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Frame Covariance and Fine Tuning in Inflationary Cosmology
FRAME COVARIANCE AND FINE TUNING IN INFLATIONARY COSMOLOGY A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2019 By Sotirios Karamitsos School of Physics and Astronomy Contents Abstract 8 Declaration 9 Copyright Statement 10 Acknowledgements 11 1 Introduction 13 1.1 Frames in Cosmology: A Historical Overview . 13 1.2 Modern Cosmology: Frames and Fine Tuning . 15 1.3 Outline . 17 2 Standard Cosmology and the Inflationary Paradigm 20 2.1 General Relativity . 20 2.2 The Hot Big Bang Model . 25 2.2.1 The Expanding Universe . 26 2.2.2 The Friedmann Equations . 29 2.2.3 Horizons and Distances in Cosmology . 33 2.3 Problems in Standard Cosmology . 34 2.3.1 The Flatness Problem . 35 2.3.2 The Horizon Problem . 36 2 2.4 An Accelerating Universe . 37 2.5 Inflation: More Questions Than Answers? . 40 2.5.1 The Frame Problem . 41 2.5.2 Fine Tuning and Initial Conditions . 45 3 Classical Frame Covariance 48 3.1 Conformal and Weyl Transformations . 48 3.2 Conformal Transformations and Unit Changes . 51 3.3 Frames in Multifield Scalar-Tensor Theories . 55 3.4 Dynamics of Multifield Inflation . 63 4 Quantum Perturbations in Field Space 70 4.1 Gauge Invariant Perturbations . 71 4.2 The Field Space in Multifield Inflation . 74 4.3 Frame-Covariant Observable Quantities . 78 4.3.1 The Potential Slow-Roll Hierarchy . 81 4.3.2 Isocurvature Effects in Two-Field Models . 83 5 Fine Tuning in Inflation 88 5.1 Initial Conditions Fine Tuning . -
PDF Solutions
Solutions to exercises Solutions to exercises Exercise 1.1 A‘stationary’ particle in anylaboratory on theEarth is actually subject to gravitationalforcesdue to theEarth andthe Sun. Thesehelp to ensure that theparticle moveswith thelaboratory.Ifstepsweretaken to counterbalance theseforcessothatthe particle wasreally not subject to anynet force, then the rotation of theEarth andthe Earth’sorbital motionaround theSun would carry thelaboratory away from theparticle, causing theforce-free particle to followacurving path through thelaboratory.Thiswouldclearly show that the particle didnot have constantvelocity in the laboratory (i.e.constantspeed in a fixed direction) andhence that aframe fixed in the laboratory is not an inertial frame.More realistically,anexperimentperformed usingthe kind of long, freely suspendedpendulum known as a Foucaultpendulum couldreveal the fact that a frame fixed on theEarth is rotating andthereforecannot be an inertial frame of reference. An even more practical demonstrationisprovidedbythe winds,which do not flowdirectly from areas of high pressure to areas of lowpressure because of theEarth’srotation. - Exercise 1.2 TheLorentzfactor is γ(V )=1/ 1−V2/c2. (a) If V =0.1c,then 1 γ = - =1.01 (to 3s.f.). 1 − (0.1c)2/c2 (b) If V =0.9c,then 1 γ = - =2.29 (to 3s.f.). 1 − (0.9c)2/c2 Notethatitisoften convenient to write speedsinterms of c instead of writingthe values in ms−1,because of thecancellation between factorsofc. ? @ AB Exercise 1.3 2 × 2 M = Theinverse of a matrix CDis ? @ 1 D −B M −1 = AD − BC −CA. Taking A = γ(V ), B = −γ(V )V/c, C = −γ(V)V/c and D = γ(V ),and noting that AD − BC =[γ(V)]2(1 − V 2/c2)=1,wehave ? @ γ(V )+γ(V)V/c [Λ]−1 = . -
Hubble's Evidence for the Big Bang
Hubble’s Evidence for the Big Bang | Instructor Guide Students will explore data from real galaxies to assemble evidence for the expansion of the Universe. Prerequisites ● Light spectra, including graphs of intensity vs. wavelength. ● Linear (y vs x) graphs and slope. ● Basic measurement statistics, like mean and standard deviation. Resources for Review ● Doppler Shift Overview ● Students will consider what the velocity vs. distance graph should look like for 3 different types of universes - a static universe, a universe with random motion, and an expanding universe. ● In an online interactive environment, students will collect evidence by: ○ using actual spectral data to calculate the recession velocities of the galaxies ○ using a “standard ruler” approach to estimate distances to the galaxies ● After they have collected the data, students will plot the galaxy velocities and distances to determine what type of model Universe is supported by their data. Grade Level: 9-12 Suggested Time One or two 50-minute class periods Multimedia Resources ● Hubble and the Big Bang WorldWide Telescope Interactive Materials ● Activity sheet - Hubble’s Evidence for the Big Bang Lesson Plan The following represents one manner in which the materials could be organized into a lesson: Focus Question: ● How does characterizing how galaxies move today tell us about the history of our Universe? Learning Objective: ● SWBAT collect and graph velocity and distance data for a set of galaxies, and argue that their data set provides evidence for the Big Bang theory of an expanding Universe. Activity Outline: 1. Engage a. Invite students to share their ideas about these questions: i. Where did the Universe come from? ii. -
