Lost in the Tensors: Einstein's Struggles with Covariance Principles 1912-1916"
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Package ‘einsum’ May 15, 2021 Type Package Title Einstein Summation Version 0.1.0 Description The summation notation suggested by Einstein (1916) <doi:10.1002/andp.19163540702> is a concise mathematical notation that implicitly sums over repeated indices of n- dimensional arrays. Many ordinary matrix operations (e.g. transpose, matrix multiplication, scalar product, 'diag()', trace etc.) can be written using Einstein notation. The notation is particularly convenient for expressing operations on arrays with more than two dimensions because the respective operators ('tensor products') might not have a standardized name. License MIT + file LICENSE Encoding UTF-8 SystemRequirements C++11 Suggests testthat, covr RdMacros mathjaxr RoxygenNote 7.1.1 LinkingTo Rcpp Imports Rcpp, glue, mathjaxr R topics documented: einsum . .2 einsum_package . .3 Index 5 1 2 einsum einsum Einstein Summation Description Einstein summation is a convenient and concise notation for operations on n-dimensional arrays. Usage einsum(equation_string, ...) einsum_generator(equation_string, compile_function = TRUE) Arguments equation_string a string in Einstein notation where arrays are separated by ’,’ and the result is separated by ’->’. For example "ij,jk->ik" corresponds to a standard matrix multiplication. Whitespace inside the equation_string is ignored. Unlike the equivalent functions in Python, einsum() only supports the explicit mode. This means that the equation_string must contain ’->’. ... the arrays that are combined. All arguments are converted to arrays with -
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. -
Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein. -
2008. Pruning Some Branches from 'Branching Spacetimes'
CHAPTER 10 Pruning Some Branches from “Branching Spacetimes” John Earman* Abstract Discussions of branching time and branching spacetime have become com- mon in the philosophical literature. If properly understood, these concep- tions can be harmless. But they are sometimes used in the service of debat- able and even downright pernicious doctrines. The purpose of this chapter is to identify the pernicious branching and prune it back. 1. INTRODUCTION Talk of “branching time” and “branching spacetime” is wide spread in the philo- sophical literature. Such expressions, if properly understood, can be innocuous. But they are sometimes used in the service of debatable and even downright per- nicious doctrines. The purpose of this paper is to identify the pernicious branching and prune it back. Section 2 distinguishes three types of spacetime branching: individual branch- ing, ensemble branching, and Belnap branching. Individual branching, as the name indicates, involves a branching structure in individual spacetime models. It is argued that such branching is neither necessary nor sufficient for indeterminism, which is explicated in terms of the branching in the ensemble of spacetime mod- els satisfying the laws of physics. Belnap branching refers to the sort of branching used by the Belnap school of branching spacetimes. An attempt is made to sit- uate this sort of branching with respect to ensemble branching and individual branching. Section 3 is a sustained critique of various ways of trying to imple- ment individual branching for relativistic spacetimes. Conclusions are given in Section 4. * Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, USA The Ontology of Spacetime II © Elsevier BV ISSN 1871-1774, DOI: 10.1016/S1871-1774(08)00010-7 All rights reserved 187 188 Pruning Some Branches from “Branching Spacetimes” 2. -
Notes on Manifolds
Notes on Manifolds Justin H. Le Department of Electrical & Computer Engineering University of Nevada, Las Vegas [email protected] August 3, 2016 1 Multilinear maps A tensor T of order r can be expressed as the tensor product of r vectors: T = u1 ⊗ u2 ⊗ ::: ⊗ ur (1) We herein fix r = 3 whenever it eases exposition. Recall that a vector u 2 U can be expressed as the combination of the basis vectors of U. Transform these basis vectors with a matrix A, and if the resulting vector u0 is equivalent to uA, then the components of u are said to be covariant. If u0 = A−1u, i.e., the vector changes inversely with the change of basis, then the components of u are contravariant. By Einstein notation, we index the covariant components of a tensor in subscript and the contravariant components in superscript. Just as the components of a vector u can be indexed by an integer i (as in ui), tensor components can be indexed as Tijk. Additionally, as we can view a matrix to be a linear map M : U ! V from one finite-dimensional vector space to another, we can consider a tensor to be multilinear map T : V ∗r × V s ! R, where V s denotes the s-th-order Cartesian product of vector space V with itself and likewise for its algebraic dual space V ∗. In this sense, a tensor maps an ordered sequence of vectors to one of its (scalar) components. Just as a linear map satisfies M(a1u1 + a2u2) = a1M(u1) + a2M(u2), we call an r-th-order tensor multilinear if it satisfies T (u1; : : : ; a1v1 + a2v2; : : : ; ur) = a1T (u1; : : : ; v1; : : : ; ur) + a2T (u1; : : : ; v2; : : : ; ur); (2) for scalars a1 and a2. -
The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics
The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand. -
Multilinear Algebra and Applications July 15, 2014
