Non-Abelian and gravitational Chern-Simons densities
|
|
- Luke Green
- 5 years ago
- Views:
Transcription
1 Non-Abelian and gravitational Chern-Simons densities Tigran Tchrakian School of Theoretical Physics, Dublin nstitute for Advanced Studies (DAS) and Department of Computer Science, Maynooth University, Maynooth, reland Fourth Summer School on High Energy Physics and Quantum Field Theory, August 2016, Yerevan, Armenia
2 Plan Chern-Simons densities Chern-Simons densities of Abelian gauge fields Chern-Simons densities of non-abelian gauge fields Gauge transformation of Chern-Simons densities Einstein-Cartan formulation of gravity Equivalence with Metric formalism p-einstein-hilbert systems in d-dimensions p-einstein equations Gravitational Chern-Pontryagin densities Gravitational Chern-Simons densities Chern-Simons gravity
3 Abelian Chern-Simons densities Abelian curvature of Abelian connection A i F ij = i A j j A i Definition of Abelian (gauge invariant) Chern-Pontryagin density in 2n + 2 dimensions Ω CP = ε i1 j 1 i 2 j 2...i n+1 j n+1 F i1 j 1 F i2 j 2... F in+1 j n+1 = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 µn+1 ( Aνn+1 F µ1 ν 1 F µ2 ν 2... F µnν n ) = µn+1 Ω µn+1 implies definition of Abelian (gauge variant) Chern-Simons density in 2n + 1 dimensions Ω CS = ε i1 j 1 i 2 j 2...i nj nj n+1 A jn+1 F i1 j 1 F i2 j 2... F inj n Ω CS is not a total divergence and leads to the variational equations which is gauge covariant. ε i1 j 1 i 2 j 2...i nj nj n+1 F i1 j 1 F i2 j 2... F inj n = 0
4 Non-Abelian (na) Chern-Simons densities non-abelian curvature of Abelian connection A i F ij = i A j j A i + [A i, A j ] Definition of non-abelian (gauge invariant) Chern-Pontryagin density in 2n + 2 dimensions Ω CP = ε µ1 ν 1 µ 2 ν 2...µ n+1 ν n+1 Tr F µ1 ν 1 F µ2 ν 2... F µn+1 ν n+1 = µn+1 Ω µn+1 which is a total divergence and likewise implies definition of non-abelian (gauge variant) Chern-Simons density in 2n + 1 dimensions. For n = 1, D = 3, it is ( Ω (1) CS = ε ijktr A k F ij 2 ) 3 F if j. For n = 2, D = 5, it is Ω (2) CS = ε ijklm Tr A m ( F ij F kl F ij A k A l A ia j A k A l ).
5 For n 3 there are multiple distinct definitions for the CS density, each characterised by the number of traces in the definition. For n = 3, D = 7, there are two possibilities; a definition with a single trace and another one with double trace. These are Ω (3) CS = ε ijklmnp Tr A p (F ij F kl F mn 4 5 F ijf kl A m A n 2 5 F ija k F lm A n F ija k A l A m A n 8 ) 35 A ia j A k A l A m A n, ( Ω (3) CS = ε ijklmnp Tr A p F mn 2 ) 3 A ma n (Tr F ij F kl )
6 As in the Abelian case, here too the equations of motion of (the usual) non-abelian CS densities are gauge covariant. For the examples listed above they are for d = 3, 5 respectively, and for d = 7 ε ijk F ij = 0 ε ijklm F ij F kl = 0 ε ijklmnp F ij F kl F mn = 0 ε ijklmnp (Tr F ij F kl ) F mn = 0
7 Gauge transformation of non-abelian CS densities Abelian CS densities in all dimensions transform as Ω CP Ω CP + Ω Non-Abelian CS densities also transform with a total divergence term, plus a winding number term featuring the group element α µ = µ g g 1. n d = 3, 5 explicit expressions for this are Ω (2) (2) CS Ω CS = Ω(2) CS 2 3 ε λµνtr α λ α µ α ν 2ε λµν λ Tr α µ A ν Ω (3) (3) CS Ω CS = Ω(3) CS 2 5 ε λµνρσtr α λ α µ α ν α ρ α σ [ ( +2 ε λµνρσ λ Tr α µ A ν F ρσ 1 ) 2 A ρa σ + (F ρσ 12 A ρa σ ) A ν 1 2 A ν α ρ A σ α ν α ρ A σ ]
8 Metric, covariant derivative and curvature Covariant vector V M and contravariant vector V M are related by the space(-time) metric g MN V M = g MN V N, V M = g MN V N, g MN = g NM, g MK g KN = δ N M The signature of the metric can be chosen to be Euclidean, Minkowskian, etc. For gravity, we choose Minkowskian. The covariant derivative of V M is defined as M V N = M V N + Γ MN K V K in terms of the Christoffel symbol Γ MN K The metric tensor is covariantly constant K g MN = 0. The Riemann curvature tensor is defined as R MNK L = ( [M N] ) K L = [M Γ N]K L + (Γ MK Γ N L Γ NK Γ M L ) and using the Leibniz rule it satisfies the Bianchi identity L R MNR S + (cycl. L, M, N) = 0.
