DOCSLIB.ORG

ÇZ-26-1 of 3È

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
ÇZ-26-1 of 3È AN OVERVIEW OF VARIOUS OCCURRENCES OF GENERAL EXPRESSIONS FOR THE COEFFICIENTS OF LOVELOCK LAGRANGIANS AND FOR LOVELOCK TENSORS FROM THE 0th TO THE 5th ORDER IN CURVATURE C. C. Briggs Center for Academic Computing, Penn State University, University Park, PA 16802 Wednesday, March 22, 2000 Abstract. An overview is given of various occurrences of general expressions for the coefficients of Lovelock Lagrangians and for Lovelock tensors from the 0th to the 5th order in curvature in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general linear connection. PACS numbers: 02.40.-k, 04.20.Cv, 04.20.Fy “The quintic Lovelock tensor has only recently been evaluated.” curvature tensors and Riemann curvature scalar for n-dimensional — Davis, S., “Symmetric Variations of the Metric and Extrema differentiable manifolds having a general linear connection, where the of the Action for Pure Gravity,” Gen. Rel. Grav., 30 (1998) 345. d 80 Riemann-Christoffel curvature tensor Rabc is given by INTRODUCTION d ≡ ∂ Γ d Γ d Γ e Ω e Γ d Rabc 2 ( [a b] c + [a |e| b] c + a b e c), (1) This paper presents an overview of various occurrences of general b the Ricci curvature tensor Ra by 1-3 expressions for the coefficients L(p) of Lovelock Lagrangians (also called b ≡ bc = − bc = − cb + ∇ bc + d bc Ra Rca Rac Rca 2 ( [c Qa] Sca Qd ), (2) “Euler densities,”4-5 “dimensionally continued Euler forms,”6-8 “sectional and the Riemann curvature scalar R by curvatures,”9-13 “Gauss curvature forms,”14-16 “Gauss-Bonnet forms,”17-19 ≡ a = ab = − ab = ba = − ba “Gauss-Kronecker curvatures,”20-52 “Lipschitz-Killing curvatures,”53-74 each R Ra Rba Rab Rab Rba , (3) 75-76 77 ∂ Γ b Ω b of the latter two as distinct from the other, etc.) and for Lovelock tensors where a is the Pfaffian derivative, a c the connection coefficient, a c the b 78-79 bc c G(p)a (also called “generalized Einstein curvature tensors,” etc.) from the object of anholonomity, Qa the non-metricity tensor, and Sab the torsion 0th to the 5th order in curvature in terms of the Riemann-Christoffel and Ricci tensor. 1 Mena Marugán, G. A., “Dynamically generated four-dimensional models in Lovelock cosmology,” Phys. Rev. D, 46 (1992) 4340. 2 Demaret, J., Y. De Rop, P. Tombal, and A. Moussiaux, “Qualitative Analysis of Ten-Dimensional Lovelock Cosmological Models,” Gen. Rel. Grav., 24 (1992) 1169. 3 Verwimp, T., “Unified prescription for the generation of electroweak and gravitational gauge field Lagrangian on a principal fiber bundle,” Journ. Math. Phys., 31 (1990) 3047. 4 Euler, L., “Recherches sur la courbure des surfaces,” Mémoires de l’académie [sic] des sciences de Berlin, 1767, 16 (1760) 119. 5 Mardones, A., and J. Zanelli, “Lovelock-Cartan theory of gravity,” Class. Quantum Grav., 8 (1991) 1545. 6 Müller-Hoissen, F., “Dimensionally continued Euler forms: Kaluza-Klein cosmology and dimensional reduction,” Class. Quantum Grav., 3 (1986) 665. 7 Arik, M., E. Hizel, and A. Mostafazadeh, “The Schwarzschild solution in non-Abelian Kaluza-Klein theory,” Class. Quantum Grav., 7 (1990) 1425. 8 Dereli, T., and G. Üçoluk, “Direct-curvature Yang-Mills field couplings induced by the Kaluza-Klein reduction of Euler form actions in seven dimensions,” Class. Quantum Grav., 7 (1990) 533. 