2.9 Bending the Hyperbolic Surface CH(∂D)

Total Page:16

File Type:pdf, Size:1020Kb

2.9 Bending the Hyperbolic Surface CH(∂D) Abstract On geometry along grafting rays in Teichm¨uller space by Renee A. Laverdiere In this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm¨uller space. The main technique is to describe the hyperbolic metric σt at a point along the grafting ray in terms of a conformal factor gt times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays. Acknowledgments First and foremost, this thesis would not have been possible without the support, encouragement, and patience of my advisor, Mike Wolf. With his guidance, I have learned the rhythm of dancing between seeing the forest and seeing the trees. Thank you. In my daily work, I have been fortunate to be surrounded by others both gener- ous with their time and with enthusiasm for my work. In particular, I would like to thank Casey Douglas, Andy Huang, Frank Jones, Evelyn Lamb, and Qiongling Li for listening, nodding, smiling, and asking insightful questions. I am grateful to Bernard and Carolyn Aresu, Matteo Pasquali, and Marie-Nathalie Contou-Carrere for helping me establish a sense of community at Rice that bouyed me through darker days. I owe many smiles, meals, and the belief that there is a light at the end of the tunnel to Anna Grassini and John McDevitt. I will always be grateful to my parents for fostering an appreciation of mathe- matics in our home and for encouraging me to pursue my dreams. I am especially thankful for the gift of Father Nelligan, the canine love of my life. Nelligan deserves his own thanks, for reminding me on a daily basis that a dose of love, a few treats, and a breath of fresh air are the foundation for happiness. I owe my deepest gratitude to my husband, David. The words “thank you” are iv insufficient for managing life when my mind has been otherwise occupied, coping with uncapped pens and scattered stacks of loose paper, and (most importantly) believing in me when I did not believe in myself. Dedication For David. Without you, and the promise that you kept, this dream may have remained precisely that - a dream. Contents Abstract..................................... ii Acknowledgments . iii Dedication.................................... v List of Figures . 1 1 Introduction 2 2 Background 5 2.1 Klein’s perspective . 5 2.2 Fundamentals: Teichm¨uller space and complex projective structures . 8 2.2.1 (X, G)-structures ......................... 8 2.2.2 Complex and complex projective structures . 9 2.3 Parameterization of (S)........................12 P 2.3.1 Analytic parameterization . 12 2.3.2 Grafting parameterization . 16 3 Grafting applications 27 3.1 Polar Coordinates for Teichm¨uller space . 27 vi vii 3.2 Recent developments . 30 4 A surface geometry approach to grafting 36 4.1 Introduction . 36 4.2 Introduction to technique . 38 4.2.1 Representing the metrics along the grafting ray: two approaches. 39 4.2.2 Length............................... 44 4.2.3 Prescribed curvature equation. 46 4.2.4 Applying Formula 4.10 . 49 4.3 A geometric formula . 56 4.3.1 First variation of length . 57 4.3.2 Variational vector field . 61 4.3.3 Relating the variational formula to the complement of the ↵-collar 63 4.3.4 A general formula . 66 4.4 Observations . 70 4.5 An application . 72 4.5.1 Positive terms . 73 4.5.2 Small terms . 74 4.5.3 Summary . 87 List of Figures 2.1 Triangle bounded by circular arcs with angles ↵,β,⌫.......... 6 2.2 Polar attachment on the left; lateral attachment on the right. 7 2.3 Inserting a Euclidean annulus in the grafting construction. 17 2.4 The region A˜t ............................... 18 2.5 Locally, a geodesic lamination can look like a cantor set on the surface. 19 2.6 Grafting along a weighted multicurve. 21 2.7 Three possibilities for C(@ Dz)..................... 23 1 2.8 The Thurston metric is flat on the intersection D00. This picture is precisely the image for grafting. 24 2.9 Bending the hyperbolic surface CH(@D) along `............ 25 3.1 Grafting by t + s along the top line versus iterated grafting. 28 4.1 The surface with cylinder of length t + t0 (bottom) di↵ers from that of length t0 (top) by a linear stretch along the cylinder. 40 4.2 The boundary curves, ↵±, and the geodesic as imagined on the grafted (top image) and hyperbolic (bottom image) surfaces. 