9 Frame vector formalism of gravity: Einstein-Cartan theory Covariant frame-vector φ a and contravariant vector φ a are related by the flat metric η ab φ a = η ab φ b, φ a = η ab φ b, η ab = η ba, η ac η bc = δ b a (Since η ab is constant, henceforth ignore co- and contra-variace of frame indices.) The covariant derivative of the (co- or contra-variant) frame vector is defined in terms of the the spin-connection ω ab M as D M φ a = M φ a + ω ab M φb and the corresponding curvature is R ab MN = (D [MD N] ) ab = [M ω ab N] + ωac [M ωcb N] and applying the Leibniz rule the Bianchi identity is D L RMN ab + (cycl. L, M, N) = 0.
10 The dynamical quantities replacing the metric tensor are the Vielbeine e a M and ema (or e M a ) with one spacetime and one (flat) frame index, such that e a M = g MNe N a, e M a = g MN e a N, and satisfying the orthonormality properties e M a e N a = g MN, e a M ea N = g MN, e a M en a = δ N M, ea M em b = δa b. The covariant derivative of the Vielbein e a M is the spin connection ω ab M The Torsion T a MN D M e a N = Me a N + ωab M eb N acting only on the frame index b. is the antisymmetrised covariant derivative T a MN = D [Me a N].
11 Equivalence with metric formalism The spacetime-vector φ M and frame-vector φ a are related by φ M = e a M φ a, such that the covariant derivative M φ N and (D M φ) a are related by M φ N = e a N (D Mφ) a. This results in the expression for the Christoffel symbol, Γ MN K = e K a D M e a N which results in the covariant constancy of the metric K g MN = 0, AS REQURED! When the Christoffel symbol is symmetric Γ MN K = Γ NM K Γ [MN] K = e K a T a MN = 0 T a MN = 0. This is called a Levi-Civita connection.
12 p-einstein-hilbert systems in all d-dimensions Employing the Vielbein field em a the definitions of the gravitational curvature and torsion are RMN ab = (D [M D N] ) ab φ b = ( M ωn ab NωM ab M ωcb N ωac N ωcb M ) φb TMN a = D [MeN] a = MeN a NeM a + ωac M ec N ωac N ec M, with M = 1, 2,..., d ; a = 1, 2,..., d. Splitting the indices on the Levi-Civita symbols as ε M 1M 2...M 2p M 2p+1...M d and ε a1 a 2...a 2p a 2p+1...a d d-dimensional p-einsten-hilbert (p-eh) Lagrangians are L (p,d) EH = ε M 1M 2...M 2p M 2p+1...M d e a 2p+1 M 2p+1 e a 2p+2 M 2p+2... e a d M d ε a1 a 2...a 2p a 2p+1...a d R a 1a 2 R a 3a 4 M 3 M 4... R a 2p 1a 2p M 2p 1 M 2p For p = 0 this is a total divergence, for p = 1 it is the usual Einstein-Hilbert Lagrangian in d-dimensions, for p = 2 it is the usual Gauss-Bonnet Lagrangian in d-dimensions, etc.
