Complex Spacetimes and the Newman-Janis Trick
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Chapter 13 the Geometry of Circles
Chapter 13 TheR. Connelly geometry of circles Math 452, Spring 2002 Math 4520, Fall 2017 CLASSICAL GEOMETRIES So far we have been studying lines and conics in the Euclidean plane. What about circles, 14. The geometry of circles one of the basic objects of study in Euclidean geometry? One approach is to use the complex numbersSo far .we Recall have thatbeen the studying projectivities lines and of conics the projective in the Euclidean plane overplane., whichWhat weabout call 2, circles, Cone of the basic objects of study in Euclidean geometry? One approachC is to use CP are given by 3 by 3 matrices, and these projectivities restricted to a complex projective the complex numbers 1C. Recall that the projectivities of the projective plane over C, line,v.rhich which we wecall call Cp2, a CPare, given are the by Moebius3 by 3 matrices, functions, and whichthese projectivities themselves correspondrestricted to to a 2-by-2 matrices.complex Theprojective Moebius line, functions which we preservecall a Cpl the, are cross the Moebius ratio. This functions, is where which circles themselves come in. correspond to a 2 by 2 matrix. The Moebius functions preserve the cross ratio. This is 13.1where circles The come cross in. ratio for the complex field We14.1 look The for anothercross ratio geometric for the interpretation complex field of the cross ratio for the complex field, or better 1 yet forWeCP look= Cfor[f1g another. Recall geometric the polarinterpretation decomposition of the cross of a complexratio for numberthe complexz = refield,iθ, where r =orjz betterj is the yet magnitude for Cpl = ofCz U, and{ 00 } .Recallθ is the the angle polar that decomposition the line through of a complex 0 and znumbermakes with thez real -rei9, axis. -
1 Electromagnetic Fields and Charges In
Electromagnetic fields and charges in ‘3+1’ spacetime derived from symmetry in ‘3+3’ spacetime J.E.Carroll Dept. of Engineering, University of Cambridge, CB2 1PZ E-mail: [email protected] PACS 11.10.Kk Field theories in dimensions other than 4 03.50.De Classical electromagnetism, Maxwell equations 41.20Jb Electromagnetic wave propagation 14.80.Hv Magnetic monopoles 14.70.Bh Photons 11.30.Rd Chiral Symmetries Abstract: Maxwell’s classic equations with fields, potentials, positive and negative charges in ‘3+1’ spacetime are derived solely from the symmetry that is required by the Geometric Algebra of a ‘3+3’ spacetime. The observed direction of time is augmented by two additional orthogonal temporal dimensions forming a complex plane where i is the bi-vector determining this plane. Variations in the ‘transverse’ time determine a zero rest mass. Real fields and sources arise from averaging over a contour in transverse time. Positive and negative charges are linked with poles and zeros generating traditional plane waves with ‘right-handed’ E, B, with a direction of propagation k. Conjugation and space-reversal give ‘left-handed’ plane waves without any net change in the chirality of ‘3+3’ spacetime. Resonant wave-packets of left- and right- handed waves can be formed with eigen frequencies equal to the characteristic frequencies (N+½)fO that are observed for photons. 1 1. Introduction There are many and varied starting points for derivations of Maxwell’s equations. For example Kobe uses the gauge invariance of the Schrödinger equation [1] while Feynman, as reported by Dyson, starts with Newton’s classical laws along with quantum non- commutation rules of position and momentum [2]. -
Cracking the Einstein Code: Relativity and the Birth of Black Hole Physics, with an Afterword by Roy Kerr / Fulvio Melia
CRA C K I N G T H E E INSTEIN CODE @SZObWdWbgO\RbVS0W`bV]T0ZOQY6]ZS>VgaWQa eWbVO\/TbS`e]`RPg@]gS`` fulvio melia The University of Chicago Press chicago and london fulvio melia is a professor in the departments of physics and astronomy at the University of Arizona. He is the author of The Galactic Supermassive Black Hole; The Black Hole at the Center of Our Galaxy; The Edge of Infinity; and Electrodynamics, and he is series editor of the book series Theoretical Astrophysics published by the University of Chicago Press. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2009 by The University of Chicago All rights reserved. Published 2009 Printed in the United States of America 18 17 16 15 14 13 12 11 10 09 1 2 3 4 5 isbn-13: 978-0-226-51951-7 (cloth) isbn-10: 0-226-51951-1 (cloth) Library of Congress Cataloging-in-Publication Data Melia, Fulvio. Cracking the Einstein code: relativity and the birth of black hole physics, with an afterword by Roy Kerr / Fulvio Melia. p. cm. Includes bibliographical references and index. isbn-13: 978-0-226-51951-7 (cloth: alk. paper) isbn-10: 0-226-51951-1 (cloth: alk. paper) 1. Einstein field equations. 