A Schrödinger approach to Newton-Cartan gravity. Miami 2015
|
|
- Piers Hodges
- 6 years ago
- Views:
Transcription
1 A Schrödinger approach to Newton-Cartan gravity Eric Bergshoeff Groningen University Miami 2015 A topical conference on elementary particles, astrophysics, and cosmology Fort Lauderdale, December
2 why non-relativistic gravity? Newton-Cartan (NC) gravity is Newtonian gravity in arbitrary frame Cartan (1923)
3 Motivation condensed matter physics Son et al. ( ) gauge-gravity duality Christensen, Hartong, Kiritsis Obers and Rollier ( ) Hořava-Lifshitz gravity Hořava (2009); Hartong, Obers (2015) non-relativistic strings/branes Gomis, Ooguri (2000); Gomis, Kamimura, Townsend (2004)
4 How do we construct (Non-)relativistic Gravity? (1) gauging a (non-)relativistic algebra (2) taking a non-relativistic limit (3) using a nonrelativistic version of the conformal tensor calculus
5 Outline NC Gravity from gauging Bargmann
6 Outline NC Gravity from gauging Bargmann The Schrödinger Method
7 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion
8 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
9 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
10 Einstein Gravity In the relativistic case free-falling frames are connected by the Poincare symmetries: space-time translations: δx µ = ξ µ Lorentz transformations: δx µ = λ µ ν x ν In free-falling frames there is no gravitational force in arbitrary frames the gravitational force is described by an invertable Vierbein field e µ A (x) µ = 0,1,2,3; A=0,1,2,3
11 Non-relativistic Gravity In the non-relativistic case free-falling frames are connected by the Galilean symmetries: time translations: δt = ξ 0 space translations: δx i = ξ i i = 1,2,3 spatial rotations: Galilean boosts: δx i = λ i j x j δx i = λ i t In free-falling frames there is no gravitational force
12 Newtonian gravity versus Newton-Cartan gravity in frames with constant acceleration (δx i = 1 2 ai t 2 ) the gravitational force is described by the Newton potential Φ( x) Newtonian gravity in arbitrary frames the gravitational force is described by a temporal Vierbein τ µ (x), spatial Vierbein e µ a (x) plus a vector m µ (x) µ = 0,1,2,3; a=1,2,3 Newton-Cartan (NC) gravity
13 The Galilei Algebra versus the Bargmann algebra Einstein gravity follows from gauging the Poincare algebra The Galilei algebra is the contraction of the Poincare algebra does NC gravity follow from gauging the Galilei algebra? Can NC gravity be obtained by taking the non-relativistic limit of Einstein gravity? No! one needs Bargmann instead of Galilei and Poincare U(1)!
14 Gauging the Bargmann algebra [J ab,p c ] = 2δ c[a P b], [J ab,g c ] = 2δ c[a G b], [G a,h] = P a, [G a,p b ] = δ ab Z, a = 1,2,...,d symmetry generators gauge field parameters curvatures time translations H τ µ ζ(x ν ) R µν (H) space translations P a a e µ ζ a (x ν ) R a µν (P) Galilean boosts G a a ω µ λ a (x ν ) R a µν (G) spatial rotations J ab ab ω µ λ ab (x ν ) R ab µν (J) central charge transf. Z m µ σ(x ν ) R µν (Z)
15 Imposing Constraints R µν a (P) = 0, R µν (Z) = 0 : solve for spin-connection fields R µν (H) = [µ τ ν] = 0 τ µ = µ τ : foliation of Newtonian spacetime ( zero torsion ) R µν ab (J) = 0 (flat space) : optional R 0(a,b) (G) 0 : only non-zero components left
16 The Final Result The independent NC fields {τ µ,e µ a,m µ } transform as follows: δτ µ = 0, δe µ a = λ a be µ b +λ a τ µ, δm µ = µ σ +λ a e µ a The spin-connection fields ω µ ab and ω µ a are functions of e,τ and m There are two Galilean-invariant metrics: τ µν = τ µ τ ν, h µν = e µ ae ν bδ ab
17 The NC Equations of Motion Taking the non-relativistic limit of the Einstein equations τ µ e ν ar µν a (G) = 0 e ν ar µν ab (J) = 0 after gauge-fixing and assuming flat space the first NC e.o.m. becomes Φ = 0 note: there is no action that gives rise to these equations of motion
18 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
