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Teleparallel Gravity in Five Dimensional Theories

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Teleparallel Gravity in Five Dimensional Theories Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary RESCEU APCosPA Summer School 2015 Teleparallel Gravity in Five Dimensional Theories Reference: Phys. Lett. B 737, 248 (2014), Class. Quant. Grav. 31 (2014) 185004 Ling-Wei Luo Institute of Physics, National Chiao Tung University Collaborators: Chao-Qiang Geng (NTHU/NCTS), Chang Lai (CQUPT, China), Huan Hsin Tseng (NTHU/Tufts Univ.), August 2, 2015 @ Nikko city, Tochigi Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 0/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Summary Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 1/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Summary Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 1/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Absolute Parallelism The orthonormal frame in Weitzenb¨ockgeometry W4 i j gµν = ηij eµ eν with ηij = diag(+1; −1; −1; −1) : Metric compatible condition r gµν = 0: i ρ i i j d eµ − Γµ eρ + ! j eµ = 0 and !ij = − !ji : Absolute parallelism for parallel vectors (Cartan 1922/Eisenhart 1925) i i i ρ rν eµ = @ν eµ − eρ Γµν = 0 : w ρ ρ i =) Weitzenb¨ockconnection Γµν = ei @ν eµ − !ijµ = 0. σ σ j i Curvature-free R ρµν (Γ) = ei eρR jµν (!) = 0. w w i i i i i Torsion tensor T µν ≡ Γνµ − Γµν = @µeν − @ν eµ. Contorsion tensor 1 Kρ = − (T ρ − T ρ − T ρ ) = −K ρ : µν 2 µν µ ν ν µ µ ν Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 2/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Geometrical Meaning of Torsion 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) T (u; v) = ruv − rvu − [u; v] | {z } vanished in coordinate space Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 3/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Teleparallel Equivalent to GR in W4 Decomposition of the Weitzenb¨ockconnection w ρ ρ ρ Γµν = fµν g + K µν ; Teleparallel Equivalent to GR (GRk or TEGR) in W4 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 . The TEGR action 1 Z p S = d4x e T (e = −g) : TEGR 2κ Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 4/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Motivation Fundamental fields inGR: Metric tensor gµν ρ 1 ρσ =) Levi-Civita connection fµν g = 2 g @µgσν + @ν gµσ − @σgµν Fundamental field in Teleparallel Gravity: i Veierbein fields eµ ρ ρ i =) Weitzenb¨ockconnection Γµν = ei @ν eµ. Question Does there exist any different effect coming from the extra dimension? Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 5/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Summary Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 5/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Hypersurface in GR Gauss normal coordinate with the signature 5 _ 5 e 5 _ _ e (+ − − − "). e The tensor BMN := −r~ M nN is defined by the unit normal vector of the hypersurface n V M=M_ MN expansion θ = h BMN ; 1 shear σ = B − θh ; MN (MN) 3 MN twist !MN = B[MN] −! 0 (Hypersurface orthogonal) ; where hMN =g ¯MN − " nM nN the projection operator. Gauss's equation µ µ µ µ R¯ νρσ = R νρσ + "(K σ Kνρ − K ρ Kνσ) Extrinsic curvature ~ 1 5 Kµν = B(µν) = −" rµn · eν = −" 2 Lngµν = " fµν g n5 Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 6/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary 5-Dimension Teleparallelism An embedding: W4 −! W5. In Gauss normal coordinate g (xµ; y) 0 g¯ = µν : MN 0 εφ2(xµ; y) The 5D torsion scalar in the orthonormal frame (5) 1 i5^j j5^i j i 5^ j k5^ T = T¯ + T¯ ^ T¯ + T¯ ^ T¯ + 2 T¯ j T¯ ^ − T¯ ^ T¯ k : |{z} 2 i5j i5j i5 5j induced 4D torsion scalar i 5^ The non-vanishing components of vielbein are eµ and e5 Projection of the torsion tensor ρ ρ T¯ µν = T µν (purely 4-dimensional object) Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 7/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary i −! µ ; i −! 