2. Kerr, R. P. (Roy P.). 3. Kerr black holes—Mathematical models. 4. Black holes (Astronomy)—Mathematical models. I. Title. qc173.6.m434 2009 530.11—dc22 2008044006 To natalina panaia and cesare melia, in loving memory CONTENTS preface ix 1 Einstein’s Code 1 2 Space and Time 5 3 Gravity 15 4 Four Pillars and a Prayer 24 5 An Unbreakable Code 39 6 Roy Kerr 54 7 The Kerr Solution 69 8 Black Hole 82 9 The Tower 100 10 New Zealand 105 11 Kerr in the Cosmos 111 12 Future Breakthrough 121 afterword 125 references 129 index 133 PREFACE Something quite remarkable arrived in my mail during the summer of 2004. -
SPINORS and SPACE–TIME ANISOTROPY
Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics. -
SEMISIMPLE WEAKLY SYMMETRIC PSEUDO–RIEMANNIAN MANIFOLDS 1. Introduction There Have Been a Number of Important Extensions of Th
SEMISIMPLE WEAKLY SYMMETRIC PSEUDO{RIEMANNIAN MANIFOLDS ZHIQI CHEN AND JOSEPH A. WOLF Abstract. We develop the classification of weakly symmetric pseudo{riemannian manifolds G=H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G=H and the signature of the invariant pseudo{riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature (n − 1; 1) and trans{lorentzian signature (n − 2; 2). 1. Introduction There have been a number of important extensions of the theory of riemannian symmetric spaces. Weakly symmetric spaces, introduced by A. Selberg [8], play key roles in number theory, riemannian geometry and harmonic analysis. See [11]. Pseudo{riemannian symmetric spaces, including semisimple symmetric spaces, play central but complementary roles in number theory, differential geometry and relativity, Lie group representation theory and harmonic analysis. Much of the activity there has been on the Lorentz cases, which are of particular interest in physics. Here we study the classification of weakly symmetric pseudo{ riemannian manifolds G=H where G is a semisimple Lie group and H is a reductive subgroup. We apply the result to obtain explicit listings for metrics of riemannian (Table 5.1), lorentzian (Table 5.2) and trans{ lorentzian (Table 5.3) signature. The pseudo{riemannian symmetric case was analyzed extensively in the classical paper of M. Berger [1]. Our analysis in the weakly symmetric setting starts with the classifications of Kr¨amer[6], Brion [2], Mikityuk [7] and Yakimova [16, 17] for the weakly symmetric riemannian manifolds. -
The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 4, October 2016, Pages 513–554 http://dx.doi.org/10.1090/bull/1544 Article electronically published on August 2, 2016 THE EMERGENCE OF GRAVITATIONAL WAVE SCIENCE: 100 YEARS OF DEVELOPMENT OF MATHEMATICAL THEORY, DETECTORS, NUMERICAL ALGORITHMS, AND DATA ANALYSIS TOOLS MICHAEL HOLST, OLIVIER SARBACH, MANUEL TIGLIO, AND MICHELE VALLISNERI In memory of Sergio Dain Abstract. On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein’s prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detec- tion of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole bina- ries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. -
ROBERT L. FOOTE CURRICULUM VITA March 2017 Address
ROBERT L. FOOTE CURRICULUM VITA March 2017 Address/Telephone/E-Mail Department of Mathematics & Computer Science Wabash College Crawfordsville, Indiana 47933 (765) 361-6429 [email protected] Personal Date of Birth: December 2, 1953 Citizenship: USA Education Ph.D., Mathematics, University of Michigan, April 1983 Dissertation: Curvature Estimates for Monge-Amp`ere Foliations Thesis Advisor: Daniel M. Burns Jr. M.A., Mathematics, University of Michigan, April 1978 B.A., Mathematics, Kalamazoo College, June 1976 Magna cum Laude with Honors in Mathematics, Phi Beta Kappa, Heyl Science Scholarship Employment 1989–present, Wabash College Department Chair, 1997–2001, 2009–2012 Full Professor since 2004 Associate Professor, 1993–2004 Assistant Professor, 1991–1993 Byron K. Trippet Assistant Professor, 1989–1991 1983–1989, Texas Tech University, Assistant Professor (granted tenure) 1983, Kalamazoo College, Visiting Instructor 1976–1982, University of Michigan, Graduate Student Teaching Assistant 1976, 1977, The Upjohn Company, Mathematical Analyst Research Visits 2009, Korea Institute for Advanced Study (KIAS), Visiting Scholar Three weeks at the invitation of C. K. Han. 2009, Pennsylvania State Univ., Shapiro Visitor Four weeks at the invitation of Sergei Tabachnikov. 2008–2009, Univ. of Georgia, Visiting Scholar (sabbatical leave) 1996–1997, 2003–2004, Univ. of Illinois at Urbana Champaign, Visiting Scholar (sabbatical leave) 1991, Pohang Institute of Science and Technology, Pohang, Korea Three months at the invitation of C. K. Han. 1990, Texas Tech University Ten weeks at the invitation of Lance D. Drager. Current Fields of Interest Primary: Differential Geometry, Integral Geometry Professional Affiliations American Mathematical Society, Mathematical Association of America. Teaching Experience Graduate courses Differentiable manifolds, real analysis, complex analysis. -