19 The Relativistic Conformal Method Conformal = Poincare + D (dilatations) + K µ (special conf. transf.) conformal gravity gauging of conformal algebra δb µ = Λ a K(x)e µ a, f µ a = f µ a (e,ω,b) Poincare invariant CFT of real scalar
20 An example P 1 : e 1 L = 1 κ 2 R STEP 1 STEP 2 (e µ A ) P = κ 2 D 2 ϕ(eµ A ) C δφ = Λ D φ, with δ(e µ A ) C = Λ D (e µ A ) C (e µ A ) C = δ µ A µ ξ ν +Λ νµ +Λ D δ µ ν = 0 make redefinition ϕ = φ 2 D 2, D > 2 CFT 1 : L = 4 D 1 D 2 φ φ with δφ = ξµ µ φ 1 2 (D 2)Λ Dφ
21 from CFT 1 back to P 1 CFT 1 : L φ φ or L µ φ µ φ with δφ = ξ µ µ φ+wλ D φ STEP 1 require that δl = µ (ξ µ L) and replace derivatives by conformal-covariant derivatives LCFT 1 : e 1 L = 4 D 1 D 2 φ C φ STEP 2 gauge-fix dilatations by imposing φ = 1 κ P 1 : e 1 L = 1 κ 2 R
22 Three Different Invariants 1. Potential terms Example: cosmological constant (κ = 1) P 0 : e 1 L = Λ CFT 0 : L = Λφ 2, w = D 2 2. Kinetic terms Example: L φ φ e 1 L = R includes all CFT s with time derivatives 3. Curvature terms Example: Weyl tensor squared e 1 L φ 2D 4 D 2 ( C µν AB ) 2 D 4
23 The Schrödinger Method The contraction of the conformal Algebra is the Galilean Conformal Algebra (GCA) which has no central extension! z = 2 Schrödinger = Bargmann + D (dilatations) + K (special conf.) [H,D] = zh, [P a,d] = P a z = 1: conformal algebra, z 2 : no special conf. transf.
24 Schrödinger Gravity Hartong, Rosseel + E.B. (2014) Gauging the z = 2 Schrödinger algebra we find that the independent gauge fields {τ µ,e µ a,m µ } transform as follows: δτ µ = 2Λ D τ µ, δe µ a = Λ a be µ b +Λ a τ µ +Λ D e µ a, δm µ = µ σ +Λ a e µ a The time projection τ µ b µ of b µ transforms under K as a a shift while the spatial projection b a e a µ b µ is dependent b a (e,τ) represents (twistless) torsion!
25 Schrödinger Field Theories (SFT s) the Schrödinger action for a complex scalar Ψ with weights (w,m) SFT : S = dtd d xψ ( i 0 1 2M a a )Ψ is invariant under the rigid Schrödinger transformations δψ = ( b 2λ D t +λ K t 2) 0 Ψ+ ( b a λ ab x b λ a t λ D x a +λ K tx a) a Ψ +w ( λ D λ K t ) Ψ+iM ( σ λ a x a λ Kx 2) Ψ for w(ψ) = d/2
26 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
27 Case 1: zero torsion: b a = 0 foliation constraint : µ (τ ν ) G ν (τ µ ) G = 0, Gal E.O.M. : (τ µ ) G (e ν a) G R µν a (G) = 0, (e ν a) G R µν ab (J) = 0. Schrödinger method leads to (Ψ = ϕe iχ ) SFT E.O.M. : 0 0 ϕ = 0 and a ϕ = 0 with w = 1
28 Case 2: twistless torsion: b a 0 foliation constraint is conformal invariant use the second compensating scalar χ to restore Schrödinger invariance: 0 0 ϕ 2 M ( 0 a ϕ) a χ+ 1 M 2( a b ϕ) a χ b χ = 0 Φ+ ˆτ µ µ K +K ab K ab 8Φb b 2ΦD b 6b a D a Φ = 0 plus e ν ar µν ab (J) = 0 Afshar, Mehra, Parekh, Rollier + E.B. (2015), to appear
29 Outline NC Gravity from gauging Bargmann The Schrödinger Method NC Gravity with Torsion Future Directions
30 Outlook matter-coupled NC gravity extension to z 2 and Galilean conformal symmetries relation to Hořava-Lifshitz gravity Hartong and Obers (2015) Afshar, Mehra, Parekh, Rollier + E.B. (2015), to appear non-relativistic supergravity localization techniques Knodel, Lisbao, Liu (2015)
Applied Newton-Cartan Geometry: A Pedagogical Review
Applied Newton-Cartan Geometry: A Pedagogical Review Eric Bergshoeff Groningen University 10th Nordic String Meeting Bremen, March 15 2016 Newtonian Gravity Free-falling frames: Galilean symmetries Earth-based
More informationRunning at Non-relativistic Speed
Running at Non-relativistic Speed Eric Bergshoeff Groningen University Symmetries in Particles and Strings A Conference to celebrate the 70th birthday of Quim Gomis Barcelona, September 4 2015 Why always
More informationNewton-Cartan Geometry and Torsion
Newton-Cartan Geometry and Torsion Eric Bergshoeff Groningen University Supergravity, Strings and Dualities: A Meeting in Celebration of Chris Hull s 60th Birthday London, April 29 2017 why non-relativistic
More informationString Theory and Nonrelativistic Gravity
String Theory and Nonrelativistic Gravity Eric Bergshoeff Groningen University work done in collaboration with Ceyda Şimşek, Jaume Gomis, Kevin Grosvenor and Ziqi Yan A topical conference on elementary
