DOCSLIB.ORG

Getting the Measure of Einstein's Space and Time

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Getting the Measure of Einstein's Space and Time UNIVERSITY OF NEVADA, LAS VEGAS Getting the Measure of Einstein’s Space and Time An Introduction to Special Relativity Len Zane 12/12/2014 The space and time introduced by Albert Einstein in 1905 is explained by examining a series of simple thought or “gedanken” experiments. The development makes extensive use of spacetime diagrams to help readers appreciate the full extent of these changes. Table of Contents Introduction ................................................................................................................................................ 4 Chapter One: Space and Time Before 1905 ............................................................................................ 6 Speed and Velocity ................................................................................................................................ 6 The Experimenters are Introduced ...................................................................................................... 6 The First Set of Experiments: Earth-Based ........................................................................................ 8 The Second Set of Experiments: Bus-Based ..................................................................................... 11 Bev and Anne Carefully Observe the Experiment Done on the Bus ............................................ 12 The Addition of Velocities Formula .................................................................................................. 15 A Picture is Worth a Thousand Words ............................................................................................. 17 The Galilean Transformation Equations ........................................................................................... 22 Summary of Chapter One ................................................................................................................... 26 The Principle of Relativity and the Isotropy of Space................................................................. 26 Space and Time pre-1905 ................................................................................................................ 27 Chapter Two: The Speed of Light ......................................................................................................... 28 Chapter Three: Space and Time After 1905 ......................................................................................... 31 A New Set of Experiments are Proposed.......................................................................................... 31 New Stop Watches are distributed .................................................................................................... 31 Anne and Bev Measure the Speed of Light ...................................................................................... 31 The Super Bus Rolls into the Story .................................................................................................... 34 Anne and Bev Observe the Light Flash on the Bus: I ..................................................................... 36 Anne and Bev Observe the Light Flash on the Bus: II .................................................................... 41 The Effect of Motion on Space ............................................................................................................ 48 The Effect of Motion on Time ............................................................................................................. 51 Light Moves through Space and Time in a Very Strange Way ..................................................... 54 Chapter Four: Generalizing the Observations .................................................................................... 56 Defining the New Problem ................................................................................................................. 57 Finding the Shrinkage Factor ............................................................................................................. 58 The Ticking Rate of Moving Clocks .................................................................................................. 62 1 Quantifying the Disagreement over Synchronization .................................................................... 63 Testing the Theory ............................................................................................................................... 66 A Question about Synchronization ................................................................................................... 68 Global Positioning System (GPS) ....................................................................................................... 70 Chapter Five: The Relationship between Bus and Earth Observers ................................................ 72 Combining Spacetime Graphs ........................................................................................................... 72 Bev Generalizes Chuck’s Spacetime Diagram ................................................................................. 79 A Simple Example for Dean ............................................................................................................... 83 The Revised Equation for the Addition of Velocities ..................................................................... 84 Chapter Six: Chuck Suggests an Experiment and Anne has a Dream ............................................ 89 Experiment I.......................................................................................................................................... 90 Experiment II ........................................................................................................................................ 93 Answers for Experiment I ................................................................................................................... 97 Answers for Experiment II ............................................................................................................... 100 Anne has a Dream about a Superluminal Pigeon ......................................................................... 104 Chapter Seven: The Bus and the Garage ........................................................................................... 108 Anne’s Proposal ................................................................................................................................. 115 The Collision between Theory and Reality .................................................................................... 116 R.I.P. Super Bus .................................................................................................................................. 118 Chapter Eight: The Solar System and Beyond .................................................................................. 120 Synchronizing Watches Separated by Large Distances ................................................................ 121 Chapter Nine: Chuck and Dean Travel to Alpha Centauri ............................................................. 123 May 1, Thirteen Years Later.............................................................................................................. 124 Using the Lorentz Equations ............................................................................................................ 130 Looking at a Moving Watch – The Doppler Effect ........................................................................ 132 Chapter Ten: Space Travel ................................................................................................................... 136 The Equivalence Principle ................................................................................................................ 136 Designing the Space Adventure ...................................................................................................... 139 The Results of Chuck’s Calculations ............................................................................................... 143 2 How does a Rocket Accelerate? ....................................................................................................... 148 Bonus Topic: A Peek at General Relativity ..................................................................................... 152 Chapter Eleven: Does it Really Happen? ........................................................................................... 154 The Far-Away Observer (FAO) ........................................................................................................ 154 A Dynamic Explanation for the Behavior of Watches and Rulers .............................................. 158 A Model for a Hydrogen Atom ........................................................................................................ 159 Chapter Twelve: E = mc2 ...................................................................................................................... 162 Appendix A: Graphing ......................................................................................................................... 169 Appendix B: Scientific Notation.......................................................................................................... 172 Appendix C: The Light Clock .............................................................................................................. 173 Appendix D: The Gravitational Clock Effect ...................................................................................
