Annual Report 2000
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The Schwarzschild Metric and Applications 1
The Schwarzschild Metric and Applications 1 Analytic solutions of Einstein's equations are hard to come by. It's easier in situations that e hibit symmetries. 1916: Karl Schwarzschild sought the metric describing the static, spherically symmetric spacetime surrounding a spherically symmetric mass distribution. A static spacetime is one for which there exists a time coordinate t such that i' all the components of g are independent of t ii' the line element ds( is invariant under the transformation t -t A spacetime that satis+es (i) but not (ii' is called stationary. An example is a rotating azimuthally symmetric mass distribution. The metric for a static spacetime has the form where xi are the spatial coordinates and dl( is a time*independent spatial metric. -ross-terms dt dxi are missing because their presence would violate condition (ii'. 23ote: The Kerr metric, which describes the spacetime outside a rotating ( axisymmetric mass distribution, contains a term ∝ dt d.] To preser)e spherical symmetry& dl( can be distorted from the flat-space metric only in the radial direction. In 5at space, (1) r is the distance from the origin and (2) 6r( is the area of a sphere. Let's de+ne r such that (2) remains true but (1) can be violated. Then, A,xi' A,r) in cases of spherical symmetry. The Ricci tensor for this metric is diagonal, with components S/ 10.1 /rimes denote differentiation with respect to r. The region outside the spherically symmetric mass distribution is empty. 9 The vacuum Einstein equations are R = 0. To find A,r' and B,r'# (. -
18Th International Conference on General
18TH INTERNATIONAL CONFERENCE ON GENERAL RELATIVITY AND GRAVITATION (GRG18) 8 – 13 July 2007 7TH EDOARDO AMALDI CONFERENCE ON GRAVITATIONAL WAVES (AMALDI7) www.grg18.com 8 – 14 July 2007 www.Amaldi7.com Sydney Convention & Exhibition Centre • Darling Harbour • Sydney Australia INSPIRALLING BLACK HOLES RAPID EPANSION OF AN UNSTABLE NEUTRON STAR Credit: Seidel (LSU/AEI) / Kaehler (ZIB) Credit: Rezzolla (AEI) / Benger (ZIB) GRAZING COLLISION OF TWO BLACK HOLES VISUALISATION OF A GRAVITATIONAL WAVE HAVING Credit: Seidel (AEI) / Benger (ZIB) TEUKOLSKY'S SOLUTION AS INITIAL DATA Credit: Seidel (AEI) / Benger (ZIB) ROLLER COASTER DISTORTED BY SPECIAL RELATIVISTIC EFFECTS Credit: Michael Hush, Department of Physics, The Australian National University The program for GRG18 will incorporate all areas of General Relativity and Gravitation including Classical General Relativity; Numerical Relativity; Relativistic Astrophysics and Cosmology; Experimental Work on Gravity and Quantum Issues in Gravitation. The program for Amaldi7 will cover all aspects of Gravitational Wave Physics and Detection. PLENARY LECTURES Peter Schneider (Bonn) Donald Marolf USA Bernard Schutz Germany Bernd Bruegman (Friedrich-Schiller-University Jena) Gravitational lensing David McClelland Australia Robin Stebbins United States Numerical relativity Daniel Shaddock (JPL California Institute of Technology) Jorge Pullin USA Kimio Tsubono Japan Daniel Eisenstein (University of Arizona) Gravitational wave detection from space: technology challenges Norna Robertson USA Stefano Vitale Italy Dark energy Stan Whitcomb (California Institute of Technology) Misao Sasaki Japan Clifford Will United States Ground-based gravitational wave detection: now and future Bernard F Schutz Germany Francis Everitt (Stanford University) Susan M. Scott Australia LOCAL ORGANISING COMMITTEE Gravity Probe B and precision tests of General Relativity Chair: Susan M. -
Physics 305, Fall 2008 Problem Set 8 Due Thursday, December 3