More informationarxiv: v1 [hep-th] 19 Dec 2015
UG 15 20 arxiv:1512.06277v1 [hep-th] 19 Dec 2015 A Schrödinger approach to Newton-Cartan and Hořava-Lifshitz gravities Hamid R. Afshar 1, Eric A. Bergshoeff 2, Aditya Mehra 3, Pulastya Parekh 3 and Blaise
More informationMassive and Newton-Cartan Gravity in Action
Massive and Newton-Cartan Gravity in Action Eric Bergshoeff Groningen University The Ninth Aegean Summer School on Einstein s Theory of Gravity and its Modifications: From Theory to Observations based
More informationA Short Note on D=3 N=1 Supergravity
A Short Note on D=3 N=1 Supergravity Sunny Guha December 13, 015 1 Why 3-dimensional gravity? Three-dimensional field theories have a number of unique features, the massless states do not carry helicity,
More informationN-C from the Noether Procedure and Galilean Electrodynamics
gen. Vertical version with logotype under the N-C from the Noether Procedure and Galilean Electrodynamics Dennis Hansen 10th Nordic String Theory Meeting, Bremen 2016 The Niels Bohr Institute and Niels
More informationApplications of AdS/CFT correspondence to cold atom physics
Applications of AdS/CFT correspondence to cold atom physics Sergej Moroz in collaboration with Carlos Fuertes ITP, Heidelberg Outline Basics of AdS/CFT correspondence Schrödinger group and correlation
More informationScale-invariance from spontaneously broken conformal invariance
Scale-invariance from spontaneously broken conformal invariance Austin Joyce Center for Particle Cosmology University of Pennsylvania Hinterbichler, Khoury arxiv:1106.1428 Hinterbichler, AJ, Khoury arxiv:1202.6056
More informationNew Massive Dual Gravity: beyond 3D
New Massive Dual Gravity: beyond 3D Eric Bergshoeff Groningen University Work in progress together with Jose Juan Fernandez, Jan Rosseel and Paul Townsend Miami, December 18 2011 Outline Introduction Outline
More informationAspects of Spontaneous Lorentz Violation
Aspects of Spontaneous Lorentz Violation Robert Bluhm Colby College IUCSS School on CPT & Lorentz Violating SME, Indiana University, June 2012 Outline: I. Review & Motivations II. Spontaneous Lorentz Violation
More informationNon-relativistic holography
University of Amsterdam AdS/CMT, Imperial College, January 2011 Why non-relativistic holography? Gauge/gravity dualities have become an important new tool in extracting strong coupling physics. The best
More informationHolography for non-relativistic CFTs
Holography for non-relativistic CFTs Herzog, Rangamani & SFR, 0807.1099, Rangamani, Son, Thompson & SFR, 0811.2049, SFR & Saremi, 0907.1846 Simon Ross Centre for Particle Theory, Durham University Liverpool
More informationA brief introduction to modified theories of gravity
(Vinc)Enzo Vitagliano CENTRA, Lisboa May, 14th 2015 IV Amazonian Workshop on Black Holes and Analogue Models of Gravity Belém do Pará The General Theory of Relativity dynamics of the Universe behavior
More informationDilaton gravity at the brane with general matter-dilaton coupling
Dilaton gravity at the brane with general matter-dilaton coupling University of Würzburg, Institute for Theoretical Physics and Astrophysics Bielefeld, 6. Kosmologietag May 5th, 2011 Outline introduction
More informationRecent Progress on Curvature Squared Supergravities in Five and Six Dimensions
Recent Progress on Curvature Squared Supergravities in Five and Six Dimensions Mehmet Ozkan in collaboration with Yi Pang (Texas A&M University) hep-th/1301.6622 April 24, 2013 Mehmet Ozkan () April 24,
More informationA rotating charged black hole solution in f (R) gravity
PRAMANA c Indian Academy of Sciences Vol. 78, No. 5 journal of May 01 physics pp. 697 703 A rotating charged black hole solution in f R) gravity ALEXIS LARRAÑAGA National Astronomical Observatory, National
More informationGiinter Ludyk. Einstein in Matrix. Form. Exact Derivation of the Theory of Special. without Tensors. and General Relativity.