Load more

Recommended publications
  • 26-2 Spacetime and the Spacetime Interval We Usually Think of Time and Space As Being Quite Different from One Another
    Answer to Essential Question 26.1: (a) The obvious answer is that you are at rest. However, the question really only makes sense when we ask what the speed is measured with respect to. Typically, we measure our speed with respect to the Earth’s surface. If you answer this question while traveling on a plane, for instance, you might say that your speed is 500 km/h. Even then, however, you would be justified in saying that your speed is zero, because you are probably at rest with respect to the plane. (b) Your speed with respect to a point on the Earth’s axis depends on your latitude. At the latitude of New York City (40.8° north) , for instance, you travel in a circular path of radius equal to the radius of the Earth (6380 km) multiplied by the cosine of the latitude, which is 4830 km. You travel once around this circle in 24 hours, for a speed of 350 m/s (at a latitude of 40.8° north, at least). (c) The radius of the Earth’s orbit is 150 million km. The Earth travels once around this orbit in a year, corresponding to an orbital speed of 3 ! 104 m/s. This sounds like a high speed, but it is too small to see an appreciable effect from relativity. 26-2 Spacetime and the Spacetime Interval We usually think of time and space as being quite different from one another. In relativity, however, we link time and space by giving them the same units, drawing what are called spacetime diagrams, and plotting trajectories of objects through spacetime.
    [Show full text]
  • A Producer's Handbook
    DEVELOPMENT AND OTHER CHALLENGES A PRODUCER’S HANDBOOK by Kathy Avrich-Johnson Edited by Daphne Park Rehdner Summer 2002 Introduction and Disclaimer This handbook addresses business issues and considerations related to certain aspects of the production process, namely development and the acquisition of rights, producer relationships and low budget production. There is no neat title that encompasses these topics but what ties them together is that they are all areas that present particular challenges to emerging producers. In the course of researching this book, the issues that came up repeatedly are those that arise at the earlier stages of the production process or at the earlier stages of the producer’s career. If not properly addressed these will be certain to bite you in the end. There is more discussion of various considerations than in Canadian Production Finance: A Producer’s Handbook due to the nature of the topics. I have sought not to replicate any of the material covered in that book. What I have sought to provide is practical guidance through some tricky territory. There are often as many different agreements and approaches to many of the topics discussed as there are producers and no two productions are the same. The content of this handbook is designed for informational purposes only. It is by no means a comprehensive statement of available options, information, resources or alternatives related to Canadian development and production. The content does not purport to provide legal or accounting advice and must not be construed as doing so. The information contained in this handbook is not intended to substitute for informed, specific professional advice.
    [Show full text]
  • Thermodynamics of Spacetime: the Einstein Equation of State
    gr-qc/9504004 UMDGR-95-114 Thermodynamics of Spacetime: The Einstein Equation of State Ted Jacobson Department of Physics, University of Maryland College Park, MD 20742-4111, USA [email protected] Abstract The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with δQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air. arXiv:gr-qc/9504004v2 6 Jun 1995 The four laws of black hole mechanics, which are analogous to those of thermodynamics, were originally derived from the classical Einstein equation[1]. With the discovery of the quantum Hawking radiation[2], it became clear that the analogy is in fact an identity. How did classical General Relativity know that horizon area would turn out to be a form of entropy, and that surface gravity is a temperature? In this letter I will answer that question by turning the logic around and deriving the Einstein equation from the propor- tionality of entropy and horizon area together with the fundamental relation δQ = T dS connecting heat Q, entropy S, and temperature T .