Multilinear Algebra and Applications July 15, 2014. Contents Chapter 1. Introduction 1 Chapter 2. Review of Linear Algebra 5 2.1. Vector Spaces and Subspaces 5 2.2. Bases 7 2.3. The Einstein convention 10 2.3.1. Change of bases, revisited 12 2.3.2. The Kronecker delta symbol 13 2.4. Linear Transformations 14 2.4.1. Similar matrices 18 2.5. Eigenbases 19 Chapter 3. Multilinear Forms 23 3.1. Linear Forms 23 3.1.1. Definition, Examples, Dual and Dual Basis 23 3.1.2. Transformation of Linear Forms under a Change of Basis 26 3.2. Bilinear Forms 30 3.2.1. Definition, Examples and Basis 30 3.2.2. Tensor product of two linear forms on V 32 3.2.3. Transformation of Bilinear Forms under a Change of Basis 33 3.3. Multilinear forms 34 3.4. Examples 35 3.4.1. A Bilinear Form 35 3.4.2. A Trilinear Form 36 3.5. Basic Operation on Multilinear Forms 37 Chapter 4. Inner Products 39 4.1. Definitions and First Properties 39 4.1.1. Correspondence Between Inner Products and Symmetric Positive Definite Matrices 40 4.1.1.1. From Inner Products to Symmetric Positive Definite Matrices 42 4.1.1.2. From Symmetric Positive Definite Matrices to Inner Products 42 4.1.2. Orthonormal Basis 42 4.2. Reciprocal Basis 46 4.2.1. Properties of Reciprocal Bases 48 4.2.2. Change of basis from a basis to its reciprocal basis g 50 B B III IV CONTENTS 4.2.3. -
Abstract Tensor Systems As Monoidal Categories
Abstract Tensor Systems as Monoidal Categories Aleks Kissinger Dedicated to Joachim Lambek on the occasion of his 90th birthday October 31, 2018 Abstract The primary contribution of this paper is to give a formal, categorical treatment to Penrose’s abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories. 1 Introduction This paper formalises the connection between monoidal categories and the ab- stract index notation developed by Penrose in the 1970s, which has been used by physicists directly, and category theorists implicitly, via the diagrammatic languages for traced symmetric monoidal and compact closed categories. This connection is given as a representation theorem for the free traced symmet- ric monoidal category as a syntactically-defined strict monoidal category whose morphisms are equivalence classes of certain kinds of terms called Einstein ex- pressions. Representation theorems of this kind form a rich history of coherence results for monoidal categories originating in the 1960s [17, 6]. Lambek’s con- arXiv:1308.3586v1 [math.CT] 16 Aug 2013 tribution [15, 16] plays an essential role in this history, providing some of the earliest examples of syntactically-constructed free categories and most of the key ingredients in Kelly and Mac Lane’s proof of the coherence theorem for closed monoidal categories [11]. -
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. -
Math, Physics, and Calabi–Yau Manifolds
Math, Physics, and Calabi–Yau Manifolds Shing-Tung Yau Harvard University October 2011 Introduction I’d like to talk about how mathematics and physics can come together to the benefit of both fields, particularly in the case of Calabi-Yau spaces and string theory. This happens to be the subject of the new book I coauthored, THE SHAPE OF INNER SPACE It also tells some of my own story and a bit of the history of geometry as well. 2 In that spirit, I’m going to back up and talk about my personal introduction to geometry and how I ended up spending much of my career working at the interface between math and physics. Along the way, I hope to give people a sense of how mathematicians think and approach the world. I also want people to realize that mathematics does not have to be a wholly abstract discipline, disconnected from everyday phenomena, but is instead crucial to our understanding of the physical world. 3 There are several major contributions of mathematicians to fundamental physics in 20th century: 1. Poincar´eand Minkowski contribution to special relativity. (The book of Pais on the biography of Einstein explained this clearly.) 2. Contributions of Grossmann and Hilbert to general relativity: Marcel Grossmann (1878-1936) was a classmate with Einstein from 1898 to 1900. he was professor of geometry at ETH, Switzerland at 1907. In 1912, Einstein came to ETH to be professor where they started to work together. Grossmann suggested tensor calculus, as was proposed by Elwin Bruno Christoffel in 1868 (Crelle journal) and developed by Gregorio Ricci-Curbastro and Tullio Levi-Civita (1901). -
Marcel Grossmann Awards
MG15 MARCEL GROSSMANN AWARDS ROME 2018 ICRANet and ICRA MG XV MARCEL GROSSMANN AWARDS ROME 2018 and TEST The 15th Marcel Grossmann Meeting – MG XV 2nd July 2018, Rome (Italy) Aula Magna – University “Sapienza” of Rome Institutional Awards Goes to: PLANCK SCIENTIFIC COLLABORATION (ESA) “for obtaining important constraints on the models of inflationary stage of the Universe and level of primordial non-Gaussianity; measuring with unprecedented sensitivity gravitational lensing of Cosmic Microwave Background fluctuations by large-scale structure of the Universe and corresponding B- polarization of CMB, the imprint on the CMB of hot gas in galaxy clusters; getting unique information about the time of reionization of our Universe and distribution and properties of the dust and magnetic fields in our Galaxy” - presented to Jean-Loup Puget, the Principal Investigator of the High Frequency Instrument (HFI) HANSEN EXPERIMENTAL PHYSICS LABORATORY AT STANFORD UNIVERSITY “to HEPL for having developed interdepartmental activities at Stanford University at the frontier of fundamental physics, astrophysics and technology” - presented to Research Professor Leo Hollberg, HEPL Assistant Director Individual Awards Goes to LYMAN PAGE “for his collaboration with David Wilkinson in realizing the NASA Explorer WMAP mission and as founding director of the Atacama Cosmology Telescope” Goes to RASHID ALIEVICH SUNYAEV “for the development of theoretical tools in the scrutinising, through the CMB, of the first observable electromagnetic appearance of our Universe” Goes to SHING-TUNG YAU “for the proof of the positivity of total mass in the theory of general relativity and perfecting as well the concept of quasi-local mass, for his proof of the Calabi conjecture, for his continuous inspiring role in the study of black holes physics” Each recipient is presented with a silver casting of the TEST sculpture by the artist A.