Giinter Ludyk Einstein in Matrix Form Exact Derivation of the Theory of Special and General Relativity without Tensors ^ Springer Contents 1 Special Relativity 1 1.1 Galilei Transformation 1 1.1.1 Relativity
More informationStatus of Hořava Gravity
Status of Institut d Astrophysique de Paris based on DV & T. P. Sotiriou, PRD 85, 064003 (2012) [arxiv:1112.3385 [hep-th]] DV & T. P. Sotiriou, JPCS 453, 012022 (2013) [arxiv:1212.4402 [hep-th]] DV, arxiv:1502.06607
More informationAn all-scale exploration of alternative theories of gravity. Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste
An all-scale exploration of alternative theories of gravity Thomas P. Sotiriou SISSA - International School for Advanced Studies, Trieste General Outline Beyond GR: motivation and pitfalls Alternative
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationFefferman Graham Expansions for Asymptotically Schroedinger Space-Times
Copenhagen, February 20, 2012 p. 1/16 Fefferman Graham Expansions for Asymptotically Schroedinger Space-Times Jelle Hartong Niels Bohr Institute Nordic String Theory Meeting Copenhagen, February 20, 2012
More informationCurved Spacetime III Einstein's field equations
Curved Spacetime III Einstein's field equations Dr. Naylor Note that in this lecture we will work in SI units: namely c 1 Last Week s class: Curved spacetime II Riemann curvature tensor: This is a tensor
More informationExercise 1 Classical Bosonic String
Exercise 1 Classical Bosonic String 1. The Relativistic Particle The action describing a free relativistic point particle of mass m moving in a D- dimensional Minkowski spacetime is described by ) 1 S
More informationFirst structure equation
First structure equation Spin connection Let us consider the differential of the vielbvein it is not a Lorentz vector. Introduce the spin connection connection one form The quantity transforms as a vector
More informationQuestion 1: Axiomatic Newtonian mechanics
February 9, 017 Cornell University, Department of Physics PHYS 4444, Particle physics, HW # 1, due: //017, 11:40 AM Question 1: Axiomatic Newtonian mechanics In this question you are asked to develop Newtonian
More informationEinstein Double Field Equations
Einstein Double Field Equations Stephen Angus Ewha Woman s University based on arxiv:1804.00964 in collaboration with Kyoungho Cho and Jeong-Hyuck Park (Sogang Univ.) KIAS Workshop on Fields, Strings and
More informationTensor Calculus, Relativity, and Cosmology
Tensor Calculus, Relativity, and Cosmology A First Course by M. Dalarsson Ericsson Research and Development Stockholm, Sweden and N. Dalarsson Royal Institute of Technology Stockholm, Sweden ELSEVIER ACADEMIC
More informationarxiv: v3 [hep-th] 30 Jan 2019
An Action for Extended String Newton-Cartan Gravity Eric A. Bergshoeff a, Kevin T. Grosvenor b, Ceyda Şimşek a, Ziqi Yan c a Van Swinderen Institute, University of Groningen Nijenborgh 4, 9747 AG Groningen,
More information1 Canonical quantization conformal gauge
Contents 1 Canonical quantization conformal gauge 1.1 Free field space of states............................... 1. Constraints..................................... 3 1..1 VIRASORO ALGEBRA...........................
