Lorentz Surfaces with Constant Curvature and Their Physical
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Line Element in Noncommutative Geometry
Line element in noncommutative geometry P. Martinetti G¨ottingenUniversit¨at Wroclaw, July 2009 . ? ? - & ? !? The line element p µ ν ds = gµν dx dx is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - . ? !? The line element p µ ν ds = gµν dx dx & ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - !? The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). ? - The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). The line element p µ ν ds = gµν dx dx . & ? ? is mainly useful to measure distance ? -Z y !? d(x; y) = inf ds: x If, for some quantum gravity reasons, [x µ; x ν ] 6= 0 is one losing the notion of distance ? (annoying then to speak of noncommutative geo-metry). -
The Floating Body in Real Space Forms
THE FLOATING BODY IN REAL SPACE FORMS Florian Besau & Elisabeth M. Werner Abstract We carry out a systematic investigation on floating bodies in real space forms. A new unifying approach not only allows us to treat the important classical case of Euclidean space as well as the recent extension to the Euclidean unit sphere, but also the new extension of floating bodies to hyperbolic space. Our main result establishes a relation between the derivative of the volume of the floating body and a certain surface area measure, which we called the floating area. In the Euclidean setting the floating area coincides with the well known affine surface area, a powerful tool in the affine geometry of convex bodies. 1. Introduction Two important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body. The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant δ. Taking the right-derivative of the volume of the floating body gives rise to the affine surface area. This was established for all convex bodies in all dimensions by Schütt and Werner in [62]. The affine surface area was introduced by Blaschke in 1923 [8]. Due to its important properties, which make it an effective and powerful tool, it is omnipresent in geometry. The affine surface area and its generalizations in the rapidly developing Lp and Orlicz Brunn–Minkowski theory are the focus of intensive investigations (see e.g. [14,18, 20,21, 45, 46,65, 67,68, 70,71]). -
Matrices for Fenchel–Nielsen Coordinates
Annales Academi½ Scientiarum Fennic½ Mathematica Volumen 26, 2001, 267{304 MATRICES FOR FENCHEL{NIELSEN COORDINATES Bernard Maskit The University at Stony Brook, Mathematics Department Stony Brook NY 11794-3651, U.S.A.; [email protected] Abstract. We give an explicit construction of matrix generators for ¯nitely generated Fuch- sian groups, in terms of appropriately de¯ned Fenchel{Nielsen (F-N) coordinates. The F-N coordi- nates are de¯ned in terms of an F-N system on the underlying orbifold; this is an ordered maximal set of simple disjoint closed geodesics, together with an ordering of the set of complementary pairs of pants. The F-N coordinate point consists of the hyperbolic sines of both the lengths of these geodesics, and the lengths of arc de¯ning the twists about them. The mapping from these F-N coordinates to the appropriate representation space is smooth and algebraic. We also show that the matrix generators are canonically de¯ned, up to conjugation, by the F-N coordinates. As a corol- lary, we obtain that the TeichmullerÄ modular group acts as a group of algebraic di®eomorphisms on this Fenchel{Nielsen embedding of the TeichmullerÄ space. 1. Introduction There are several di®erent ways to describe a closed Riemann surface of genus at least 2; these include its representation as an algebraic curve; its representation as a period matrix; its representation as a Fuchsian group; its representation as a hyperbolic manifold, in particular, using Fenchel{Nielsen (F-N) coordinates; its representation as a Schottky group; etc. One of the major problems in the overall theory is that of connecting these di®erent visions. -
ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof
ASSIGNMENTS Week 4 (F. Saueressig) Cosmology 13/14 (NWI-NM026C) Prof. A. Achterberg, Dr. S. Larsen and Dr. F. Saueressig Exercise 1: Flat space in spherical coordinates For constant time t the line element of special relativity reduces to the flat three-dimensional Euclidean metric Euclidean 2 SR 2 2 2 2 (ds ) = (ds ) =const = dx + dy + dz . (1) − |t Use the coordinate transformation x = r sin θ cos φ, y = r sin θ sin φ , z = r cos θ , (2) with θ [0,π] and φ [0, √2 π[ to express the line element in spherical coordinates. ∈ ∈ Exercise 2: Distances, areas and volumes For constant time the metric describing the spatial part of a homogeneous closed Friedmann- Robertson Walker (FRW) universe is given by 2 2 dr 2 2 2 2 ds = + r dθ + sin θ dφ , r [ 0 , a ] . (3) 1 (r/a)2 ∈ − The scale factor a(t) depends on time only so that, for t = const, it can be considered as a fixed number. Compute from this line element a) The proper circumference of the sphere around the equator. b) The proper distance from the center of the sphere up to the coordinate radius a. c) The proper area of the two-surface at r = a. d) The proper volume inside r = a. Compare your results to the ones expected from flat Euclidean space. Which of the quantities are modified by the property that the line element (3) is curved? Exercise 3: The metric is covariantly constant In the lecture we derived a formula for the Christoffel symbol Γα 1 gαβ g + g g . -
19710025511.Pdf
ORBIT PERTLJFEATION THEORY USING HARMONIC 0SCII;LCITOR SYSTEPE A DISSERTATION SUBMITTED TO THE DEPARTbIEllT OF AJ3RONAUTICS AND ASTROW-UTICS AND THE COMMITTEE ON GRADUATE STUDDS IN PAEMlIAL FULFI-NT OF THE mQUIREMENTS I FOR THE DEGREE OF DOCTOR OF PHILOSOPHY By William Byron Blair March 1971 1;"= * ..--"".-- -- -;.- . ... - jlgpsoijed f~:;c2ilc e distribution unlinl:;&. --as:.- ..=-*.;- .-, . - - I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. d/c fdd&- (Principal Adviser) I certify that I have read this thesis and that in my opinion it is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosoph~ ., ., I certify that I have read this thesis and that in my opinion it is fully adequate, in scop: and quality, as a dissertation for the degree of Doctor of Philosophy. Approved for the University Cormnittee on Graduate Studies: Dean of Graduate Studies ABSTRACT Unperturbed two-body, or Keplerian motion is transformed from the 1;inie domain into the domains of two ur,ique three dimensional vector harmonic oscillator systems. One harmoriic oscillator system is fully regularized and hence valid for all orbits including the rectilinear class up to and including periapsis passage, The other system is fdly as general except that the solution becomes unbounded at periapsis pas- sage of rectilinear orbits. The natural frequencies of the oscillator systems are related to certain Keplerian orbit scalar constants, while the independent variables are related to well-known orbit angular measurements, or anomalies. -
Relativistic Spacetime Crystals
research papers Relativistic spacetime crystals Venkatraman Gopalan* ISSN 2053-2733 Department of Materials Science and Engineering, Department of Physics, Department of Engineering Science and Mechanics, and the Materials Research Institute, Pennsylvania State University, University Park, PA 16802, USA. *Correspondence e-mail: [email protected] Received 30 November 2020 Periodic space crystals are well established and widely used in physical sciences. Accepted 26 March 2021 Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity Edited by S. J. L. Billinge, Columbia University, USA through Lorentz transformation, and have been listed only in 2D. This work shows that there exists a transformation between the conventional Minkowski Keywords: spacetime; special relativity; renor- spacetime (MS) and what is referred to here as renormalized blended spacetime malized blended spacetime; relativistic (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime crystals. spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other’s Supporting information: this article has clocks as their own, while retaining their original space coordinates, the supporting information at journals.iucr.org/a observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. -
Some Applications of the Spectral Theory of Automorphic Forms
