DOCSLIB.ORG

Physics 116 Relativity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Physics 116 Relativity Physics 116 Session 29 Relativity Nov 17, 2011 R. J. Wilkes Email: [email protected] Announcements: •! Updated quiz score totals will be posted on WebAssign tomorrow •! Nice series on PBS covering topics we will discuss in class: Brian Greene’s Fabric of the Cosmos http://www.pbs.org/wgbh/nova/physics/fabric-of-cosmos.html Lecture Schedule (up to exam 3) Today 3 Last time: Paradox? •! Each one says the other’s clock is slow: contradiction •! Einstein says: both are correct 1899, Henri Poincare: “…The simultaneity of two events, or the order of their succession, as well as the equality of two time intervals, must be defined in such a way that the statements of the natural laws be as simple as possible.” Einstein took Poincare seriously! •! Paradox does not occur if we reject relativity Galileo/Newton/Maxwell “classical physics” universe: –! Time is absolute, and universal (same in all reference frames) –! Ether rest frame = absolute universal reference frame •! Speed of vehicle adds/subtracts from light speed •! So, Q: Why believe in such a silly concept as relativity? 4 Einstein’s insight: we live in 4 dimensions, not 3 Last time: A: worse contradictions and complications arise if we don’t! •! The world makes sense only if we treat time in the same way as space coordinates - then –! Maxwell’s equations work in any (non-accelerated) frame –! Michelson’s experiment is explained (And many, many other things…) •! 3 space dimensions (up-down, N-S, E-W) + time = 4 dimensions –! Universe occupies a 4-dimensional space-time continuum –! Time is also relative: it’s just another coordinate •! We don’t find it peculiar that Bill and Phil measure different location coordinates for the same event - why not different times? –! Objects trace out “world-lines” in 4D spacetime •! “Event” = something that occurs at some point in spacetime –! Emission of light pulse, detection of light pulse: some interaction •! Events are what physics observations must agree upon –! Not their coordinates! Any coordinate frame is OK 5 Example of worldlines •! Can’t draw pictures in 4D, but can sketch motion in 1 space coord: time Bill Plots of time vs position = worldlines time Phil Me Bill Me (at rest (moving (at rest on earth) backward) on train) Phil (moving forward) x coordinate (earth) x’ coordinate (train) P Jill time Me B J (at rest 0.1 c on bike) Bill Jill’s picture of the same situation: (moving Phil She’s riding a bike (slower than train). backward) (moving She is at rest relative to her bike. forward) Notice: slopes of worldlines = relative speed Vertical = 0 speed x’’ coordinate (bike) More slanted = faster speed 6 Your worldline in spacetime •! Every object has a worldline – for example, you! Your worldline, in rest frame of room A-118 time Recall: slope of worldline ~ speed Future leaving class “Elsewhere” “Light cone” = worldline of light YOU You can’t influence events here moving outward from here/now – and they can’t affect you! Nothing can go faster! Here, right now position Sitting in class Past coming to class 7 Time dilation Use light clock to analyze difference in time between frames: D D P d P B d 0.1 c Train Earth A B •! In Earth frame: Train frame viewed from Earth: This is what Bill says about Phil’s clock – it ticks every Notice: Exactly the same calculation for Earth frame viewed from train: Phil says Bill’s clock ticks at 10.05 ns intervals 8 Cosmic ray muons: time dilation confirmed •!Cosmic rays = high energy protons and nuclei that circulate in our Galaxy for millions of years. •!When they strike our atmosphere, they smash air nuclei and make pion particles which in turn produce muons. •!Muons decay after 2.2 microsec in their own rest frame. Note: time in your own rest frame = your “proper time” •!The muons are “relativistic” – they have v ~ c •!They are typically produced at 15 km altitude •!They are detected abundantly at sea level How far could a muon travel in 2.2 microsec of Earth time? Suppose a muon has v = 0.999c in our (Earth = rest frame) coordinates: We would detect almost no muons at sea level if this were true But actually, relative to our frame, 2.2 microsec in the muon’s proper time is dilated to a much longer time, so it can travel much further: Postulates of Special Relativity 1. The laws of physics are the same in any inertial reference frame •! Inertial frame = coordinate system where Newton’s Laws apply •! In general: coordinate origin has no acceleration •! New idea: this applies not just to mechanics but to all physics 2. The speed of light in vacuum c is