DOCSLIB.ORG

A New Approach to Differential Geometry Using Clifford's Geometric

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A New Approach to Differential Geometry Using Clifford's Geometric John Snygg A New Approach to Differential Geometry Using Clifford’s Geometric Algebra John Snygg 433 Prospect St. East Orange, NJ 07017 [email protected] ISBN 978-0-8176-8282-8 e-ISBN 978-0-8176-8283-5 DOI 10.1007/978-0-8176-8283-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011940217 Mathematics Subject Classification (2010): 11E88, 15A66, 53-XX, 53-03 c Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) To Pertti Lounesto Preface This book was written with the intention that it be used as a text for an undergraduate course. The end result is not only suitable for an undergraduate course but also ideal for a masters level course directed toward future high school math teachers. It is also appropriate for anyone who wants to acquaint himself or herself with the usefulness of Clifford algebra. In that context, instructors teaching Ph.D. students may want to use it as a source book. Most introductory books on differential geometry are restricted to three dimen- sions. I use the notation used by geometers in n-dimensions. Admittedly, most of my examples are in three dimensions. However in Chap. 3, I present some aspects of the four-dimensional theory of special relativity. One can present some fun aspects of special relativity without mention of such concepts as “force”, “momentum”, and “energy.” For example, adding speeds possibly near the speed of light results in a sum that is always less than the speed of light. The capstone topic for this book is Einstein’s general theory of relativity. Here again, knowledge of Newtonian physics is not a prerequisite. Because of the geometric nature of Einstein’s theory, some interesting aspects can be presented without knowledge of Newtonian physics. In particular, I discuss the possibility of twins aging at different rates, the precession of Mercury, and the bending of light rays passing near the Sun or some other massive body. The only topic that does require some knowledge of Newtonian physics is Huygen’s isochronous pendulum clock, and the relevant section should be consid- ered optional. My strategy for writing this book had three steps: Step 1: Steal as many good pedagogical ideas from as many authors as possible. Step 2: Improve on them if I could. Step 3: Choose topics that are fun for me and fit together in a coherent manner. Frequently I was able to improve on the presentation of others using Clifford algebra. The selection of this book may introduce a hurdle for some instructors. It is likely they will have to learn something new – Clifford algebra. Paradoxically, vii viii Preface the use of Clifford algebra will make differential geometry more accessible to students who have completed a course in linear algebra. That is because in this book, Clifford algebra replaces the more complicated and less powerful formalism of differential forms. Anyone who is familiar with the concept of non-commutative matrix multiplication will find it easy to master the Clifford algebra presented in this text. Using Clifford algebra, it becomes unnecessary to discuss mappings back and forth between the space of tangent vectors and the space of differential forms. With Clifford algebra, everything takes place in one space. The fact that Clifford algebra (otherwise known as “geometric algebra”) is not deeply embedded in our current curriculum is an accident of history. William Kingdon Clifford wrote two papers on the topic shortly before his early death in 1879 at the age of 33. Although Clifford was recognized worldwide as one of England’s most distinguished mathematicians, he chose to have the first paper published in what must have been a very obscure journal at the time. Quite possibly it was a gesture of support for the efforts of James Joseph Sylvester to establish the first American graduate program in mathematics at Johns Hopkins University. As part of his endeavors, Sylvester founded the American Journal of Mathematics and Clifford’s first paper on what is now known as Clifford algebra appeared in the very first volume of that journal. The second paper was published after his death in unfinished form as part of his collected papers. Both of these papers were ignored and soon forgotten. As late as 1923, math historian David Eugene Smith discussed Clifford’s achievements without mentioning “geometric algebra” (Smith, David Eugene 1923). In 1928, P.A.M. Dirac reinvented Clifford algebra to formulate his equation for the electron. This equation enabled him to predict the discovery of the positron in 1931. In 1946 and 1958, Marcel Riesz published some results on Clifford algebra that stimulated David Hestenes to investigate the subject. In 1966, David Hestenes published a thin volume entitled Space-time Algebra (Hestenes 1966). And 18 years later, with his student Garret Sobezyk, he wrote a more extensive book entitled Clifford Algebra to Geometric Calculus – A Unified Language for Mathematics and Physics (Hestenes and Sobczyk 1984). Since then, extensive research has been carried out in Clifford algebra with a multitude of applications. Had Clifford lived longer, “geometric algebra” would probably have become mainstream mathematics near the beginning of the twentieth century. In the decades following Clifford’s death, a battle broke out between those who wanted to use quaternions