Should Metric Signature Matter in Clifford Algebra Formulations Of
Should Metric Signature Matter in Clifford Algebra Formulations of Physical Theories? ∗ William M. Pezzaglia Jr. † Department of Physics Santa Clara University Santa Clara, CA 95053 John J. Adams ‡ P.O. Box 808, L-399 Lawrence Livermore National Laboratory Livermore, CA 94550 February 4, 2008 Abstract Standard formulation is unable to distinguish between the (+++−) and (−−−+) spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformu- lations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear if this is mere artifact, or if there is a right/wrong choice as to which one describes reality. However, recently it has been shown that one can map from one signature to the other using a tilt transformation[8]. The broader question is that if the universe is arXiv:gr-qc/9704048v1 17 Apr 1997 signature blind, then perhaps a complete theory should be manifestly tilt covariant. A generalized multivector wave equation is proposed which is fully signature invariant in form, because it includes all the components of the algebra in the wavefunction (instead of restricting it to half) as well as all the possibilities for interaction terms. Summary of talk at the Special Session on Octonions and Clifford Algebras, at the 1997 Spring Western Sectional Meeting of the American Mathematical Society, Oregon State University, Corvallis, OR, 19-20 April 1997. ∗anonymous ftp://www.clifford.org/clf-alg/preprints/1995/pezz9502.latex †Email: [email protected] or [email protected] ‡Email: [email protected] 1 I. -
Spacetime Warps and the Quantum World: Speculations About the Future∗
Spacetime Warps and the Quantum World: Speculations about the Future∗ Kip S. Thorne I’ve just been through an overwhelming birthday celebration. There are two dangers in such celebrations, my friend Jim Hartle tells me. The first is that your friends will embarrass you by exaggerating your achieve- ments. The second is that they won’t exaggerate. Fortunately, my friends exaggerated. To the extent that there are kernels of truth in their exaggerations, many of those kernels were planted by John Wheeler. John was my mentor in writing, mentoring, and research. He began as my Ph.D. thesis advisor at Princeton University nearly forty years ago and then became a close friend, a collaborator in writing two books, and a lifelong inspiration. My sixtieth birthday celebration reminds me so much of our celebration of Johnnie’s sixtieth, thirty years ago. As I look back on my four decades of life in physics, I’m struck by the enormous changes in our under- standing of the Universe. What further discoveries will the next four decades bring? Today I will speculate on some of the big discoveries in those fields of physics in which I’ve been working. My predictions may look silly in hindsight, 40 years hence. But I’ve never minded looking silly, and predictions can stimulate research. Imagine hordes of youths setting out to prove me wrong! I’ll begin by reminding you about the foundations for the fields in which I have been working. I work, in part, on the theory of general relativity. Relativity was the first twentieth-century revolution in our understanding of the laws that govern the universe, the laws of physics. -
Listening for Einstein's Ripples in the Fabric of the Universe
Listening for Einstein's Ripples in the Fabric of the Universe UNCW College Day, 2016 Dr. R. L. Herman, UNCW Mathematics & Statistics/Physics & Physical Oceanography October 29, 2016 https://cplberry.com/2016/02/11/gw150914/ Gravitational Waves R. L. Herman Oct 29, 2016 1/34 Outline February, 11, 2016: Scientists announce first detection of gravitational waves. Einstein was right! 