Hyperbolic Trigonometry in Two-Dimensional Space-Time Geometry
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Einstein's Velocity Addition Law and Its Hyperbolic Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 53 (2007) 1228–1250 www.elsevier.com/locate/camwa Einstein’s velocity addition law and its hyperbolic geometry Abraham A. Ungar∗ Department of Mathematics, North Dakota State University, Fargo, ND 58105, USA Received 6 March 2006; accepted 19 May 2006 Abstract Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the introduction of these concepts into hyperbolic geometry leads to hyperbolic vector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. c 2007 Elsevier Ltd. All rights reserved. Keywords: Special relativity; Einstein’s velocity addition law; Thomas precession; Hyperbolic geometry; Hyperbolic trigonometry 1. Introduction The hyperbolic law of cosines is nearly a century old result that has sprung from the soil of Einstein’s velocity addition law that Einstein introduced in his 1905 paper [1,2] that founded the special theory of relativity. It was established by Sommerfeld (1868–1951) in 1909 [3] in terms of hyperbolic trigonometric functions as a consequence of Einstein’s velocity addition of relativistically admissible velocities. Soon after, Varicakˇ (1865–1942) established in 1912 [4] the interpretation of Sommerfeld’s consequence in the hyperbolic geometry of Bolyai and Lobachevski. -
Old-Fashioned Relativity & Relativistic Space-Time Coordinates
Relativistic Coordinates-Classic Approach 4.A.1 Appendix 4.A Relativistic Space-time Coordinates The nature of space-time coordinate transformation will be described here using a fictional spaceship traveling at half the speed of light past two lighthouses. In Fig. 4.A.1 the ship is just passing the Main Lighthouse as it blinks in response to a signal from the North lighthouse located at one light second (about 186,000 miles or EXACTLY 299,792,458 meters) above Main. (Such exactitude is the result of 1970-80 work by Ken Evenson's lab at NIST (National Institute of Standards and Technology in Boulder) and adopted by International Standards Committee in 1984.) Now the speed of light c is a constant by civil law as well as physical law! This came about because time and frequency measurement became so much more precise than distance measurement that it was decided to define the meter in terms of c. Fig. 4.A.1 Ship passing Main Lighthouse as it blinks at t=0. This arrangement is a simplified model for a 1Hz laser resonator. The two lighthouses use each other to maintain a strict one-second time period between blinks. And, strict it must be to do relativistic timing. (Even stricter than NIST is the universal agency BIGANN or Bureau of Intergalactic Aids to Navigation at Night.) The simulations shown here are done using RelativIt. Relativistic Coordinates-Classic Approach 4.A.2 Fig. 4.A.2 Main and North Lighthouses blink each other at precisely t=1. At p recisel y t=1 sec. -
Geodesy. Application of the Theory of Least Squares to the Adjustment Of
SI No Serial ^) DEPARTME]S'T OF COMMERCE 0. S. COAST AND GEODETIC SURVEY it E. LESTER JONES, Supei^intendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION RY OSCAR S. ADAM^ COMPUTER UNITED STATES COAST AND GEODETIC SURJ^Y No. 28 Special Publication { WASHINGTON GOVERNMENT I'RINTING OFFICE 1915 Serial No. 9 DEPARTMEISTT OF COMMERCE U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, Superintendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION BY OSCAR S. ADAMS COMPUTER UNITED STATES COAST AND GEODETIC SURVEY Special Publication No. 28 WASHINGTON GOVERNMENT PRINTING OFFICE 1815 ADDITIONAL COPIES OF THIS PUBLICATION MAY BE PROCURED FROM THE SUPERINTENDENT OF DOCUMENTS GOVERNMENT PRINTING OFFICE WASHINGTON, D. C. AT 25 CENTS PER COPY QB 32/ CONTENTS Page. General statement 7 Station adjustment 7 Observed angles 8 List of directions 8 Condition equations , 9 Formation of normal equations by differentiation 9 Correlate equations 11 Formation of normal equations 11 Normal equations 12 Discussion of method of solution of normal equations 12 Solution of normal equations 13 Back solution 13 Computation of corrections 13 Adjustment of a quadrilateral 14 General statement 14 Lists of directions 16 Figure 16 Angle equations 17 Side equation 17 Formation of normal equations by differentiation 17 Correlate equations 18 Normal equations 18 Solution of normal equations 19 Back solution 19 Computation of corrections 19 Adjustment of a quadrilateral by the use of two angle and two -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. -
Using Differentials to Differentiate Trigonometric and Exponential Functions Tevian Dray
Using Differentials to Differentiate Trigonometric and Exponential Functions Tevian Dray Tevian Dray ([email protected]) received his B.S. in mathematics from MIT in 1976, his Ph.D. in mathematics from Berkeley in 1981, spent several years as a physics postdoc, and is now a professor of mathematics at Oregon State University. A Fellow of the American Physical Society for his early work in general relativity, his current research interests include the octonions as well as science education. He directs the Vector Calculus Bridge Project. (http://www.math.oregonstate.edu/bridge) Differentiating a polynomial is easy. To differentiate u2 with respect to u, start by computing d.u2/ D .u C du/2 − u2 D 2u du C du2; and then dropping the last term, an operation that can be justified in terms of limits. Differential notation, in general, can be regarded as a shorthand for a formal limit argument. Still more informally, one can argue that du is small compared to u, so that the last term can be ignored at the level of approximation needed. After dropping du2 and dividing by du, one obtains the derivative, namely d.u2/=du D 2u. Even if one regards this process as merely a heuristic procedure, it is a good one, as it always gives the correct answer for a polynomial. (Physicists are particularly good at knowing what approximations are appropriate in a given physical context. A physicist might describe du as being much smaller than the scale imposed by the physical situation, but not so small that quantum mechanics matters.) However, this procedure does not suffice for trigonometric functions. -