9 Thorpe, J. A., “Sectional Curvatures and Characteristic Classes,” Annals of Mathematics, 80 (1964) 429; “On the Curvatures of Riemannian Manifolds,” Illinois Journ. Math., 10 (1966) 412; “Some Remarks on the Gauss-Bonnet Integral,” Journ. Math. Mech., 18 (1969) 779. 10 Gray, A., “Some Relations Between Curvature and Characteristic Classes,” Mathematische Annalen, 184 (1970) 257. 11 Bishop, R. L., and S. I. Goldberg, “Some Implications of the Generalized Gauss-Bonnet Theorem,” Trans. Amer. Math. Soc., 112 (1964) 508. 12 Cheung, Y. K., and C. C. Hsiung, “Curvature and Characteristic Classes of Compact Riemannian Manifolds,” Journ. Diff. Geom., 1 (1967) 89. 13 Kulkarni, R. S., “Curvature and Metric,” Annals of Mathematics, 91 (1970) 311; “Curvature Structures and Conformal Transformations,” Journ. Diff. Geom., 4 (1970) 425; “On a Theorem of F. Schur,” ibid., 4 (1970) 453. 14 Gauss, C. F., “Disquisitiones Generales circa Superficies Curvas,” Commentationes societatis regiae scientiarum Gottingensis recentiores [Nouveaux Mémoires de Gottingue], vol. 6 (ad a. 1823-1827), Gottingae, Germany, 1828, Commentationes classis mathematicae, p. 99. 15 Eells, J., “A generalization of the Gauss-Bonnet Theorem,” Trans. Amer. Math. Soc., 92 (1959) 142. 16 Thorpe, J. A., (1964), op. cit. 17 Bonnet, P. O., “Mémoire sur la théorie générale des surfaces,” Journal de l’École Polytechnique, 19/Cahier 32 (1848) 1; cf. “Sur quelques propriétés générales des surfaces et des lignes tracées sur les surfaces,” Comptes rendus hebdomadaires des séances de l’Academie des Sciences (Paris), 19 (1844) 980. 18 Müller-Hoissen, F., “Spontaneous Compactification with Quadratic and Cubic Curvature Terms,” Phys. Lett., 163B (1985) 106; “Gravity Actions, Boundary Terms and Second-Order Field Equations,” Nucl. Phys., B337 (1990) 709; “From Chern-Simons to Gauss-Bonnet,” ibid., B346 (1990) 235. 19 Mardones, A., and J. Zanelli, op. cit. 20 Kronecker, L., “Über Systeme von Functionen [sic] mehrer Variabeln [Zweite Abhandlung] [sic],” Monatsberichte der königlich preussischen Akademie der Wissenschaften zu Berlin, Gesammtsitzung [sic] vom 5. August 1869, (1869) 688. 21 Chen, B.-Y., “Some Integral Formulas of the Gauss-Kronecker Curvature,” Kódai Math. Sem. Rep., 20 (1968) 410. 22 4 Cheng, Q.-M., “Complete Maximal Spacelike Hypersurfaces of H1(C),” Manuscripta Math., 82 (1994) 149. 23 de Almeida, S. C., and F. G. B. Brito, “Minimal Hypersurfaces of S4 with Constant Gauss-Kronecker Curvature,” Mathematische Zeitschrift, 195 (1987) 99. 24 Fu, J. H. G., “Curvature Measures and Generalized Morse Theory,” Journ. Diff. Geom., 30 (1989) 619. 25 Griffiths, P. A., “Complex Differential and Integral Geometry and Curvature Integrals Associated to Singularities of Complex Analytic Varieties,” Duke Mathematical Journal, 45 (1978) 427. 26 Guan, B., and J. Spruck, “Boundary-value problems on Sn for surfaces of constant Gauss curvature,” Annals of Mathematics, 138 (1993) 601. 27 Hsiung, C.-C., and S. S. Mittra, “Isometries of Compact Hypersurfaces with Boundary in a Riemannian Space,” in Kobayashi, S., M. Obata, and T. Takahashi (eds.), Differential Geometry, in Honor of Kentaro Yano, Kinokuniya Book-Store Co., Ltd., Tokyo, Japan (1972), p. 145. 28 Huang, W.-H., “Superharmonicity of Curvatures for Surfaces of Constant Mean Curvature,” Pacific Journal of Mathematics, 152 (1992) 291. 