the same in all inertial frames, independent of motion of source or observer •! Really new idea! Seems bizarre at first glance •! Not so crazy if rate of passage of time differs between frames That’s all ! Simple, but revolutionized our picture of the universe •! Light clock comparison shows that operational definition of time does differ between observers in different inertial frames •! Remember the meaning of what we found: •! !t = a time interval in frame where clock is at rest (its proper time) •! !t’ = !t as measured from a frame with speed v relative to clock frame 10 Time dilation: Lorentz factors Time dilation factor depends on v relative to c: Remember, !t = a time interval in frame where clock is at rest (its proper time) !t’ = !t as measured from a frame with speed v relative to clock frame So if 1 year passes in light clock’s frame, 7 years have passed in frame where v = 0.990 c relative to clock We can go the other way: So if 1 year passes in frame where v = 0.990 c, only 0.14 years have passed in light clock’s frame 11 Length contraction too! Fitzgerald contraction •! We can also use light as a measuring stick: the distance from A to B is measured in units of (light speed)*(time for light to go from A to B): Distance light travels in 1 year = 1 light-year (ly) ~ 1015 meters •! “Proper length” of an object = its length measured in a frame where the object is at rest •! What is L’ = length of a meter stick as measured by observer moving with speed v relative to the rest frame of the object? Notice: v (relative speed) is the same in both frames Length in meter stick’s rest frame (observer 1) is L1 =xB - xA = c !t1 But observer 2 (on train) sees meter stick moving toward him with speed v Points A and B move past him in time L2 =v !t2 , but he knows time ticks more slowly for observer 1, and c is the same for both 2 1 A B v “Twin paradox” •! Famous example – but there is no real “paradox”! •! Twins 1 and 2 are 20 years old when 2 travels to a star 25.3 light-years away, with constant speed* v=0.990c •! How long does it take by twin 1’s clock? •! How long does it take according to twin 2’s clock? we know !t’ = 25.6 yr (time according to twin 1 who is moving backward at speed v=0.990 c relative to spaceship), we want to find !t (in spaceship clock’s rest frame) Notice: v (relative speed) is the same in both frames Q: How can it take only 3.6 yrs? Star A: the star is closer for twin 2 ! 1 2 25.3 ly in Earth frame = v Earth Relativistic momentum and energy •! We find that Newtonian momentum p = mv also needs a Lorentz factor, if the particle is moving at significant speed compared to c Notice that p ~ Newtonian for small v/c p blows up as v gets closer to c ! •! We can view this as meaning that mass, in effect, grows with v •! We can apply Newtonian mechanics calculations using this version of m –! Notice: a = F/m -- this means we need an ever-increasing force to maintain constant acceleration of an object – and can never reach v=c if m > 0 •! Einstein showed that the total energy of an object is given by –! Einstein’s most famous result! Notice contradiction of classical physics: any object has non-zero energy at rest, and mass itself is a form of energy Then since E = rest energy + kinetic energy, must have 14 General relativity •! Einstein, 1915: extended relativity to accelerated frames: general relativity –! GR really describes the geometry of spacetime: gravity of massive objects warps spacetime in their vicinity –! Equivalence Principle: Observations cannot distinguish a uniformly accelerated frame from a uniform gravity field –! Eddington, 1919: GR predictions matched observed anomalies in orbit of Mercury, Newtonian predictions do not – Einstein is right* •! More predictions and consequences of GR: *“If relativity is proved right, the Germans will –! Gravitational time dilation and redshift call me a great German, the Swiss will call me a –! Deflection of light by gravity great Swiss, and the French will call me a great –! Gravitational waves citizen of the world. If relativity is proved wrong, the French will call –! Black holes me a Swiss, the Swiss will call me a German, •! Applications confirming GR today and the Germans will call me a Jew.” -Einstein –! GPS satellite orbits: precision needed requires GR calculations –! Gravitational lensing, black holes: astronomical observations confirm –! Gravitational wave astronomy: see http://www.ligo-la.caltech.edu/LLO/overviewsci.htm –! Notice: LIGO is a variety of Michelson apparatus! •! We’re still looking for unexplained anomalies: UW is a center for this work –! See http://www.npl.washington.edu/eotwash/index.html 15 .