Cheryl Jaeger Balm Hyperbolic Function Project
Math 43 Fall 2016 Instructor: Cheryl Jaeger Balm Hyperbolic Function Project Circles are part of a family of curves called conics. The various conic sections can be derived by slicing a plane through a double cone. A hyperbola is a conic with two basic forms: y y 4 4 2 2 • (0; 1) (−1; 0) (1; 0) • • x x -4 -2 2 4 -4 -2 2 4 (0; −1) • -2 -2 -4 -4 x2 − y2 = 1 y2 − x2 = 1 x2 − y2 = 1 is the unit hyperbola. Hyperbolic Functions: Similar to how the trigonometric functions, cosine and sine, correspond to the x and y values of the unit circle (x2 + y2 = 1), there are hyperbolic functions, hyperbolic cosine (cosh) and hyperbolic sine (sinh), which correspond to the x and y values of the right side of the unit hyperbola (x2 − y2 = 1). y y -1 1 x x π π 3π 2π π π 3π 2π 2 2 2 2 -1 -1 cos x sin x y y 6 2 4 x 2 -2 2 • (0; 1) -2 x -2 2 cosh x sinh x Hyperbolic Angle: Just like how the argument for the trigonometric functions is an angle, the argument for the hyperbolic functions is something called a hyperbolic angle. Instead of being defined by arc length, the hyperbolic angle is defined by area. If that seems confusing, consider the area of the circular sector of the unit circle. The r2θ equation for the area of a circular sector is Area = 2 , so because the radius of the unit circle is 1, any sector of the unit circle will have an area equal to half of the sector's central θ angle, A = 2 . -
7 TRIGONOMETRY Objectives in This Chapter You Will Learn About • Revision of Solution of Triangles
7 TRIGONOMETRY Objectives In this chapter you will learn about • Revision of solution of triangles. • Solving problems in 2–dimensions and in 3-dimensions by constructing and interpreting models. 1 | P a g e 7.1 Revision In previous grades, we dealt with the types of triangles named according to side and angles. In trigonometry, we are able to find the unknown sides and angles when given the magnitudes of the minimum required number of angles and sides. We looked at two main types of triangles: • Right angled triangles. • Non-right angled triangles Right angled triangles When a right angled triangle is given, the following may be useful: • Pythagoras theorem • Definitions of Trigonometric ratios Non-right angled triangles • Sine rule • Cosine rule • Area rule NOTE: • Not all triangles that you need to solve are right-angled triangles. In grade 11 you learned about rules which can be used to solve non-right-angled triangles. • Theorems/axioms involving lines, triangles, quadrilateral and circles are also useful. 7.1.1 The Sine Rule The sine rule expresses the relationship between the two of the sides of the triangles and two of the angles. In any triangle, a is the side opposite angle A, b is the side opposite angle B and c is the side opposite angle C, then the lengthof any side the length of any other side = the sine of the angle opposite that side the sine of the angle opposite that side That is: 2 | P a g e In ABC, a b c = = sin A sinB sinC This can also be written as: sin A sinB sinC = = a b c The sine rule can be used when the following information about a triangle is given: • Two sides and an angle opposite to one of the two sides • One side and any two angles Worked Example 1 Solve the triangle given below. -
Hyperbolic Geometry
Flavors of Geometry MSRI Publications Volume 31, 1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a ThisworkwassupportedinpartbyTheGeometryCenter,UniversityofMinnesota,anSTC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry. This has not been true of the mathematicians and physicists of our generation. -
Using Differentials to Differentiate Trigonometric and Exponential Functions
Using differentials to differentiate trigonometric and exponential functions Tevian Dray Department of Mathematics Oregon State University Corvallis, OR 97331 [email protected] 3 April 2012 Differentiating a polynomial is easy. To differentiate u2 with respect to u, start by computing d(u2)=(u + du)2 − u2 = 2u du + du2 then dropping the last term, an operation that can be justified in terms of limits. Differential notation, in general, can be regarded as a shorthand for a formal limit argument. Still more informally, one can argue that du is small compared to u, so that the last term can be ignored at the level of approximation needed. After dropping du2 and dividing by du, one obtains the derivative, namely d(u2)/du = 2u. Even if one regards this process as merely a heuristic procedure, it is a good one, as it always gives the correct answer for a polynomial. (Physicists are particularly good at knowing what approximations are appropriate in a given physical context. A physicist might describe du as being much smaller than the scale imposed by the physical situation, but not so small that quantum mechanics matters.) However, this procedure does not suffice for trigonometric functions. For example, we may write d(sin θ) = sin(θ + dθ) − sin θ = sin θ cos(dθ) − 1 + cos θ sin(dθ), ³ ´ 1 but to go further we must know something about sin θ and cos θ for small values of θ. Exponential functions offer a similar challenge, since d(eβ)= eβ+dβ − eβ = eβ(edβ − 1), and again we need additional information, in this case about eβ for small values of β. -