29 Hug, D., “Curvature Relations and Affine Surface Area for a General Convex Body and its [sic] Polar,” Results in Mathematics, 29 (1996) 233; “Contributions to Affine Surface Area,” Manuscripta Mathematica, 91 (1996) 283. 30 Kojima, M., “q-Conformally flat hypersurfaces,” Differential Geometry and its Applications, 5 (1995) 51. 31 Kowalski, O., “On the Gauss-Kronecker Curvature Tensors,” Mathematische Annalen, 203 (1973) 335. 32 Kulkarni, R. S., “On the Bianchi identities,” Mathematische Annalen, 199 (1972) 175. 33 Langevin, R., and H. Rosenberg, “Fenchel type theorems for submanifolds of Sn,” Comment. Math. Helvetici, 71 (1996) 594. 34 Li, A.-M., “Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space,” Arch. Math. (Basel), 64 (1995) 534. 35 Li, Y. Y., “Group invariant convex hypersurfaces with prescribed Gauss-Kronecker curvature,” in “Multidimensional Complex Analysis and Partial Differential Equations; A Collection of Papers in Honor of François Treves,” Contemporary Mathematics, 205 (1997) 203. 36 Martínez, A., and F. Milán, “On the Affine Bernstein Problem,” Geometriae Dedicata, 37 (1991) 295. 37 Milán, F., “Pick Invariant and Affine Gauss-Kronecker Curvature,” Geometriae Dedicata, 45 (1993) 41. 38 Nelli, B., and H. Rosenberg, “Some Remarks on Positive Scalar and Gauss-Kronecker Curvature Hypersurfaces of Rn + 1 and Hn + 1,” Ann. Inst. Fourier, Grenoble, 47 (1997) 1209. 39 Qingming, C., “Hypersurfaces With [sic] Constant Quasi-Gauss-Kronecker Curvature in S4(1),” Advances in Mathematics (China), 22 (1993) 125. 40 Ramanathan, J., “Minimal hypersurfaces in S4 with vanishing Gauss-Kronecker curvature,” Mathematische Zeitschrift, 205 (1990) 645. 41 Ros, A., “Compact Hypersurfaces with Constant Scalar Curvature and a Congruence Theorem,” Journ. Diff. Geom., 27 (1988) 215. 42 Rosenberg, H., and J. Spruck, “On the Existence of Convex Hypersurfaces of Constant Gauss Curvature in Hyperbolic Space,” Journ. Diff. Geom., 40 (1994) 379. 43 Schmuckenschlaeger, M., “A Simple Proof of an Approximation Theorem of H. Minkowski,” Geometriae Dedicata, 48 (1993) 319. 44 Schneider, R., “Polyhedral Approximation of Smooth Convex Bodies,” Journal of Mathematical Analysis and Applications, 128 (1987) 470. 2 Wednesday, March 22, 2000 Z-26 81 δb = In accordance with various general definitions given by Müller-Hoissen a,ifp 0 G b = (5) 82 b (p)a (2 p + 1)! and Verwimp, L and G are given, using anholonomic coordinates, by δ b i1i2 i3i4 i2 p − 1i2 p > (p) (p)a p + 1 [a Ri i Ri i … Ri i ] ,ifp 0 = 2 p 1 2 3 4 2 p − 1 2 p 1, if p 0 cd = ce d δb = respectively, where Rab g Rabe and a is the Kronecker delta. L(p) (2p)! (4) i1i2 i3i4 i2p − 1i2p > b p R R … R ,ifp 0 Some numerical properties of L and G for 0 ≤ p ≤ 5 appear in Tables 2 [i1i2 i3i4 i2p − 1i2p] (p) (p)a and 1 and 2 (see below). Complete expressions and the overview follow. ≤ ≤ b ≤ ≤ TABLE 1. SOME NUMERICAL PROPERTIES OF L(p) FOR 0 p 5. TABLE 2. SOME NUMERICAL PROPERTIES OF G(p)a FOR 0 p 5. ORDER QUANTITY NUMBER OF NUMBER OF PERMUTA- SUM OF NUMERICAL ORDER QUANTITY NUMBER OF NUMBER OF PERMUTA- SUM OF NUMERICAL TERMS TIONS COMPREHENDED FACTORS TERMS TIONS COMPREHENDED FACTORS b 0 L(0) 1 110G(0)a 1 11 b 1 1 L(1) 1 211G(1)a 2 61 2 b 1 2 L(2) 3 24 6 2 G(2)a 7 120 7 2 b 3 L(3) 8 720 90 3 G(3)a 26 5040 105 b 4 L(4) 25 40,320 2520 4 G(4)a 115 362,880 2835 b 5 L(5) 85 3,628,800 113,400 5 G(5)a 596 39,916,800 124,740 45 Schneider, R., and J.