Load more

Recommended publications
  • The Multiverse: Conjecture, Proof, and Science
    The multiverse: conjecture, proof, and science George Ellis Talk at Nicolai Fest Golm 2012 Does the Multiverse Really Exist ? Scientific American: July 2011 1 The idea The idea of a multiverse -- an ensemble of universes or of universe domains – has received increasing attention in cosmology - separate places [Vilenkin, Linde, Guth] - separate times [Smolin, cyclic universes] - the Everett quantum multi-universe: other branches of the wavefunction [Deutsch] - the cosmic landscape of string theory, imbedded in a chaotic cosmology [Susskind] - totally disjoint [Sciama, Tegmark] 2 Our Cosmic Habitat Martin Rees Rees explores the notion that our universe is just a part of a vast ''multiverse,'' or ensemble of universes, in which most of the other universes are lifeless. What we call the laws of nature would then be no more than local bylaws, imposed in the aftermath of our own Big Bang. In this scenario, our cosmic habitat would be a special, possibly unique universe where the prevailing laws of physics allowed life to emerge. 3 Scientific American May 2003 issue COSMOLOGY “Parallel Universes: Not just a staple of science fiction, other universes are a direct implication of cosmological observations” By Max Tegmark 4 Brian Greene: The Hidden Reality Parallel Universes and The Deep Laws of the Cosmos 5 Varieties of Multiverse Brian Greene (The Hidden Reality) advocates nine different types of multiverse: 1. Invisible parts of our universe 2. Chaotic inflation 3. Brane worlds 4. Cyclic universes 5. Landscape of string theory 6. Branches of the Quantum mechanics wave function 7. Holographic projections 8. Computer simulations 9. All that can exist must exist – “grandest of all multiverses” They can’t all be true! – they conflict with each other.
    [Show full text]
  • Modern Physics 436: Course Outline
    Modern Physics 436: Course Outline 1. The book that I'm going to use for most part of the course will be • Modern Physics: J. Bernstein, P. Fishbane, S. Gasiorowicz (Prentice Hall) (It will be available at the Paragraphe Bookstore) 2. The course generally assumes that you are familiar with basics of integrations, differen- tiations, classical and quantum mechanics. Your pre-requisite are Phys 446, Phys 333. In this course I plan to cover the following topics: • Single particle atom: Basic review of Phys 446 • Multi-particle atoms: Molecules, molecular bondings • Statistical mechanics: Classical and quantum aspects, Maxwell-Boltzmann distribution, Fermi-Dirac distribution, Bose-Einstein distribution, Bose-Einstein condensation • Lasers: Induced radiations, coherent and squeezed states of light • Condensed matter Physics: Band structure, semiconductors, superconductivity, BCS theory, Hall effects (classical and quantum), Laughlin function • Nuclear Physics: Nuclear structure, nuclear interactions, nuclear models • Elementary Particle Physics: Quantum field theory (QFT), virtual particles, Feynman path-integrals, Feynman diagrams, Quantum electrodynamics (QED), particle zoo, weak interactions, leptons, hadrons, Quantum chromodynamics (QCD), quarks Wish-list: • General Relativity and Cosmology: Basic GTR, cosmological inflation, big-bang nucle- osynthesis • Superstring Theory: Very basic ideas, types of string theories, supersymmetry, super- gravity 3. Although I'll try to stick with one book, you're always advised to read other books for more details. A few other important books on the above topics are: • Quantum Mechanics: You may have already been using the book by David Griffiths. Another very good book is by Claude Cohen-Tannoudji, Bernard Diu and Frank Laloe. It comes in two volumes and does everything in full details.