The Big-Bang Theory AST-101, Ast-117, AST-602
AST-101, Ast-117, AST-602 The Big-Bang theory Luis Anchordoqui Thursday, November 21, 19 1 17.1 The Expanding Universe! Last class.... Thursday, November 21, 19 2 Hubbles Law v = Ho × d Velocity of Hubbles Recession Distance Constant (Mpc) (Doppler Shift) (km/sec/Mpc) (km/sec) velocity Implies the Expansion of the Universe! distance Thursday, November 21, 19 3 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 4 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 5 To a first approximation, a rough maximum age of the Universe can be estimated using which of the following? A. the age of the oldest open clusters B. 1/H0 the Hubble time C. the age of the Sun D. -
The Discovery of the Expansion of the Universe
galaxies Review The Discovery of the Expansion of the Universe Øyvind Grøn Faculty of Technology, Art and Design, Oslo Metropolitan University, PO Box 4 St. Olavs Plass, NO-0130 Oslo, Norway; [email protected]; Tel.: +047-90-94-64-60 Received: 2 November 2018; Accepted: 29 November 2018; Published: 3 December 2018 Abstract: Alexander Friedmann, Carl Wilhelm Wirtz, Vesto Slipher, Knut E. Lundmark, Willem de Sitter, Georges H. Lemaître, and Edwin Hubble all contributed to the discovery of the expansion of the universe. If only two persons are to be ranked as the most important ones for the general acceptance of the expansion of the universe, the historical evidence points at Lemaître and Hubble, and the proper answer to the question, “Who discovered the expansion of the universe?”, is Georges H. Lemaître. Keywords: cosmology history; expansion of the universe; Lemaitre; Hubble 1. Introduction The history of the discovery of the expansion of the universe is fascinating, and it has been thoroughly studied by several historians of science. (See, among others, the contributions to the conference Origins of the expanding universe [1]: 1912–1932). Here, I will present the main points of this important part of the history of the evolution of the modern picture of our world. 2. Einstein’s Static Universe Albert Einstein completed the general theory of relativity in December 1915, and the theory was presented in an impressive article [2] in May 1916. He applied [3] the theory to the construction of a relativistic model of the universe in 1917. At that time, it was commonly thought that the universe was static, since one had not observed any large scale motions of the stars. -
Cosmology and Religion — an Outsider's Study
Cosmology and Religion — An Outsider’s Study (Big Bang and Creation) Dezs˝oHorváth [email protected] KFKI Research Institute for Particle and Nuclear Physics (RMKI), Budapest and Institute of Nuclear Research (ATOMKI), Debrecen Dezs˝oHorváth: Cosmology and religion Wien, 10.03.2010 – p. 1/46 Outline Big Bang, Inflation. Lemaître and Einstein. Evolution and Religion. Big Bang and Hinduism, Islam, Christianity. Saint Augustine on Creation. Saint Augustine on Time. John Paul II and Stephen Hawking. Dezs˝oHorváth: Cosmology and religion Wien, 10.03.2010 – p. 2/46 Warning Physics is an exact science (collection of formulae) It is based on precise mathematical formalism. A theory is valid if quantities calculated with it agree with experiment. Real physical terms are measurable quantities, words are just words. Behind the words there are precise mathematics and experimental evidence What and how: Physics Why: philosopy? And theology? Dezs˝oHorváth: Cosmology and religion Wien, 10.03.2010 – p. 3/46 What is Cosmology? Its subject is the Universe as a whole. How did it form? (Not why?) Static, expanding or shrinking? Open or closed? Its substance, composition? Its past and future? Dezs˝oHorváth: Cosmology and religion Wien, 10.03.2010 – p. 4/46 The Story of the Big Bang Theory Red Shift of Distant Galaxies Henrietta S. Leavitt Vesto Slipher Distances to galaxies Red shift of galaxies 1908–1912 1912 Dezs˝oHorváth: Cosmology and religion Wien, 10.03.2010 – p. 5/46 Expanding Universe Cosmological principle: if the expansion linear A. Friedmann v(B/A) = v(C/B) ⇒ v(C/A)=2v(B/A) the Universe is homogeneous, it has no special point. -
Dark Universe: Science & Literacy Activity for Grades 9-12
