Albert Einstein’S “Miraculous Year” in Which He Published Three Papers, All Within a Few Months, Describing Ideas That Have Since Influenced All of Modern Physics
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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 ................. -
Hendrik Antoon Lorentz's Struggle with Quantum Theory A. J
Hendrik Antoon Lorentz’s struggle with quantum theory A. J. Kox Archive for History of Exact Sciences ISSN 0003-9519 Volume 67 Number 2 Arch. Hist. Exact Sci. (2013) 67:149-170 DOI 10.1007/s00407-012-0107-8 1 23 Your article is published under the Creative Commons Attribution license which allows users to read, copy, distribute and make derivative works, as long as the author of the original work is cited. You may self- archive this article on your own website, an institutional repository or funder’s repository and make it publicly available immediately. 1 23 Arch. Hist. Exact Sci. (2013) 67:149–170 DOI 10.1007/s00407-012-0107-8 Hendrik Antoon Lorentz’s struggle with quantum theory A. J. Kox Received: 15 June 2012 / Published online: 24 July 2012 © The Author(s) 2012. This article is published with open access at Springerlink.com Abstract A historical overview is given of the contributions of Hendrik Antoon Lorentz in quantum theory. Although especially his early work is valuable, the main importance of Lorentz’s work lies in the conceptual clarifications he provided and in his critique of the foundations of quantum theory. 1 Introduction The Dutch physicist Hendrik Antoon Lorentz (1853–1928) is generally viewed as an icon of classical, nineteenth-century physics—indeed, as one of the last masters of that era. Thus, it may come as a bit of a surprise that he also made important contribu- tions to quantum theory, the quintessential non-classical twentieth-century develop- ment in physics. The importance of Lorentz’s work lies not so much in his concrete contributions to the actual physics—although some of his early work was ground- breaking—but rather in the conceptual clarifications he provided and his critique of the foundations and interpretations of the new ideas. -
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. -
Hypercomplex Algebras and Their Application to the Mathematical
Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H. -
Ripples in Spacetime
editorial Ripples in spacetime The 2017 Nobel prize in Physics has been awarded to Rainer Weiss, Barry C. Barish and Kip S. Thorne “for decisive contributions to the LIGO detector and the observation of gravitational waves”. It is, frankly, difficult to find something original to say about the detection of gravitational waves that hasn’t been said already. The technological feat of measuring fluctuations in the fabric of spacetime less than one-thousandth the width of an atomic nucleus is quite simply astonishing. The scientific achievement represented by the confirmation of a century-old prediction by Albert Einstein is unique. And the collaborative effort that made the discovery possible — the Laser Interferometer Gravitational-Wave Observatory (LIGO) — is inspiring. Adapted from Phys. Rev. Lett. 116, 061102 (2016), under Creative Commons Licence. Rainer Weiss and Kip Thorne were, along with the late Ronald Drever, founders of the project that eventually became known Barry Barish, who was the director Last month we received a spectacular as LIGO. In the 1960s, Thorne, a black hole of LIGO from 1997 to 2005, is widely demonstration that talk of a new era expert, had come to believe that his objects of credited with transforming it into a ‘big of gravitational astronomy was no interest should be detectable as gravitational physics’ collaboration, and providing the exaggeration. Cued by detections at LIGO waves. Separately, and inspired by previous organizational structure required to ensure and Virgo, an interferometer based in Pisa, proposals, Weiss came up with the first it worked. Of course, the passion, skill and Italy, more than 70 teams of researchers calculations detailing how an interferometer dedication of the thousand or so scientists working at different telescopes around could be used to detect them in 1972. -
Introduction to General Relativity