    [Show full text]
  • Dear Brian Greene, When Friends Come to Visit at My House, They See
    Dear Brian Greene, When friends come to visit at my house, they see warning signs on my door saying “High Voltage! Neutron Radiation!” This is because last spring, I took on the task of building a fusion reactor which is now operational and in a research phase. Thanks to the inspiration that I received from reading your book, The Elegant Universe, I have committed my life to a future in nuclear physics. Last year, I was searching for informational books on particle physics since this was an interest of mine; however, I found it difficult to find higher level materials that didn’t require college mathematics knowledge. The solution was your book, which I quickly read and have since been fascinated by the subject. The way that advanced topics are portrayed in a simple to understand and intriguing fashion make the book a brilliant piece of work that any aspiring physicist should read. I vividly recall two nights regarding my fascination with high energy physics. The first of those was the evening that I obtained a copy of The Elegant Universe after it was suggested to me as an excellent resource in string theory and quantum mechanics. I don’t think I ever went to bed that night while I devoured as much literature as I could. Within a week, I had completed the book and realized that my future was in this field. The other event that I recall was when I was searching for a science project online and stumbled upon fusor.net, a research consortium dedicated to amateur nuclear fusion.
    [Show full text]
  • At NYC Sci Fest, Asking 'What If We're Holograms?' 30 May 2010, by SAMANTHA GROSS , Associated Press Writer
    At NYC sci fest, asking 'What if we're holograms?' 30 May 2010, By SAMANTHA GROSS , Associated Press Writer of the universe to scores of non-geniuses through his book "The Elegant Universe" and the PBS specials by the same name. The physicist founded the festival in 2008 with his wife, Tracy Day. In a way, they say, it's an extension of his work translating into layman's terms the fundamentals of string theory - the idea that the universe and its most fundamental forces could be best explained if everything around us were made up of minuscule, vibrating strings. Brian Greene, a string theorist known for bringing his Greene is not the only scientist working to show complex field of science to the masses, and Tracy Day, Americans the relevance of the field, and hoping to his wife and organizing partner behind World Science make it cooler for U.S. youth. Despite the recent Festival, pose in Times Square, New York, Wednesday murmurings about the era of "geek chic," many May 19, 2010. (AP Photo/Bebeto Matthews) teenagers still largely see science as a dorky pursuit, says Michio Kaku, a presenter at the festival and another string theorist who's built a career bringing his science to the public. (AP) -- Brian Greene works in a world where scientific reasoning rules all and imagination leads The numbers in the National Science Board's to the most unlikely truths. Greene and other yearly examination of science and engineering "string theorists" are exploring a possible scenario indicators paint a mixed picture for American in which people and the world around us are students.
    [Show full text]
  • String Theory and Our Relationship with Nature: the Convergence of Science, Curriculum Theory, and the Environment
    Georgia Southern University Digital Commons@Georgia Southern Electronic Theses and Dissertations Graduate Studies, Jack N. Averitt College of Fall 2010 String Theory and Our Relationship with Nature: The Convergence of Science, Curriculum Theory, and the Environment Virginia Therese Bennett Follow this and additional works at: https://digitalcommons.georgiasouthern.edu/etd Recommended Citation Bennett, Virginia Therese, "String Theory and Our Relationship with Nature: The Convergence of Science, Curriculum Theory, and the Environment" (2010). Electronic Theses and Dissertations. 530. https://digitalcommons.georgiasouthern.edu/etd/530 This dissertation (open access) is brought to you for free and open access by the Graduate Studies, Jack N. Averitt College of at Digital Commons@Georgia Southern. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of Digital Commons@Georgia Southern. For more information, please contact [email protected]. STRING THEORY AND OUR RELATIONSHIP WITH NATURE: THE CONVERGENCE OF SCIENCE, CURRICULUM THEORY, AND THE ENVIRONMENT by VIRGINIA THERESE BENNETT (Under the Direction of John A. Weaver) ABSTRACT Curriculum Theory affords us the opportunity to examine education from a multitude of directions. This work takes advantage of that opportunity to explore the relationships between science, nature, and curriculum using string theory and our ideas about the environment as a backdrop. Both the energy and multiple possibilities created by strings and the rich history leading up to the theory help to illustrate the many opportunities we have to advance discussions in alternative ways of looking at science. By considering the multiple dimensions inherent in string theory as multiple pathways and interweaving metaphors from Deleuze and Guattari, Michel Serres, and Donna Haraway, our approach to environmental issues and environmental education allow us to include alternative ways of looking at the world.