More informationLecture 9: RR-sector and D-branes
Lecture 9: RR-sector and D-branes José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 6, 2013 José D. Edelstein (USC) Lecture 9: RR-sector and D-branes 6-mar-2013
More informationThe Geometric Scalar Gravity Theory
The Geometric Scalar Gravity Theory M. Novello 1 E. Bittencourt 2 J.D. Toniato 1 U. Moschella 3 J.M. Salim 1 E. Goulart 4 1 ICRA/CBPF, Brazil 2 University of Roma, Italy 3 University of Insubria, Italy
More informationClassical aspects of Poincaré gauge theory of gravity
Classical aspects of Poincaré gauge theory of gravity Jens Boos jboos@perimeterinstitute.ca Perimeter Institute for Theoretical Physics Wednesday, Nov 11, 2015 Quantum Gravity group meeting Perimeter Institute
More informationPresentation by: Filip Tolovski March 2019
Special and General Relativity Theory Presentation by: Filip Tolovski March 2019 The Special Theory of Relativity 01/03/2019 By: Filip Tolovski 2 Classical Mechanics and the restricted Principle of Relativity
More informationMetric-affine theories of gravity
Introduction Einstein-Cartan Poincaré gauge theories General action Higher orders EoM Physical manifestation Summary and the gravity-matter coupling (Vinc) CENTRA, Lisboa 100 yy, 24 dd and some hours later...
More informationQuantum Fields in Curved Spacetime
Quantum Fields in Curved Spacetime Lecture 3 Finn Larsen Michigan Center for Theoretical Physics Yerevan, August 22, 2016. Recap AdS 3 is an instructive application of quantum fields in curved space. The
More informationCold atoms and AdS/CFT
Cold atoms and AdS/CFT D. T. Son Institute for Nuclear Theory, University of Washington Cold atoms and AdS/CFT p.1/20 What is common for strong coupled cold atoms and QGP? Cold atoms and AdS/CFT p.2/20
More informationSymmetries of curved superspace
School of Physics, University of Western Australia Second ANZAMP Annual Meeting Mooloolaba, November 27 29, 2013 Based on: SMK, arxiv:1212.6179 Background and motivation Exact results (partition functions,
More informationOne-loop renormalization in a toy model of Hořava-Lifshitz gravity
1/0 Università di Roma TRE, Max-Planck-Institut für Gravitationsphysik One-loop renormalization in a toy model of Hořava-Lifshitz gravity Based on (hep-th:1311.653) with Dario Benedetti Filippo Guarnieri
More informationCold atoms and AdS/CFT
Cold atoms and AdS/CFT D. T. Son Institute for Nuclear Theory, University of Washington Cold atoms and AdS/CFT p.1/27 History/motivation BCS/BEC crossover Unitarity regime Schrödinger symmetry Plan Geometric
More informationSupersymmetric Randall-Sundrum Scenario
CIT-USC/00-015 Supersymmetric Randall-Sundrum Scenario arxiv:hep-th/000117v May 000 Richard Altendorfer, Jonathan Bagger Department of Physics and Astronomy The Johns Hopkins University 400 North Charles
More informationClass Notes Introduction to Relativity Physics 375R Under Construction
Class Notes Introduction to Relativity Physics 375R Under Construction Austin M. Gleeson 1 Department of Physics University of Texas at Austin Austin, TX 78712 March 20, 2007 1 gleeson@physics.utexas.edu
More informationA Higher Derivative Extension of the Salam-Sezgin Model from Superconformal Methods
A Higher Derivative Extension of the Salam-Sezgin Model from Superconformal Methods Frederik Coomans KU Leuven Workshop on Conformal Field Theories Beyond Two Dimensions 16/03/2012, Texas A&M Based on
More information2 Post-Keplerian Timing Parameters for General Relativity
1 Introduction General Relativity has been one of the pilars of modern physics for over 100 years now. Testing the theory and its consequences is therefore very important to solidifying our understand
More informationHIGHER SPIN PROBLEM IN FIELD THEORY
HIGHER SPIN PROBLEM IN FIELD THEORY I.L. Buchbinder Tomsk I.L. Buchbinder (Tomsk) HIGHER SPIN PROBLEM IN FIELD THEORY Wroclaw, April, 2011 1 / 27 Aims Brief non-expert non-technical review of some old
More informationChapter 3: Duality Toolbox
3.: GENEAL ASPECTS 3..: I/UV CONNECTION Chapter 3: Duality Toolbox MIT OpenCourseWare Lecture Notes Hong Liu, Fall 04 Lecture 8 As seen before, equipped with holographic principle, we can deduce N = 4
More informationQuantum Field Theory
Quantum Field Theory PHYS-P 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S-1 A proper description of particle physics
More informationÜbungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.
Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση
More informationParticles and Deep Inelastic Scattering
Particles and Deep Inelastic Scattering Heidi Schellman University HUGS - JLab - June 2010 June 2010 HUGS 1 Course Outline 1. Really basic stuff 2. How we detect particles 3. Basics of 2 2 scattering 4.