Load more

Recommended publications
  • Curvature Tensors in a 4D Riemann–Cartan Space: Irreducible Decompositions and Superenergy
    Curvature tensors in a 4D Riemann–Cartan space: Irreducible decompositions and superenergy Jens Boos and Friedrich W. Hehl [email protected] [email protected]"oeln.de University of Alberta University of Cologne & University of Missouri (uesday, %ugust 29, 17:0. Geometric Foundations of /ravity in (artu Institute of 0hysics, University of (artu) Estonia Geometric Foundations of /ravity Geometric Foundations of /auge Theory Geometric Foundations of /auge Theory ↔ Gravity The ingredients o$ gauge theory: the e2ample o$ electrodynamics ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: redundancy conserved e2ternal current 5 ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current Complete, gauge-theoretical description: 9 local #7,) invariance ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: iers Complex spinor 6eld: rce carr ry of fo mic theo rrent rosco rnal cu m pic en exte att desc gredundancyiv er; N ript oet ion o conserved e2ternal current 5 invariance her f curr conserved #7,8 current e n t s Complete, gauge-theoretical description: gauge theory = complete description of matter and 9 local #7,) invariance how it interacts via gauge bosons ,3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincar; gauge theory *3,.
    [Show full text]
  • Arxiv:1108.0677V2 [Hep-Th]
    DAMTP-2011-59 MIFPA-11-34 THE SHEAR VISCOSITY TO ENTROPY RATIO: A STATUS REPORT Sera Cremonini ♣,♠ ∗ ♣ Centre for Theoretical Cosmology, DAMTP, CMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK ♠ George and Cynthia Mitchell Institute for Fundamental Physics and Astronomy Texas A&M University, College Station, TX 77843–4242, USA (Dated: October 22, 2018) This review highlights some of the lessons that the holographic gauge/gravity duality has taught us regarding the behavior of the shear viscosity to entropy density in strongly coupled field theories. The viscosity to entropy ratio has been shown to take on a very simple universal value in all gauge theories with an Einstein gravity dual. Here we describe the origin of this universal ratio, and focus on how it is modified by generic higher derivative corrections corresponding to curvature corrections on the gravity side of the duality. In particular, certain curvature corrections are known to push the viscosity to entropy ratio below its universal value. This disproves a longstanding conjecture that such a universal value represents a strict lower bound for any fluid in nature. We discuss the main developments that have led to insight into the violation of this bound, and consider whether the consistency of the theory is responsible for setting a fundamental lower bound on the viscosity to entropy ratio. arXiv:1108.0677v2 [hep-th] 23 Aug 2011 ∗ Electronic address: [email protected] 2 Contents I. Introduction 3 A. The quark gluon plasma and the shear viscosity to entropy ratio 4 II. The universality of η/s 6 A.
    [Show full text]
  • Classical Field Theories from Hamiltonian Constraint: Local
    Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields V´aclav Zatloukal∗ Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Bˇrehov´a7, 115 19 Praha 1, Czech Republic and Max Planck Institute for the History of Science, Boltzmannstrasse 22, 14195 Berlin, Germany We consider the Hamiltonian constraint formulation of classical field theories, which treats space- time and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local trans- formations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article. Throughout, we employ the mathematical language of geometric algebra and calculus. I. INTRODUCTION The Hamiltonian constraint is a concept useful for Hamiltonian formulation not only of general relativity [1], but, in fact, of a generic field theory, as pointed out in Ref. [2, Ch. 3], and exploited further in [3] and [4]. Characteristic features of this formulation are: finite-dimensional configu- ration space, and multivector-valued momentum variable. In this respect it is congruent with the covariant (or De Donder-Weyl) Hamiltonian formalism [5{13] and should be contrasted with the canonical (or instantaneous) Hamiltonian formalism [14], which utilizes an infinite-dimensional space of field configurations defined at a given instant of time.