Academic year 2009-2010 Department of Mathematics Some applications of the spectral theory of automorphic forms Research in Mathematics M.Phil Thesis Francois Crucifix Supervisor : Doctor Yiannis Petridis 2 I, Francois Nicolas Bernard Crucifix, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. SIGNED CONTENTS 3 Contents Introduction 4 1 Brief overview of general theory 5 1.1 Hyperbolic geometry and M¨obius transformations . ............. 5 1.2 Laplaceoperatorandautomorphicforms. ......... 7 1.3 Thespectraltheorem.............................. .... 8 2 Farey sequence 14 2.1 Fareysetsandgrowthofsize . ..... 14 2.2 Distribution.................................... 16 2.3 CorrelationsofFareyfractions . ........ 18 2.4 Good’sresult .................................... 22 3 Multiplier systems 26 3.1 Definitionsandproperties . ...... 26 3.2 Automorphicformsofnonintegralweights . ......... 27 3.3 Constructionofanewseries. ...... 30 4 Modular knots and linking numbers 37 4.1 Modularknots .................................... 37 4.2 Linkingnumbers .................................. 39 4.3 TheRademacherfunction . 40 4.4 Ghys’result..................................... 41 4.5 SarnakandMozzochi’swork. ..... 43 Conclusion 46 References 47 4 Introduction The aim of this M.Phil thesis is to present my research throughout the past academic year. My topics of interest ranged over a fairly wide variety of subjects. I started with the study of automorphic forms from an analytic point of view by applying spectral methods to the Laplace operator on Riemann hyperbolic surfaces. I finished with a focus on modular knots and their linking numbers and how the latter are related to the theory of well-known analytic functions. My research took many more directions, and I would rather avoid stretching the extensive list of applications, papers and books that attracted my attention. -
Appendix a Tensor Mathematics
Appendix A Tensor Mathematics A.l Introduction In physics we are accustomed to relating quantities to other quantities by means of mathematical expressions. The quantities that represent physical properties of a point or of an infinitesimal volume in space may be of a single numerical value without direction, such as temperature, mass, speed, dist ance, specific gravity, energy, etc.; they are defined by a single magnitude and we call them scalars. Other physical quantities have direction as well and their magnitude is represented by an array of numerical values, their number corresponding to the number of dimensions of the space, three numerical values in a three-dimensional space; such quantities are called vectors, for example velocity, acceleration, force, etc., and the three numerical values are the components of the vector in the direction of the coordinates of the space. Still other physical quantities such as moment of inertia, stress, strain, permeability coefficients, electric flux, electromagnetic flux, etc., are repre sented in a three-dimensional space by nine numerical quantities, called components, and are known as tensors. We will introduce a slightly different definition. We shall call the scalars tensors of order zero, vectors - tensors of order one and tensors with nine components - tensors of order two. There are, of course, in physics and mathematics, tensors of higher order. Tensors of order three have an array of 27 components and tensors of order four have 81 components and so on. Couple stresses that arise in materials with polar forces are examples of tensors of order three, and the Riemann curvature tensor that appears in geodesics is an example of a tensor of order four. -
Hilbert Geometry of the Siegel Disk: the Siegel-Klein Disk Model
Hilbert geometry of the Siegel disk: The Siegel-Klein disk model Frank Nielsen Sony Computer Science Laboratories Inc, Tokyo, Japan Abstract We study the Hilbert geometry induced by the Siegel disk domain, an open bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel-Klein disk model to differentiate it with the classical Siegel upper plane and disk domains. In the Siegel-Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data-structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel-Poincar´edisk and in the Siegel-Klein disk: We demonstrate that geometric computing in the Siegel-Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel-Poincar´edisk model, and (ii) to approximate fast and numerically the Siegel-Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries. Keywords: Hyperbolic geometry; symmetric positive-definite matrix manifold; symplectic group; Siegel upper space domain; Siegel disk domain; Hilbert geometry; Bruhat-Tits space; smallest enclosing ball. 1 Introduction German mathematician Carl Ludwig Siegel [106] (1896-1981) and Chinese mathematician Loo-Keng Hua [52] (1910-1985) have introduced independently the symplectic geometry in the 1940's (with a preliminary work of Siegel [105] released in German in 1939). -
