Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary NTNU TGWG Group Seminar Teleparallelism with Applications Ling-Wei Luo National Tsing Hua University (NTHU) August 11, 2017@NTNU Ling-Wei Luo TGWG Group Seminar@NTNU 0/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Main References: Teleparallel Conformal Invariant Models Induced by Kaluza-Klein Reduction Chao-Qiang Geng, LWL, published on Class. Quant. Grav. 34 185004 (2017). Kaluza{Klein theory for teleparallel gravity Chao-Qiang Geng, Chang Lai, LWL and Huan Hsin Tseng, published on Phys. Lett. B 737, 248 (2014). Teleparallel gravity in five dimensional theories Chao-Qiang Geng, LWL and Huan Hsin Tseng, published on Class. Quantum Grav. 31 (2014) 185004. Ling-Wei Luo TGWG Group Seminar@NTNU 1/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Specific Models 6 Weak Field Approximation 7 Summary Ling-Wei Luo TGWG Group Seminar@NTNU 2/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Specific Models 6 Weak Field Approximation 7 Summary Ling-Wei Luo TGWG Group Seminar@NTNU 2/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Standard Gravity Theory General Relativity Einstein equation 1 G = 8πGT with G := R − Rg µν µν µν µν 2 µν (Einstein, Nov. 25, 1915) Hilbert action −1 Z p d4x −gR + S 2κ m (Hilbert, Nov. 20, 1915) \Spacetime tells matter how to move; matter tells spacetime how to curve." | John Wheeler. Ling-Wei Luo TGWG Group Seminar@NTNU 3/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Alternative Gravitational Theory Riemannian Geometry with Maintaining the Notion of Distant Parallelism (Teleparallelism, Einstein, 1928) Torsion scalar (Einstein, 1929) =) symmetric EoM Equivalent to the Hilbert action (Lanczos, 1929) Generalization: New General Relativity (NGR) 1 1 ; ; −1 −! (a; b; c) 4 2 (Hayashi & Shirafuji, 1979) Ling-Wei Luo TGWG Group Seminar@NTNU 4/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Trajectory of a particle: Frenet-Serret formula 2 Circle on E 3 Helix in E 0 0 1 0 1 p 1 0 ! e1 Be0 C B 0 κC @ 1A = @ A 0 e2 e2 −κ 0 Curvature ∼ Acceleration: 0 0 1 0 1 p 100 0 1 e1 2 0 de d p Be1C B 0 κ 0 C κ = je0 j = 1 = ; B C = B C Be C 1 2 B 0 C B C @ 2A ds ds @e2A @ −κ 0 τ A e3 e0 0 −τ 0 where s = R pdx2 + dy2 is the arc 3 length. τ: torsion =) the trajectory would Osculating plane: spanned by e1 and not lie on the osculating plane. e2. Ling-Wei Luo TGWG Group Seminar@NTNU 5/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Torsion-free: A tangent vector does not rotate when we parallel transport it. (P.371, John Baez and Javier P. Muniain, \Gauge Fields, Knots and Gravity," 1994) Ling-Wei Luo TGWG Group Seminar@NTNU 6/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Cartan's Structure Equations i The infinitesimal translation of a point p: drp = # ⊗ ei j The change of the basis: drei = rei = ! i ⊗ ej. Torsion is associated with the point translation: drp i i j i dr ◦ drp = ( d# + ! j ^ # ) ⊗ ei := T ei = T : The infinitesimal contour integral at a small region D with boundary @D gives I Z Z drp = dr ◦ drp = T : @D D D Curvature is associated with the rotation of the basis vector: drei j j k j dr ◦ drei = ( d! i + ! k ^ ! i ) ⊗ ej := R i ej = Ri : Similarly, I Z Z drei = dr ◦ drei = Ri : @D D D Ling-Wei Luo TGWG Group Seminar@NTNU 7/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Different gravitational theories with geometry (arXiv:9602013[gr-qc]). Ling-Wei Luo TGWG Group Seminar@NTNU 8/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Absolute Parallelism The orthonormal frame in Weitzenb¨ockgeometry T4 i j gµν = ηij e µ e ν with ηij = diag(+1; −1; −1; −1) : Parallel vectors (absolute parallelism) (Cartan, 1922/Eisenhart, 1925) w ν ρ µ ρ ν ρ rei = dx (@ν ei + ei Γ µν )@ρ := dx (rν ei )@ρ = 0 : w ρ ρ i Weitzenb¨ockconnection: Γ µν = ei @ν e µ − !ijµ = 0. σ σ j i Curvature-free R ρµν (Γ) = ei e ρR jµν (!) = 0. w w i i i i i Torsion tensor T µν ≡ Γ νµ − Γ µν = @µe ν − @ν e µ. ρ 1 ρ ρ ρ ρ Contorsion tensor K µν = − 2 (T µν − Tµ ν − Tν µ) = −Kµ ν . ( i i j i i k T = K j ^ # with K j := K jk# ; i i i !e j := ! j − K j the Levi-Civita connection form : Ling-Wei Luo TGWG Group Seminar@NTNU 9/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Teleparallel Equivalent to GR in T4 Decomposition of the Weitzenb¨ockconnection w ρ ρ ρ Γ µν = fµν g + K µν ; Teleparallel Equivalent to GR (GRk or TEGR) in T4 based on the ν the relation (Tµ := T νµ) µ µ R(Γ) = R~(e)+ T − 2 r~ µT = 0 =)− R~(e)= T − 2 r~ µT : Torsion Scalar (Einstein, 1929) 1 1 1 T ≡ T ρ T µν + T ρ T νµ − T ν T σµ = T i S µν 4 µν ρ 2 µν ρ µν σ 2 µν i µν µν µ σν ν σµ νµ Sρ ≡ K ρ + δρ T σ − δρ T σ = −Sρ is superpotential . TEGR action Z 1 4 p STEGR = d x e T (e = −g) : 2κ Ling-Wei Luo TGWG Group Seminar@NTNU 10/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Shortest and Straitest Curves Shortest lines ! geodesic 2 µ ν ρ Z δS d x dx dx S = ds −! + f µ g = 0 dτ 2 νρ dτ dτ µ dxµ Straitest lines ! autoparallel curves for V = dτ µ ρ ? dV µ ν dx Action principle −! + Γ νρV = 0 dτ dτ µ µ µ Symm. part: Γ (νρ) = f(νρ)g + K (νρ) =) torsion involved in! New action principle with torsion (Kleinert & Pelster, 1996) Teleparallelism: Straitest lines ! loxodromic curves (rhumb lines in navigation) Meridians at a constant angle. Path everywhere orthogonal to torsion vector =) Straitest lines = Shortest lines Ling-Wei Luo TGWG Group Seminar@NTNU 11/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Problems with Torsion Curved spacetime or twisted spacetime Coupling with gauge field (violation of Gauge invariance) ? Fµν = @µAν − @ν Aµ −! rµAν − rν Aµ Modified covariant derivative D = d + Γ − K =) Same as GR (Andrade & Pereira, 1997) Equivalent principle: Normal coordinate =) Γ = 0 Only for symmetric connection (Veblen, 1973) First law of thermodynamics violation in f(T ) (Miao, Li, Miao, 2011) Gravity ! Black Hole Thermodynamics (Bardeen, Carter, Hawking 1973) µ µ µ Translational gauge field (ei = δi @µ −! Di = δi rµ + Bi rµ) µ µ µ µ ei or ei = δi + Bi : Local Lorentz violation: only axial vector torsion is invariant. Lagrangian is inv. =) EoM is not inv. (i) Dirac equation is not local Lorentz inv. in T4. (ii) Energy-momentum asymmetry in TEGR Ling-Wei Luo TGWG Group Seminar@NTNU 12/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Coupling of fermion =) chiral anomaly (index theorem) contribution of torsion (?) Gravitational chiral anomaly inGR (Kimura, 1969) 1 2 d ? j = R~ ^ R~ ij = d Ω~ ^ d Ω~ + Ω~ ^ Ω~ ^ Ω~ : A 384 π2 ij 3 Dirac Lagrangian with torsion is NOT Hermitian i i L = e ¯ i γm e µ @ − !~ (e) + K σjk with σjk = [γj ; γk] : D m µ 2 jkµ jkµ 4 | {z } Dµ Dirac equation in U4 i µ 1 i γ ei Dµ − m = 0 with Dµ := Dµ − Tµ : ~ 2 Nieh-Yan term (Nieh & Yan, 1981) i i j i Ti ^ T − Rij ^# ^ # = d Ti ^ # |{z} vanished in teleparallelism Teleparallel SUGRA? Ling-Wei Luo TGWG Group Seminar@NTNU 13/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Applications with Torsion Quasi local energy of GR in teleparallel formulation (Nester,1989) Cosmological magnetic field (magnetogenesis) (Sabbata & Gasperini, 1980) Nonsingular, big-bounce cosmology in Einstein-Cartan-Sciama-Kibble (ECSK) theory (Pop lawski,2012) Born-Infeld type inflation (Ferraro & Fiorini, 2007) r eλ 2T L = 1 + − 1 16πG λ Teleparallel dark energy (Geng, Lee, Saridakis, Wu, 2011) 1 1 µ 2 L = e T + @µφ∂ φ + ξT φ − V (φ) 16πG 2 Ling-Wei Luo TGWG Group Seminar@NTNU 14/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Constraints on Torsion? Black hole in teleparallelism Y.