    [Show full text]
  • Applications Gauge Theory/Gravity
    APPLICATIONS OF THE GAUGE THEORY/GRAVITY CORRESPONDENCE Andrea Helen Prinsloo A thesis submitted for the degree of Doctor of Philosophy in Applied Mathematics, Department of Mathematics and Applied Mathematics, University of Cape Town. May, 2010 i Abstract The gauge theory/gravity correspondence encompasses a variety of different specific dualities. We study examples of both Super Yang-Mills/type IIB string theory and Super Chern-Simons-matter/type IIA string theory dualities. We focus on the recent ABJM correspondence as an example of the latter. We conduct a detailed investigation into the properties of D-branes and their operator 3 duals. The D2-brane dual giant graviton on AdS4×CP is initially studied: we calculate its spectrum of small fluctuations and consider open string excitations in both the short pp-wave and long semiclassical string limits. We extend Mikhailov's holomorphic curve construction to build a giant graviton on 1;1 AdS5 × T . This is a non-spherical D3-brane configuration, which factorizes at maxi- mal size into two dibaryons on the base manifold T1;1. We present a fluctuation analysis and also consider attaching open strings to the giant's worldvolume. We finally propose 3 an ansatz for the D4-brane giant graviton on AdS4 × CP , which is embedded in the complex projective space. 2 The maximal D4-brane giant factorizes into two CP dibaryons. A comparison is made 2 between the spectrum of small fluctuations about one such CP dibaryon and the conformal dimensions of BPS excitations of the dual determinant operator in ABJM theory. We conclude with a study of the thermal properties of an ensemble of pp-wave strings under a Lunin-Maldacena deformation.
    [Show full text]
  • General Relativity Vs. Gauge Theory Gravity
    Experimental Test of Quantum Gravity: General Relativity vs. Gauge Theory Gravity Peter Cameron and Michaele Suisse∗ PO Box 1030 Mattituck, NY USA 11952 (Dated: September 16, 2017) With recent detection of gravitational waves[1, 2], the possibility exists that orientation-dependent detector re- sponses might permit distinguishing between General Relativity (GR) and Gauge Theory Gravity (GTG)[3]. The classical equivalence of these two models was established over twenty years ago.[4{7]. The question is whether this equivalence persists in their respective quantum theories. While such a theory is not yet known to exist for the curved spacetime of GR, the task is not so difficult in the flat Minkowski spacetime of GTG. The language of GTG is geometric Clifford alge- bra, the background-independent[8] interaction lan- guage of fundamental geometric objects of space - Eu- clid's point, line, plane, and volume elements, the ge- ometric objects of Pauli algebra of three-dimensional space. In quantized GTG they are taken to comprise the vacuum wavefunction. Their interactions gener- ate the Dirac algebra of four-dimensional Minkowski spacetime[9]. They permit one to define a geometric wavefunction at the Planck length, and when endowed with experimentally observed quantized electric and FIG. 1. Classical GR says interferometer response is optimal for magnetic fields reveal an exact relation between elec- orientation (A) and less so for (B)[20], whereas quantized GTG tromagnetism and gravity, yielding a naturally finite, is optimal for (B) and less so for (A). confined, and gauge invariant quantum theory that has no free parameters and contains gravity[10{16].
    [Show full text]
  • Applications of the Gauge/Gravity Duality (DRAFT: July 30, 2013)
    Applications of the Gauge/Gravity Duality by Kevin Robert Leslie Whyte B. Applied Science, University of Waterloo, 2004 B. Mathematics, University of Waterloo, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in THE FACULTY OF GRADUATE STUDIES (Physics) The University Of British Columbia (Vancouver) August 2013 c Kevin Robert Leslie Whyte, 2013 Abstract While varied applications of gauge/gravity duality have arisen in literature from studies of condensed matter systems including superconductivity to studies of quenched Quantum Chromodynamics (QCD), this thesis focuses on applications of the dual- ity to holographic QCD-like field theories and to inflationary model that uses a QCD-like field theory. In particular the first half of the thesis examines a holographic QCD-like field theory with scalar quarks closely related to the Sakai-Sugimoto model of holo- graphic QCD. The behaviour of baryons and mesons in the model is examined to find a continuous mass spectrum for the mesons, and a baryon that can identified with a topological charge. It then slightly modifies the theory to further study the behaviour of holographic field theories. The second half of the thesis presents a novel model for early Universe infla- tion, using an SU(N) gauge field theory as the inflaton. The inflation model is studied at both weak coupling and strong coupling using the gauge/gravity dual- ity. The robustness of model’s predictions to exciting multiple inflationary fields beyond the single field of its original proposal, and its possible role in breaking the supersymmetry of the Minimal Supersymmetric Standard Model (MSSM) is also explored.
