THE MATHEMATICS of GRAVITATIONAL WAVES Courtesy of LIGO/T
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
On the Status of the Geodesic Principle in Newtonian and Relativistic Physics1
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilSci Archive On the Status of the Geodesic Principle in Newtonian and Relativistic Physics1 James Owen Weatherall2 Logic and Philosophy of Science University of California, Irvine Abstract A theorem due to Bob Geroch and Pong Soo Jang [\Motion of a Body in General Relativ- ity." Journal of Mathematical Physics 16(1), (1975)] provides a sense in which the geodesic principle has the status of a theorem in General Relativity (GR). I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton- Cartan theory) [Weatherall, J. O. \The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52(3), (2011)]. Here I compare the interpretations of these two the- orems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case. 1 Introduction The geodesic principle is the central principle of General Relativity (GR) that describes the inertial motion of test particles. It states that free massive test point particles traverse timelike geodesics. There is a long-standing view, originally due to Einstein, that the geodesic principle has a special status in GR that arises because it can be understood as a theorem, rather than a postulate, of the theory. (It turns out that capturing the geodesic principle as a theorem in GR is non-trivial, but a result due to Bob Geroch and Pong Soo Jang (1975) 1Thank you to David Malament and Jeff Barrett for helpful comments on a previous version of this paper and for many stimulating conversations on this topic. -
A Mathematical Derivation of the General Relativistic Schwarzschild
A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components ................. -
Aspects of Black Hole Physics
Aspects of Black Hole Physics Andreas Vigand Pedersen The Niels Bohr Institute Academic Advisor: Niels Obers e-mail: [email protected] Abstract: This project examines some of the exact solutions to Einstein’s theory, the theory of linearized gravity, the Komar definition of mass and angular momentum in general relativity and some aspects of (four dimen- sional) black hole physics. The project assumes familiarity with the basics of general relativity and differential geometry, but is otherwise intended to be self contained. The project was written as a ”self-study project” under the supervision of Niels Obers in the summer of 2008. Contents Contents ..................................... 1 Contents ..................................... 1 Preface and acknowledgement ......................... 2 Units, conventions and notation ........................ 3 1 Stationary solutions to Einstein’s equation ............ 4 1.1 Introduction .............................. 4 1.2 The Schwarzschild solution ...................... 6 1.3 The Reissner-Nordstr¨om solution .................. 18 1.4 The Kerr solution ........................... 24 1.5 The Kerr-Newman solution ..................... 28 2 Mass, charge and angular momentum (stationary spacetimes) 30 2.1 Introduction .............................. 30 2.2 Linearized Gravity .......................... 30 2.3 The weak field approximation .................... 35 2.3.1 The effect of a mass distribution on spacetime ....... 37 2.3.2 The effect of a charged mass distribution on spacetime .. 39 2.3.3 The effect of a rotating mass distribution on spacetime .. 40 2.4 Conserved currents in general relativity ............... 43 2.4.1 Komar integrals ........................ 49 2.5 Energy conditions ........................... 53 3 Black holes ................................ 57 3.1 Introduction .............................. 57 3.2 Event horizons ............................ 57 3.2.1 The no-hair theorem and Hawking’s area theorem .... -
Measurement of the Speed of Gravity
Measurement of the Speed of Gravity Yin Zhu Agriculture Department of Hubei Province, Wuhan, China Abstract From the Liénard-Wiechert potential in both the gravitational field and the electromagnetic field, it is shown that the speed of propagation of the gravitational field (waves) can be tested by comparing the measured speed of gravitational force with the measured speed of Coulomb force. PACS: 04.20.Cv; 04.30.Nk; 04.80.Cc Fomalont and Kopeikin [1] in 2002 claimed that to 20% accuracy they confirmed that the speed of gravity is equal to the speed of light in vacuum. Their work was immediately contradicted by Will [2] and other several physicists. [3-7] Fomalont and Kopeikin [1] accepted that their measurement is not sufficiently accurate to detect terms of order , which can experimentally distinguish Kopeikin interpretation from Will interpretation. Fomalont et al [8] reported their measurements in 2009 and claimed that these measurements are more accurate than the 2002 VLBA experiment [1], but did not point out whether the terms of order have been detected. Within the post-Newtonian framework, several metric theories have studied the radiation and propagation of gravitational waves. [9] For example, in the Rosen bi-metric theory, [10] the difference between the speed of gravity and the speed of light could be tested by comparing the arrival times of a gravitational wave and an electromagnetic wave from the same event: a supernova. Hulse and Taylor [11] showed the indirect evidence for gravitational radiation. However, the gravitational waves themselves have not yet been detected directly. [12] In electrodynamics the speed of electromagnetic waves appears in Maxwell equations as c = √휇0휀0, no such constant appears in any theory of gravity. -
Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
Legal "Black Hole"? Extraterritorial State Action and International Treaty Law on Civil and Political Rights
Michigan Journal of International Law Volume 26 Issue 3 2005 Legal "Black Hole"? Extraterritorial State Action and International Treaty Law on Civil and Political Rights Ralph Wilde University of London Follow this and additional works at: https://repository.law.umich.edu/mjil Part of the Human Rights Law Commons, Military, War, and Peace Commons, and the National Security Law Commons Recommended Citation Ralph Wilde, Legal "Black Hole"? Extraterritorial State Action and International Treaty Law on Civil and Political Rights, 26 MICH. J. INT'L L. 739 (2005). Available at: https://repository.law.umich.edu/mjil/vol26/iss3/1 This Article is brought to you for free and open access by the Michigan Journal of International Law at University of Michigan Law School Scholarship Repository. It has been accepted for inclusion in Michigan Journal of International Law by an authorized editor of University of Michigan Law School Scholarship Repository. For more information, please contact [email protected]. LEGAL "BLACK HOLE"? EXTRATERRITORIAL STATE ACTION AND INTERNATIONAL TREATY LAW ON CIVIL AND POLITICAL RIGHTSt Ralph Wilde* I. INTRODUCTION ......................................................................... 740 II. EXTRATERRITORIAL STATE ACTIVITIES ................................... 741 III. THE NEED FOR GREATER SCRUTINY ........................................ 752 A. Ignoring ExtraterritorialActivity ...................................... 753 B. GreaterRisks of Rights Violations in the ExtraterritorialContext .................................................... -
Einstein's Mistakes
Einstein’s Mistakes Einstein was the greatest genius of the Twentieth Century, but his discoveries were blighted with mistakes. The Human Failing of Genius. 1 PART 1 An evaluation of the man Here, Einstein grows up, his thinking evolves, and many quotations from him are listed. Albert Einstein (1879-1955) Einstein at 14 Einstein at 26 Einstein at 42 3 Albert Einstein (1879-1955) Einstein at age 61 (1940) 4 Albert Einstein (1879-1955) Born in Ulm, Swabian region of Southern Germany. From a Jewish merchant family. Had a sister Maja. Family rejected Jewish customs. Did not inherit any mathematical talent. Inherited stubbornness, Inherited a roguish sense of humor, An inclination to mysticism, And a habit of grüblen or protracted, agonizing “brooding” over whatever was on its mind. Leading to the thought experiment. 5 Portrait in 1947 – age 68, and his habit of agonizing brooding over whatever was on its mind. He was in Princeton, NJ, USA. 6 Einstein the mystic •“Everyone who is seriously involved in pursuit of science becomes convinced that a spirit is manifest in the laws of the universe, one that is vastly superior to that of man..” •“When I assess a theory, I ask myself, if I was God, would I have arranged the universe that way?” •His roguish sense of humor was always there. •When asked what will be his reactions to observational evidence against the bending of light predicted by his general theory of relativity, he said: •”Then I would feel sorry for the Good Lord. The theory is correct anyway.” 7 Einstein: Mathematics •More quotations from Einstein: •“How it is possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” •Questions asked by many people and Einstein: •“Is God a mathematician?” •His conclusion: •“ The Lord is cunning, but not malicious.” 8 Einstein the Stubborn Mystic “What interests me is whether God had any choice in the creation of the world” Some broadcasters expunged the comment from the soundtrack because they thought it was blasphemous. -
“Geodesic Principle” in General Relativity∗
A Remark About the “Geodesic Principle” in General Relativity∗ Version 3.0 David B. Malament Department of Logic and Philosophy of Science 3151 Social Science Plaza University of California, Irvine Irvine, CA 92697-5100 [email protected] 1 Introduction General relativity incorporates a number of basic principles that correlate space- time structure with physical objects and processes. Among them is the Geodesic Principle: Free massive point particles traverse timelike geodesics. One can think of it as a relativistic version of Newton’s first law of motion. It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein’s ∗I am grateful to Robert Geroch for giving me the basic idea for the counterexample (proposition 3.2) that is the principal point of interest in this note. Thanks also to Harvey Brown, Erik Curiel, John Earman, David Garfinkle, John Manchak, Wayne Myrvold, John Norton, and Jim Weatherall for comments on an earlier draft. 1 ab equation (or of the conservation principle ∇aT = 0 that is, itself, a conse- quence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic prin- ciple can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). -
Orbital Resonances and GPU Acceleration of Binary Black Hole Inspiral Simulations