Physics 305, Fall 2008 Problem Set 8 due Thursday, December 3 1. Einstein A and B coefficients (25 pts): This problem is to make sure that you have read and understood Griffiths 9.3.1. Consider a system that consists of atoms with two energy levels E1 and E2 and a thermal gas of photons. There are N1 atoms with energy E1, N2 atoms with energy E2 and the energy density of photons with frequency ! = (E2 − E1)=~ is W (!). In thermal equilbrium at temperature T , W is given by the Planck distribution: ~!3 1 W (!) = 2 3 : π c exp(~!=kBT ) − 1 According to Einstein, this formula can be understood by assuming the following rules for the interaction between the atoms and the photons • Atoms with energy E1 can absorb a photon and make a transition to the excited state with energy E2; the probability per unit time for this transition to take place is proportional to W (!), and therefore given by Pabs = B12W (!) for some constant B12. • Atoms with energy E2 can make a transition to the lower energy state via stimulated emission of a photon. The probability per unit time for this to happen is Pstim = B21W (!) for some constant B21. • Atoms with energy E2 can also fall back into the lower energy state via spontaneous emission. The probability per unit time for spontaneous emission is independent of W (!). Let's call this probability Pspont = A21 : A21, B21, and B12 are known as Einstein coefficients. a. Write a differential equation for the time dependence of the occupation numbers N1 and N2. -
Inis: Terminology Charts
IAEA-INIS-13A(Rev.0) XA0400071 INIS: TERMINOLOGY CHARTS agree INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, AUGUST 1970 INISs TERMINOLOGY CHARTS TABLE OF CONTENTS FOREWORD ... ......... *.* 1 PREFACE 2 INTRODUCTION ... .... *a ... oo 3 LIST OF SUBJECT FIELDS REPRESENTED BY THE CHARTS ........ 5 GENERAL DESCRIPTOR INDEX ................ 9*999.9o.ooo .... 7 FOREWORD This document is one in a series of publications known as the INIS Reference Series. It is to be used in conjunction with the indexing manual 1) and the thesaurus 2) for the preparation of INIS input by national and regional centrea. The thesaurus and terminology charts in their first edition (Rev.0) were produced as the result of an agreement between the International Atomic Energy Agency (IAEA) and the European Atomic Energy Community (Euratom). Except for minor changesq the terminology and the interrela- tionships btween rms are those of the December 1969 edition of the Euratom Thesaurus 3) In all matters of subject indexing and ontrol, the IAEA followed the recommendations of Euratom for these charts. Credit and responsibility for the present version of these charts must go to Euratom. Suggestions for improvement from all interested parties. particularly those that are contributing to or utilizing the INIS magnetic-tape services are welcomed. These should be addressed to: The Thesaurus Speoialist/INIS Section Division of Scientific and Tohnioal Information International Atomic Energy Agency P.O. Box 590 A-1011 Vienna, Austria International Atomic Energy Agency Division of Sientific and Technical Information INIS Section June 1970 1) IAEA-INIS-12 (INIS: Manual for Indexing) 2) IAEA-INIS-13 (INIS: Thesaurus) 3) EURATOM Thesaurusq, Euratom Nuclear Documentation System. -
Science & ROGER PENROSE
Science & ROGER PENROSE Live Webinar - hosted by the Center for Consciousness Studies August 3 – 6, 2021 9:00 am – 12:30 pm (MST-Arizona) each day 4 Online Live Sessions DAY 1 Tuesday August 3, 2021 9:00 am to 12:30 pm MST-Arizona Overview / Black Holes SIR ROGER PENROSE (Nobel Laureate) Oxford University, UK Tuesday August 3, 2021 9:00 am – 10:30 am MST-Arizona Roger Penrose was born, August 8, 1931 in Colchester Essex UK. He earned a 1st class mathematics degree at University College London; a PhD at Cambridge UK, and became assistant lecturer, Bedford College London, Research Fellow St John’s College, Cambridge (now Honorary Fellow), a post-doc at King’s College London, NATO Fellow at Princeton, Syracuse, and Cornell Universities, USA. He also served a 1-year appointment at University of Texas, became a Reader then full Professor at Birkbeck College, London, and Rouse Ball Professor of Mathematics, Oxford University (during which he served several 1/2-year periods as Mathematics Professor at Rice University, Houston, Texas). He is now Emeritus Rouse Ball Professor, Fellow, Wadham College, Oxford (now Emeritus Fellow). He has received many awards and honorary degrees, including knighthood, Fellow of the Royal Society and of the US National Academy of Sciences, the De Morgan Medal of London Mathematical Society, the Copley Medal of the Royal Society, the Wolf Prize in mathematics (shared with Stephen Hawking), the Pomeranchuk Prize (Moscow), and one half of the 2020 Nobel Prize in Physics, the other half shared by Reinhard Genzel and Andrea Ghez. -