    [Show full text]
  • The Quantum Structure of Space and Time
    QcEntwn Structure &pace and Time This page intentionally left blank Proceedings of the 23rd Solvay Conference on Physics Brussels, Belgium 1 - 3 December 2005 The Quantum Structure of Space and Time EDITORS DAVID GROSS Kavli Institute. University of California. Santa Barbara. USA MARC HENNEAUX Universite Libre de Bruxelles & International Solvay Institutes. Belgium ALEXANDER SEVRIN Vrije Universiteit Brussel & International Solvay Institutes. Belgium \b World Scientific NEW JERSEY LONOON * SINGAPORE BElJlNG * SHANGHAI HONG KONG TAIPEI * CHENNAI Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofJice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice; 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-PublicationData A catalogue record for this hook is available from the British Library. THE QUANTUM STRUCTURE OF SPACE AND TIME Proceedings of the 23rd Solvay Conference on Physics Copyright 0 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoi may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-952-9 ISBN 981-256-953-7 (phk) Printed in Singapore by World Scientific Printers (S) Pte Ltd The International Solvay Institutes Board of Directors Members Mr.
    [Show full text]
  • Gravity, Gauge Theories and Geometric Algebra
    GRAVITY, GAUGE THEORIES AND GEOMETRIC ALGEBRA Anthony Lasenby1, Chris Doran2 and Stephen Gull3 Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK. Abstract A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of ‘intrinsic’ relations between physical fields. For a wide range of phenomena, including all present experimental tests, the theory repro- duces the predictions of general relativity. Differences do emerge, however, through the first-order nature of the equations and the global properties of the gauge fields, and through the relationship with quantum theory. The properties of the gravitational gauge fields are derived from both classi- cal and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure se- lects an action that is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interac- tion. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These equations are then applied to the study of cosmology, and to the formation and properties of black holes. Insistence on find- ing global solutions, together with the first-order nature of the equations, arXiv:gr-qc/0405033v1 6 May 2004 leads to a new understanding of the role played by time reversal.
    [Show full text]
  • A Curious Story of Quantum Gravity in the Ultraviolet
    A Curious Story of Quantum Gravity in the Ultraviolet MHV@30 Fermilab March 17, 2016 Zvi Bern UCLA ZB, Carrasco, Johansson, arXiv:1004.0476 ZB, T. Dennen, S. Davies, V. Smirnov and A. Smirnov, arXiv:1309.2496 ZB, T. Dennen, S. Davies, arXiv:1409.3089; arXiv:1412.2441 ZB, C. Cheung, H.H. Chi, S. Davies, L. Dixon, J. Nohle. arXiv:1507.06118 ZB, S. Davies, J. Nohle, arXiv:1510.03448 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA Simplicity in Scattering Amplitudes For the history, see other talks: Kunszt, Kosower, Hodges and others. Here I will only talk about history directly relevant for the rest of my talk. 28 years ago David Kosower mentioned the “Parke-Taylor formula”. I said, “What’s that?” (Words to be forgotten!) David Kosower’s response should be immortalized: “Everyone needs to know the Parke-Taylor formula!” David was right. 28 years later everyone does indeed know it! MHV amplitude in spinor notation: Mangano, Parke and Xu (1988) 4 + + 12 A(1−, 2−, 3 ,...,n )=i h i 12 23 n1 2 h ih i ···h i MHV Amplitudes (educated guess) 4 + + 12 A(1−, 2−, 3 ,...,n )=i h i 12 23 n1 h ih i ···h i It wasn’t so obvious this formula would be important: • It’s too special. Limited helicities. • No masses. • Not directly applicable to phenomenology. • No obvious generalization to loops. • Etc. But those of us who were young at the time did not know we were supposed to worry about the above problems.