13 The p-einstein equations Zero Torsion: The Levi-Civita connection is not an independent field since T a MN = D [Me a N] = 0 εmn... ω ab M eb N = εmn... M e a N can be inverted to give a closed form expression for the spin connactin ωm ab ωab M [e, e 1, e]. The p-einstein equations follow from the variation of the free-standing Vielbeine in the p-eh Lagrangians L (p,d) EH. (Note that the curvature dependant terms are total divergence.) Non-zero Torsion: n this case the p-eh Lagrangians must be varied separately w.r.t. the spin-connection yielding the Torsion equation, e.g. varying L (2,5) EH w.r.t. ωpq 2ε LMNRS ε pqabc RRS bc T MN a = matter current L
14 The p-einstein-hilbert systems in d-dimensions, for = 1, 2, 3 are L (1,d) EH = ε M 1M 2 M 3...M d e a 3 M 3 e a 4 M 4... e a d M d ε a1 a 2 a 3...a d R a 1a 2 L (2,d) EH = ε M 1M 2 M 3 M 4 M 5...M d e a 5 M 7 e a 6 M 6... e a d M d ε a1 a 2 a 3 a 4 a 5...a d R a 1a 2 R a 3a 4 M 3 M 4 L (3,d) EH = ε M 1M 2...M 7...M d e a 7 M 7... e a d M d ε a1 a 2...a 7...a d R a 1a 2 R a 3a 4 M 3 M 4 R a 5a 6 M 5 M 6 with the resulting Einstein equations E M d a d = 0 ε M 1M 2 M 3...M d e a 3 M 3 e a 4 M 4... e a d 1 M d 1 ε a1 a 2 a 3...a d R a 1a 2 = 0 ε M 1M 2 M 3 M 4 M 5...M d e a 5 M 7 e a 6 M 6... e a d 1 M d 1 ε a1 a 2 a 3 a 4 a 5...a d R a 1a 2 R a 3a 4 M 3 M 4 = 0 ε M 1M 2...M 7...M d e a 7 M 7... e a d 1 M d 1 ε a1 a 2...a 7...a d R a 1a 2 R a 3a 4 M 3 M 4 R a 5a 6 M 5 M 6 = 0 R M d a d (2p) 1 2p R(2p) em d a d = 0 R M d a d (2p) = R M d N d (2p) e N d a d, R(2p) = R a d M d (2p) e M d a d the p-ricci tensor and the p-ricci scalar.
15 Gravitational Chern-Pontryagin densities: two choices Passage from non-abelian gauge (YM) fields to gravity is achieved by replacing the YM connection and curvature in evaluating the Traces defining the (YM) CP densities: A M = 1 2 ωab M γ ab, F MN = 1 2 Rab MN γ ab, a = 1, 2,... D ; where γ ab are the generators of SO(D) γ ab = 1 4 [γ a, γ b ] D = 2n and γ a are the Dirac matrices in D = 2n dimensions. Since D = 2n is even, there exists a chiral matrix γ D+1, γ 2 D+1 = 1.
16 n evaluating the Traces there are two choices, leading to two types of gravitational CP (GCP) densities, Type- without, and Type- with, γ 2n+1 C (n) = ε M 1M 2...M 2n 1 M 2n R a 1a 2... R a 2n 1a 2n M 2n 1 M 2n Tr γ a1 a 2... γ a2n 1 a 2n C (n) = ε M 1M 2...M 2n 1 M 2n R a 1a 2... R a 2n 1a 2n M 2n 1 M 2n Tr γ D+1 γ a1 a 2... γ a2n 1 a 2n. C (n) C (n) are called (gravitational) Chern-Pontryagin densities, are called Euler-Hirzbruch densities. t follows from the prperties of the Clifford algebras that: C (n) = 0, for odd n C (n) = ε M 1M 2...M 2n 1 M 2n ˆδ a n+1...a 2n a 1 a 2...a n R a 1a 2... R a n 1a n M n 1 M n, for even n C (n) = ε M 1M 2...M 2n 1 M 2n ε a1 a 2...a 2n 1 a 2n R a 1a 2... R a 2n 1a 2n M 2n 1 M 2n, for all n.
17 Gravitational Chern-Simons densities: one choice These are constructed by evaluating the Traces in the YM (na) Chern-Simons densities in D 1 dimensions. The na-cs densities are extracted from the na-cp densities which are defined for SO(D) gauge group. The SO(D) gauge group is not the required group for the frame indices in D 1 = 2n 1 dimensions. Again, there are two choices, leading to two types of gravitational CP (GCP) densities, Type- without, and Type- with, γ D+1. There are two steps in this process: Evaluate the Traces yielding gravitational densities in D 1 dimensions with SO(D) frame index, Contract the gauge group SO(D) SO(D 1).