Dark Universe Science & Literacy Activity GRADES 9-12 OVERVIEW Common Core State Standards: This activity, which is aligned to the Common Core State Standards (CCSS) WHST.9-12.2, WHST.9-12.8, WHST.9-12.9, for English Language Arts, introduces students to scientific knowledge and RST.9-12.1, RST.9-12.2, RST.9-12.4, RST.9-12.7, language related to the study of cosmology. Students will read content-rich RST.9-12.10 texts, view the Dark Universe space show, and use what they have learned New York State Science Core Curriculum: to complete a CCSS-aligned writing task, creating an illustrated text about PS 1.2a how scientists study the history of the universe. Next Generation Science Standards: Materials in this activity include: PE HS-ESS1-2 • Teacher instructions for: DCI ESS1.A: The Universe and Its Stars o Pre-visit student reading The Big Bang theory is supported by o Viewing the Dark Universe space show observations of distant galaxies receding o Post-visit writing task from our own, of the measured composition • Text for student reading: “Case Study: The Cosmic Microwave Background ” of stars and non-stellar gases, and of the • Student Writing Guidelines maps of spectra of the primordial radiation (cosmic microwave background) that still • Teacher rubric for writing assessment fills the universe. SUPPORTS FOR DIVERSE LEARNERS: An Overview This resource has been designed to engage all learners with the principles of Universal Design for Learning in mind. It represents information in multiple ways and offers multiple ways for your students to engage with content as they read about, discuss, view, and write about scientific concepts. -
A the Highland Park Enews August 1, 2016 221 S
, A The Highland Park eNews August 1, 2016 www.hpboro.com 221 S. Fifth Avenue, Highland Park, NJ, 08904 732-572-3400 Events & Meetings Tuesday, August 2 Book Break 3-5 yr. old Story Time 11:30 a.m., Public Library Tuesday, August 2 Teen Sidewalk Chalk Graffiti and Games 3:00 p.m., Public Library Tuesday, August 2 National Night Out rd 5:00 p.m., North 3 /Raritan Ave Tuesday, August 2 The NJ Youth Corps of Middlesex County was on-hand Wednesday, July 27th working the International Film Festival – Second Mother 1st stage of our Senior/Youth Center beautification project. Weeds were pulled, shrubs trimmed, and beds mulched. We can't wait to see how the project progresses. Thank 6:30 p.m., Public Library you to all involved. Wednesday, August 3 Borough Council Meeting Re: Buck Woods Having a Blast at the Highland Park Recreation Summer Camp 7:00 p.m., Borough Council Wednesday, August 3 Community Food Pantry 7:00 p.m., Senior/Youth Center Tuesday, August 9 Book Break 3-5 yr. old Story Time 11:30 a.m., Public Library Tuesday, August 9 Summer Afternoon Film Series –Rudy (PG) 3:00 p.m., Public Library Summer in Highland Park is always fun, especially for the many kids who attend our Tuesday, August 9 Recreation Summer Camp. This year we are averaging 175 campers per week in our day International Film Series – Coming Home camp, sports camp, teen camp, and new special needs camp. This week’s sports camp is 6:30 p.m., Public Library soccer, and next week will be tennis or wrestling. -
Georges Lemaître & the Big Bang
Monsignor Georges Lemaître & Albert Einstein, 1933 In the winter of 1998, two separate teams of astronomers in Berkeley, California, made a similar, startling discovery. They were both observing supernovae – exploding stars visible over great distances – to see how fast the universe is expanding. In accordance with prevailing scientific wisdom, the astronomers expected to find the rate of expansion to be decreasing, Instead they found it to be increasing – a discovery which has since "shaken astronomy to its core" (Astronomy, October 1999). This discovery would have come as no surprise to Georges Lemaitre (1894-1966), a Belgian mathematician and Catholic priest who developed the theory of the Big Bang. Lemaitre described the beginning of the universe as a burst of fireworks, comparing galaxies to the burning embers spreading out in a growing sphere from the center of the burst. He believed this burst of fireworks was the beginning of time, taking place on "a day without yesterday." After decades of struggle, other scientists came to accept the Big Bang as fact. But while most scientists – including the mathematician Stephen Hawking -- predicted that gravity would eventually slow down the expansion of the universe and make the universe fall back toward its center, Lemaitre believed that the universe would keep expanding. He argued that the Big Bang was a unique event, while other scientists believed that the universe would shrink to the point of another Big Bang, and so on. The observations made in Berkeley supported Lemaitre's contention that the Big Bang was in fact "a day without yesterday." When Georges Lemaitre was born in Charleroi, Belgium, most scientists thought that the universe was infinite in age and constant in its general appearance. -
Hubble's Law and the Expanding Universe