1 Gravitation 2 General Relativity 3 Search for Gravitational Waves 4 Detection of GWs - LIGO 5 GWs Detected! Figure: Person of the Century. Gravitational Waves R. L. Herman Oct 29, 2016 2/34 Isaac Newton (1642-1727) In 1680s Newton sought to present derivation of Kepler's planetary laws of motion. Principia 1687. Took 18 months. Laws of Motion. Law of Gravitation. Space and time absolute. Figure: The Principia. Gravitational Waves R. L. Herman Oct 29, 2016 3/34 James Clerk Maxwell (1831-1879) - EM Waves Figure: Equations of Electricity and Magnetism. Gravitational Waves R. L. Herman Oct 29, 2016 4/34 Special Relativity - 1905 ... and then came A. Einstein! Annus mirabilis papers. Special Relativity. Inspired by Maxwell's Theory. Time dilation. Length contraction. Space and Time relative - Flat spacetime. Brownian motion. Photoelectric effect. E = mc2: Figure: Einstein (1879-1955) Gravitational Waves R. L. Herman Oct 29, 2016 5/34 General Relativity - 1915 Einstein generalized special relativity for Curved Spacetime. Einstein's Equation Gravity = Geometry Gµν = 8πGTµν: Mass tells space how to bend and space tell mass how to move. Predictions. Perihelion of Mercury. Bending of Light. Time dilation. Inspired by his "happiest thought." Gravitational Waves R. L. Herman Oct 29, 2016 6/34 The Equivalence Principle Bodies freely falling in a gravitational field all accelerate at the same rate. -
The Dirac Equation in General Relativity, a Guide for Calculations
The Dirac equation in general relativity A guide for calculations Peter Collas1 and David Klein2 (September 7, 2018) Abstract In these informal lecture notes we outline different approaches used in doing calculations involving the Dirac equation in curved spacetime. We have tried to clarify the subject by carefully pointing out the various con- ventions used and by including several examples from textbooks and the existing literature. In addition some basic material has been included in the appendices. It is our hope that graduate students and other re- searchers will find these notes useful. arXiv:1809.02764v1 [gr-qc] 8 Sep 2018 1Department of Physics and Astronomy, California State University, Northridge, Northridge, CA 91330-8268. Email: [email protected]. 2Department of Mathematics and Interdisciplinary Research Institute for the Sci- ences, California State University, Northridge, Northridge, CA 91330-8313. Email: [email protected]. Contents 1 The spinorial covariant derivative3 1.1 The Fock-Ivanenko coefficients . .3 1.2 The Ricci rotation coefficient approach . .8 1.3 The electromagnetic interaction . .9 1.4 The Newman-Penrose formalism . 10 2 Examples 14 2.1 Schwarzschild spacetime N-P . 14 2.2 Schwarzschild spacetime F-I . 16 2.3 Nonfactorizable metric . 18 2.4 de Sitter spacetime, Fermi coordinates . 19 3 The Dirac equation in (1+1) GR 23 3.1 Introduction to (1+1) . 23 3.2 The Dirac equation in the Milne universe . 24 4 Scalar product 28 4.1 Conservation of j in SR . 28 4.2 The current density in GR . 29 4.3 The Scalar product in SR . -
Complex Analysis and Complex Geometry
Complex Analysis and Complex Geometry Finnur Larusson,´ University of Adelaide Norman Levenberg, Indiana University Rasul Shafikov, University of Western Ontario Alexandre Sukhov, Universite´ des Sciences et Technologies de Lille May 1–6, 2016 1 Overview of the Field Complex analysis and complex geometry form synergy through the geometric ideas used in analysis and an- alytic tools employed in geometry, and therefore they should be viewed as two aspects of the same subject. The fundamental objects of the theory are complex manifolds and, more generally, complex spaces, holo- morphic functions on them, and holomorphic maps between them. Holomorphic functions can be defined in three equivalent ways as complex-differentiable functions, convergent power series, and as solutions of the homogeneous Cauchy-Riemann equation. The threefold nature of differentiability over the complex numbers gives complex analysis its distinctive character and is the ultimate reason why it is linked to so many areas of mathematics. Plurisubharmonic functions are not as well known to nonexperts as holomorphic functions. They were first explicitly defined in the 1940s, but they had already appeared in attempts to geometrically describe domains of holomorphy at the very beginning of several complex variables in the first decade of the 20th century. Since the 1960s, one of their most important roles has been as weights in a priori estimates for solving the Cauchy-Riemann equation. They are intimately related to the complex Monge-Ampere` equation, the second partial differential equation of complex analysis. There is also a potential-theoretic aspect to plurisubharmonic functions, which is the subject of pluripotential theory. In the early decades of the modern era of the subject, from the 1940s into the 1970s, the notion of a complex space took shape and the geometry of analytic varieties and holomorphic maps was developed.