18 1st step: n = 2, 3, 4 Employing the index notation µ = 1, 2,... D 1, (D = 2n), list the two types Traces Ω (n) and Ω (n) for n = 2, 3, 4. Tr (2) = ε λµν δ a b ab ω ab λ Tr (2) = ε λµν ε abcd ωλ ab Tr (3) = 0 Tr (3) = ε λµνρσ ε abcdef ωλ ab [ R a b µν 2 3 (ω µω ν ) a b ] [ R cd µν 2 3 (ω µω ν ) cd ], [ R cd Tr (4) = ε λµνρστκˆδa b c d abcd ωλ ab, µνrρσ ef Rµν cd (ω ρ ω σ ) ef (ω µω ν ) cd (ω ρ [ RµνR cd a b ρσ R c d τκ 4 5 Rcd µνr a b ρσ (ω τ ω κ ) c d Rcd µν (ω ρ ω σ ) a b (ω τ ω κ ) c d 8 35 (ω µω ν ) cd ( [ Tr (4) = ε λµνρστκ ε abcdefgh ωλ ab RµνR cd ρσr ef τκ gh 4 5 Rcd µνrρσ ef (ω τ ω κ ) gh 2 5 Rcd µ + 4 R cd µν (ω ρ ω σ ) ef (ω τ ω κ ) gh 8 (ω µ ω ν ) cd (ω
19 2nd step: Group contraction Note that the frame indices on ω ab µ, have the range a = 1, 2,..., D. The correct range for the frame indices of gravity in d = D 1 dimensions is α = 1, 2,..., d, with d = D 1. This contraction is effected by truncating the components of the spin connection ωµ ab = (ωµ αβ, ωµ αd ) according to Ω (2) = 1 2 2! ελµν δ α β αβ Ω (2) = 0, Ω (3) = 0, Ω (3) = 0, Ω (4) = ˆΩ (4) = 0, ωµ αd = 0 Rµν αd = 0. ω αβ λ 1 2 6! ελµνρστκˆδ α β γ δ αβγδ ω αβ λ [ R α β µν 2 3 (ω µω ν ) α β ] [ RµνR γδ α β ρσ R γ δ τκ 4 5 Rγδ µνr α β ρσ (ω τ ω κ ) Rγδ µν (ω ρ ω σ ) α β (ω τ ω κ ) γ δ 8 35 (ω µω ν ) γδ (ω
20 Chern-Simons gravity Not gravitational Chern-Simons density! Extend the Ansatz used above: A µ = 1 2 ωab µ γ ab, F µν = 1 2 Rab µν γ ab, a = 1, 2,... D ; by incorporating the 2nd step above: D = 2n a = (α, D), α = 1, 2,..., d, d = D 1 = 2n 1 such that now leading to F µν = 1 2 A µ = 1 2 ωαβ µ γ αβ + κ e α µ γ αd ( ) Rµν αβ κ 2 e[µ α eβ ν] γ αβ + κ Tµνγ α αd
21 Substituting this Ansatz for A µ in the non-abelian SO(4) CS density of Type- in d = 2n 1 = 3 (n = 2) results in the Chern-Simons gravitational Lagrangian ( L (2) CSG = κ εµνλ ε abc eλ c Rab µν 2 ) 3 κ2 eµe a ν b eλ c and in d = 2n 1 = 3 (n = 2) ( L (3) 3 CSG = κ εµνρσλ ε abcde 4 ee λ Rab µν Rρσ cd eρe c σ d eλ e Rab µν + 3 ) 5 κ4 eµe a ν b eρe c σ d eλ e
Abelian and non-abelian Hopfions in all odd dimensions
Abelian and non-abelian Hopfions in all odd dimensions Tigran Tchrakian Dublin Institute for Advanced Studies (DIAS) National University of Ireland Maynooth, Ireland 7 December, 2012 Table of contents
More informationChern-Simons Gravities (CSG) and Gravitational Chern-Simons (GCS) densities in all dimensions
Chern-Simons Gravities (G) and Gravitational Chern-Simons (G) densities in all dimensions D. H. Tchrakian arxiv:1712.05190v2 [gr-qc 1 Jan 2018 School of Theoretical Physics, Dublin Institute for Advanced
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 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 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 informationA note on the principle of least action and Dirac matrices
AEI-2012-051 arxiv:1209.0332v1 [math-ph] 3 Sep 2012 A note on the principle of least action and Dirac matrices Maciej Trzetrzelewski Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,
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 informationGRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are.