to do physics and geometry and those who wanted to use vectors. Quaternions were superior for dealing with rotations, but they are useless in dimensions higher than three or four without grafting on some extra structure. Eventually vectors won out. Since the structure of both quaternions and vectors are contained in the formalism of Clifford algebra, the debate would have taken a different direction had Clifford lived longer. While alive, Clifford was an articulate spokesman and his writing for popular consumption still gets published from time to time. Had Clifford participated in the quaternion–vector debate, “geometric algebra” would have received more serious consideration. Preface ix The advantage that quaternions have for dealing with rotations in three dimensions can be generalized to higher dimensions using Clifford algebra. This is important for dealing with the most important feature of a surface in any dimension – namely its curvature. Suppose you were able to walk from the North Pole along a curve of constant longitude to the equator, then walk east along the equator for 37ı and finally return to the North Pole along another curve of constant longitude. In addition, suppose at the start of your trek, you picked up a spear, pointed it in the south direction and then avoided any rotation of the spear with respect to the surface of the earth during your long journey. If you were careful, the spear would remain pointed south during the entire trip. However, on your return to the North Pole, you would discover that your spear had undergone a 37ı rotation from its initial position. This rotation is a measure of the curvature of the Earth’s surface. The components of the Riemann tensor, used to measure curvature, are somewhat abstract in the usual formalism. Using Clifford algebra, the components of the Riemann tensor can be interpreted as components of an infinitesimal rotation operator that indicates what happens when a vector is “parallel transported” around an infinitesimal loop in a curved space. In many courses on differential geometry, the Gauss–Bonnet Formula is the cap- stone result. Exploiting the power of Clifford algebra, a proof appears slightly less than halfway through this book. If optional intervening historical digressions were eliminated, the proof of the Gauss–Bonnet Formula would appear on approximately p. 115. This should leave time to cover other topics that interest the instructor or the instructor’s students. I hope that instructors endeavor to cover enough of the theory of general relativity to discuss the precession of Mercury. The general theory of relativity is essentially geometric in nature. Whatever topics are chosen, I hope people have fun. Acknowledgments An author would like to have the illusion that he or she has accomplished something without the aid of others. However, a little introspection, at least in my case, makes such an illusion evaporate. To begin with as I prepared to write various sections of this book, I read and reread the relevant sections in Dirk J. Struik’s Lectures in Classical Differential Geometry 2nd Ed. (1988), and Barrett O’Neill’s Differential Geometry 2nd Ed. (1997). These two texts are restricted to three dimensions. For higher dimensions, I devoted many hours to Johan C.H. Gerretsen’s Lectures on Tensor Calculus and Differential Geometry (1962). For my chapter on non- Euclidean geometry, I found a lot of very useful ideas in Geometry 1st Ed. (1999) by David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray. From time to time I used other geometry books, which appear in my bibliography. In addition, I have benefited from those who have developed the tools of Clifford algebra.
Load more

Recommended publications
  • Philosophy and the Physicists Preview
    L. Susan Stebbing Philosophy and the Physicists Preview il glifo ebooks ISBN: 9788897527466 First Edition: December 2018 (A) Copyright © il glifo, December 2018 www.ilglifo.it 1 Contents FOREWORD FROM THE EDITOR Note to the 2018 electronic edition PHILOSOPHY AND THE PHYSICISTS Original Title Page PREFACE NOTE PART I - THE ALARMING ASTRONOMERS Chapter I - The Common Reader and the Popularizing Scientist Chapter II - THE ESCAPE OF SIR JAMES JEANS PART II - THE PHYSICIST AND THE WORLD Chapter III - ‘FURNITURE OF THE EARTH’ Chapter IV - ‘THE SYMBOLIC WORLD OF PHYSICS’ Chapter V - THE DESCENT TO THE INSCRUTABLE Chapter VI - CONSEQUENCES OF SCRUTINIZING THE INSCRUTABLE PART III - CAUSALITY AND HUMAN FREEDOM Chapter VII - THE NINETEENTH-CENTURY NIGHTMARE Chapter VIII - THE REJECTION OF PHYSICAL DETERMINISM Chapter IX - REACTIONS AND CONSEQUENCES Chapter X - HUMAN FREEDOM AND RESPONSIBILITY PART IV - THE CHANGED OUTLOOK Chapter XI - ENTROPY AND BECOMING Chapter XII - INTERPRETATIONS BIBLIOGRAPHY INDEX BACK COVER Susan Stebbing 2 Foreword from the Editor In 1937 Susan Stebbing published Philosophy and the Physicists , an intense and difficult essay, in reaction to reading the works written for the general public by two physicists then at the center of attention in England and the world, James Jeans (1877-1946) and Arthur Eddington (1882- 1944). The latter, as is known, in 1919 had announced to the Royal Society the astronomical observations that were then considered experimental confirmations of the general relativity of Einstein, and who by that episode had managed to trigger the transformation of general relativity into a component of the mass and non-mass imaginary of the twentieth century.