54 4.3 ↵+ is a natural boundary for comp+................... 64 viii 1 4.4 If A = 0 then the pre-image of the curve ⇠ = A must intersect γ⇤... 78 6 0 4.5 The ↵-rectangle is the smallest Euclidean sub-cylinder containing the ↵-collar. .................................. 82 4.6 The skinny tubes T↵± bound the wandering of the ⇠=constant curves. 85 Chapter 1 Introduction In this paper, we examine a natural geometric deformation of hyperbolic structures, grafting.LetS be a compact, orientable surface of genus g 2. Then a complex ≥ structure X on S is a maximal atlas of charts from S to C with biholomorphic transi- tion functions. A complex projective structure (or CP 1-structure) Z on S is a maximal atlas of charts from open subsets of S to CP 1 that are restrictions of M¨obius maps. Of primary interest is the Teichm¨uller space (S) and the space of CP 1-structures T (S). These are isotopy classes of complex and CP 1-structures, respectively. In par- P ticular, because every M¨obius map is holomorphic, every CP 1-structure is in fact a complex structure, and this underlying structure makes S a Riemann surface. This defines the forgetful projection map ⇡ : (S) (S). P !T There are two parameterizations of (S). The first, essentially complex-analytic P in nature, associates to every Z (S) a holomorophic quadratic di↵erential via the 2P Schwarzian derivative of the developing map. The second, due to Thurston, shows 2 3 that (S) is homeomorphic to (S) (S), where (S) is the space of measured P ML ⇥P ML geodesic laminations. This homeomorphism Gr : (S) (S) (S) is called ML ⇥T !P the projective grafting map. The composition gr = ⇡ Gr : (S)x (S) (S) ◦ ML T !T is called the conformal grafting map. The simplest picture of conformal grafting is as follows: choose a hyperbolic metric X (S) and γ a simple closed X-geodesic. 2T t t Then the grafting of X by tγ is obtained by replacing γ by the flat cylinder [ − , ] γ. 2 2 ⇥ This gives a new conformal structure, whic we denote by grtγX. Via uniformization, there is a new hyperbolic metric in the conformal class of gr X. Fixing γ (S), tγ 2ML we call the set of conformal structures grtγX : t R a grafting ray. We are inter- { 2 } ested in the evolution of geometric invariants of the hyperbolic surfaces along grafting rays in Teichm¨uller space. In first chapter, we develop the background material and sketch Thurston’s proof that the projective grafting map Gr : (S) (S) (S) is a homeomorphism. ML ⇥T !P The second chapter is a review of current research involving grafting. We begin the third chapter by introducing a technique for studying geometry at any point along a grafting ray generated by a simple closed geodesic, assuming that the geodesic in the homotopy class of the grafting locus does not escape the grafting cylinder. In section 3.2, we derive a formula that expresses the change in two geometric values in terms of the geometry of the hyperbolic surface in the conformal class of the grafted surface. More specifically, for any curve δ on the surface, denote by `(t, δ) the length of the δ with respect to the hyperbolic metric in the conformal class of gr(t+t0)γX. Then the formula of section 3.2 can be paraphrased to say the following: 4 + Theorem 4.1. Let γ be a separating curve on S. Choose t0 R . Let σ0 be 2 the hyperbolic metric in the conformal class of grt0γX. Then if γ0 is the hyperbolic + geodesic in the homotopy class of γ, and ↵0 and ↵0− are curves that are parallel to γ0 with respect to σ0, then + `(0,↵0 ) d + `(0,↵0−) d d 2 `(t, ↵ ) 2 `(t, ↵−)+4 `(t, γ ) − ` dt 0 − ` dt 0 dt t t=0 t=0 t=0 is given in terms of t0, the Euclidean length of the grafting cylinder, the conformal 2 + factor g, where σ = g(z) dz , distance of ↵ and ↵− from γ , and the average 0 | | 0 0 0 angle between ⇠=constant, a curve of constant σ0 (hyperbolic) geodesic curvature, and x=constant, a curve of constant Euclidean geodesic curvature. In section 3.3, we observe that if the surface S is actually a cylinder and γ is the core geodesic for a hyperbolic metric X on S, then the geometry along the grafting ray is particularly simple to understand. We restate the formula given in section 3.2 in such a way that shows the defect of the grafting operation on the general surface from being like that of grafting on a cylinder.