INTRODUCTION TO GENERAL RELATIVITY Gerard 't Hooft Institute for Theoretical Physics Utrecht University and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail: [email protected] internet: http://www.phys.uu.nl/~thooft/ Version November 2010 1 Prologue General relativity is a beautiful scheme for describing the gravitational ¯eld and the equations it obeys. Nowadays this theory is often used as a prototype for other, more intricate constructions to describe forces between elementary particles or other branches of fundamental physics. This is why in an introduction to general relativity it is of importance to separate as clearly as possible the various ingredients that together give shape to this paradigm. After explaining the physical motivations we ¯rst introduce curved coordinates, then add to this the notion of an a±ne connection ¯eld and only as a later step add to that the metric ¯eld. One then sees clearly how space and time get more and more structure, until ¯nally all we have to do is deduce Einstein's ¯eld equations. These notes materialized when I was asked to present some lectures on General Rela- tivity. Small changes were made over the years. I decided to make them freely available on the web, via my home page. Some readers expressed their irritation over the fact that after 12 pages I switch notation: the i in the time components of vectors disappears, and the metric becomes the ¡ + + + metric. Why this \inconsistency" in the notation? There were two reasons for this. The transition is made where we proceed from special relativity to general relativity. -
General Relativity Fall 2019 Lecture 11: the Riemann Tensor
General Relativity Fall 2019 Lecture 11: The Riemann tensor Yacine Ali-Ha¨ımoud October 8th 2019 The Riemann tensor quantifies the curvature of spacetime, as we will see in this lecture and the next. RIEMANN TENSOR: BASIC PROPERTIES α γ Definition { Given any vector field V , r[αrβ]V is a tensor field. Let us compute its components in some coordinate system: σ σ λ σ σ λ r[µrν]V = @[µ(rν]V ) − Γ[µν]rλV + Γλ[µrν]V σ σ λ σ λ λ ρ = @[µ(@ν]V + Γν]λV ) + Γλ[µ @ν]V + Γν]ρV 1 = @ Γσ + Γσ Γρ V λ ≡ Rσ V λ; (1) [µ ν]λ ρ[µ ν]λ 2 λµν where all partial derivatives of V µ cancel out after antisymmetrization. σ Since the left-hand side is a tensor field and V is a vector field, we conclude that R λµν is a tensor field as well { this is the tensor division theorem, which I encourage you to think about on your own. You can also check that explicitly from the transformation law of Christoffel symbols. This is the Riemann tensor, which measures the non-commutation of second derivatives of vector fields { remember that second derivatives of scalar fields do commute, by assumption. It is completely determined by the metric, and is linear in its second derivatives. Expression in LICS { In a LICS the Christoffel symbols vanish but not their derivatives. Let us compute the latter: 1 1 @ Γσ = @ gσδ (@ g + @ g − @ g ) = ησδ (@ @ g + @ @ g − @ @ g ) ; (2) µ νλ 2 µ ν λδ λ νδ δ νλ 2 µ ν λδ µ λ νδ µ δ νλ since the first derivatives of the metric components (thus of its inverse as well) vanish in a LICS. -
I. I. Rabi Papers [Finding Aid]. Library of Congress. [PDF Rendered Tue Apr
I. I. Rabi Papers A Finding Aid to the Collection in the Library of Congress Manuscript Division, Library of Congress Washington, D.C. 1992 Revised 2010 March Contact information: http://hdl.loc.gov/loc.mss/mss.contact Additional search options available at: http://hdl.loc.gov/loc.mss/eadmss.ms998009 LC Online Catalog record: http://lccn.loc.gov/mm89076467 Prepared by Joseph Sullivan with the assistance of Kathleen A. Kelly and John R. Monagle Collection Summary Title: I. I. Rabi Papers Span Dates: 1899-1989 Bulk Dates: (bulk 1945-1968) ID No.: MSS76467 Creator: Rabi, I. I. (Isador Isaac), 1898- Extent: 41,500 items ; 105 cartons plus 1 oversize plus 4 classified ; 42 linear feet Language: Collection material in English Location: Manuscript Division, Library of Congress, Washington, D.C. Summary: Physicist and educator. The collection documents Rabi's research in physics, particularly in the fields of radar and nuclear energy, leading to the development of lasers, atomic clocks, and magnetic resonance imaging (MRI) and to his 1944 Nobel Prize in physics; his work as a consultant to the atomic bomb project at Los Alamos Scientific Laboratory and as an advisor on science policy to the United States government, the United Nations, and the North Atlantic Treaty Organization during and after World War II; and his studies, research, and professorships in physics chiefly at Columbia University and also at Massachusetts Institute of Technology. Selected Search Terms The following terms have been used to index the description of this collection in the Library's online catalog. They are grouped by name of person or organization, by subject or location, and by occupation and listed alphabetically therein. -
JOHN EARMAN* and CLARK GL YMUURT the GRAVITATIONAL RED SHIFT AS a TEST of GENERAL RELATIVITY: HISTORY and ANALYSIS
