Introduction Can Only Be Influenced by Its Immediate Sur- Roundings
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Refraction and Reflexion According to the Principle of General Covariance
REFRACTION AND REFLEXION ACCORDING TO THE PRINCIPLE OF GENERAL COVARIANCE PATRICK IGLESIAS-ZEMMOUR Abstract. We show how the principle of general covariance introduced by Souriau in smoothly uniform contexts, can be extended to singular situ- ations, considering the group of diffeomorphisms preserving the singular locus. As a proof of concept, we shall see how we get again this way, the laws of reflection and refraction in geometric optics, applying an extended general covariance principle to Riemannian metrics, discontinuous along a hypersurface. Introduction In his paper “Modèle de particule à spin dans le champ électromagnétique et gravitationnel” published in 1974 [Sou74], Jean-Marie Souriau suggests a pre- cise mathematical interpretation of the principle of General Relativity. He names it the Principle of General Covariance. Considering only gravitation field1, he claimed that any material presence in the universe is characterized by a covector defined on the quotient of the set of the Pseudo-Riemannian metrics on space-time, by the group of diffeomorphisms. This principle be- ing, according to Souriau, the correct statement of the Einsteins’s principle of invariance with respect to any change of coordinates. Actually, Souriau’s general covariance principle can be regarded as the active version of Einstein invariance statement, where change of coordinates are interpreted from the active point of view as the action of the group of diffeomorphisms. Now, for reasons relative to the behavior at infinity and results requirement, the group of diffeomorphisms of space-time is reduced to the subgroup of compact supported diffeomorphisms. Date: April 6, 2019. 1991 Mathematics Subject Classification. 83C10, 78A05, 37J10. -
Group Theory
Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk. -
Council for Innovative Research Peer Review Research Publishing System
ISSN 2347-3487 Einstein's gravitation is Einstein-Grossmann's equations Alfonso Leon Guillen Gomez Independent scientific researcher, Bogota, Colombia E-mail: [email protected] Abstract While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say, Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993) [1]. Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996) [2]. -
Quantum Mechanics from General Relativity : Particle Probability from Interaction Weighting
Annales de la Fondation Louis de Broglie, Volume 24, 1999 25 Quantum Mechanics From General Relativity : Particle Probability from Interaction Weighting Mendel Sachs Department of Physics State University of New York at Bualo ABSTRACT. Discussion is given to the conceptual and mathemat- ical change from the probability calculus of quantum mechanics to a weighting formalism, when the paradigm change takes place from linear quantum mechanics to the nonlinear, holistic eld theory that accompanies general relativity, as a fundamental theory of matter. This is a change from a nondeterministic, linear theory of an open system of ‘particles’ to a deterministic, nonlinear, holistic eld the- ory of the matter of a closed system. RESUM E. On discute le changement conceptuel et mathematique qui fait passer du calcul des probabilites de la mecanique quantique a un formalisme de ponderation, quand on eectue le changement de paradigme substituantalam ecanique quantique lineairelatheorie de champ holistique non lineaire qui est associee alatheorie de la relativitegenerale, prise comme theorie fondamentale de la matiere. Ce changement mene d’une theorie non deterministe et lineaire decrivant un systeme ouvert de ‘particules’ a une theorie de champ holistique, deterministe et non lineaire, de la matiere constituant un systeme ferme. 1. Introduction. In my view, one of the three most important experimental discov- eries of 20th century physics was the wave nature of matter. [The other two were 1) blackbody radiation and 2) the bending of a beam of s- tarlight as it propagates past the vicinity of the sun]. The wave nature of matter was predicted in the pioneering theoretical studies of L. -
Covariance in Physics and Convolutional Neural Networks
Covariance in Physics and Convolutional Neural Networks Miranda C. N. Cheng 1 2 3 Vassilis Anagiannis 2 Maurice Weiler 4 Pim de Haan 5 4 Taco S. Cohen 5 Max Welling 5 Abstract change of coordinates. In his words, the general principle In this proceeding we give an overview of the of covariance states that “The general laws of nature are idea of covariance (or equivariance) featured in to be expressed by equations which hold good for all sys- the recent development of convolutional neural tems of coordinates, that is, are covariant with respect to networks (CNNs). We study the similarities and any substitutions whatever (generally covariant)” (Einstein, differencesbetween the use of covariance in theo- 1916). The rest is history: the incorporation of the math- retical physics and in the CNN context. Addition- ematics of Riemannian geometry in order to achieve gen- ally, we demonstrate that the simple assumption eral covariance and the formulation of the general relativity of covariance, together with the required proper- (GR) theory of gravity. It is important to note that the seem- ties of locality, linearity and weight sharing, is ingly innocent assumption of general covariance is in fact sufficient to uniquely determine the form of the so powerful that it determines GR as the unique theory of convolution. gravity compatible with this principle, and the equivalence principle in particular, up to short-distance corrections1. In a completely different context, it has become clear 1. Covariance and Uniqueness in recent years that a coordinate-independent description It is well-known that the principle of covariance, or coordi- is also desirable for convolutional networks. -
[Physics.Hist-Ph] 14 Dec 2011 Poincaré and Special Relativity
Poincar´eand Special Relativity Emily Adlam Department of Philosophy, The University of Oxford Abstract Henri Poincar´e’s work on mathematical features of the Lorentz transfor- mations was an important precursor to the development of special relativity. In this paper I compare the approaches taken by Poincar´eand Einstein, aim- ing to come to an understanding of the philosophical ideas underlying their methods. In section (1) I assess Poincar´e’s contribution, concluding that al- though he inspired much of the mathematical formalism of special relativity, he cannot be credited with an overall conceptual grasp of the theory. In section (2) I investigate the origins of the two approaches, tracing differences to a disagreement about the appropriate direction for explanation in physics; I also discuss implications for modern controversies regarding explanation in the philosophy of special relativity. Finally, in section (3) I consider the links between Poincar´e’s philosophy and his science, arguing that apparent inconsistencies in his attitude to special relativity can be traced back to his acceptance of a ‘convenience thesis’ regarding conventions. Keywords Poincare; Einstein; special relativity; explanation; conventionalism; Lorentz transformations arXiv:1112.3175v1 [physics.hist-ph] 14 Dec 2011 1 1 Did Poincar´eDiscover Special Relativity? Poincar´e’s work introduced many ideas that subsequently became impor- tant in special relativity, and on a cursory inspection it may seem that his 1905 and 1906 papers (written before Einstein’s landmark paper was pub- lished) already contain most of the major features of the theory: he had the correct equations for the Lorentz transformations, articulated the relativity principle, derived the correct relativistic transformations for force and charge density, and found the rule for relativistic composition of velocities. -
Einstein's Equations for Spin 2 Mass 0 from Noether's Converse Hilbertian
Einstein’s Equations for Spin 2 Mass 0 from Noether’s Converse Hilbertian Assertion October 4, 2016 J. Brian Pitts Faculty of Philosophy, University of Cambridge [email protected] forthcoming in Studies in History and Philosophy of Modern Physics Abstract An overlap between the general relativist and particle physicist views of Einstein gravity is uncovered. Noether’s 1918 paper developed Hilbert’s and Klein’s reflections on the conservation laws. Energy-momentum is just a term proportional to the field equations and a “curl” term with identically zero divergence. Noether proved a converse “Hilbertian assertion”: such “improper” conservation laws imply a generally covariant action. Later and independently, particle physicists derived the nonlinear Einstein equations as- suming the absence of negative-energy degrees of freedom (“ghosts”) for stability, along with universal coupling: all energy-momentum including gravity’s serves as a source for gravity. Those assumptions (all but) imply (for 0 graviton mass) that the energy-momentum is only a term proportional to the field equations and a symmetric curl, which implies the coalescence of the flat background geometry and the gravitational potential into an effective curved geometry. The flat metric, though useful in Rosenfeld’s stress-energy definition, disappears from the field equations. Thus the particle physics derivation uses a reinvented Noetherian converse Hilbertian assertion in Rosenfeld-tinged form. The Rosenfeld stress-energy is identically the canonical stress-energy plus a Belinfante curl and terms proportional to the field equations, so the flat metric is only a convenient mathematical trick without ontological commitment. Neither generalized relativity of motion, nor the identity of gravity and inertia, nor substantive general covariance is assumed. -
Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916"
JOHN EARMAN and CLARK GL YMOUR LOST IN THE TENSORS: EINSTEIN'S STRUGGLES WITH COVARIANCE PRINCIPLES 1912-1916" Introduction IN 1912 Einstein began to devote a major portion of his time and energy to an attempt to construct a relativistic theory of gravitation. A strong intimation of the struggle that lay ahead is contained in a letter to Arnold Sommerfeld dated October 29, 1912: At the moment I am working solely on the problem of gravitation and believe 1 will be able to overcome all difficulties with the help of a local, friendly mathemat- ician. But one thing is certain, that I have never worked so hard in my life, and that I have been injected with a great awe of mathematics, which in my naivet~ until now I only viewed as a pure luxury in its subtler forms! Compared to this problem the original theory of relativity is mere child's play.' Einstein's letter contained only a perfunctory reply to a query from Sommerfeld about the Debye-Born theory of specific heats. Obviously disappointed, Som- merfeld wrote to Hilbert: 'My letter to Einstein was in vain . Einstein is evidently so deeply mired in gravitation that he is deaf to everything else? Sommerfeld's words were more prophetic than he could possibly have known; the next three years were to see Einstein deeply mired in gravitation, sometimes seemingly hopelessly so. In large measure, Einstein's struggle resulted from his use and his misuse, his understanding and his misunderstanding of the nature and implications of covariance principles. In brief, considerations of general covariance were bound up with Einstein's motive for seeking a 'generalized' theory of relativity; mis- understandings about the meaning and implementation of this motivation threatened to wreck the search; and in the end, the desire for general covariance helped to bring Einstein back onto the track which led to what we now recognize *Present address c/o Department of Philosophy, University of Minnesota, Minneapolis, Minn, U.S.A. -
Ngit `500&2-256
NGiT `500&2-256 Foundations for a Theory of Gravitation Theories* 9 KIP S. THORNE, DAVID L. LEE,t and ALAN P. LIGHTMAN$ California Institute of Technology, Pasadena, California 91109 ABSTRACT A foundation is laid for future analyses of gravitation theories. This foundation is applicable to any theory formulated in terms of geometric objects defined on a 4-dimensional spacetime manifold. The foundation consists of (i) a glossary of fundamental concepts; (ii) a theorem that delineates the overlap between Lagrangian-based theories and metric theories; (iii) a conjecture (due to Schiff) that the Weak Equivalence Principle implies the Einstein Equivalence Principle; and (iv) a plausibility argument supporting this conjecture for the special case of relativistic, Lagrangian-based theories. Reproduced by NATIONAL TECHNICAL INFORMATION SERVICE US Department of Commerce Springfield, VA. 22151 Supported in part by the National Aeronautics and Space Administration [NGR 05-002-256] and the National Science Foundation [GP-27304, GP-28027]. t Imperial Oil Predoctoral Fellow. NSF Predoctoral Fellow during part of the period of this research. , (NASA-CR-130753) FOUNDATIONS FOR A N73-17499 I THEORY OF GRAVITATION THEORIES (California Inst. of Tech.) 45 p HC. $4.25 CSCL 08N Unclas V_._ __ _ ~ ___ __ _-G3/13 172 52 -' / I. INTRODUCTION Several years ago our group initiatedl a project of constructing theo- retical foundations for experimental tests of gravitation theories. The results of that project to date (largely due to Clifford M. Will and Wei-Tou Ni), and the results of a similar project being carried out by the group of Kenneth Nordtvedt at Montana State University, are summarized in several 2-4 recent review articles. -
Introduction