Recommended publications
  • 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.
    [Show full text]
  • Relativistic Rocket Dynamics and Accelerating Reference Frames (Rindler Coordinates)
    Supplemental Lecture 3 Acceleration: Relativistic Rocket Dynamics and Accelerating Reference Frames (Rindler Coordinates) Abstract An accelerating rocket is studied in special relativity and its equation of motion is contrasted to its non-relativistic treatment. Uniformly accelerating reference frames are formulated and the transformation between measurements in this non-inertial reference frame and an inertial frame is obtained and applied. The space-time metric inside an accelerating rocket is derived and the accelerating frame is found to be locally equivalent to the environment in a uniform gravitational field. The global properties of the accelerating frame (Rindler Wedge) are different from those of a uniform gravitational field: there is a past and future horizon of the accelerating frame but not for the uniform gravitational field. Analogies to the Schwarzschild black hole are drawn. This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work. Keywords: Special relativity, acceleration, inertial frames of reference, accelerating grid, Rindler coordinates, Equivalence Principle, Horizons, Black Holes. Rockets in Special Relativity Let’s study rocket dynamics as an illustration of relativistic dynamics and an introduction to accelerating reference frames and general relativity [1,2]. Along the way we will discuss the practical challenges to interstellar travel via “conventional” rocketry. Consider inertial frames and in the usual orientation as shown in Fig. 1. ′ 1 Fig. 1 Reference frames and , with moving in the direction at velocity . ′ ′ Frame moves to the right in the -direction at velocity relative to the frame .
    [Show full text]
  • Chapter 14: Geometry and Coordinates
    14. GEOMETRY AND COORDINATES We define . Given we say that the x-coordinate is while the y-coordinate is . We can view the coordinates as mappings from to : Coordinates take in a point in the plane and output a real number. Our goal in this chapter go beyond Cartesian coordinates for the plane. Are there other useful descriptions of ? How do those other coordinate systems relate to the standard Cartesian coordinate system? Geometrically, the Cartesian coordinates of a point correspond to a horizontal displacement of -units and a vertical displacement of -units that reach from the origin to the point . Basically, the Cartesian coordinates are built from travelling lines from the origin. Moreover, these lines are perpendicular. We’ll learn that we can construct other coordinate systems which are not based just on perpendicular lines. We can use circles, hyperbolas, parabolas, ellipses and many other curves as well as lines. The coordinates will be prescribed according to how the point is reached by travelling from the (new) origin along those curves. We define the origin for a particular coordinate system to be the point at which the coordinates are both zero. This may or may not line up with the Cartesian coordinate origin. For example, Puts the barred-origin at the point . In other words, . Obviously there is danger of confusion here, what is meant by ? Is the origin in the -coordinates or is this the origin in the -coordinate system? The solution is context, when dealing with several coordinate systems at once it is important to say which coordinates are used for a particular statement.
    [Show full text]
  • (Electronic) Conversion of Hyperbolic Into Rectangular Coordinates
    (ELECTRONIC) CONVERSION OF HYPERBOLIC INTO RECTANGULAR COORDINATES by J. Th.VERSTELLE Senior Civil Hydrographic Officer Hydrographic Office Royal Netherlands Navy June 1963 1. — Introduction 1.1. — Radionavigation — and (the more accurate) survey receivers — measure in the patterns radiated by a “ chain ” of radionavigation (survey) transmitters. The measured information is displayed in a suitable form on indicators. The information required by the navigator (surveyor) is the coordinates of the location of his receiver. 1.2. — Most navigation and survey systems radiate hyperbolic patterns of Lines Of Position = L. O. P.s.; examples are Loran and Decca. Some survey systems measure only distances and their L. O.P. s. are consequently circular; of this group Shoran (Hiran) is mentioned as an example. Other survey systems, such as for example Raydist, Hi-Fix, etc., can operate as well on the hyperbolic as on the circular principle. 1.3. — The usual nautical and survey practice is to provide the user with charts, on which a lattice of the L. 0. P. s. is overprinted together w ith a grid (X, Y rectangular coordinate systems) an d /o r a graticule (system of parallels and meridians). Provided care is taken that the lattice L. O. P. s. correspond with the radiated L.O.P.s., the navigator or surveyor can plot his observed L.O.P.s. in the latticed chart and can scale off his position in the graticule or grid, thus obtaining the desired geographical or rectangular coordinates. This is in fact a graphical method of conversion of hyperbolic (or circular) coordinates into geographical and/or Cartesian coordinates.
