NTNU TGWG Group Seminar [0.2Cm] Teleparallelism with Applications

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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

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 √ 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    p 1 0 ! e1 e0   0 κ  1 =   0 e2 e2 −κ 0 Curvature ∼ Acceleration:  0    p 100   e1 2 0 de d p e1  0 κ 0  κ = |e0 | = 1 = ,   =   e  1 2  0     2 ds ds e2  −κ 0 τ  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: d∇p = ϑ ⊗ ei j The change of the basis: d∇ei = ∇ei = ω i ⊗ ej. Torsion is associated with the point translation: d∇p

i i j i d∇ ◦ d∇p = ( dϑ + ω j ∧ ϑ ) ⊗ ei := T ei = T .

The inﬁnitesimal contour integral at a small region D with boundary ∂D gives I Z Z d∇p = d∇ ◦ d∇p = T . ∂D D D

Curvature is associated with the rotation of the basis vector: d∇ei

j j k j d∇ ◦ d∇ei = ( dω i + ω k ∧ ω i ) ⊗ ej := R i ej = Ri .

Similarly, I Z Z d∇ei = d∇ ◦ d∇ei = 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 ∇ei = dx (∂ν ei + ei Γ µν )∂ρ := dx (∇ν ei )∂ρ = 0 .

w ρ ρ i Weitzenb¨ockconnection: Γ µν = ei ∂ν e µ ←− ωijµ = 0. σ σ j i Curvature-free R ρµν (Γ) = ei e ρR jµν (ω) = 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 ∧ ϑ with K j := K jkϑ , i i i ωe j := ω 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 = {µν } + 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 ∇˜ µT = 0 =⇒− R˜(e)= T − 2 ∇˜ 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 √ 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 −→ + { µ } = 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) = {(νρ)} + 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 −→ ∇µAν − ∇ν 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 ∇µ + Bi ∇µ) 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 ψ¯ i γm e µ ∂ − ω˜ (e) + K σjk ψ 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 ψ = 0 with Dµ := Dµ − Tµ . ~ 2

Nieh-Yan term (Nieh & Yan, 1981)

i i j i Ti ∧ T − Rij ∧ϑ ∧ ϑ = d Ti ∧ ϑ |{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. N. Obukhov and J. G. Pereira, “Metric affine approach to teleparallel gravity,” Phys. Rev. D 67, 044016 (2003). J. B. Formiga, “Comment on “Metric-affine approach to teleparallel gravity”,” Phys. Rev. D 88, no. 6, 068501 (2013). Searches for torsion Y. Mao, M. Tegmark, A. H. Guth and S. Cabi, “Constraining Torsion with Gravity Probe B,” Phys. Rev. D 76, 104029 (2007) F. W. Hehl and W. T. Ni, “Inertial effects of a Dirac particle,” Phys. Rev. D 42, 2045 (1990). The translational gauge theory deviates from Einstein’s theory in fifth order of the post-newtonian approximation. J. Nitsch and F. W. Hehl, “Translational Gauge Theory of Gravity: Postnewtonian Approximation and Spin Precession,” Phys. Lett. 90B, 98 (1980). Review: W. T. Ni, “Searches for the role of spin and polarization in gravity,” Rept. Prog. Phys. 73, 056901 (2010).

Ling-Wei Luo TGWG Group Seminar@NTNU 15/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Motivation

Fundamental ﬁelds inGR:

Metric tensor gµν ρ 1 ρσ =⇒ Levi-Civita connection {µν } = 2 g ∂µgσν + ∂ν gµσ − ∂σgµν Fundamental field in Teleparallelism: i Veierbein fields e µ ρ ρ i =⇒ Weitzenböckconnection Γ µν = ei ∂ν e µ.

What will be arised from the extra dimensions? (i) Any new interaction? (ii) New classes of gravity theory?

Ling-Wei Luo TGWG Group Seminar@NTNU 16/ 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

Gauss normal coordinate with the signature 5 _ 5 e 5 _ _ e (+ − − − ε). e

The tensor Bµν := −∇˜ ν n · eµ is deﬁned by the unit normal vector of the hypersurface n =e ¯5.