    [Show full text]
  • The 29 of May Or Sir Arthur and The
    International Journal of Management and Applied Science, ISSN: 2394-7926 Volume-4, Issue-4, Apr.-2018 http://iraj.in THE 29TH OF MAY OR SIR ARTHUR AND THE BENDING LIGHT (TEACHING SCIENCE AND LITERATURE JOINTLY) MICHAEL KATZ Faculty of Education, Haifa University, Mount Carmel, Haifa, Israel E-mail: [email protected] Abstract - In this paper I present an instance illustrating a program of joint teaching of science and literature. The instance sketched here rests on recognition that at the age when children read books like Kästner's "The 35th of May" they can already taste notions of Relativity Theory through the story of Eddington's expedition to Principe Island. Eddington's story is briefly recounted here with references to Kästner's book. The relationships thus exhibited between science and fiction, fantasy and reality, theory and actuality, humor and earnestness, should help evoke school age children's interest in science and literature at early stages in their studies' endeavor. Keywords - Bending Light, Fantasy, Gravitation, Relativity, Science and Literature. I. INTRODUCTION South Seas" – a story of adventure and humor by Erich Kästner, the well known German author of The idea that teaching science and humanities in more than a few dearly loved novels, mostly for primary and secondary schools needn't necessarily be children and adolescents. The book was first seen as disjoint objectives is prevalent in the thought published in German in 1931 and in English shortly and practice of more than a few scholars and teachers after. in recent years. Carole Cox [1], for example, unfolds forty strategies of literature based teaching of various Twelve years earlier, in 1919, another journey topics in arts and sciences.
    [Show full text]
  • Revised Theory of Gravity Doesn't Predict a Big Bang 12 July 2010, by Lisa Zyga
    Revised theory of gravity doesn't predict a Big Bang 12 July 2010, By Lisa Zyga decades he became more interested in finding a theory to unify gravity and quantum mechanics - a task that is still being studied today. In 1924, Eddington proposed a new “gravitational action” as an alternative to the Einstein-Hilbert action, which could serve as an alternative starting point to general relativity. In astrophysics, a gravitational action is the mechanism that describes how gravity can emerge from space-time being curved by matter and energy. However, Eddington’s theory of gravity only worked for empty space and didn’t Illustration: Time Line of the Universe Credit: include any source of energy such as matter, NASA/WMAP making it an incomplete theory. Since Eddington’s proposal, scientists have attempted various ways of including matter into the (PhysOrg.com) -- The Big Bang theory has formed theory, although they have run into problems. In the basis of our understanding of the universe's this study, Banados and Ferreira have tried a new origins since it was first proposed in 1927 by way to extend the theory to include matter by using Georges Lemaitre. And for good reason: the theory a gravitational action called the Born-Infeld action. is supported by scientists' latest observations and experiments, and is based on Einstein's widely In their analysis, the scientists found that a key accepted theory of general relativity. But scientists characteristic of Eddington’s revised theory of are always on the lookout for any evidence that gravity is that it reproduces Einstein gravity might suggest an alternative to the Big Bang.