JOHN EARMAN* and CLARK GL YMUURT THE GRAVITATIONAL RED SHIFT AS A TEST OF GENERAL RELATIVITY: HISTORY AND ANALYSIS CHARLES St. John, who was in 1921 the most widely respected student of the Fraunhofer lines in the solar spectra, began his contribution to a symposium in Nncure on Einstein’s theories of relativity with the following statement: The agreement of the observed advance of Mercury’s perihelion and of the eclipse results of the British expeditions of 1919 with the deductions from the Einstein law of gravitation gives an increased importance to the observations on the displacements of the absorption lines in the solar spectrum relative to terrestrial sources, as the evidence on this deduction from the Einstein theory is at present contradictory. Particular interest, moreover, attaches to such observations, inasmuch as the mathematical physicists are not in agreement as to the validity of this deduction, and solar observations must eventually furnish the criterion.’ St. John’s statement touches on some of the reasons why the history of the red shift provides such a fascinating case study for those interested in the scientific reception of Einstein’s general theory of relativity. In contrast to the other two ‘classical tests’, the weight of the early observations was not in favor of Einstein’s red shift formula, and the reaction of the scientific community to the threat of disconfirmation reveals much more about the contemporary scientific views of Einstein’s theory. The last sentence of St. John’s statement points to another factor that both complicates and heightens the interest of the situation: in contrast to Einstein’s deductions of the advance of Mercury’s perihelion and of the bending of light, considerable doubt existed as to whether or not the general theory did entail a red shift for the solar spectrum. -
The Theory of Relativity and Applications: a Simple Introduction
The Downtown Review Volume 5 Issue 1 Article 3 December 2018 The Theory of Relativity and Applications: A Simple Introduction Ellen Rea Cleveland State University Follow this and additional works at: https://engagedscholarship.csuohio.edu/tdr Part of the Engineering Commons, and the Physical Sciences and Mathematics Commons How does access to this work benefit ou?y Let us know! Recommended Citation Rea, Ellen. "The Theory of Relativity and Applications: A Simple Introduction." The Downtown Review. Vol. 5. Iss. 1 (2018) . Available at: https://engagedscholarship.csuohio.edu/tdr/vol5/iss1/3 This Article is brought to you for free and open access by the Student Scholarship at EngagedScholarship@CSU. It has been accepted for inclusion in The Downtown Review by an authorized editor of EngagedScholarship@CSU. For more information, please contact [email protected]. Rea: The Theory of Relativity and Applications What if I told you that time can speed up and slow down? What if I told you that everything you think you know about gravity is a lie? When Albert Einstein presented his theory of relativity to the world in the early 20th century, he was proposing just that. And what’s more? He’s been proven correct. Einstein’s theory has two parts: special relativity, which deals with inertial reference frames and general relativity, which deals with the curvature of space- time. A surface level study of the theory and its consequences followed by a look at some of its applications will provide an introduction to one of the most influential scientific discoveries of the last century. -
A Brief History of Gravitational Waves
universe Review A Brief History of Gravitational Waves Jorge L. Cervantes-Cota 1, Salvador Galindo-Uribarri 1 and George F. Smoot 2,3,4,* 1 Department of Physics, National Institute for Nuclear Research, Km 36.5 Carretera Mexico-Toluca, Ocoyoacac, C.P. 52750 Mexico, Mexico; [email protected] (J.L.C.-C.); [email protected] (S.G.-U.) 2 Helmut and Ana Pao Sohmen Professor at Large, Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077 Hong Kong, China 3 Université Sorbonne Paris Cité, Laboratoire APC-PCCP, Université Paris Diderot, 10 rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France 4 Department of Physics and LBNL, University of California; MS Bldg 50-5505 LBNL, 1 Cyclotron Road Berkeley, 94720 CA, USA * Correspondence: [email protected]; Tel.:+1-510-486-5505 Academic Editors: Lorenzo Iorio and Elias C. Vagenas Received: 21 July 2016; Accepted: 2 September 2016; Published: 13 September 2016 Abstract: This review describes the discovery of gravitational waves. We recount the journey of predicting and finding those waves, since its beginning in the early twentieth century, their prediction by Einstein in 1916, theoretical and experimental blunders, efforts towards their detection, and finally the subsequent successful discovery. Keywords: gravitational waves; General Relativity; LIGO; Einstein; strong-field gravity; binary black holes 1. Introduction Einstein’s General Theory of Relativity, published in November 1915, led to the prediction of the existence of gravitational waves that would be so faint and their interaction with matter so weak that Einstein himself wondered if they could ever be discovered. -
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.