GRAVITATION F0 S. G. RAJEEV Lecture. Maxwell s Equations in Curved Space-Time.. Recall that Maxwell equations in Lorentz covariant form are. µ F µν = j ν, F µν = µ A ν ν A µ... They follow from the variational
More informationPAPER 52 GENERAL RELATIVITY
MATHEMATICAL TRIPOS Part III Monday, 1 June, 2015 9:00 am to 12:00 pm PAPER 52 GENERAL RELATIVITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
More informationInequivalence of First and Second Order Formulations in D=2 Gravity Models 1
BRX TH-386 Inequivalence of First and Second Order Formulations in D=2 Gravity Models 1 S. Deser Department of Physics Brandeis University, Waltham, MA 02254, USA The usual equivalence between the Palatini
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 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 informationRepresentation theory and the X-ray transform
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 1/18 Representation theory and the X-ray transform Differential geometry on real and complex projective
More informationQuantising Gravitational Instantons
Quantising Gravitational Instantons Kirill Krasnov (Nottingham) GARYFEST: Gravitation, Solitons and Symmetries MARCH 22, 2017 - MARCH 24, 2017 Laboratoire de Mathématiques et Physique Théorique Tours This
More informationIntroduction to Chern-Simons forms in Physics - II
Introduction to Chern-Simons forms in Physics - II 7th Aegean Summer School Paros September - 2013 Jorge Zanelli Centro de Estudios Científicos CECs - Valdivia z@cecs.cl Lecture I: 1. Topological invariants
More informationWeek 9: Einstein s field equations
Week 9: Einstein s field equations Riemann tensor and curvature We are looking for an invariant characterisation of an manifold curved by gravity. As the discussion of normal coordinates showed, the first
More informationObserver dependent background geometries arxiv:
Observer dependent background geometries arxiv:1403.4005 Manuel Hohmann Laboratory of Theoretical Physics Physics Institute University of Tartu DPG-Tagung Berlin Session MP 4 18. März 2014 Manuel Hohmann
More informationThe Divergence Myth in Gauss-Bonnet Gravity. William O. Straub Pasadena, California November 11, 2016
The Divergence Myth in Gauss-Bonnet Gravity William O. Straub Pasadena, California 91104 November 11, 2016 Abstract In Riemannian geometry there is a unique combination of the Riemann-Christoffel curvature
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 informationA Generally Covariant Field Equation For Gravitation And Electromagnetism
3 A Generally Covariant Field Equation For Gravitation And Electromagnetism Summary. A generally covariant field equation is developed for gravitation and electromagnetism by considering the metric vector
More informationThe spectral action for Dirac operators with torsion
The spectral action for Dirac operators with torsion Christoph A. Stephan joint work with Florian Hanisch & Frank Pfäffle Institut für athematik Universität Potsdam Tours, ai 2011 1 Torsion Geometry, Einstein-Cartan-Theory
More informationPAPER 309 GENERAL RELATIVITY
MATHEMATICAL TRIPOS Part III Monday, 30 May, 2016 9:00 am to 12:00 pm PAPER 309 GENERAL RELATIVITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
More informationmatter The second term vanishes upon using the equations of motion of the matter field, then the remaining term can be rewritten
9.1 The energy momentum tensor It will be useful to follow the analogy with electromagnetism (the same arguments can be repeated, with obvious modifications, also for nonabelian gauge theories). Recall
More informationON CHERN-SIMONS GAUGE THEORY
ON CHERN-SIMONS GAUGE THEORY XIAO XIAO June 27, 200 Abstract In the chiral anomaly of the Abelian and non-abelian gauge theory in even dimensional spacetime, it is a curious fact that the anomalous breaking
More informationNonlinear wave-wave interactions involving gravitational waves
Nonlinear wave-wave interactions involving gravitational waves ANDREAS KÄLLBERG Department of Physics, Umeå University, Umeå, Sweden Thessaloniki, 30/8-5/9 2004 p. 1/38 Outline Orthonormal frames. Thessaloniki,
More informationA Gravity, U(4) U(4) Yang-Mills and Matter Unification in Clifford Spaces
A Gravity, U(4) U(4) Yang-Mills and Matter Unification in Clifford Spaces Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, GA. 30314, perelmanc@hotmail.com
More informationA GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM. Institute for Advanced Study Alpha Foundation
A GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM Myron W. Evans Institute for Advanced Study Alpha Foundation E-mail: emyrone@aol.com Received 17 April 2003; revised 1 May 2003
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 informationLorentz-covariant spectrum of single-particle states and their field theory Physics 230A, Spring 2007, Hitoshi Murayama