Orbital resonances and GPU acceleration of binary black hole inspiral simulations by Adam Gabriel Marcel Lewis A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Physics University of Toronto c Copyright 2018 by Adam Gabriel Marcel Lewis Abstract Orbital resonances and GPU acceleration of binary black hole inspiral simulations Adam Gabriel Marcel Lewis Doctor of Philosophy Department of Physics University of Toronto 2018 Numerical relativity, the direct numerical integration of the Einstein field equations, is now a mature subfield of computational physics, playing a critical role in the generation of signal templates for comparison with data from ground-based gravitational wave de- tectors. The application of numerical relativity techniques to new problems is at present complicated by long wallclock times and intricate code. In this thesis we lay groundwork to improve this situation by presenting a GPU port of the numerical relativity code SpEC. Our port keeps code maintenance feasible by relying on various layers of automation, and achieves high performance across a variety of GPUs. We secondly introduce a C++ soft- ware package, TLoops, which allows numerical manipulation of tensors using single-line C++ source-code expressions resembling familiar tensor calculus notation. These expres- sions may be compiled and executed immediately, but also can be used to automatically generate equivalent GPU or low-level CPU code, which then executes in their place. The GPU code in particular achieves near-peak performance. Finally, we present simulations of eccentric binary black holes. We develop new methods to extract the fundamental fre- quencies of these systems. -
Comments on the 2011 Shaw Prize in Mathematical Sciences - - an Analysis of Collectively Formed Errors in Physics by C
Global Journal of Science Frontier Research Physics and Space Science Volume 12 Issue 4 Version 1.0 June 2012 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896 Comments on the 2011 Shaw Prize in Mathematical Sciences - - An Analysis of Collectively Formed Errors in Physics By C. Y. Lo Applied and Pure Research Institute, Nashua, NH Abstract - The 2011 Shaw Prize in mathematical sciences is shared by Richard S. Hamilton and D. Christodoulou. However, the work of Christodoulou on general relativity is based on obscure errors that implicitly assumed essentially what is to be proved, and thus gives misleading results. The problem of Einstein’s equation was discovered by Gullstrand of the 1921 Nobel Committee. In 1955, Gullstrand is proven correct. The fundamental errors of Christodoulou were due to his failure to distinguish the difference between mathematics and physics. His subsequent errors in mathematics and physics were accepted since judgments were based not on scientific evidence as Galileo advocates, but on earlier incorrect speculations. Nevertheless, the Committee for the Nobel Prize in Physics was also misled as shown in their 1993 press release. Here, his errors are identified as related to accumulated mistakes in the field, and are illustrated with examples understandable at the undergraduate level. Another main problem is that many theorists failed to understand the principle of causality adequately. It is unprecedented to award a prize for mathematical errors. Keywords : Nobel Prize; general relativity; Einstein equation, Riemannian Space; the non- existence of dynamic solution; Galileo. GJSFR-A Classification : 04.20.-q, 04.20.Cv Comments on the 2011 Shaw Prize in Mathematical Sciences -- An Analysis of Collectively Formed Errors in Physics Strictly as per the compliance and regulations of : © 2012. -
Curriculum Overview Physics/Pre-AP 2018-2019 1St Nine Weeks
Curriculum Overview Physics/Pre-AP 2018-2019 1st Nine Weeks RESOURCES: Essential Physics (Ergopedia – online book) Physics Classroom http://www.physicsclassroom.com/ PHET Simulations https://phet.colorado.edu/ ONGOING TEKS: 1A, 1B, 2A, 2B, 2C, 2D, 2F, 2G, 2H, 2I, 2J,3E 1) SAFETY TEKS 1A, 1B Vocabulary Fume hood, fire blanket, fire extinguisher, goggle sanitizer, eye wash, safety shower, impact goggles, chemical safety goggles, fire exit, electrical safety cut off, apron, broken glass container, disposal alert, biological hazard, open flame alert, thermal safety, sharp object safety, fume safety, electrical safety, plant safety, animal safety, radioactive safety, clothing protection safety, fire safety, explosion safety, eye safety, poison safety, chemical safety Key Concepts The student will be able to determine if a situation in the physics lab is a safe practice and what appropriate safety equipment and safety warning signs may be needed in a physics lab. The student will be able to determine the proper disposal or recycling of materials in the physics lab. Essential Questions 1. How are safe practices in school, home or job applied? 2. What are the consequences for not using safety equipment or following safe practices? 2) SCIENCE OF PHYSICS: Glossary, Pages 35, 39 TEKS 2B, 2C Vocabulary Matter, energy, hypothesis, theory, objectivity, reproducibility, experiment, qualitative, quantitative, engineering, technology, science, pseudo-science, non-science Key Concepts The student will know that scientific hypotheses are tentative and testable statements that must be capable of being supported or not supported by observational evidence. The student will know that scientific theories are based on natural and physical phenomena and are capable of being tested by multiple independent researchers.