More informationGauge Theory of Gravitation: Electro-Gravity Mixing
Gauge Theory of Gravitation: Electro-Gravity Mixing E. Sánchez-Sastre 1,2, V. Aldaya 1,3 1 Instituto de Astrofisica de Andalucía, Granada, Spain 2 Email: sastre@iaa.es, es-sastre@hotmail.com 3 Email: valdaya@iaa.es
More informationOutline. 1 Relativistic field theory with variable space-time. 3 Extended Hamiltonians in field theory. 4 Extended canonical transformations
Outline General Relativity from Basic Principles General Relativity as an Extended Canonical Gauge Theory Jürgen Struckmeier GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany j.struckmeier@gsi.de,
More informationIs there any torsion in your future?
August 22, 2011 NBA Summer Institute Is there any torsion in your future? Dmitri Diakonov Petersburg Nuclear Physics Institute DD, Alexander Tumanov and Alexey Vladimirov, arxiv:1104.2432 and in preparation
More information8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline
8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline 1 Main Headings I Introduction and relativity pre Einstein II Einstein s principle of relativity and a new concept of spacetime III
More informationOutline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up 2.2. The Relativistic Point Particle 2.3. The
Classical String Theory Proseminar in Theoretical Physics David Reutter ETH Zürich April 15, 2013 Outline 1. Introduction 1.1. Historical Overview 1.2. The Theory 2. The Relativistic String 2.1. Set Up
More informationGravitation: Gravitation
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationAnisotropic Interior Solutions in Hořava Gravity and Einstein-Æther Theory
Anisotropic Interior Solutions in and Einstein-Æther Theory CENTRA, Instituto Superior Técnico based on DV and S. Carloni, arxiv:1706.06608 [gr-qc] Gravity and Cosmology 2018 Yukawa Institute for Theoretical
More informationNon-relativistic AdS/CFT
Non-relativistic AdS/CFT Christopher Herzog Princeton October 2008 References D. T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry, Phys. Rev. D 78,
More informationChapter 7 Curved Spacetime and General Covariance
Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME
More informationFrom Gravitation Theories to a Theory of Gravitation
From Gravitation Theories to a Theory of Gravitation Thomas P. Sotiriou SISSA/ISAS, Trieste, Italy based on 0707.2748 [gr-qc] in collaboration with V. Faraoni and S. Liberati Sep 27th 2007 A theory of
More informationExperimental Tests and Alternative Theories of Gravity
Experimental Tests and Alternative Theories of Gravity Gonzalo J. Olmo Alba gonzalo.olmo@uv.es University of Valencia (Spain) & UW-Milwaukee Experimental Tests and Alternative Theories of Gravity p. 1/2
More informationBackreaction effects of matter coupled higher derivative gravity
Backreaction effects of matter coupled higher derivative gravity Lata Kh Joshi (Based on arxiv:1409.8019, work done with Ramadevi) Indian Institute of Technology, Bombay DAE-HEP, Dec 09, 2014 Lata Joshi
More informationPoincaré gauge theory and its deformed Lie algebra mass-spin classification of elementary particles
Poincaré gauge theory and its deformed Lie algebra mass-spin classification of elementary particles Jens Boos jboos@perimeterinstitute.ca Perimeter Institute for Theoretical Physics Friday, Dec 4, 2015
More informationarxiv:hep-th/ v1 10 Apr 2006
Gravitation with Two Times arxiv:hep-th/0604076v1 10 Apr 2006 W. Chagas-Filho Departamento de Fisica, Universidade Federal de Sergipe SE, Brazil February 1, 2008 Abstract We investigate the possibility
More informationRelativity, Gravitation, and Cosmology
Relativity, Gravitation, and Cosmology A basic introduction TA-PEI CHENG University of Missouri St. Louis OXFORD UNIVERSITY PRESS Contents Parti RELATIVITY Metric Description of Spacetime 1 Introduction
More informationExact solutions in supergravity
Exact solutions in supergravity James T. Liu 25 July 2005 Lecture 1: Introduction and overview of supergravity Lecture 2: Conditions for unbroken supersymmetry Lecture 3: BPS black holes and branes Lecture
More informationHorava-Lifshitz Theory of Gravity & Applications to Cosmology
Horava-Lifshitz Theory of Gravity & Applications to Cosmology Anzhong Wang Phys. Dept., Baylor Univ. Waco, Texas 76798 Presented to Texas Cosmology Network Meeting, Austin, TX Oct. 30, 2009 Collaborators
More informationSymmetry and Duality FACETS Nemani Suryanarayana, IMSc
Symmetry and Duality FACETS 2018 Nemani Suryanarayana, IMSc What are symmetries and why are they important? Most useful concept in Physics. Best theoretical models of natural Standard Model & GTR are based
More informationCovariance of the Schrödinger equation under low velocity boosts.