    [Show full text]
  • Eddington, Lemaître and the Hypothesis of Cosmic Expansion in 1927
    Eddington, Lemaître and the hypothesis of cosmic expansion in 1927 Cormac O’Raifeartaigh School of Science and Computing, Waterford Institute of Technology, Cork Road, Waterford, Ireland Author for correspondence: [email protected] 1 1. Introduction Arthur Stanley Eddington was one of the leading astronomers and theorists of his generation (Smart et al. 1945; McCrea 1982; Chandrasekhar, S. 1983). An early and important proponent of the general theory of relativity, his 1918 ‘Report on the Relativity Theory of Gravitation’ for the Physical Society (Eddington 1918) provided an early authoritative exposition of the subject for English-speaking physicists (Vibert Douglas 1956 p42; Chandrasekhar 1983 p24). He played a leading role in the eclipse observations of 1919 that offered early astronomical evidence in support of the theory (Vibert Douglas 1956 pp 39-41; Chandrasekhar 1983 pp 24- 29; Kennefick, this volume) while his book ‘Space, Time and Gravitation’ (Eddington 1920) was one of the first popular treatises on general relativity for an English-speaking audience. In addition, Eddington’s textbook ‘The Mathematical Theory of Relativity’ (Eddington 1923) became a classic reference for English-speaking physicists with an interest in relativity (McCrea 1982; Chandrasekhar 1983 p32). Indeed, the book provided one of the first textbook accounts of relativistic models of the cosmos, complete with a discussion of possible links to one of the greatest astronomical puzzles of the age, the redshifts of the spiral nebulae (Eddington 1923 pp 155-170). It seems therefore quite surprising that, when Eddington’s former student Georges Lemaître suggested in a seminal article of 1927 (Lemaître 1927) that a universe of expanding radius could be derived from general relativity, and that the phenomenon could provide a natural explanation for the redshifts of the spiral nebulae, Eddington (and others) paid no attention.
    [Show full text]
  • 90 Years on – the 1919 Eclipse Expedition at Príncipe
    ELLIS ET AL.: 1919 ECLIPSE REVISITED ELLIS ET AL.: 1919 ECLIPSE REVISITED 90 years on – the 1919 ecli pse expedition at Príncipe Richard Ellis, Pedro G Ferreira, Richard Massey and Gisa Weszkalnys return to Príncipe in the International Year of Astronomy to celebrate the 1919 RAS expedition led by Sir Arthur Eddington. he first experiment to observationally stars near the Sun are confirm Einstein’s General Theory of not normally visible, TRelativity was carried out in May 1919, they would be during on a Royal Astronomical Society expedition to a total solar eclipse. observe a total solar eclipse. Sir Arthur Edding- Einstein attempted ton travelled to Príncipe, a small island off the to convince observers west coast of Africa, and sent another team to to measure the gravi- Sobral, Brazil, from where the eclipse would tational deflection. He also be visible. This year, in a new RAS-funded found an enthusiastic expedition organized for the International Year colleague in Erwin of Astronomy, we returned to Príncipe to cel- Findlay-Freundlich, 1: The old plantation buildings at Roça Sundy, Eddington’s base in 1919. ebrate this key experiment that shook the foun- whose expedition to dations of 20th-century science. Russia in August 1914 was scuppered by the America and Africa, but it was important to Since 1687, Sir Isaac Newton’s law of gravity outbreak of war in the very month of the solar be as close as possible to the sub-solar point, had been the workhorse of celestial mechanics. eclipse: as a German national in Russia, he where the Earth’s rotation keeps observers in Newtonian gravity could be used to explain the was arrested.
    [Show full text]
  • A Short and Transparent Derivation of the Lorentz-Einstein Transformations
    A brief and transparent derivation of the Lorentz-Einstein transformations via thought experiments Bernhard Rothenstein1), Stefan Popescu 2) and George J. Spix 3) 1) Politehnica University of Timisoara, Physics Department, Timisoara, Romania 2) Siemens AG, Erlangen, Germany 3) BSEE Illinois Institute of Technology, USA Abstract. Starting with a thought experiment proposed by Kard10, which derives the formula that accounts for the relativistic effect of length contraction, we present a “two line” transparent derivation of the Lorentz-Einstein transformations for the space-time coordinates of the same event. Our derivation make uses of Einstein’s clock synchronization procedure. 1. Introduction Authors make a merit of the fact that they derive the basic formulas of special relativity without using the Lorentz-Einstein transformations (LET). Thought experiments play an important part in their derivations. Some of these approaches are devoted to the derivation of a given formula whereas others present a chain of derivations for the formulas of relativistic kinematics in a given succession. Two anthological papers1,2 present a “two line” derivation of the formula that accounts for the time dilation using as relativistic ingredients the invariant and finite light speed in free space c and the invariance of distances measured perpendicular to the direction of relative motion of the inertial reference frame from where the same “light clock” is observed. Many textbooks present a more elaborated derivation of the time dilation formula using a light clock that consists of two parallel mirrors located at a given distance apart from each other and a light signal that bounces between when observing it from two inertial reference frames in relative motion.3,4 The derivation of the time dilation formula is intimately related to the formula that accounts for the length contraction derived also without using the LET.