COMMENTARY COMMENTARY Hubble’s Law and the expanding universe Neta A. Bahcall1 the expansion rate is constant in all direc- Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 tions at any given time, this rate changes with time throughout the life of the uni- verse. When expressed as a function of cos- In one of the most famous classic papers presented the observational evidence for one H t in the annals of science, Edwin Hubble’s of science’s greatest discoveries—the expand- mic time, ( ), it is known as the Hubble 1929 PNAS article on the observed relation inguniverse.Hubbleshowedthatgalaxiesare Parameter. The expansion rate at the pres- between distance and recession velocity of receding away from us with a velocity that is ent time, Ho, is about 70 km/s/Mpc (where 1 Mpc = 106 parsec = 3.26 × 106 light-y). galaxies—the Hubble Law—unveiled the proportional to their distance from us: more The inverse of the Hubble Constant is the expanding universe and forever changed our distant galaxies recede faster than nearby gal- Hubble Time, tH = d/v = 1/H ; it reflects understanding of the cosmos. It inaugurated axies. Hubble’s classic graph of the observed o the time since a linear cosmic expansion has the field of observational cosmology that has velocity vs. distance for nearby galaxies is begun (extrapolating a linear Hubble Law uncovered an amazingly vast universe that presented in Fig. 1; this graph has become back to time t = 0); it is thus related to has been expanding and evolving for 14 bil- a scientific landmark that is regularly repro- the age of the Universe from the Big-Bang lion years and contains dark matter, dark duced in astronomy textbooks. -
The Universe of General Relativity, Springer 2005.Pdf
Einstein Studies Editors: Don Howard John Stachel Published under the sponsorship of the Center for Einstein Studies, Boston University Volume 1: Einstein and the History of General Relativity Don Howard and John Stachel, editors Volume 2: Conceptual Problems of Quantum Gravity Abhay Ashtekar and John Stachel, editors Volume 3: Studies in the History of General Relativity Jean Eisenstaedt and A.J. Kox, editors Volume 4: Recent Advances in General Relativity Allen I. Janis and John R. Porter, editors Volume 5: The Attraction of Gravitation: New Studies in the History of General Relativity John Earman, Michel Janssen and John D. Norton, editors Volume 6: Mach’s Principle: From Newton’s Bucket to Quantum Gravity Julian B. Barbour and Herbert Pfister, editors Volume 7: The Expanding Worlds of General Relativity Hubert Goenner, Jürgen Renn, Jim Ritter, and Tilman Sauer, editors Volume 8: Einstein: The Formative Years, 1879–1909 Don Howard and John Stachel, editors Volume 9: Einstein from ‘B’ to ‘Z’ John Stachel Volume 10: Einstein Studies in Russia Yuri Balashov and Vladimir Vizgin, editors Volume 11: The Universe of General Relativity A.J. Kox and Jean Eisenstaedt, editors A.J. Kox Jean Eisenstaedt Editors The Universe of General Relativity Birkhauser¨ Boston • Basel • Berlin A.J. Kox Jean Eisenstaedt Universiteit van Amsterdam Observatoire de Paris Instituut voor Theoretische Fysica SYRTE/UMR8630–CNRS Valckenierstraat 65 F-75014 Paris Cedex 1018 XE Amsterdam France The Netherlands AMS Subject Classification (2000): 01A60, 83-03, 83-06 Library of Congress Cataloging-in-Publication Data The universe of general relativity / A.J. Kox, editors, Jean Eisenstaedt. p. -
22. Big-Bang Cosmology
1 22. Big-Bang Cosmology 22. Big-Bang Cosmology Revised August 2019 by K.A. Olive (Minnesota U.) and J.A. Peacock (Edinburgh U.). 22.1 Introduction to Standard Big-Bang Model The observed expansion of the Universe [1–3] is a natural (almost inevitable) result of any homogeneous and isotropic cosmological model based on general relativity. However, by itself, the Hubble expansion does not provide sufficient evidence for what we generally refer to as the Big-Bang model of cosmology. While general relativity is in principle capable of describing the cosmology of any given distribution of matter, it is extremely fortunate that our Universe appears to be homogeneous and isotropic on large scales. Together, homogeneity and isotropy allow us to extend the Copernican Principle to the Cosmological Principle, stating that all spatial positions in the Universe are essentially equivalent. The formulation of the Big-Bang model began in the 1940s with the work of George Gamow and his collaborators, Ralph Alpher and Robert Herman. In order to account for the possibility that the abundances of the elements had a cosmological origin, they proposed that the early Universe was once very hot and dense (enough so as to allow for the nucleosynthetic processing of hydrogen), and has subsequently expanded and cooled to its present state [4,5]. In 1948, Alpher and Herman predicted that a direct consequence of this model is the presence of a relic background radiation with a temperature of order a few K [6,7]. Of course this radiation was observed 16 years later as the Cosmic Microwave Background (CMB) [8].