Apeiron, Vol. 13, No. 2, April 2006 449 Covariance of the Schrödinger equation under low velocity boosts. A. B. van Oosten, Theor. Chem.& Mat. Sci. Centre, University of Groningen, Nijenborgh 4, Groningen
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.81 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.81 F008 Lecture 1: Boundary of AdS;
More informationNew Fundamental Wave Equation on Curved Space-Time and its Cosmological Applications
New Fundamental Wave Equation on Curved Space-Time and its Cosmological Applications Z.E. Musielak, J.L. Fry and T. Chang Department of Physics University of Texas at Arlington Flat Space-Time with Minkowski
More informationD. f(r) gravity. φ = 1 + f R (R). (48)
5 D. f(r) gravity f(r) gravity is the first modified gravity model proposed as an alternative explanation for the accelerated expansion of the Universe [9]. We write the gravitational action as S = d 4
More informationA solution in Weyl gravity with planar symmetry
Utah State University From the SelectedWorks of James Thomas Wheeler Spring May 23, 205 A solution in Weyl gravity with planar symmetry James Thomas Wheeler, Utah State University Available at: https://works.bepress.com/james_wheeler/7/
More informationHolography and phase-transition of flat space
Holography and phase-transition of flat space Daniel Grumiller Institute for Theoretical Physics Vienna University of Technology Workshop on Higher-Spin and Higher-Curvature Gravity, São Paulo, 4. November
More informationQuantum Field Theory for hypothetical fifth force
June 18, 2012 Quantum Field Theory for hypothetical fifth force Patrick Linker Contact information: Rheinstrasse 13 61206 Woellstadt Germany Phone: +49 (0)6034 905296 E-Mail: Patrick.Linker@t-online.de
More informationThe N = 2 Gauss-Bonnet invariant in and out of superspace
The N = 2 Gauss-Bonnet invariant in and out of superspace Daniel Butter NIKHEF College Station April 25, 2013 Based on work with B. de Wit, S. Kuzenko, and I. Lodato Daniel Butter (NIKHEF) Super GB 1 /
More informationLecture IX: Field equations, cosmological constant, and tides
Lecture IX: Field equations, cosmological constant, and tides Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: October 28, 2011) I. OVERVIEW We are now ready to construct Einstein
More informationGENERAL RELATIVITY: THE FIELD THEORY APPROACH
CHAPTER 9 GENERAL RELATIVITY: THE FIELD THEORY APPROACH We move now to the modern approach to General Relativity: field theory. The chief advantage of this formulation is that it is simple and easy; the
More informationConnection Variables in General Relativity
Connection Variables in General Relativity Mauricio Bustamante Londoño Instituto de Matemáticas UNAM Morelia 28/06/2008 Mauricio Bustamante Londoño (UNAM) Connection Variables in General Relativity 28/06/2008
More informationClassification theorem for the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes possessing a photon sphere
Classification theorem for the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes possessing a photon sphere Boian Lazov and Stoytcho Yazadjiev Varna, 2017 Outline 1 Motivation 2 Preliminaries
More informationConformal Symmetry Breaking in Einstein-Cartan Gravity Coupled to the Electroweak Theory
Conformal Symmetry Breaking in Einstein-Cartan Gravity Coupled to the Electroweak Theory J. Lee Montag, Ph.D. December 018 Abstract We develop an alternative to the Higgs mechanism for spontaneously breaking
More informationNew Geometric Formalism for Gravity Equation in Empty Space
New Geometric Formalism for Gravity Equation in Empty Space Xin-Bing Huang Department of Physics, Peking University, arxiv:hep-th/0402139v2 23 Feb 2004 100871 Beijing, China Abstract In this paper, complex
More informationAttempts at relativistic QM
Attempts at relativistic QM based on S-1 A proper description of particle physics should incorporate both quantum mechanics and special relativity. However historically combining quantum mechanics and
More informationCovariant Equations of Motion of Extended Bodies with Mass and Spin Multipoles
Covariant Equations of Motion of Extended Bodies with Mass and Spin Multipoles Sergei Kopeikin University of Missouri-Columbia 1 Content of lecture: Motivations Statement of the problem Notable issues
More informationDilaton: Saving Conformal Symmetry
Dilaton: Saving Conformal Symmetry Alexander Monin Ecole Polytechnique Fédérale de Lausanne December 2, 2013 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry
More informationNew Geometric Formalism for Gravity Equation in Empty Space
New Geometric Formalism for Gravity Equation in Empty Space Xin-Bing Huang Department of Physics, Peking University, arxiv:hep-th/0402139v3 10 Mar 2004 100871 Beijing, China Abstract In this paper, complex
More informationNumber-Flux Vector and Stress-Energy Tensor
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Number-Flux Vector and Stress-Energy Tensor c 2000, 2002 Edmund Bertschinger. All rights reserved. 1 Introduction These
More informationCurved spacetime and general covariance
Chapter 7 Curved spacetime and general covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 219 220 CHAPTER 7. CURVED SPACETIME
More informationWHY BLACK HOLES PHYSICS?