A Geometric Introduction to Spacetime and Special Relativity
A GEOMETRIC INTRODUCTION TO SPACETIME AND SPECIAL RELATIVITY. WILLIAM K. ZIEMER Abstract. A narrative of special relativity meant for graduate students in mathematics or physics. The presentation builds upon the geometry of space- time; not the explicit axioms of Einstein, which are consequences of the geom- etry. 1. Introduction Einstein was deeply intuitive, and used many thought experiments to derive the behavior of relativity. Most introductions to special relativity follow this path; taking the reader down the same road Einstein travelled, using his axioms and modifying Newtonian physics. The problem with this approach is that the reader falls into the same pits that Einstein fell into. There is a large difference in the way Einstein approached relativity in 1905 versus 1912. I will use the 1912 version, a geometric spacetime approach, where the differences between Newtonian physics and relativity are encoded into the geometry of how space and time are modeled. I believe that understanding the differences in the underlying geometries gives a more direct path to understanding relativity. Comparing Newtonian physics with relativity (the physics of Einstein), there is essentially one difference in their basic axioms, but they have far-reaching im- plications in how the theories describe the rules by which the world works. The difference is the treatment of time. The question, \Which is farther away from you: a ball 1 foot away from your hand right now, or a ball that is in your hand 1 minute from now?" has no answer in Newtonian physics, since there is no mechanism for contrasting spatial distance with temporal distance. -
4 Trigonometry and Complex Numbers
4 Trigonometry and Complex Numbers Trigonometry developed from the study of triangles, particularly right triangles, and the relations between the lengths of their sides and the sizes of their angles. The trigono- metric functions that measure the relationships between the sides of similar triangles have far-reaching applications that extend far beyond their use in the study of triangles. Complex numbers were developed, in part, because they complete, in a useful and ele- gant fashion, the study of the solutions of polynomial equations. Complex numbers are useful not only in mathematics, but in the other sciences as well. Trigonometry Most of the trigonometric computations in this chapter use six basic trigonometric func- tions The two fundamental trigonometric functions, sine and cosine, can be de¿ned in terms of the unit circle—the set of points in the Euclidean plane of distance one from the origin. A point on this circle has coordinates +frv w> vlq w,,wherew is a measure (in radians) of the angle at the origin between the positive {-axis and the ray from the ori- gin through the point measured in the counterclockwise direction. The other four basic trigonometric functions can be de¿ned in terms of these two—namely, vlq { 4 wdq { @ vhf { @ frv { frv { frv { 4 frw { @ fvf { @ vlq { vlq { 3 ?w? For 5 , these functions can be found as a ratio of certain sides of a right triangle that has one angle of radian measure w. Trigonometric Functions The symbols used for the six basic trigonometric functions—vlq, frv, wdq, frw, vhf, fvf—are abbreviations for the words cosine, sine, tangent, cotangent, secant, and cose- cant, respectively.You can enter these trigonometric functions and many other functions either from the keyboard in mathematics mode or from the dialog box that drops down when you click or choose Insert + Math Name. -
1 Hyperbolic Geometry
1 Hyperbolic Geometry The purpose of this chapter is to give a bare bones introduction to hyperbolic geometry. Most of material in this chapter can be found in a variety of sources, for example: Alan Beardon’s book, The Geometry of Discrete Groups, • Bill Thurston’s book, The Geometry and Topology of Three Manifolds, • Svetlana Katok’s book, Fuchsian Groups, • John Ratcliffe’s book, Hyperbolic Geometry. • The first 2 sections of this chapter might not look like geometry at all, but they turn out to be very important for the subject. 1.1 Linear Fractional Transformations Suppose that a b A = c d isa2 2 matrix with complex number entries and determinant 1. The set of × these matrices is denoted by SL2(C). In fact, this set forms a group under matrix multiplication. The matrix A defines a complex linear fractional transformation az + b T (z)= . A cz + d We will sometimes omit the word complex from the name, though we will always have in mind a complex linear fractional transformation when we say linear fractional transformation. Such maps are also called M¨obius transfor- mations, Note that the denominator of T (z) is nonzero as long as z = d/c. It is A 6 − convenient to introduce an extra point and define TA( d/c) = . This definition is a natural one because of the∞ limit − ∞ lim TA(z) = . z d/c →− | | ∞ 1 The determinant condition guarantees that a( d/c)+ b = 0, which explains − 6 why the above limit works. We define TA( ) = a/c. This makes sense because of the limit ∞ lim TA(z)= a/c.