    [Show full text]
  • Gravitational Dynamics—A Novel Shift in the Hamiltonian Paradigm
    universe Article Gravitational Dynamics—A Novel Shift in the Hamiltonian Paradigm Abhay Ashtekar 1,* and Madhavan Varadarajan 2 1 Institute for Gravitation and the Cosmos & Physics Department, The Pennsylvania State University, University Park, PA 16802, USA 2 Raman Research Institute, Bangalore 560 080, India; [email protected] * Correspondence: [email protected] Abstract: It is well known that Einstein’s equations assume a simple polynomial form in the Hamil- tonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that time evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in spatial directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomor- phism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the ‘integrable’ (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the ‘double copy’ ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored. Keywords: general relativity dynamics; gauge theory; hamiltonian framework Citation: Ashtekar, A.; Varadarajan, M.
    [Show full text]
  • Relations Between Gravity and Gauge Theory and Implications for UV Properties April 16, 2012 Loops and Legs in Quantum Field Theory
    Relations Between Gravity and Gauge Theory and Implications for UV Properties April 16, 2012 Loops and Legs in Quantum Field Theory. Zvi Bern, UCLA Based on papers with John Joseph Carrasco, Scott Davies, Tristan Dennen, Lance Dixon, Yu-tin Huang, Harald Ita, Henrik Johansson and Radu Roiban. 1 Multiloop Non-Planar Amplitudes In this talk we will explain how we carry out high-loop-order calculations in supergravity theories, proving supergravity is much better behaved in the UV than had been anticipated. Update on latest developments in 2012. Nonplanar is less developed than planar but we do have powerful general purpose tools (not limited to susy): • Unitarity method ZB, Dixon, Dunbar, Kosower • Method of maximal cuts ZB, Carrasco, Johansson , Kosower • Duality between color and kinematics ZB, Carrasco and Johansson 2 Gravity vs Gauge Theory Consider the gravity Lagrangian flat metric graviton field metric Infinite number of complicated interactions + … terrible mess Non-renormalizable Compare to Yang-Mills Lagrangian on which QCD is based Only three and four point interactions Gravity seems so much more complicated than gauge theory. 3 Three Vertices Standard Feynman diagram approach. Three-gluon vertex: Three-graviton vertex: G3¹®;º¯;¾°(k1; k2; k3) = About 100 terms in three vertex Naïve conclusion: Gravity is a nasty mess. Definitely not a good approach. 4 Simplicity of Gravity Amplitudes People were looking at gravity the wrong way. On-shell viewpoint more powerful. On-shell three vertices contains all information: gauge theory: double copy gravity: of Yang-Mills vertex. • Using modern on-shell methods, any gravity scattering amplitude constructible solely from on-shell 3 vertices.
    [Show full text]
  • Quadratic Lagrangians and Topology in Gauge Theory Gravity
    Quadratic Lagrangians and Topology in Gauge Theory Gravity Antony Lewis,∗1 Chris Doran,†1 and Anthony Lasenby1 October 16, 2018 Abstract We consider topological contributions to the action integral in a gauge theory formulation of gravity. Two topological invariants are found and are shown to arise from the scalar and pseudoscalar parts of a single integral. Neither of these action integrals contribute to the classical field equations. An identity is found for the invariants that is valid for non-symmetric Riemann tensors, generalizing the usual GR expression for the topological invariants. The link with Yang-Mills instantons in Euclidean gravity is also explored. Ten independent quadratic terms are constructed from the Riemann tensor, and the topological invariants reduce these to eight possible independent terms for a quadratic Lagrangian. The resulting field equations for the parity non-violating terms are presented. Our derivations of these results are considerably simpler that those found in the literature. KEY WORDS: Quadratic Lagrangians, topology, instantons, ECKS theory. 1 Introduction In the construction of a gravitational field theory there is considerable freedom in the choice of Lagrangian. Einstein’s theory is obtained when just the Ricci scalar is used, but there is no compelling reason to believe that this is anything other than a good approximation. Since quadratic terms will be small when the curvature is small one would expect them to have a small effect at low energies. However they may have a considerable effect in cosmology or on arXiv:gr-qc/9910039v1 12 Oct 1999 singularity formation when the curvature gets larger. Quadratic terms may also be necessary to formulate a sensible quantum theory.
    [Show full text]