    [Show full text]
  • Brian Greene
    BOOK REVIEW Staff undergraduate Editor-in-Chief BRIAN GREENE James Stankowicz physics The Fabric of the Cosmos: Space, Time, and the Texture of Reality Layout Director George CB Ling newsletter Brian Greene is a string theorist The final part of the book focuses Volume VI, Issue #1, September 2008 U P at Colombia University. His first on the experimental efforts (with Online Editor UP News Online: http://www.phys.ufl.edu/~upnews general public book, The Elegant many references to the LHC) Universe, was a Pulitzer Prize ongoing, the possibility of time Harold Rodriguez finalist, and served as the basis travel, and what might be waiting Production Manager as we delve ever further down LARGE HADRON who we are for a three hour PBS TV program Steven Hochman of the same name. into the fabric of reality. Assistant Editor COLLIDER UNVEILED! The Fabric of the Cosmos in some As The Fabric of the Cosmos is UP is a monthly undergraduate Alicia Swift It’s the world’s biggest and most in huge caves, track and analyze ways picks up where The Elegant written for the general public, physics newsletter sponsored ambitious scientific experiment exactly what happens. With more there are no mathematical Universe left off, but it is also a Staff Writers in human history. From revealing than 600 million collisions every by the University of Florida’s stand-alone book that doesn’t formulas in the main text, Arthur Ianuzzi what happened in the moments second, staggering volumes of chapter of the Society of Physics require any knowledge of The although the notes in the back of Bryce Bolin after the Big Bang, to probing the data are generated, hundreds of Elegant Universe.
    [Show full text]
  • The Elegant Universe
    The Elegant Universe Teacher’s Guide On the Web It’s the holy grail of physics—the search for the ultimate explanation of how the universe works. And in the past few years, excitement has grown among NOVA has developed a companion scientists in pursuit of a revolutionary approach to unify nature’s four Web site to accompany “The Elegant fundamental forces through a set of ideas known as superstring theory. NOVA Universe.” The site features interviews unravels this intriguing theory in its three-part series “The Elegant Universe,” with string theorists, online activities based on physicist Brian Greene’s best-selling book of the same name. to help clarify the concepts of this revolutionary theory, ways to view the The first episode introduces string theory, traces human understanding of the program online, and more. Find it at universe from Newton’s laws to quantum mechanics, and outlines the quest www.pbs.org/nova/elegant/ for and challenges of unification. The second episode traces the development of string theory and the Standard Model and details string theory’s potential to bridge the gap between quantum mechanics and the general theory of relativity. The final episode explores what the universe might be like if string theory is correct and discusses experimental avenues for testing the theory. Throughout the series, scientists who have made advances in the field share personal stories, enabling viewers to experience the thrills and frustrations of physicists’ search for the “theory of everything.” Program Host Brian Greene, a physicist who has made string theory widely accessible to public audiences, hosts NOVA’s three-part series “The Elegant Universe.” A professor of physics and mathematics at Columbia University in New York, Greene received his undergraduate degree from Harvard University and his doctorate from Oxford University, where he was a Rhodes Scholar.