    [Show full text]
  • The Eclipse to Confirm the General Theory of Relativity - Openmind Search Private Area
    8/9/2015 The Eclipse to Confirm the General Theory of Relativity - OpenMind Search Private area Sharing knowledge for a better future Home The Eclipse to Confirm the General Theory of Relativity The Eclipse to Confirm the General Theory of Relativity Share 20 July 2015 Physics, Science 1 Sign in or register to rate this publication One of the milestones of the science of light commemorated during this International Year of Light and Light-based Technologies is “the embedding of light in cosmology through general relativity in 1915,” that is, the celebration of the centenary of Albert Einstein’s general theory of relativity. As Adolfo de Azcárraga, president of the Spanish Royal Society of Physics (RSEF), points out in his book titled Albert Einstein, His Science and His Time, Einstein’s theory contained a spectacular prediction: “light also possessed ‘weight’, i.e., it should be attracted and deflected by celestial bodies.” Since the equivalence between acceleration and gravity extends to electromagnetic phenomena and light is an electromagnetic wave, light rays should bend in the presence of a gravitational field. Einstein had already realized that the only way to experimentally verify this theoretical prediction was for a total solar eclipse to take place since this would make it possible to photograph a star near the Sun when observed from Earth without the presence of strong sunlight. Well, on May 29, 1919 there would be a solar eclipse, which would be total on some parts of the Earth’s surface and would make it possible to verify that light rays are bent by gravity.
    [Show full text]
  • Massachusetts Institute of Technology Midterm
    Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Special Relativity January 18, 2005 7:30–9:30 pm Midterm Instructions • This exam contains SIX problems – pace yourself accordingly! • You have two hours for this test. Papers will be picked up at 9:30 pm sharp. • The exam is scored on a basis of 100 points. • Please do ALL your work in the white booklets that have been passed out. There are extra booklets available if you need a second. • Please remember to put your name on the front of your exam booklet. • If you have any questions please feel free to ask the proctor(s). 1 2 Information Lorentz transformation (along the x­axis) and its inverse x� = γ(x − βct) x = γ(x� + βct�) y� = y y = y� z� = z z = z� ct� = γ(ct − βx) ct = γ(ct� + βx�) � where β = v/c, and γ = 1/ 1 − β2. Velocity addition (relative motion along the x­axis): � ux − v ux = 2 1 − uxv/c � uy uy = 2 γ(1 − uxv/c ) � uz uz = 2 γ(1 − uxv/c ) Doppler shift Longitudinal � 1 + β ν = ν 1 − β 0 Quadratic equation: ax2 + bx + c = 0 1 � � � x = −b ± b2 − 4ac 2a Binomial expansion: b(b − 1) (1 + a)b = 1 + ba + a 2 + . 2 3 Problem 1 [30 points] Short Answer (a) [4 points] State the postulates upon which Einstein based Special Relativity. (b) [5 points] Outline a derviation of the Lorentz transformation, describing each step in no more than one sentence and omitting all algebra. (c) [2 points] Define proper length and proper time.
    [Show full text]
  • Minkowski's Proper Time and the Status of the Clock Hypothesis
    Minkowski’s Proper Time and the Status of the Clock Hypothesis1 1 I am grateful to Stephen Lyle, Vesselin Petkov, and Graham Nerlich for their comments on the penultimate draft; any remaining infelicities or confusions are mine alone. Minkowski’s Proper Time and the Clock Hypothesis Discussion of Hermann Minkowski’s mathematical reformulation of Einstein’s Special Theory of Relativity as a four-dimensional theory usually centres on the ontology of spacetime as a whole, on whether his hypothesis of the absolute world is an original contribution showing that spacetime is the fundamental entity, or whether his whole reformulation is a mere mathematical compendium loquendi. I shall not be adding to that debate here. Instead what I wish to contend is that Minkowski’s most profound and original contribution in his classic paper of 100 years ago lies in his introduction or discovery of the notion of proper time.2 This, I argue, is a physical quantity that neither Einstein nor anyone else before him had anticipated, and whose significance and novelty, extending beyond the confines of the special theory, has become appreciated only gradually and incompletely. A sign of this is the persistence of several confusions surrounding the concept, especially in matters relating to acceleration. In this paper I attempt to untangle these confusions and clarify the importance of Minkowski’s profound contribution to the ontology of modern physics. I shall be looking at three such matters in this paper: 1. The conflation of proper time with the time co-ordinate as measured in a system’s own rest frame (proper frame), and the analogy with proper length.