    [Show full text]
  • Chapter 5 the Relativistic Point Particle
    Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story.
    [Show full text]
  • ELEVATION Songs from Afar
    ELEVATION Songs from Afar Transylvanian expat pianist & composer Lucian Ban and ELEVATION celebrate their 2 n d album for Sunnyside records - Songs from Afar . Released in 2016 the album immediately garnered rave reviews and honors – DOWNBEAT BEST ALBUM OF THE YEAR , a 5* review in DB march 2016 issue, ALL ABOUT JAZZ calls it “a triumph of emotional and musical communication” and New York City Jazz Record talks of an “Inspiring and touching journey, that seamlessly blends the traditions of jazz and folk songs”. ”Alluring timelessness . a strong life-force that seems to flow through this music . like many of the great masters, pianist Lucian Ban makes personal art that feels universal” 5* STARS! “Songs From Afar is a triumph of emotional and musical communication, and is not to be missed” Streams of influence from the past and cultural identities sometimes merge with one’s quest for forms of expression. Songs From Afar represents two traditions, two musical worlds, and many musical elements coming together to fashion a unique identity that truly spans continents and styles. "Songs from afar is very personal for me because the album is intimately tied to my Romanian cultural heritage and to the jazz influences that help me find out more about where I come from – and where I’m going. It's not only the ancient Transylvanian folk songs that we approach in this recording, it's also how the other pieces and improvisations reflect the constant search for musical meaning" notes Transylvanian expat pianist and composer Lucian Ban talking about his 2 n d album with Elevation quartet in his 3 r d appearance for Sunnyside.
    [Show full text]
  • Recorded Jazz in the 20Th Century
    Recorded Jazz in the 20th Century: A (Haphazard and Woefully Incomplete) Consumer Guide by Tom Hull Copyright © 2016 Tom Hull - 2 Table of Contents Introduction................................................................................................................................................1 Individuals..................................................................................................................................................2 Groups....................................................................................................................................................121 Introduction - 1 Introduction write something here Work and Release Notes write some more here Acknowledgments Some of this is already written above: Robert Christgau, Chuck Eddy, Rob Harvilla, Michael Tatum. Add a blanket thanks to all of the many publicists and musicians who sent me CDs. End with Laura Tillem, of course. Individuals - 2 Individuals Ahmed Abdul-Malik Ahmed Abdul-Malik: Jazz Sahara (1958, OJC) Originally Sam Gill, an American but with roots in Sudan, he played bass with Monk but mostly plays oud on this date. Middle-eastern rhythm and tone, topped with the irrepressible Johnny Griffin on tenor sax. An interesting piece of hybrid music. [+] John Abercrombie John Abercrombie: Animato (1989, ECM -90) Mild mannered guitar record, with Vince Mendoza writing most of the pieces and playing synthesizer, while Jon Christensen adds some percussion. [+] John Abercrombie/Jarek Smietana: Speak Easy (1999, PAO) Smietana
    [Show full text]
  • Lecture 17 Relativity A2020 Prof. Tom Megeath What Are the Major Ideas
    4/1/10 Lecture 17 Relativity What are the major ideas of A2020 Prof. Tom Megeath special relativity? Einstein in 1921 (born 1879 - died 1955) Einstein’s Theories of Relativity Key Ideas of Special Relativity • Special Theory of Relativity (1905) • No material object can travel faster than light – Usual notions of space and time must be • If you observe something moving near light speed: revised for speeds approaching light speed (c) – Its time slows down – Its length contracts in direction of motion – E = mc2 – Its mass increases • Whether or not two events are simultaneous depends on • General Theory of Relativity (1915) your perspective – Expands the ideas of special theory to include a surprising new view of gravity 1 4/1/10 Inertial Reference Frames Galilean Relativity Imagine two spaceships passing. The astronaut on each spaceship thinks that he is stationary and that the other spaceship is moving. http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ Which one is right? Both. ClassMechanics/Relativity/Relativity.html Each one is an inertial reference frame. Any non-rotating reference frame is an inertial reference frame (space shuttle, space station). Each reference Speed limit sign posted on spacestation. frame is equally valid. How fast is that man moving? In contrast, you can tell if a The Solar System is orbiting our Galaxy at reference frame is rotating. 220 km/s. Do you feel this? Absolute Time Absolutes of Relativity 1. The laws of nature are the same for everyone In the Newtonian universe, time is absolute. Thus, for any two people, reference frames, planets, etc, 2.