    [Show full text]
  • 15.Geometry and Coordinates
    15. GEOMETRY AND COORDINATES We define . Given we say that the x-coordinate is while the y- coordinate is . We can view the coordinates as mappings from to : Coordinates take in a point in the plane and output a real number. Polar coordinates are based on partitioning the plane into circles or rays from the origin. Each point in the plane will be found on a unique circle and ray hence we can label the point by the radius of the circle and the angle of the ray. Some ambiguity arises in assigning the angle because there is a natural duplicity in the assignment. We can always add an integer multiple of 360 degrees or radians. This means if we are to be open minded about angles then we have to accept there is more than one correct answer to a given question. Once the basics of polar coordinates are complete we study parametric curves and the calculation of areas and arclength in polar coordinates (currently there are no examples or discussion of area or arclength in polars, see Section 11.4 of Stewart for examples, I will probably work additional examples in lecture). Another major topic in geometry is that of conic sections. We review the major equations and the geometric definitions which define the ellipse, parabola and hyperbola. If time permits we may also examine the polar form of their equations in lecture (although it is not found in these notes presently) Finally, I conclude this chapter by exploring the existence of other coordinate systems in the plane.
    [Show full text]
  • Hyperbolic Mapping of Human Proximity Networks Marco A
    www.nature.com/scientificreports OPEN Hyperbolic mapping of human proximity networks Marco A. Rodríguez‑Flores & Fragkiskos Papadopoulos* Human proximity networks are temporal networks representing the close‑range proximity among humans in a physical space. They have been extensively studied in the past 15 years as they are critical for understanding the spreading of diseases and information among humans. Here we address the problem of mapping human proximity networks into hyperbolic spaces. Each snapshot of these networks is often very sparse, consisting of a small number of interacting (i.e., non‑zero degree) nodes. Yet, we show that the time‑aggregated representation of such systems over sufciently large periods can be meaningfully embedded into the hyperbolic space, using methods developed for traditional (non‑mobile) complex networks. We justify this compatibility theoretically and validate it experimentally. We produce hyperbolic maps of six diferent real systems, and show that the maps can be used to identify communities, facilitate efcient greedy routing on the temporal network, and predict future links with signifcant precision. Further, we show that epidemic arrival times are positively correlated with the hyperbolic distance from the infection sources in the maps. Thus, hyperbolic embedding could also provide a new perspective for understanding and predicting the behavior of epidemic spreading in human proximity systems. Understanding the time-varying proximity patterns among humans in a physical space is crucial for better understanding the transmission of airborne diseases, the efciency of information dissemination, social behavior, and infuence1–8. To this end, human proximity networks have been captured in diferent environments over days, weeks or months2,4,5,9–13.
    [Show full text]
  • Hyperbolic Trees in Complex Networks
    Hyperbolic Trees in Complex Networks Zalan´ Heszberger1;2 Andras´ Gulyas´ 1;2 Andras´ Biro´1 Laszl´ o´ Balazs´ 1 Jozsef´ B´ıro´1 1Budapest University of Technology and Economics 2MTA-BME Information Systems Modelling Research Group Budapest, Hungary, [email protected] Abstract—The two-dimensional hyperbolic space turned out II. THE HYPERBOLIC PLANE to be an efficient geometry for generative models of complex networks. The networks generated with this hyperbolic metric The hyperbolic plane (the two-dimensional hyperbolic space share their basic structural properties (like small diameter space) is a metric space, hence there is a well-defined distance or scale-free degree distribution) with several real networks. In calculation such as (hyperbolic cosine theorem) this paper, we present a new model for generating trees in the two-dimensional hyperbolic plane. The generative model is not cosh d(u; v) = cosh ru cosh rv − sinh ru sinh rv cos φ (1) based on known hyperbolic network models: the trees are not inferred from the existing links of any network; instead, the where ru and rv are the radial components of the polar hyperbolic tree is generated from scratch purely based on the coordinates of points u and v, and φ = φu − φv is the hyperbolic coordinates of nodes. We show that these hyperbolic difference of the angular components of the polar coordinates trees have scale-free degree distributions and are present to [2], [5]. Note that it significantly differs from the Euclidean a large extent both in synthetic hyperbolic complex networks and real ones (Internet autonomous system topology, US flight cosine theorem.