    [Show full text]
  • 2021 WSFB Program
    March 24–28 WHY WORLD of SCIENCE ONDER W FESTIVAL The BRISBANE 2021 PROGRAM QUEENSLAND MUSEUM Prehistoric Ocean Predators Until 3 May 2021 BOOK NOW seamonsters.qm.qld.gov.au Queensland Museum Network Exhibition Producer and Tour Manager Exhibition Partner Exhibition Development Partner Love science? Buy an Annual Pass and get 12 months of unlimited entry to Queensland Museum’s Queensland Museum, South Bank qm.qld.gov.au The Wonder of Why From devastating bushfires and COVID-19 to global civil unrest and mass protests – there is no doubt 2020 was a challenging year and the world has changed in the past 12 months. World Science Festival Brisbane has had to change also, so some things will be different in 2021 – we have made Contents more of our program available online, we’re avoiding mass gatherings and everyone will have to check in at events. But this is the new normal, and we’ll get there together. 2 Welcome No matter what, now is the time to celebrate science! In many cases what seemed like abstract ideas last year 4 About the festival are now our everyday stark reality. What have we learnt? How will science lead us out of our current dilemmas? What were the warning signs? 6 Digital program More than ever, we turn to science, technology and human 8 City of Science calendar of events curiosity to help us understand our circumstances and learn how we and nature might adapt to the new reality 10 City of Science map confronting us. This year’s festival celebrates the wonder of ‘why’, with a program placing the spotlight on insightful and 12 Wednesday 24 March challenging conversations and an escape to new and 14 Thursday 25 March entertaining experiences.
    [Show full text]
  • Brian Green: the Fabric of the Cosmos. Space Time and the Texture Of
    Brian Greene - The Fabric Of The Cosmos.htm THIS IS A BORZOI BOOK PUBLISHED BY ALFRED A. KNOPF Copyright © 2004 by Brian R. Greene All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Alfred A. Knopf, a division of Random House, Inc., New York, and in Canada by Random House of Canada Limited, Toronto. Distributed by Random House, Inc., New York. www.aaknopf.com Knopf, Borzoi Books, and the colophon are registered trademarks of Random House, Inc. Library of Congress Cataloging-in-Publication Data Greene, B. (Brian). The fabric of the cosmos : space, time, and the texture of reality / Brian Greene. p. cm. Includes bibliographical references (pp. 543-44). ISBN 0-375-41288-3 1. Cosmology—Popular works. I. Title. QB982.G74 2004 523.1—dc22 2003058918 Manufactured in the United States of America First Edition file:///E|/temp/greene/Brian%20Greene%20-%20The%20Fabric%20Of%20The%20Cosmos-htm/Brian%20Greene%20-%20The%20Fabric%20Of%20The%20Cosmos.htm (1 de 740)24/05/2005 21:29:47 Brian Greene - The Fabric Of The Cosmos.htm To Tracy file:///E|/temp/greene/Brian%20Greene%20-%20The%20Fabric%20Of%20The%20Cosmos-htm/Brian%20Greene%20-%20The%20Fabric%20Of%20The%20Cosmos.htm (2 de 740)24/05/2005 21:29:47 Brian Greene - The Fabric Of The Cosmos.htm Contents file:///E|/temp/greene/Brian%20Greene%20-%20The%20Fabric%20Of%20The%20Cosmos-htm/Brian%20Greene%20-%20The%20Fabric%20Of%20The%20Cosmos.htm (3 de 740)24/05/2005 21:29:47 Brian Greene - The Fabric Of The Cosmos.htm Preface ix Parti REALITY'S ARENA 1.
    [Show full text]
  • Holographic Universe
    Wednesday, April 30, 2014 Fifth exam and sky watch, FRIDAY, May 2. Review sheet today. Review session Thursday, 5:00 PM, RLM 7.104. Reading for Exam 5: Chapter 9 – Sections 9.6.1, 9.6.2, 9.7, 9.8; Chapter 10 - Sections 10.1-10.4, 10.9; Chapter 11 - all except Section 11.6 (abbreviated, focus on lectures); Chapter 12 - all; SKIP Chapter 13; Chapter 14 – very abbreviated version. Electronic class reviews now available. Astronomy in the news: Exhibit on 75th anniversary of McDonald Observatory at the Bullock Museum, across MLK from the Blanton Museum. Update on new “nearby” supernova SN 2014J in M82 I just received yet another new draft paper to edit and comment upon. This paper focuses on the dust along the line of sight to SN 2014 and associated observations with the Hubble Space Telescope made a few months ago. Can use current theories to “predict” the conditions for which the theoretical collision occurs, where the theory of quantum gravity is most crucially needed, effectively the scale of length where quantum uncertainty and space-time curvature are equal. Planck length - about 10-33 centimeters, vastly smaller than any particle, but still not zero! Planck density - about 1093 grams/cubic centimeter, huge, but not infinite! Planck time - about 10-43 seconds, short, but not zero! Cannot predict earlier times in the Big Bang. On the Planck scale, not just that the position of things would be uncertain in space, but that space and time themselves could be quantum uncertain, “up” “down” “before” “after” difficult if not impossible to define.