    [Show full text]
  • HANDPICKED July 30-August 4 • Blue Note
    WWW.JAZZINSIDEMAGAZINE.COM AUGUST 2013 Interviews Andrew Cyrille David Chesky Satoko Fujii Francisco Mela Dizzy’s Club, August 19 Roy Hargrove Big Band Blue Note, August 20-25 Marcus Strickland EARL Jazz Standard, August 13-14 Comprehensive Directory of NY Club Concert KLUGH & Event Listings HANDPICKED E view Section! July 30-August 4 • Blue Note xpanded CD Re The Jazz Music Dashboard — Smart Listening Experiences www.ConcordMusicGroup.com www.Chesky.com www.WhalingCitySound.com www.SatokoFujii.com The Stone August 20-25 Like Us facebook.com/JazzInsideMedia Follow Us twitter.com/JazzInsideMag Watch Us youtube.com/JazzInsideMedia TJC14_Ad_Jazz Inside.pdf 1 4/26/13 5:31 PM 13TH ANNUAL SAILING OF THE JAZZ CRUISE WHERE EVERY PERFORMANCE IS SPECIAL Ernie Adams Tony Kadleck John Allred Tom Kennedy Shelly Berg Joe LaBarbera MUSIC DIRECTOR Christoph Luty Alonzo Bodden COMEDIAN Dennis Mackrel Randy Brecker Manhattan Transfer Ann Hampton Callaway Marcus Miller Quartet Quartet Bill Charlap Trio Bob Mintzer Clayton Brothers Lewis Nash Trio Quintet Dick Oatts C Freddy Cole Trio M Ken Peplowski Kurt Elling Quartet SHOW HOST Y Robin Eubanks Houston Person CM Quartet John Fedchock MY BIG BAND DIRECTOR John Pizzarelli CY Quartet JAN. 26-FEB. 2 David Finck Gregory Porter Quartet CMY Chuck Findley 2014 K Poncho Sanchez Bruce Forman Arturo Sandoval Nnenna Freelon Trio Gary Smulyan Wycliffe Gordon GOSPEL SHOW HOST Cedar Walton Trio Jimmy Greene Jennifer Wharton Jeff Hamilton Niki Haris Antonio Hart Tamir Hendelman Dick Hyman Tommy Igoe Sextet Sean Jones TO L L -FREE US & C ANADA FT LAUDERDALE • TURKS & CAICOS • SAN JUAN 888.852.9987 ST.
    [Show full text]
  • Chapter 2: Minkowski Spacetime
    Chapter 2 Minkowski spacetime 2.1 Events An event is some occurrence which takes place at some instant in time at some particular point in space. Your birth was an event. JFK's assassination was an event. Each downbeat of a butterfly’s wingtip is an event. Every collision between air molecules is an event. Snap your fingers right now | that was an event. The set of all possible events is called spacetime. A point particle, or any stable object of negligible size, will follow some trajectory through spacetime which is called the worldline of the object. The set of all spacetime trajectories of the points comprising an extended object will fill some region of spacetime which is called the worldvolume of the object. 2.2 Reference frames w 1 w 2 w 3 w 4 To label points in space, it is convenient to introduce spatial coordinates so that every point is uniquely associ- ated with some triplet of numbers (x1; x2; x3). Similarly, to label events in spacetime, it is convenient to introduce spacetime coordinates so that every event is uniquely t associated with a set of four numbers. The resulting spacetime coordinate system is called a reference frame . Particularly convenient are inertial reference frames, in which coordinates have the form (t; x1; x2; x3) (where the superscripts here are coordinate labels, not powers). The set of events in which x1, x2, and x3 have arbi- x 1 trary fixed (real) values while t ranges from −∞ to +1 represent the worldline of a particle, or hypothetical ob- x 2 server, which is subject to no external forces and is at Figure 2.1: An inertial reference frame.