Recent Observations of Gravitational Waves by LIGO and Virgo Detectors
universe Review Recent Observations of Gravitational Waves by LIGO and Virgo Detectors Andrzej Królak 1,2,* and Paritosh Verma 2 1 Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland 2 National Centre for Nuclear Research, 05-400 Otwock, Poland; [email protected] * Correspondence: [email protected] Abstract: In this paper we present the most recent observations of gravitational waves (GWs) by LIGO and Virgo detectors. We also discuss contributions of the recent Nobel prize winner, Sir Roger Penrose to understanding gravitational radiation and black holes (BHs). We make a short introduction to GW phenomenon in general relativity (GR) and we present main sources of detectable GW signals. We describe the laser interferometric detectors that made the first observations of GWs. We briefly discuss the first direct detection of GW signal that originated from a merger of two BHs and the first detection of GW signal form merger of two neutron stars (NSs). Finally we present in more detail the observations of GW signals made during the first half of the most recent observing run of the LIGO and Virgo projects. Finally we present prospects for future GW observations. Keywords: gravitational waves; black holes; neutron stars; laser interferometers 1. Introduction The first terrestrial direct detection of GWs on 14 September 2015, was a milestone Citation: Kro´lak, A.; Verma, P. discovery, and it opened up an entirely new window to explore the universe. The combined Recent Observations of Gravitational effort of various scientists and engineers worldwide working on the theoretical, experi- Waves by LIGO and Virgo Detectors. -
MATTERS of GRAVITY, a Newsletter for the Gravity Community, Number 3
MATTERS OF GRAVITY Number 3 Spring 1994 Table of Contents Editorial ................................................... ................... 2 Correspondents ................................................... ............ 2 Gravity news: Open Letter to gravitational physicists, Beverly Berger ........................ 3 A Missouri relativist in King Gustav’s Court, Clifford Will .................... 6 Gary Horowitz wins the Xanthopoulos award, Abhay Ashtekar ................ 9 Research briefs: Gamma-ray bursts and their possible cosmological implications, Peter Meszaros 12 Current activity and results in laboratory gravity, Riley Newman ............. 15 Update on representations of quantum gravity, Donald Marolf ................ 19 Ligo project report: December 1993, Rochus E. Vogt ......................... 23 Dark matter or new gravity?, Richard Hammond ............................. 25 Conference Reports: Gravitational waves from coalescing compact binaries, Curt Cutler ........... 28 Mach’s principle: from Newton’s bucket to quantum gravity, Dieter Brill ..... 31 Cornelius Lanczos international centenary conference, David Brown .......... 33 Third Midwest relativity conference, David Garfinkle ......................... 36 arXiv:gr-qc/9402002v1 1 Feb 1994 Editor: Jorge Pullin Center for Gravitational Physics and Geometry The Pennsylvania State University University Park, PA 16802-6300 Fax: (814)863-9608 Phone (814)863-9597 Internet: [email protected] 1 Editorial Well, this newsletter is growing into its third year and third number with a lot of strength. In fact, maybe too much strength. Twelve articles and 37 (!) pages. In this number, apart from the ”traditional” research briefs and conference reports we also bring some news for the community, therefore starting to fulfill the original promise of bringing the gravity/relativity community closer together. As usual I am open to suggestions, criticisms and proposals for articles for the next issue, due September 1st. Many thanks to the authors and the correspondents who made this issue possible. -
Fundamentals of Radiative Transfer