    [Show full text]
  • Length Contraction the Proper Length of an Object Is Longest in the Reference Frame in Which It Is at Rest
    Newton’s Principia in 1687. Galilean Relativity Velocities add: V= U +V' V’= 25m/s V = 45m/s U = 20 m/s What if instead of a ball, it is a light wave? Do velocities add according to Galilean Relativity? Clocks slow down and rulers shrink in order to keep the speed of light the same for all observers! Time is Relative! Space is Relative! Only the SPEED OF LIGHT is Absolute! A Problem with Electrodynamics The force on a moving charge depends on the Frame. Charge Rest Frame Wire Rest Frame (moving with charge) (moving with wire) F = 0 FqvB= sinθ Einstein realized this inconsistency and could have chosen either: •Keep Maxwell's Laws of Electromagnetism, and abandon Galileo's Spacetime •or, keep Galileo's Space-time, and abandon the Maxwell Laws. On the Electrodynamics of Moving Bodies 1905 EinsteinEinstein’’ss PrinciplePrinciple ofof RelativityRelativity • Maxwell’s equations are true in all inertial reference frames. • Maxwell’s equations predict that electromagnetic waves, including light, travel at speed c = 3.00 × 108 m/s =300,000km/s = 300m/µs . • Therefore, light travels at speed c in all inertial reference frames. Every experiment has found that light travels at 3.00 × 108 m/s in every inertial reference frame, regardless of how the reference frames are moving with respect to each other. Special Theory: Inertial Frames: Frames do not accelerate relative to eachother; Flat Spacetime – No Gravity They are moving on inertia alone. General Theory: Noninertial Frames: Frames accelerate: Curved Spacetime, Gravity & acceleration. Albert Einstein 1916 The General Theory of Relativity Postulates of Special Relativity 1905 cxmsms==3.0 108 / 300 / μ 1.
    [Show full text]
  • Physical Time and Physical Space in General Relativity Richard J
    Physical time and physical space in general relativity Richard J. Cook Department of Physics, U.S. Air Force Academy, Colorado Springs, Colorado 80840 ͑Received 13 February 2003; accepted 18 July 2003͒ This paper comments on the physical meaning of the line element in general relativity. We emphasize that, generally speaking, physical spatial and temporal coordinates ͑those with direct metrical significance͒ exist only in the immediate neighborhood of a given observer, and that the physical coordinates in different reference frames are related by Lorentz transformations ͑as in special relativity͒ even though those frames are accelerating or exist in strong gravitational fields. ͓DOI: 10.1119/1.1607338͔ ‘‘Now it came to me:...the independence of the this case, the distance between FOs changes with time even gravitational acceleration from the nature of the though each FO is ‘‘at rest’’ (r,␪,␾ϭconstant) in these co- falling substance, may be expressed as follows: ordinates. In a gravitational field (of small spatial exten- sion) things behave as they do in space free of III. PHYSICAL SPACE gravitation,... This happened in 1908. Why were another seven years required for the construction What is the distance dᐉ between neighboring FOs sepa- of the general theory of relativity? The main rea- rated by coordinate displacement dxi when the line element son lies in the fact that it is not so easy to free has the general form oneself from the idea that coordinates must have 2 ␣ ␤ an immediate metrical meaning.’’ 1 Albert Ein- ds ϭg␣␤ dx dx ? ͑1͒ stein Einstein’s answer is simple.3 Let a light ͑or radar͒ pulse be transmitted from the first FO ͑observer A) to the second FO I.
    [Show full text]
  • Section 5 Special Relativity
    North Berwick High School Higher Physics Department of Physics Unit 1 Our Dynamic Universe Section 5 Special Relativity Section 5 Special Relativity Note Making Make a dictionary with the meanings of any new words. Special Relativity 1. What is meant by a frame of reference? 2. Write a short note on the history leading up to Einstein's theories (just include the main dates). The principles 1. State Einstein's starting principles. Time dilation 1. State what is meant by time dilation. 2. Copy the time dilation formula and take note of the Lorentz factor. 3. Copy the example. 4. Explain why we do not notice these events in everyday life. 5. Explain how muons are able to be detected at the surface of the Earth. Length contraction (also known as Lorentz contraction) 1. Copy the length contraction formula. Section 5 Special Relativity Contents Content Statements................................................. 1 Special Relativity........................................................ 2 2 + 2 = 3................................................................... 2 History..................................................................... 2 The principles.......................................................... 4 Time Dilation........................................................... 4 Why do we not notice these time differences in everyday life?.................................... 7 Length contraction................................................... 8 Ladder paradox........................................................ 9 Section 5:
    [Show full text]