WHY BLACK HOLES PHYSICS? Nicolò Petri 13/10/2015 Nicolò Petri 13/10/2015 1 / 13 General motivations I Find a microscopic description of gravity, compatibile with the Standard Model (SM) and whose low-energy
More informationExtensions of Lorentzian spacetime geometry
Extensions of Lorentzian spacetime geometry From Finsler to Cartan and vice versa Manuel Hohmann Teoreetilise Füüsika Labor Füüsika Instituut Tartu Ülikool LQP33 Workshop 15. November 2013 Manuel Hohmann
More informationCausal RG equation for Quantum Einstein Gravity
Causal RG equation for Quantum Einstein Gravity Stefan Rechenberger Uni Mainz 14.03.2011 arxiv:1102.5012v1 [hep-th] with Elisa Manrique and Frank Saueressig Stefan Rechenberger (Uni Mainz) Causal RGE for
More informationINTRODUCTION TO GENERAL RELATIVITY AND COSMOLOGY
INTRODUCTION TO GENERAL RELATIVITY AND COSMOLOGY Living script Astro 405/505 ISU Fall 2004 Dirk Pützfeld Iowa State University 2004 Last update: 9th December 2004 Foreword This material was prepared by
More informationSome simple exact solutions to d = 5 Einstein Gauss Bonnet Gravity
Some simple exact solutions to d = 5 Einstein Gauss Bonnet Gravity Eduardo Rodríguez Departamento de Matemática y Física Aplicadas Universidad Católica de la Santísima Concepción Concepción, Chile CosmoConce,
More informationParticle Notes. Ryan D. Reece
Particle Notes Ryan D. Reece July 9, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation that
More informationA Supergravity Dual for 4d SCFT s Universal Sector
SUPERFIELDS European Research Council Perugia June 25th, 2010 Adv. Grant no. 226455 A Supergravity Dual for 4d SCFT s Universal Sector Gianguido Dall Agata D. Cassani, G.D., A. Faedo, arxiv:1003.4283 +
More informationGeneral Relativity and Cosmology Mock exam
Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers
More informationA Brief Introduction to Relativistic Quantum Mechanics
A Brief Introduction to Relativistic Quantum Mechanics Hsin-Chia Cheng, U.C. Davis 1 Introduction In Physics 215AB, you learned non-relativistic quantum mechanics, e.g., Schrödinger equation, E = p2 2m
More informationRelativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION. Wolfgang Rindler. Professor of Physics The University of Texas at Dallas
Relativity SPECIAL, GENERAL, AND COSMOLOGICAL SECOND EDITION Wolfgang Rindler Professor of Physics The University of Texas at Dallas OXPORD UNIVERSITY PRESS Contents Introduction l 1 From absolute space
More informationTHE MANY AVATARS OF GALILEAN CONFORMAL SYMMETRY
THE MANY AVATARS OF GALILEAN CONFORMAL SYMMETRY Arjun Bagchi Indian Strings Meet 2014 December 18, 2014. CONFORMAL FIELD THEORY Conformal field theories are amongst the most powerful tools in modern theoretical
More information8.821 F2008 Lecture 12: Boundary of AdS; Poincaré patch; wave equation in AdS
8.821 F2008 Lecture 12: Boundary of AdS; Poincaré patch; wave equation in AdS Lecturer: McGreevy Scribe: Francesco D Eramo October 16, 2008 Today: 1. the boundary of AdS 2. Poincaré patch 3. motivate boundary
More information