RADIATIVE PROCESSE S IN ASTROPHYSICS GEORGE B. RYBICKI, ALAN P. LIGHTMAN Copyright 0 2004 WY-VCHVerlag GmbH L Co. KCaA FUNDAMENTALS OF RADIATIVE TRANSFER 1.1 THE ELECTROMAGNETIC SPECTRUM; ELEMENTARY PROPERTIES OF RADIATION Electromagnetic radiation can be decomposed into a spectrum of con- stituent components by a prism, grating, or other devices, as was dis- covered quite early (Newton, 1672, with visible light). The spectrum corresponds to waves of various wavelengths and frequencies, related by Xv=c, where v is the frequency of the wave, h is its wavelength, and c-3.00~10" cm s-I is the free space velocity of light. (For waves not traveling in a vacuum, c is replaced by the appropriate velocity of the wave in the medium.) We can divide the spectrum up into various regions, as is done in Figure 1.1. For convenience we have given the energy E = hv and temperature T= E/k associated with each wavelength. Here h is Planck's constant = 6.625 X erg s, and k is Boltzmann's constant = 1.38 X erg K-I. This chart will prove to be quite useful in converting units or in getting a quick view of the relevant magnitude of quantities in a given portion of the spectrum. The boundaries between different regions are somewhat arbitrary, but conform to accepted usage. 1 2 Fundamentals of Radiatiw Transfer -6 -5 -4 -3 -2 -1 0 1 2 1 I 1 I I I I 1 1 log A (cm) Wavelength I I I I I log Y IHr) Frequency 0 -1 -2 -3 -4 -5 -6 I I I I I I I log Elev) Energy 43 21 0-1 I I 1 I I I log T("K)Temperature Y ray X-ray UV Visible IR Radio Figum 1.1 The electromagnetic spctnun. -
Complementary & the Copenhagen Interpretation
Complementarity and the Copenhagen Interpretation of Quantum Mechanics Click here to go to the UPSCALE home page. Click here to go to the Physics Virtual Bookshelf home page. Introduction Neils Bohr (1885 - 1962) was one of the giants in the development of Quantum Mechanics. He is best known for: 1. The development of the Bohr Model of the Atom in 1913. A small document on this topic is available here. 2. The principle of Complementarity, the "heart" of Bohr's search for the significance of the quantum idea. This principle led him to: 3. The Copenhagen Interpretation of Quantum Mechanics. In this document we discuss Complementarity and then the Copenhagen Interpretation. But first we shall briefly discuss the general issue of interpretations of Quantum Mechanics, and briefly describe two interpretations. The discussion assumes some knowledge of the Feynman Double Slit, such as is discussed here; it also assumes some knowledge of Schrödinger's Cat, such as is discussed here. Finally, further discussion of interpretations of Quantum Mechanics can be meaningfully given with some knowledge of Bell's Theorem; a document on that topic is here. The level of discussion in what follows is based on an upper-year liberal arts course in modern physics without mathematics given at the University of Toronto. In that context, the discussion of Bell's Theorem mentioned in the previous paragraph is deferred until later. A recommended reference on the material discussed below is: F. David Peat, Einstein's Moon (Contemporary Books, 1990), ISBN 0-8092-4512-4 (cloth), 0-8092-3965-5 (paper). Interpretations of Quantum Mechanics Although the basic mathematical formalism of Quantum Mechanics was developed independently by Heisenberg and Schrödinger in 1926, a full and accepted interpretation of what that mathematics means still eludes us. -
Acknowledgements Acknowl
1277 Acknowledgements Acknowl. A.1 The Properties of Light by Helen Wächter, Markus W. Sigrist by Richard Haglund The authors thank a number of coworkers for their The author thanks Prof. Emil Wolf for helpful discus- valuable input, notably R. Bartlome, Dr. C. Fischer, sions, and gratefully acknowledges the financial support D. Marinov, Dr. J. Rey, M. Stahel, and Dr. D. Vogler. of a Senior Scientist Award from the Alexander von The financial support by the Swiss National Science Humboldt Foundation and of the Medical Free-Electron Foundation and ETH Zurich for the isotopomer studies Laser program of the Department of Defense (Con- is gratefully acknowledged. tract F49620-01-1-0429) during the preparation of this chapter. by Jürgen Helmcke In writing the chapter on frequency-stabilized lasers, A.4 Nonlinear Optics the author has greatly benefited from fruitful coopera- by Aleksei Zheltikov, Anne L’Huillier, Ferenc Krausz tion and