to find a curvature measure in the combined (tensor plus pseudotensor) tensors space. Besides, there has been constructed some vector from the metric and the Levi-Civita density which gives new opportunities in geometry. 1 Introduction Tensor analysis which General Relativity is based on operates basically with true tensors and it is very little spoken of pseudotensors role. Although there are many objects in the world to be bound up with pseudotensors. For example most of particles and bodies in the Universe have angular momentum or spin. Here is a question: could a curvature tensor be a pseudotensor or include that in some way? It’s not clear. Anyway, symmetries of the Riemann-Christopher tensor are not compatible with those of the Levi-Civita pseudotensor. There are examples of using values in Physics to be a sum of a tensor and arXiv:0906.0647v1 [physics.gen-ph] 3 Jun 2009 a pseudotensor. -
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 -
Tensor Calculus and Differential Geometry
Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane . -
Pseudotensors
Pseudotensors Math 1550 lecture notes, Prof. Anna Vainchtein 1 Proper and improper orthogonal transfor- mations of bases Let {e1, e2, e3} be an orthonormal basis in a Cartesian coordinate system and suppose we switch to another rectangular coordinate system with the orthonormal basis vectors {¯e1, ¯e2, ¯e3}. Recall that the two bases are related via an orthogonal matrix Q with components Qij = ¯ei · ej: ¯ei = Qijej, ei = Qji¯ei. (1) Let ∆ = detQ (2) and recall that ∆ = ±1 because Q is orthogonal. If ∆ = 1, we say that the transformation is proper orthogonal; if ∆ = −1, it is an improper orthogonal transformation. Note that the handedness of the basis remains the same under a proper orthogonal transformation and changes under an improper one. Indeed, ¯ V = (¯e1 ׯe2)·¯e3 = (Q1mem ×Q2nen)·Q3lel = Q1mQ2nQ3l(em ×en)·el, (3) where the cross product is taken with respect to some underlying right- handed system, e.g. standard basis. Note that this is a triple sum over m, n and l. Now, observe that the terms in the above some where (m, n, l) is a cyclic (even) permutation of (1, 2, 3) all multiply V = (e1 × e2) · e3 because the scalar triple product is invariant under such permutations: (e1 × e2) · e3 = (e2 × e3) · e1 = (e3 × e1) · e2. Meanwhile, terms where (m, n, l) is a non-cyclic (odd) permutations of (1, 2, 3) multiply −V , e.g. for (m, n, l) = (2, 1, 3) we have (e2 × e1) · e3 = −(e1 × e2) · e3 = −V . All other terms in the sum in (3) (where two or more of the three indices (m, n, l) are the same) are zero (why?). -
Erich Kretschmann As a Proto-Logical-Empiricist: Adventures and Misadventures of the Point-Coincidence Argument
Erich Kretschmann as a Proto-Logical-Empiricist: Adventures and Misadventures of the Point-Coincidence Argument Marco Giovanelli Abstract The present paper attempts to show that a 1915 article by Erich Kretschmann must be credited not only for being the source of Einstein’s point-coincidence remark, but also for having anticipated the main lines of the logical-empiricist interpretation of general relativity. Whereas Kretschmann was inspired by the work of Mach and Poincaré, Einstein inserted Kretschmann’s point-coincidence parlance into the context of Ricci and Levi-Civita’s absolute differential calculus. Kretschmann himself realized this and turned the point-coincidence ar- gument against Einstein in his second and more famous 1918 paper. While Einstein had taken nothing from Kretschmann but the expression “point-coincidences”, the logical empiricists, however, instinctively dragged along with it the entire apparatus of Kretschmann’s conven- tionalism. Disappointingly, in their interpretation of general relativity, the logical empiricists unwittingly replicated some epistemological remarks Kretschmann had written before General Relativity even