    [Show full text]
  • Greene, Brian. the Elegant Universe: Superstrings, Hidden Dimensions and the Quest for the Ultimate Theory
    Greene, Brian. The Elegant Universe: Superstrings, Hidden Dimensions and the Quest for the Ultimate Theory. New York: Vintage Books, 2000, 448pp. Outline Prepared by Rev. William S. Wick, Norwich University Chaplain (12/18/2013) I. Preface p.ix A. During the last thirty years of his life, Albert Einstein sought [but never realized] a so-called unified field theory capable of describing nature’s forces within a single, all-encompassing, coherent framework. B. ... physicists believe they have found a framework for stitching these insights together into a seamless whole - capable of describing all phenomena: superstring theory. p.x C. Elegant Universe is an attempt to make these insights accessible to a broad spectrum of readers. D. Superstring theory draws on many of the central discoveries in physics (and Greene focuses on the evolving understanding of space and time). E. ... cutting-edge research has integrated [Einstein’s] discoveries into a quantum universe with numerous hidden dimensions coiled into the fabric of the cosmos - dimensions whose geometry may well hold the key to some of the most profound questions ever posed. p.xi F. Greene’s has focuses on the impact superstring theory has on concepts of space and time. II. Part I: The Edge of Knowledge (Chapter One: Tied Up with String) p.1 A. Introduction p.3 1. two foundational pillars upon which modern physics rests: a. Albert Einstein’s General Relativity (a theoretical framework for understanding the universe on the largest of scales (the immense expanse of the universe itself) b. quantum mechanics (a theoretical framework for understanding the universe on the smallest of scales - (atomic/subatomic particles) 2.
    [Show full text]
  • D-Branes and Vanishing Cycles in Higher Dimensions
    Preprint typeset in JHEP style. - HYPER VERSION D-Branes and Vanishing Cycles in Higher Dimensions. Mark Raugas Department of Mathematics, Columbia University New York, NY 10027, USA [email protected] Abstract: We investigate the quantum volume of D-branes wrapped around cycles of various dimension in Calabi-Yau fourfolds and fivefolds. Examining the cases of the sextic and heptic hypersurface Calabi-Yau varieties, as well as one example in weighted projective space, we find expressions for periods which vanish at the singular point analogous to the conifold point. As in the known three-dimensional cases, it is the top dimensional cycle which attains zero quantum volume, even though lower dimensional cycles remain non-degenerate, indicating this phenomena to be a general feature of quantum geometry. arXiv:hep-th/0102133v1 20 Feb 2001 Contents 1. Introduction 1 2. Higher Dimensional Calabi-Yau Manifolds 2 2.1 Quantum Volumes 2 2.2 Meijer Periods 2 2.3 Monodromies 4 2.4 Weakly Integral Periods 5 3. Collapsing Cycles 5 3.1 The Sextic Hypersurface 5 3.2 The Heptic Hypersurface 7 3.3 A Weighted Example 9 4. Conclusions 11 1. Introduction Quantum geometry often exhibits marked differences from the expectations one de- rives from classical intuitions. These novel features have in recent years helped aid the understanding of many phenomena in string theory [1, 2]. For example, worldsheet instanton effects lead one to determine that it is the collection of all rational curves over a given Calabi-Yau manifold which can be thought of as the quantum version of the variety [3].
    [Show full text]