helpful discussions with his colleagues at PTB, We acknowledge the support of the European Com- in particular with Drs. Fritz Riehle, Harald Schnatz, munity’s Human Potential Programme under contract Uwe Sterr, and Harald Telle. Special thanks belong to HPRN-CT-2000-00133 (ATTO) and the Swedish Sci- Dr. Fritz Riehle for his careful and critical reading of the ence Council. manuscript. Part of the work discussed in this chapter was supported by the Deutsche Forschungsgemeinschaft A.5 Optical Materials and Their Properties (DFG) under SFB 407. by Klaus Bonrad The author of Sect. 5.9.2 is grateful to Dr. Thomas C.12 Femtosecond Laser Pulses: Däubler, Dr. Dirk Hertel, and Dr. -
A Model-Independent Characterisation of Strong Gravitational Lensing by Observables
universe Review A Model-Independent Characterisation of Strong Gravitational Lensing by Observables Jenny Wagner Astronomisches Rechen-Institut, Universität Heidelberg, Zentrum für Astronomie, Mönchhofstr. 12–14, 69120 Heidelberg, Germany; [email protected] Received: 31 May 2019; Accepted: 9 July 2019; Published: 23 July 2019 Abstract: When light from a distant source object, like a galaxy or a supernova, travels towards us, it is deflected by massive objects that lie in its path. When the mass density of the deflecting object exceeds a certain threshold, multiple, highly distorted images of the source are observed. This strong gravitational lensing effect has so far been treated as a model-fitting problem. Using the observed multiple images as constraints yields a self-consistent model of the deflecting mass density and the source object. As several models meet the constraints equally well, we develop a lens characterisation that separates data-based information from model assumptions. The observed multiple images allow us to determine local properties of the deflecting mass distribution on any mass scale from one simple set of equations. Their solution is unique and free of model-dependent degeneracies. The reconstruction of source objects can be performed completely model-independently, enabling us to study galaxy evolution without a lens-model bias. Our approach reduces the lens and source description to its data-based evidence that all models agree upon, simplifies an automated treatment of large datasets, and allows for an extrapolation to a global description resembling model-based descriptions. Keywords: cosmology; dark matter; gravitational lensing; strong; methods; analytical; galaxy clusters; general; galaxies; mass function; methods; data analysis; cosmology; distance scale 1. -
Astrophysical Applications of the Einstein Ring Gravitational Lens, Mg1131+0456
ASTROPHYSICAL APPLICATIONS OF THE EINSTEIN RING GRAVITATIONAL LENS, MG1131+0456 by GRACE HSIU-LING CHEN Bachelor of Science, San Francisco State University, 1990 SUBMITTED TO THE DEPARTMENT OF PHYSICS IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1995 Copyright Q1995 Massachusetts Institute of Technology All rights reserved Signature of Author: Department of Physics May 1995 Certified by: Jacqueline N. Hewitt Thesis Supervisor Accepted by: George F. Koster #AVMIIrl:·1,~-1111 YI~j7-1t Chairman, Graduate Committee JUN 2 6 1995 UtslHl~tS ASTROPHYSICAL APPLICATIONS OF THE EINSTEIN RING GRAVITATIONAL LENS, MG1131+0456 by GRACE HSIU-LING CHEN ABSTRACT MG1131+0456 has been observed extensively with the Very Large Array tele- scope, and its complex morphology has provided the opportunity for investigating three important astrophysical problems: (1) the mass distribution in the lens galaxy, (2) the structure of the rotation measure distribution in the lens, and (3) the possi- bility of using the system to constrain the Hubble parameter. We determine the mass distribution of the lens galaxy by modeling the multifre- quency VLA images of the system. Among the models we have explored, the profile of the surface mass density of the best lens model contains a substantial core radius and declines asymptotically as r- 1 2±0 2.' The mass inside the lens enclosed by the average ring is very well constrained. If the source is at redshift of 2.0 and the lens is at redshift of 0.5, the mass of the lens enclosed by the ring radius is - 2.0 x 1011 solar mass.