Travelling Randomly on the Poincar\'E Half-Plane with a Pythagorean Compass
Travelling Randomly on the Poincar´e Half-Plane with a Pythagorean Compass July 29, 2021 V. Cammarota E. Orsingher 1 Dipartimento di Statistica, Probabilit`ae Statistiche applicate University of Rome `La Sapienza' P.le Aldo Moro 5, 00185 Rome, Italy Abstract A random motion on the Poincar´ehalf-plane is studied. A particle runs on the geodesic lines changing direction at Poisson-paced times. The hyperbolic distance is analyzed, also in the case where returns to the starting point are admitted. The main results concern the mean hyper- bolic distance (and also the conditional mean distance) in all versions of the motion envisaged. Also an analogous motion on orthogonal circles of the sphere is examined and the evolution of the mean distance from the starting point is investigated. Keywords: Random motions, Poisson process, telegraph process, hyperbolic and spherical trigonometry, Carnot and Pythagorean hyperbolic formulas, Cardano for- mula, hyperbolic functions. AMS Classification 60K99 1 Introduction Motions on hyperbolic spaces have been studied since the end of the Fifties and most of the papers devoted to them deal with the so-called hyperbolic Brownian motion (see, e.g., Gertsenshtein and Vasiliev [4], Gruet [5], Monthus arXiv:1107.4914v1 [math.PR] 25 Jul 2011 and Texier [9], Lao and Orsingher [7]). More recently also works concerning two-dimensional random motions at finite velocity on planar hyperbolic spaces have been introduced and analyzed (Orsingher and De Gregorio [11]). While in [11] the components of motion are supposed to be independent, we present here a planar random motion with interacting components. Its coun- terpart on the unit sphere is also examined and discussed. -
1 Special Relativity
1 Special relativity 1.1 Minkowski space Inertial frames and the principle of relativity Newton’s Lex Prima (or the Galilean law of inertia) states: Each force-less mass point stays at rest or moves on a straight line at constant speed. In a Cartesian inertial coordinate system, Newton’s lex prima becomes d2x d2y d2z = = = 0 . (1.1) dt2 dt2 dt2 Most often, we call such a coordinate system just an inertial frame. Newton’s first law is not just a trivial consequence of its second one, but may be seen as practical definition of those reference frames for which his following laws are valid. Since the symmetries of Euclidean space, translations a and rotations R, leave Eq. (1.1) invariant, all frames connected by x′ = Rx + a to an inertial frame are inertial frames too. Additionally, proper Galilean transformations x′ = x + vt connect inertial frames moving with relative speed v. The principle of relativity states that the physics in all inertial frames is the same. If we consider two frames with relative velocity along the x direction, then the most general linear transformation between the two frames is t′ At + Bx At + Bx ′ x Dt + Ex A(x vt) ′ = = − . (1.2) y y y ′ z z z Newtonian physics assumes the existence of an absolute time, t = t′, and thus A = 1 and B = 0. This leads to the “common sense” addition law of velocities. Time differences ∆t and space differences 2 2 2 2 ∆x12 = (x1 x2) + (y1 y2) + (z1 z2) (1.3) − − − are separately invariant. Lorentz transformations In special relativity, we replace the Galilean transformations as symmetry group of space and time by Lorentz transformations Λ. -
Metric Tensor & Chriostofell Symbols
MATHEMATICAL PHYSICS UNIT – 8 Metric Tensor & Chriostofell Symbols PRESENTED BY: DR. RAJESH MATHPAL ACADEMIC CONSULTANT SCHOOL OF SCIENCES U.O.U. TEENPANI, HALDWANI UTTRAKHAND MOB:9758417736,7983713112 STRUCTURE OF UNIT 8.1. INTRODUCTION 8.2. RIEMANNIAN SPACE: METRIC TENSOR jk 풋 8.3. FUNDAMENTAL TENSORS gjk, g AND 휹풌 8.4. CHRISTOFELL’S 3-INDEX SYMBOLS 8.5. GEODESICS 8.1. INTRODUCTION In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q.