Lorentz-covariant spectrum of single-particle states and their field theory Physics 30A, Spring 007, Hitoshi Murayama 1 Poincaré Symmetry In order to understand the number of degrees of freedom we need
More informationSupergravity. Abstract
Selfdual backgrounds in N = 2 five-dimensional Chern-Simons Supergravity Máximo Bañados Departamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile mbanados@maxwell.fis.puc.cl
More informationGeneralizations of Schwarzschild and (Anti) de Sitter Metrics in Clifford Spaces
Generalizations of Schwarzschild and (Anti) de Sitter Metrics in Clifford Spaces Carlos Castro Center for Theoretical Studies of Physical Systems Clark Atlanta University, Atlanta, Georgia. 30314, perelmanc@hotmail.com
More informationVariational Principle and Einstein s equations
Chapter 15 Variational Principle and Einstein s equations 15.1 An useful formula There exists an useful equation relating g µν, g µν and g = det(g µν ) : g x α = ggµν g µν x α. (15.1) The proof is the
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 informationDimensional Reduction in Lovelock Gravity. F. Javier Moreno
Dimensional Reduction in Lovelock Gravity F. Javier Moreno Madrid 2017 Dimensional Reduction in Lovelock Gravity F. Javier Moreno Trabajo Fin de Máster at the Instituto de Física Teórica UAM/CSIC Madrid
More informationDiffeomorphism Invariant Gauge Theories
Diffeomorphism Invariant Gauge Theories Kirill Krasnov (University of Nottingham) Oxford Geometry and Analysis Seminar 25 Nov 2013 Main message: There exists a large new class of gauge theories in 4 dimensions
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 informationarxiv:gr-qc/ v1 30 Jan 2006
Torsion induces Gravity Rodrigo Aros Departamento de Ciencias Físicas, Universidad Andrés Bello, Av. Republica 252, Santiago,Chile Mauricio Contreras Facultad de ciencias y tecnología Universidad Adolfo
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 informationTwistor Strings, Gauge Theory and Gravity. Abou Zeid, Hull and Mason hep-th/
Twistor Strings, Gauge Theory and Gravity Abou Zeid, Hull and Mason hep-th/0606272 Amplitudes for YM, Gravity have elegant twistor space structure: Twistor Geometry Amplitudes for YM, Gravity have elegant
More informationDILATON-DEPENDENT α CORRECTIONS IN GAUGE THEORY OF GRAVITATION
FIELD THEORY, GRAVITATION AND PARTICLE PHYSICS DILATON-DEPENDENT α CORRECTIONS IN GAUGE THEORY OF GRAVITATION S. BABETI Department of Physics, Politehnica University, Timiºoara, România, sbabeti@etv.utt.ro
More informationCurved spacetime tells matter how to move
Curved spacetime tells matter how to move Continuous matter, stress energy tensor Perfect fluid: T 1st law of Thermodynamics Relativistic Euler equation Compare with Newton =( c 2 + + p)u u /c 2 + pg j
More informationConnections and geodesics in the space of metrics The exponential parametrization from a geometric perspective
Connections and geodesics in the space of metrics The exponential parametrization from a geometric perspective Andreas Nink Institute of Physics University of Mainz September 21, 2015 Based on: M. Demmel
More informationt, H = 0, E = H E = 4πρ, H df = 0, δf = 4πJ.
Lecture 3 Cohomologies, curvatures Maxwell equations The Maxwell equations for electromagnetic fields are expressed as E = H t, H = 0, E = 4πρ, H E t = 4π j. These equations can be simplified if we use
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 informationTheories of Massive Gravity
UNIVERSITÀ DEGLI STUDI DI PADOVA DIPARTIMENTO DI FISICA ED ASTRONOMIA GALILEO GALILEI CORSO DI LAUREA MAGISTRALE IN FISICA Theories of Massive Gravity Supervisor: Prof. Dmitri Sorokin Student: Eugenia
More informationCitation for published version (APA): Halbersma, R. S. (2002). Geometry of strings and branes. Groningen: s.n.
University of Groningen Geometry of strings and branes Halbersma, Reinder Simon IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please
More informationPerelman s Dilaton. Annibale Magni (TU-Dortmund) April 26, joint work with M. Caldarelli, G. Catino, Z. Djadly and C.
Annibale Magni (TU-Dortmund) April 26, 2010 joint work with M. Caldarelli, G. Catino, Z. Djadly and C. Mantegazza Topics. Topics. Fundamentals of the Ricci flow. Topics. Fundamentals of the Ricci flow.
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 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 informationBondi mass of Einstein-Maxwell-Klein-Gordon spacetimes
of of Institute of Theoretical Physics, Charles University in Prague April 28th, 2014 scholtz@troja.mff.cuni.cz 1 / 45 Outline of 1 2 3 4 5 2 / 45 Energy-momentum in special Lie algebra of the Killing
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 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 informationNotes on torsion. Arkadiusz Jadczyk. October 11, 2010
Notes on torsion Arkadiusz Jadczyk October 11, 2010 A torsion is a torsion of a linear connection. What is a linear connection? For this we need an n-dimensional manifold M (assumed here to be smooth).