existed. Keywords: Erich Kretschmann, Point Coincidence Argument, Moritz Schlick, General Relativity, Logical Empiricism, Conventionalism 1. Introduction In the early 1980s, John Stachel (Stachel, 1980) and John Norton (Norton, 1984) famously shed new light on Einstein’s celebrated, yet somewhat cryptic, claim that all physical measurements amount to a determination of space-time coincidences, such as the matching of a pointer with a scale, or, if the world consisted of nothing but particles in motion, the meetings of their world-lines. In Einstein’s published writings, this remark — which Stachel has success- fully labeled the “point-coincidence argument” — amounts to the requirement of “general covariance”: since all coordinate systems necessarily agree on coin- cidences, that is, in everything observable, there is no reason to privilege one coordinate system over another. -
Energy-Momentum Pseudotensor and Superpotential for Generally
Energy-momentum pseudotensor and superpotential for generally covariant theories of gravity of general form R. V. Ilin,∗ S. A. Paston† Saint Petersburg State University, Saint Petersburg, Russia Abstract The current paper is devoted to the investigation of the general form of the energy- momentum pseudotensor (pEMT) and the corresponding superpotential for the wide class of theories. The only requirement for such a theory is the general covariance of the action without any restrictions on the order of derivatives of the independent variables in it or their transformation laws. As a result of the generalized Noether procedure, we obtain a recurrent chain of the equations, which allows one to express canonical pEMT as a divergence of the superpotential. The explicit expression for this superpotential is also given. We discuss the structure of the obtained expressions and the conditions for the derived pEMT conservation laws to be satisfied independently (fully or partially) of the equations of motion. Deformations of the superpotential form for theories with a change of the independent variables in the action are also considered. We apply these results to some interesting particular cases: General Relativity and its modifications, particularly mimetic gravity and Regge-Teitelboim embedding gravity. arXiv:2010.07073v1 [gr-qc] 14 Oct 2020 ∗E-mail: [email protected] †E-mail: [email protected] 1 1 Introduction One of the traditional issues of General Relativity (GR) is the correct definition of the conserved quantities related to the space-time symmetries, especially the energy. This problem is not special for GR and in general remains for all generally covariant theories. -
Measuring the Black Hole Spin Direction in 3D Cartesian Numerical Relativity Simulations
Measuring the black hole spin direction in 3D Cartesian numerical relativity simulations Vassilios Mewes,1 Jos´eA. Font,1, 2 and Pedro J. Montero3 1Departamento de Astronom´ıa y Astrof´ısica, Universitat de Val`encia, Dr. Moliner 50, 46100, Burjassot (Val`encia), Spain 2Observatori Astron`omic, Universitat de Val`encia, C/ Catedr´atico Jos´eBeltr´an 2, 46980, Paterna (Val`encia), Spain 3Max-Planck-Institute f¨ur Astrophysik, Karl-Schwarzschild-Str. 1, 85748, Garching bei M¨unchen, Germany We show that the so-called flat-space rotational Killing vector method for measuring the Cartesian components of a black hole spin can be derived from the surface integral of Weinberg’s pseudoten- sor over the apparent horizon surface when using Gaussian normal coordinates in the integration. Moreover, the integration of the pseudotensor in this gauge yields the Komar angular momentum integral in a foliation adapted to the axisymmetry of the spacetime. As a result, the method does not explicitly depend on the evolved lapse α and shift βi on the respective timeslice, as they are fixed to Gaussian normal coordinates, while leaving the coordinate labels of the