    [Show full text]
  • Link Prediction with Hyperbolic Geometry
    PHYSICAL REVIEW RESEARCH 2, 043113 (2020) Link prediction with hyperbolic geometry Maksim Kitsak ,1,2 Ivan Voitalov,3,2 and Dmitri Krioukov 4,2 1Faculty of Electrical Engineering, Delft University of Technology, Mathematics and Computer Science, 2600 GA Delft, The Netherlands 2Network Science Institute, Northeastern University, 177 Huntington Avenue, Boston, Massachusetts 02115, USA 3Department of Physics, Northeastern University, 110 Forsyth Street, 111 Dana Research Center, Boston, Massachusetts 02115, USA 4Department of Physics, Department of Mathematics, Department of Electrical and Computer Engineering, Northeastern University, 110 Forsyth Street, 111 Dana Research Center, Boston, Massachusetts 02115, USA (Received 3 April 2019; revised 6 June 2020; accepted 18 August 2020; published 21 October 2020) Link prediction is a paradigmatic problem in network science with a variety of applications. In latent space network models this problem boils down to ranking pairs of nodes in the order of increasing latent distances between them. The network model with hyperbolic latent spaces has a number of attractive properties suggesting it must be a powerful tool to predict links, but the past work in this direction reported mixed results. Here we perform a systematic investigation of the utility of latent hyperbolic geometry for link prediction in networks. We first show that some measures of link prediction accuracy are extremely sensitive with respect to inaccuracies in the inference of latent hyperbolic coordinates of nodes. This observation leads us to the development of a hyperbolic network embedding method, the HYPERLINK embedder, which we show maximizes the accuracy of such inference, compared to existing hyperbolic embedding methods. Applying this method to synthetic and real networks, we then find that when it comes to predicting obvious missing links hyperbolic link prediction—for short, HYPERLINK—is rarely the best but often competitive, compared to a multitude of other methods.
    [Show full text]
  • JHEP09(2016)154 Springer August 4, 2016 : September 18, 2016 September 27, 2016 :
    Published for SISSA by Springer Received: August 4, 2016 Accepted: September 18, 2016 Published: September 27, 2016 Equivalence of emergent de Sitter spaces from conformal field theory JHEP09(2016)154 Curtis T. Asplund,a Nele Callebauta;b and Claire Zukowskia aDepartment of Physics, Columbia University, 538 West 120th Street, New York, NY 10027, U.S.A. bInstituut voor Theoretische Fysica, K.U. Leuven, Celestijnenlaan 200D B-3001 Leuven, Belgium E-mail: [email protected], [email protected], [email protected] Abstract: Recently, two groups have made distinct proposals for a de Sitter space that is emergent from conformal field theory (CFT). The first proposal is that, for two-dimensional holographic CFTs, the kinematic space of geodesics on a space-like slice of the asymptot- ically anti-de Sitter bulk is two-dimensional de Sitter space (dS2), with a metric that can be derived from the entanglement entropy of intervals in the CFT. In the second proposal, de Sitter dynamics emerges naturally from the first law of entanglement entropy for per- turbations around the vacuum state of CFTs. We provide support for the equivalence of these two emergent spacetimes in the vacuum case and beyond. In particular, we study the kinematic spaces of nontrivial solutions of 3d gravity, including the BTZ black string, BTZ black hole, and conical singularities. We argue that the resulting spaces are generically globally hyperbolic spacetimes that support dynamics given boundary conditions at future infinity. For the BTZ black string, corresponding to a thermal state of the CFT, we show that both prescriptions lead to an emergent hyperbolic patch of dS2.