More informationAn Introduction to Kaluza-Klein Theory
An Introduction to Kaluza-Klein Theory A. Garrett Lisi nd March Department of Physics, University of California San Diego, La Jolla, CA 993-39 gar@lisi.com Introduction It is the aim of Kaluza-Klein theory
More informationTowards a manifestly diffeomorphism invariant Exact Renormalization Group
Towards a manifestly diffeomorphism invariant Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for UK QFT-V, University of Nottingham,
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 informationIntroduction to relativistic quantum mechanics
Introduction to relativistic quantum mechanics. Tensor notation In this book, we will most often use so-called natural units, which means that we have set c = and =. Furthermore, a general 4-vector will
More informationNewman-Penrose formalism in higher dimensions
Newman-Penrose formalism in higher dimensions V. Pravda various parts in collaboration with: A. Coley, R. Milson, M. Ortaggio and A. Pravdová Introduction - algebraic classification in four dimensions
More informationLecture 10. The Dirac equation. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 10 The Dirac equation WS2010/11: Introduction to Nuclear and Particle Physics The Dirac equation The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist
More informationGeneral Relativity (225A) Fall 2013 Assignment 8 Solutions
University of California at San Diego Department of Physics Prof. John McGreevy General Relativity (5A) Fall 013 Assignment 8 Solutions Posted November 13, 013 Due Monday, December, 013 In the first two
More informationTHE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY
THE INITIAL VALUE FORMULATION OF GENERAL RELATIVITY SAM KAUFMAN Abstract. The (Cauchy) initial value formulation of General Relativity is developed, and the maximal vacuum Cauchy development theorem is
More informationGeometry of SpaceTime Einstein Theory. of Gravity II. Max Camenzind CB Oct-2010-D7
Geometry of SpaceTime Einstein Theory of Gravity II Max Camenzind CB Oct-2010-D7 Textbooks on General Relativity Geometry of SpaceTime II Connection and curvature on manifolds. Sectional Curvature. Geodetic
More informationSyllabus. May 3, Special relativity 1. 2 Differential geometry 3
Syllabus May 3, 2017 Contents 1 Special relativity 1 2 Differential geometry 3 3 General Relativity 13 3.1 Physical Principles.......................................... 13 3.2 Einstein s Equation..........................................
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 informationNotes on General Relativity Linearized Gravity and Gravitational waves
Notes on General Relativity Linearized Gravity and Gravitational waves August Geelmuyden Universitetet i Oslo I. Perturbation theory Solving the Einstein equation for the spacetime metric is tremendously
More informationOn Torsion Fields in Higher Derivative Quantum Gravity
Annales de la Fondation Louis de Broglie, Volume 3 no -3, 007 33 On Torsion Fields in Higher Derivative Quantum Gravity S.I. Kruglov University of Toronto at Scarborough, Physical and Environmental Sciences
More informationChern-Simons Theory and Its Applications. The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee
Chern-Simons Theory and Its Applications The 10 th Summer Institute for Theoretical Physics Ki-Myeong Lee Maxwell Theory Maxwell Theory: Gauge Transformation and Invariance Gauss Law Charge Degrees of
More informationRank Three Tensors in Unified Gravitation and Electrodynamics
5 Rank Three Tensors in Unified Gravitation and Electrodynamics by Myron W. Evans, Alpha Institute for Advanced Study, Civil List Scientist. (emyrone@aol.com and www.aias.us) Abstract The role of base
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 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 informationth Aegean Summer School / Paros Minás Tsoukalás (CECs-Valdivia)
013-09-4 7th Aegean Summer School / Paros Minás Tsoukalás (CECs-Valdivia Higher Dimensional Conformally Invariant theories (work in progress with Ricardo Troncoso 1 Modifying gravity Extra dimensions (string-theory,
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 information1.13 The Levi-Civita Tensor and Hodge Dualisation
ν + + ν ν + + ν H + H S ( S ) dφ + ( dφ) 2π + 2π 4π. (.225) S ( S ) Here, we have split the volume integral over S 2 into the sum over the two hemispheres, and in each case we have replaced the volume-form
More informationFourth Order Ricci Gravity
Fourth Order Ricci Gravity A. Borowiec a, M. Francaviglia b and V.I. Smirichinski c a Institute of Theoretical Physics, Wroc law University, Poland b Departimento di Matematica, Unversitá di Torino, Italy
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Due-date: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More informationGravity theory on Poisson manifold with R-flux
Gravity theory on Poisson manifold with R-flux Hisayoshi MURAKI (University of Tsukuba) in collaboration with Tsuguhiko ASAKAWA (Maebashi Institute of Technology) Satoshi WATAMURA (Tohoku University) References
More informationConstrained BF theory as gravity