spatial metric γij and the extrinsic curvature Kij unchanged. Such gauge fixing endows the method with coordi- nate invariance, which is not present in integral expressions using Weinberg’s pseudotensor, as they normally rely on the explicit use of Cartesian coordinates. PACS numbers: 04.25.Dm, 95.30.Sf, 97.60.Lf I. INTRODUCTION surface integral on the apparent horizon (AH) of the BH 1 a b JAH = ψ R Kab dS, (1) Since the binary black hole (BBH) merger break- 8π ZS through simulations of about a decade ago [1–3], ever- a growing computational resources and advances in the where ψ is an approximate rotational Killing vector on numerical methods used to simulate these systems have the horizon that has to be determined numerically (see a b made the exploration of the vast initial parameter space [8] for a method of finding ψ ), R is the outward point- possible (see e.g. -
A Selected Bibliography of Publications By, and About, Niels Bohr
A Selected Bibliography of Publications by, and about, Niels Bohr Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 09 June 2021 Version 1.308 Title word cross-reference + [VIR+08]. $1 [Duf46]. $1.00 [N.38, Bal39]. $105.95 [Dor79]. $11.95 [Bus20]. $12.00 [Kra07, Lan08]. $189 [Tan09]. $21.95 [Hub14]. $24.95 [RS07]. $29.95 [Gor17]. $32.00 [RS07]. $35.00 [Par06]. $47.50 [Kri91]. $6.95 [Sha67]. $61 [Kra16b]. $9 [Jam67]. − [VIR+08]. 238 [Tur46a, Tur46b]. ◦ [Fra55]. 2 [Som18]. β [Gau14]. c [Dar92d, Gam39]. G [Gam39]. h [Gam39]. q [Dar92d]. × [wB90]. -numbers [Dar92d]. /Hasse [KZN+88]. /Rath [GRE+01]. 0 [wB90, Hub14, Tur06]. 0-19-852049-2 [Ano93a, Red93, Seg93]. 0-19-853977-0 [Hub14]. 0-521-35366-1 [Kri91]. 0-674-01519-3 [Tur06]. 0-85224-458-4 [Hen86a]. 0-9672617-2-4 [Kra07, Lan08]. 1 2 1.5 [GRE+01]. 100-˚aret [BR+85]. 100th [BR+85, KRW05, Sch13, vM02]. 110th [Rub97a]. 121 [Boh87a]. 153 [MP97]. 16 [SE13]. 17 [Boh55a, KRBR62]. 175 [Bad83]. 18.11.1962 [Hei63a]. 1911 [Meh75]. 1915 [SE13]. 1915/16 [SE13, SE13]. 1918 [Boh21a]. 1920s [PP16]. 1922 [Boh22a]. 1923 [Ros18]. 1925 [Cla13, Bor13, Jan17, Sho13]. 1927 [Ano28]. 1929 [HEB+80, HvMW79, Pye81]. 1930 [Lin81, Whe81]. 1930/41 [Fer68, Fer71]. 1930s [Aas85b, Stu79]. 1933 [CCJ+34]. -
Observables and Hamilton-Jacobi Approaches to General Relativity. I
Observables and Hamilton-Jacobi approaches to general relativity. I. The Earlier History Donald Salisbury1,2 1Austin College, 900 North Grand Ave, Sherman, Texas 75090, USA 2Max Planck Institute for the History of Science, Boltzmannstrasse 22, 14195 Berlin, Germany Abstract The main focus is on the Hamilton-Jacobi techniques in classical general relativity that were pursued by Peter Bergmann and Arthur Komar in the 1960’s and 1970’s. They placed special emphasis on the ability to construct the factor group of canonical transformations, where the four-dimensional diffeomorphism phase space transformations were factored out. Equivalence classes were identified by a set of phase space functions that were invari- ant under the action of the four-dimensional diffeomorphism group. This is contrasted and compared with approaches of Paul Weiss, Julian Schwinger, Richard Arnowitt, Stanley Deser, Charles Misner, Karel Kuchar – and es- pecially the geometrodynamical program of John Wheeler and Bryce DeWitt where diffeomorphism symmetry is replaced by a notion of multifingered time. The origins of all of these approaches are traced to Elie Cartan’s invariant in- tegral formulation of classical dynamics. A related correspondence concerning the thin sandwich dispute is also documented. 1 Introduction Given that the classical Hamilton-Jacobi theory played such a prominent role in the formulation of Schrødinger quantum wave mechanics, it is remarkable that gen- eral relativists did not make a serious effort in applying the technique until the early 1960’s. The origins of this attempt in flat spacetime field theory can be arXiv:2106.11894v1 [physics.hist-ph] 22 Jun 2021 traced to Dirac’s groundbreaking 1933 paper, The Lagrangian in quantum mechan- ics [Dirac, 1933], where he established the link, for systems with a finite number of degrees of freedom, between initial and final classical action configuration vari- ables, Hamilton principal functions, and quantum transition amplitudes. -