note1 : August 29, 2012 Introduction Principle of locality The principle of locality states that an object can Quantum Field Theory—in the context of parti- only be influenced by its immediate surroundings. cle physics—is a theory of elementary particles and From this principle follows the finite speed of in- their interactions. The Standard Model of elemen- formation transmission. tary particles is a quantum field theory. By definition, elementary particles are the most Principle of covariance fundamental—structureless—particles (like elec- trons and photons). They exhibit wave-particle du- The principle of covariance emphasizes formulation ality: on the one hand they diffract and interfere as of physical laws using only those physical quanti- waves (fields), on the other hand they appear and ties the measurements of which the observers in disappear as whole entities, called quanta. Hence different frames of reference could unambiguously the name of the theory. correlate. A quantum field theory seeks to explain certain Mathematically speaking, the physical quantities fundamental experimental observations—the exis- must transform covariantly, that is, under a certain tence of antiparticles, the spin-statistics relation, representation of the group of coordinate transfor- the CPT symmetry—as well as predict the results mations between admissible frames of reference of of any given experiment, like the cross-section for the physical theory. This group of coordinate trans- the Compton scattering, or the value of the anoma- formations is referred to as the covariance group of lous magnetic moment of the electron. the theory. There are two popular approaches to deal with The principle of covariance does not require in- quantum fields. -
Phys 4390: General Relativity Lecture 5: Intro. to Tensor Algebra
PHYS 4390: GENERAL RELATIVITY LECTURE 5: INTRO. TO TENSOR ALGEBRA Vectors are sufficient to describe Newtonian mechanics, however General Rela- tivity will require more general objects than vectors, namely tensors. To introduce these, we will first discuss the tensor algebra through an abstract approach, and afterwards introduce the conventional approach based on indices. While much of our discussion of tensors will appear to be in Rn, tensors may be defined on any differential manifold. This is due to the fact that a differential manifold is an object that appears locally like Euclidean space but is globally different. 1. Curves and Surfaces on a Manifold 1.1. Parametric Approach. A curve may be described with one single parame- ter, u, and defines coordinates in an n-dimensional manifold, xa = xa(u); a = 1; ::n: A surface has two parameters, and may be written as xa = xa(u; v) and in general a m-dimensional surface has m degrees of freedom, and is defined by xa = xa(u1; u2; :::; um) We may call a surface a subspace, and a surface with m = n − 1 degrees of freedom a hypersurface. 1.2. I. nstead of a parametric approach, we can describe a surface as a set of constraints. To see how, consider a hypersurface, xa = xa(u1; :::; un−1), taking each coordinate xa we could eliminate u1 to un−1 leaving one function that must vanish f(x1; x2; :::; xn) = 0 Thus, the parametric approach for a hypersurface is equivalent to imposing a con- straint. As an example in R2 the constraint x2 + y2 − R2 = 0 describes a circle ( i.e., S1). -
A Brief Study on Covariance of Newtonian Mechanics
Dr Dev Raj Mishra. International Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 10, Issue 7, (Series-IV) July 2020, pp. 11-15 RESEARCH ARTICLE OPEN ACCESS A Brief Study on Covariance of Newtonian Mechanics Dr Dev Raj Mishra Department of Physics, R.H.Government Post Graduate College, Kashipur, U.S.Nagar, Uttarakhand -244713 INDIA. ABSTRACT The possibility of a covariant formulation of Newton’s second law of motion using Energy-Momentum four- vector was explored. This covariant formulation explains the relationships between the electromagnetic potentials and the electromagnetic fields. It successfully explains the observation of pseudo-forces in nature for example the observed up-thrust in a falling frame of reference or a falling lift and the observed centrifugal force in circular motion. Keywords– Covariance,Electromagnetic Field Tensor,Four- vectors, Minkowski Space time, Relativity ----------------------------------------------------------------------------------------------------------------------------- ---------- Date of Submission: 06-07-2020 Date of Acceptance: 21-07-2020 ----------------------------------------------------------------------------------------------------------------------------- ---------- I. INTRODUCTION the applied force and the momentum. The applied The invariance under Lorentz force is also the gradient of potential energy. transformation expresses the proposition that the Therefore we have two quantities – the momentum laws of physics are same for the observers in and