Constrained BF theory as gravity (Remigiusz Durka) XXIX Max Born Symposium (June 2010) 1 / 23 Content of the talk 1 MacDowell-Mansouri gravity 2 BF theory reformulation 3 Supergravity 4 Canonical analysis
More informationSpinor Formulation of Relativistic Quantum Mechanics
Chapter Spinor Formulation of Relativistic Quantum Mechanics. The Lorentz Transformation of the Dirac Bispinor We will provide in the following a new formulation of the Dirac equation in the chiral representation
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 informationSpinor Representation of Conformal Group and Gravitational Model
Spinor Representation of Conformal Group and Gravitational Model Kohzo Nishida ) arxiv:1702.04194v1 [physics.gen-ph] 22 Jan 2017 Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan Abstract
More informationA Brief Introduction to Tensors
A Brief Introduction to Tensors Jay R Walton Fall 2013 1 Preliminaries In general, a tensor is a multilinear transformation defined over an underlying finite dimensional vector space In this brief introduction,
More informationGeometry of the Universe: Cosmological Principle
Geometry of the Universe: Cosmological Principle God is an infinite sphere whose centre is everywhere and its circumference nowhere Empedocles, 5 th cent BC Homogeneous Cosmological Principle: Describes
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationarxiv: v1 [hep-th] 16 Apr 2008
UG-08-06 Higher-spin dynamics and Chern-Simons theories arxiv:0804.2627v1 [hep-th] 16 Apr 2008 Johan Engquist 1 and Olaf Hohm 2 1 Department of Physics, University of Oslo P.O. Box 1048 Blindern, N-0316
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 informationC. C. Briggs Center for Academic Computing, Penn State University, University Park, PA Tuesday, August 18, 1998
SOME POSSIBLE FEATURES OF GENERAL EXPRESSIONS FOR LOVELOCK TENSORS AND FOR THE COEFFICIENTS OF LOVELOCK LAGRANGIANS UP TO THE 15 th ORDER IN CURVATURE (AND BEYOND) C. C. Briggs Center for Academic Computing,
More informationOverthrows a basic assumption of classical physics - that lengths and time intervals are absolute quantities, i.e., the same for all observes.
Relativistic Electrodynamics An inertial frame = coordinate system where Newton's 1st law of motion - the law of inertia - is true. An inertial frame moves with constant velocity with respect to any other
More informationApplied 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 informationTensor Calculus, Part 2
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Tensor Calculus, Part 2 c 2000, 2002 Edmund Bertschinger. 1 Introduction The first set of 8.962 notes, Introduction
More informationBlack Hole Entropy in the presence of Chern-Simons terms
Black Hole Entropy in the presence of Chern-Simons terms Yuji Tachikawa School of Natural Sciences, Institute for Advanced Study Dec 25, 2006, Yukawa Inst. based on hep-th/0611141, to appear in Class.
More informationLevi-Civita Rhymes with Lolita
Levi-Civita Rhymes with Lolita William O Straub Pasadena, California 91104 February 2, 2017 Ladies and gentlemen of the jury, exhibit number one is what the seraphs, the misinformed, simple, noble-winged
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 informationGeneral-relativistic quantum theory of the electron
Allgemein-relativistische Quantentheorie des Elektrons, Zeit. f. Phys. 50 (98), 336-36. General-relativistic quantum theory of the electron By H. Tetrode in Amsterdam (Received on 9 June 98) Translated
More informationVector and Tensor Calculus
Appendices 58 A Vector and Tensor Calculus In relativistic theory one often encounters vector and tensor expressions in both three- and four-dimensional form. The most important of these expressions are
More informationUniversity of Naples FEDERICO II
University of Naples FEDERICO II Faculty of Mathematical, Physical and Natural Sciences Department of Physical Sciences Master thesis in theoretical physics: Discrete quantum field theories of the gravitational
More informationA Dyad Theory of Hydrodynamics and Electrodynamics
arxiv:physics/0502072v5 [physics.class-ph] 19 Jan 2007 A Dyad Theory of Hydrodynamics and Electrodynamics Preston Jones Department of Mathematics and Physics University of Louisiana at Monroe Monroe, LA
More informationON VARIATION OF THE METRIC TENSOR IN THE ACTION OF A PHYSICAL FIELD
ON VARIATION OF THE METRIC TENSOR IN THE ACTION OF A PHYSICAL FIELD L.D. Raigorodski Abstract The application of the variations of the metric tensor in action integrals of physical fields is examined.
More informationOn Connections of the Anti-Symmetric and Totally Anti-Symmetric Torsion Tensor
On Connections of the Anti-Symmetric and Totally Anti-Symmetric Torsion Tensor D. Lindstrom, H. Eckardt, M. W. Evans August 5, 2016 Abstract Based on the compatibility of the metric only, in general, it
More informationGeneral Relativity and Differential
Lecture Series on... General Relativity and Differential Geometry CHAD A. MIDDLETON Department of Physics Rhodes College November 1, 2005 OUTLINE Distance in 3D Euclidean Space Distance in 4D Minkowski
More informationNTNU Trondheim, Institutt for fysikk
NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 998971 Allowed tools: mathematical tables 1. Spin zero. Consider a real, scalar field
More information