5 × ; 5^ −! µ × ; 5^ −! 5 : The 5D torsion scalar in the coordinate frame 1 (5)T = T¯ + T¯ T¯ρ5ν + T¯ T¯ν5ρ + 2 T¯σ µ T¯5 − T¯ν T¯σ5 : 2 ρ5ν ρ5ν σ µ5 5ν σ Note: ρ ρ ρ In general, induced torsion T¯ µν = T µν + C¯ µν , where 5^ C¯ µν z }| { C¯ρ =e ¯ρ(@ e¯5^ − @ e¯5^ ) : µν 5^ µ ν ν µ ¯5^ 5^ 5^ M N 5^ C µν = Γνµ − Γµν = hµ hν T MN ∼ !µν is related to the extrinsic torsion or twist !µν . Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 8/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Summary Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 8/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Braneworld Theory The vielbein in the Gauss normal coordinate ei (xµ; y) 0 e¯I = µ : M 0 φ(xµ; y) The induced torsion scalar T¯ = T . The bulk action in the orthonormal frame Z 1 5 1 i5j j5i Sbulk = dvol T + (T¯i5j T¯ + T¯i5j T¯ ) 2κ5 2 2 + e (φ) T¯a − T¯ T¯5 with T¯ := T¯b : φ i 5 A bA Note: According to [Ponce de Leon 2001], the induced-matter theory (RAB = 0, Wesson 1998) can be regarded as a mathematically equivalent formulation of the braneworld theory. Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 9/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Assume the bulk metric g¯ is maximally symmetric 3-space with spatially flat (k = 0) 2 2 2 2 g¯MN = diag 1; −a (t; y); −a (t; y); −a (t; y); " φ (t; y) ; #¯0^ = dt ; #¯a = a(t; y) dxα ; #¯5^ = φ(t; y) dy : First Cartan structure equation a_ a0 φ_ T¯0^ = d¯#¯0^ = 0; T¯a = #¯0^ ^ #¯a + #¯5^ ^ #¯a; T¯5^ = #¯0^ ^ #¯5^ ; a aφ φ Torsion 5-form reads " !# 3 + 9 " a02 a_ φ_ T¯ = T + + 6 dvol5 : φ2 a2 a φ Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 10/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary ¯ ¯B ¯ The energy-momentum tensor is ΣA = TA ?¯#B ¯B ¯B ¯B TA (t; y) = TA bulk + TA brane ; δ(y) T¯B = diag(ρ(t); −P (t); −P (t); −P (t); 0) ; A brane φ ¯B Λ5 B TA bulk = ηA : κ5 We have the 0^0^-component equation ! a_ 2 a_ φ_ 1 a00 a0 φ0 1 a02 κ + − − − = 5 T¯ : a2 a φ φ2 a a φ φ2 a2 3 0^0^ Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 11/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary The scale factor a(t; y) = θ(y)a(+)(t; y) + θ(−y)a(−)(t; y) =) a00(t; y) = δ(y)[a0](t; 0) +a ~00(t; y) with [a0] = a(+)0 − a(−)0 : The junction condition: κ symmetry κ [a0](t; 0) = − 5 ρ a (t) φ (t) =======Z2 ) a0(t; 0) = − 5 ρ a (t) φ (t) : 3" 0 0 6" 0 0 Modified Friedmann equation on the brane 2 2 a_ 0(t) a¨0(t) κ5 k5 ¯ 2 + = − ρ(t)(ρ(t) + 3P (t)) − 2 T55 bulk : a0(t) a0(t) 36 3φ0(t) =) Coincides with GR! (See Binetruy, Deffayet and Langlois 2000), but the junction condition comes from torsion itself! Remark: T¯ = T a purely 4-dimensional object in Gauss normal coordinate. =) No extrinsic torsion contribution on the brane in TEGR. Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 12/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Outline 1 Teleparallel Gravity 2 Five-Dimensional Geometry 3 Braneworld Scenario 4 Kaluza-Klein Theory 5 Summary Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 12/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Kaluza-Klein Theory KK ansatz: Cylindrical condition (no y dependency) Compactify to S1 and only consider zeroKK mode 1 The manifold is M4 × S (y = r θ) M The metric is reduced to g (xµ) 0 g¯ = µν : MN 0 −φ2(xµ) ρ 5 The residual components are T µν and T¯ µ5 = @µφ/φ.
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