A Gentle Introduction to Tensors (2014)
A Gentle Introduction to Tensors Boaz Porat Department of Electrical Engineering Technion – Israel Institute of Technology [email protected] May 27, 2014 Opening Remarks This document was written for the benefits of Engineering students, Elec- trical Engineering students in particular, who are curious about physics and would like to know more about it, whether from sheer intellectual desire or because one’s awareness that physics is the key to our understanding of the world around us. Of course, anybody who is interested and has some college background may find this material useful. In the future, I hope to write more documents of the same kind. I chose tensors as a first topic for two reasons. First, tensors appear everywhere in physics, including classi- cal mechanics, relativistic mechanics, electrodynamics, particle physics, and more. Second, tensor theory, at the most elementary level, requires only linear algebra and some calculus as prerequisites. Proceeding a small step further, tensor theory requires background in multivariate calculus. For a deeper understanding, knowledge of manifolds and some point-set topology is required. Accordingly, we divide the material into three chapters. The first chapter discusses constant tensors and constant linear transformations. Tensors and transformations are inseparable. To put it succinctly, tensors are geometrical objects over vector spaces, whose coordinates obey certain laws of transformation under change of basis. Vectors are simple and well-known examples of tensors, but there is much more to tensor theory than vectors. The second chapter discusses tensor fields and curvilinear coordinates. It is this chapter that provides the foundations for tensor applications in physics. -
The Universe of General Relativity, Springer 2005.Pdf
Einstein Studies Editors: Don Howard John Stachel Published under the sponsorship of the Center for Einstein Studies, Boston University Volume 1: Einstein and the History of General Relativity Don Howard and John Stachel, editors Volume 2: Conceptual Problems of Quantum Gravity Abhay Ashtekar and John Stachel, editors Volume 3: Studies in the History of General Relativity Jean Eisenstaedt and A.J. Kox, editors Volume 4: Recent Advances in General Relativity Allen I. Janis and John R. Porter, editors Volume 5: The Attraction of Gravitation: New Studies in the History of General Relativity John Earman, Michel Janssen and John D. Norton, editors Volume 6: Mach’s Principle: From Newton’s Bucket to Quantum Gravity Julian B. Barbour and Herbert Pfister, editors Volume 7: The Expanding Worlds of General Relativity Hubert Goenner, Jürgen Renn, Jim Ritter, and Tilman Sauer, editors Volume 8: Einstein: The Formative Years, 1879–1909 Don Howard and John Stachel, editors Volume 9: Einstein from ‘B’ to ‘Z’ John Stachel Volume 10: Einstein Studies in Russia Yuri Balashov and Vladimir Vizgin, editors Volume 11: The Universe of General Relativity A.J. Kox and Jean Eisenstaedt, editors A.J. Kox Jean Eisenstaedt Editors The Universe of General Relativity Birkhauser¨ Boston • Basel • Berlin A.J. Kox Jean Eisenstaedt Universiteit van Amsterdam Observatoire de Paris Instituut voor Theoretische Fysica SYRTE/UMR8630–CNRS Valckenierstraat 65 F-75014 Paris Cedex 1018 XE Amsterdam France The Netherlands AMS Subject Classification (2000): 01A60, 83-03, 83-06 Library of Congress Cataloging-in-Publication Data The universe of general relativity / A.J. Kox, editors, Jean Eisenstaedt. p.