    [Show full text]
  • 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.
    [Show full text]
  • Chapter S3: Spacetime and Gravity
    Chapter S3 Lecture Chapter S3: Spacetime and Gravity © 2017 Pearson Education, Inc. Spacetime and Gravity © 2017 Pearson Education, Inc. S3.1 Einstein's Second Revolution • Our goals for learning: • What are the major ideas of general relativity? • What is the fundamental assumption of general relativity? © 2017 Pearson Education, Inc. What are the major ideas of general relativity? © 2017 Pearson Education, Inc. Spacetime • Special relativity showed that space and time are not absolute. • Instead, they are inextricably linked in a four-dimensional combination called spacetime. © 2017 Pearson Education, Inc. Curved Space • Travelers going in opposite directions in straight lines will eventually meet. • Because they meet, the travelers know Earth's surface cannot be flat—it must be curved. © 2017 Pearson Education, Inc. Curved Spacetime • Gravity can cause two space probes moving around Earth to meet. • General relativity says this happens because spacetime is curved. © 2017 Pearson Education, Inc. Rubber Sheet Analogy • Matter distorts spacetime in a manner analogous to how heavy weights distort a rubber sheet. © 2017 Pearson Education, Inc. Key Ideas of General Relativity • Gravity arises from distortions of spacetime. • Time runs slowly in gravitational fields. • Black holes can exist in spacetime. • The universe may have no boundaries and no center but may still have finite volume. • Rapid changes in the motion of large masses can cause gravitational waves. © 2017 Pearson Education, Inc. What is the fundamental assumption of general relativity? © 2017 Pearson Education, Inc. Relativity and Acceleration • Our thought experiments about special relativity involved spaceships moving at constant velocity. • Is all motion still relative when acceleration and gravity enter the picture? © 2017 Pearson Education, Inc.
    [Show full text]
  • Parodying Childhood Trauma in Sylvia Plath and Louise Bourgeois
    Reframing Confession: Parodying Childhood Trauma in Sylvia Plath and Louise Bourgeois Natasha Silver UCL Thesis Submitted for PhD in Comparative Literature Declaration I, Natasha Katie Silver, 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. 1 Abstract This thesis examines how childhood trauma is aesthetically mediated in the works and ego documents of Sylvia Plath and Louise Bourgeois, departing from a biographical approach in order to reconceive confession as an aesthetic genre. In particular, I explore the extent to which parody — as an intertextual, self-reflexive and playful mode — provides a means of framing an experience that resists symbolisation. Equally, I consider how the parodic mode might enable the artists to situate themselves in the male canon. Giving a cultural history of confession, Chapter 1 identifies a convergence between confessional practices and narratives of victimhood in contemporary western discourse. However, the art forms analysed in this thesis (namely, journal, poem and sculpture) actively manipulate confessional strategies, inviting critical reflection on audience enjoyment of the aestheticisation of suffering. Chapter 2 discusses how trauma can distort linear time and memory, drawing some key distinctions between traumatic temporalities that are frequently elided in trauma theory. By contrast, I propose that parody may enable the traumatised subject to establish reflective distance from overwhelmingly painful affect by localising the experience in time. Arguing that Plath’s journals produce (rather than reveal) a traumatised subject, Chapter 3 explores how they both inhabit and exceed the figure of melancholic poet, ironising the conception of trauma as a wellspring of genius.
    [Show full text]