    [Show full text]
  • Conformal Grafting and Convergence of Fenchel-Nielsen Twist Coordinates
    CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 19, Pages 1–18 (January 23, 2015) S 1088-4173(2015)00273-8 CONFORMAL GRAFTING AND CONVERGENCE OF FENCHEL-NIELSEN TWIST COORDINATES MAXIME FORTIER BOURQUE Abstract. We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge. 1. Introduction Let S be a hyperbolic Riemann surface with finite area. We want to perform a surgery on S which we call conformal grafting.LetE be an analytic multicurve in S, i.e., a set of disjoint, non-parallel, non-peripheral, essential, simple, closed, E parametrized analytic curves. For every vector t ∈ (R≥0) , we construct a new surface St as follows. We first cut S along E and then, for each curve α ∈ E, we glue a cylinder of modulus tα to S \ E using the parametrization α on each boundary circle. This construction recovers some well-studied paths in Teichm¨uller space. If each component of E is geodesic for the hyperbolic metric on S, then for every t the set { Sλt | λ ≥ 0 } is called a grafting ray [Dum]. By taking the multicurve E to be the set of core curves for a Jenkins-Strebel quadratic differential, we obtain a Strebel ray, which is a special kind of geodesic for the the Teichm¨uller metric [Mas]. From the point of view of hyperbolic geometry, the effect of conformal grafting is to pinch the surface S along the multicurve E.LetS∞ be the surface constructed by gluing a pair of half-infinite cylinders to S \ E for each α ∈ E.
    [Show full text]
  • Minkowski Space-Time and Hyperbolic Geometry
    MASSEE International Congress on Mathematics MICOM 2015 September 22/ 26, Athens, Greece. Revised corrected version April 2018 MINKOWSKI SPACE-TIME AND HYPERBOLIC GEOMETRY J F Barrett ISVR, Southampton University Southampton, United Kingdom [email protected] ABSTRACT It has become generally recognized that hyperbolic (i.e. Lobachevskian) space can be represented upon one sheet of a two-sheeted cylindrical hyperboloid in Minkowski space-time. This paper aims to clarify the derivation of this result and to describe some further related ideas. Firstly a simple justification is given of the stated property, which seems somewhat lacking in the literature. This is straightforward once it is shown that differential displacements on the hyperboloid surface are space- like elements in Minkowski space-time. This needs certain preliminary remarks on Minkowski space-time. Two other derivations are given which are valid in any pseudo-Euclidean space of the same type. An alternative view comes from regarding Minkowski space-time projectively as a velocity space. This is possible with Minkowski's original representation but is best seen when Minkowski space-time is regarded differentially as a special case of the metric of General Relativity. Here the space may also be considered as differential space-time in the sense of Minkowski'. It may be considered as a projective space and in this case, as a velocity space which is a Lobachevsky space with hyperboloid representation. Projection of the hyperboloid to a disc or spherical ball gives an associated Beltrami-Klein representation of velocity space. This geometrical representation has important application in physics being related to the hyperbolic theory of Special Relativity which was first proposed by Varićak in 1910 following Einstein's original 1905 paper.
    [Show full text]
  • H-Holomorphic Functions of Double Variable and Their Applications
    h-holomorphic functions of double variable and their applications Dmitriy G. Pavlov∗ and Sergey S. Kokarev† Research Institute of Hypercomplex system in Geometry and Physics (Fryazino), Regional Scientific-Educational Center ”Logos” , (Yaroslavl) Abstract The paper studies the complex differentiable functions of double argument and their prop- erties, which are similar to the properties of the holomorphic functions of complex variable: the Cauchy formula, the hyperbolic harmonicity, the properties of general h-conformal mappings and the properties of the mappings, which are hyperbolic analogues of complex elementary functions. We discuss the utility of h-conformal mappings to solving 2-dimensional hyperbolic problems of Mathematical Physics. Keywords: double numbers, hyperbolic Cauchy problem, h-holomorphicity, conformal mappings. 1 Introduction Double numbers (or, as they are called sometimes, hyperbolic-complex or splittable numbers) are known for quite a long time, and find applications both in Mathematics and Physics [1, 2]. Taking into account that their corresponding algebra is isomorphic to the direct sum of two real algebras, it was considered for a long time that the properties of double numbers are not interesting, and moreover, that these are even trivial, while compared to complex numbers. In this paper we try to prove the wrongness of this opinion and to show that the possibilities offered by the algebra and the analysis (i.e., the h-analysis) of double numbers, considered together with the associated geometry of the 2-dimensional plane, are still far from being exhaustively studied. arXiv:1501.03048v1 [math.GM] 3 Jan 2015 We shall see that, in many respects, double numbers are far – in any sense, from being inferior to complex numbers.
    [Show full text]