V M=M_ µν expansion θ = h Bµν , 1 shear σµν = B − θhµν , (µν) n − 1 twist ωµν = B[µν] −→ 0 (Hypersurface orthogonal) ,

where hMN =g ¯MN − ε nn the projection operator with ε = −1. Gauss’s equation

µ µ µ µ R¯ νρσ = R νρσ + ε(K σ Kνρ − K ρ Kνσ)

1 5 Extrinsic curvature Kµν = B(µν) = −ε 2 Lngµν = {µν } n

Ling-Wei Luo TGWG Group Seminar@NTNU 17/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary 5-Dimension TEGR

Embedding: T4 −→ T5. 5D Notation 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 i5ˆj j5ˆi j i 5ˆ j k5ˆ T = T¯ + T¯ ˆ T¯ + T¯ ˆ T¯ + 2 T¯ j T¯ ˆ − T¯ ˆ T¯ k . |{z} 2 i5j i5j i5 5j induced 4D torsion scalar

i 5ˆ The non-vanishing components of vielbein are e µ and e 5 Projection of the torsion tensor

ρ ρ T¯ µν = T µν (purely 4-dimensional object)

Ling-Wei Luo TGWG Group Seminar@NTNU 18/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary

i −→ µ ( ) i −→ 5 (×) 5ˆ −→ µ (×) 5ˆ −→ 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, the induced torsion is T¯ µν = T µν + C¯ µν , where

5ˆ C¯ µν z }| { ¯ρ ρ 5ˆ 5ˆ C µν =e ¯5ˆ (∂µe ν − ∂ν e µ) .

¯5ˆ 5ˆ 5ˆ M N 5ˆ C µν = Γ νµ − Γ µν = hµ hν T MN ∼ ωµν is related to the extrinsic torsion or twist ωµν .

Ling-Wei Luo TGWG Group Seminar@NTNU 19/ 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

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 a 5 b + ei(φ) T¯ − T¯5 T¯ with T¯A := T¯ bA . φ

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 TGWG Group Seminar@NTNU 20/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation 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), ε φ (t, y) ,

ˆ ˆ ϑ¯0 = dt , ϑ¯a = a(t, y) dxα , ϑ¯5 = φ(t, y) dy . First Cartan structure equation

0 ˆ ˆ a˙ ˆ a ˆ ˆ φ˙ ˆ ˆ T¯0 = d¯ϑ¯0 = 0, T¯a = ϑ¯0 ∧ ϑ¯a + ϑ¯5 ∧ ϑ¯a, T¯5 = ϑ¯0 ∧ ϑ¯5 , a aφ φ Torsion 5-form reads " !# 3 + 9 ε a02 a˙ φ˙ T¯ = T + + 6 dvol5 , φ2 a2 a φ T = −6H2 .

Ling-Wei Luo TGWG Group Seminar@NTNU 21/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary

¯ ¯B ¯ The energy-momentum tensor is ΣA = TA ?¯ϑB ¯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 0ˆ0ˆ-component equation

2 ˙ ! 00 0 0 02 a˙ a˙ φ 1 a a φ 1 a κ5 + − − − = T¯ˆˆ . a2 a φ φ2 a a φ φ2 a2 3 00

Ling-Wei Luo TGWG Group Seminar@NTNU 22/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation 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:

0 κ5 Z2 symmetry 0 κ5 [a ](t, 0) = − ρ a0(t) φ0(t) ======⇒ a (t, 0) = − ρ a0(t) φ0(t) . 3ε 6ε 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.

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 23/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Kaluza-Klein (KK) Theory KK ansatz: Cylindrical condition (no y dependency) Compactiﬁcation: G = S1 and only consider zero KK 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 1 The residual components are T µν and T µ5 = φ ∂µφ. The 5D torsion scalar with κ4 = κ5/(2πr) (5) σ µ 5 T = T +2 T σ T¯ µ5

Eﬀective action of 5D TEGR Z 1 4 µ Seﬀ = d x e (φ T +2 T ∂µφ) 2κ4 (C.Q. Geng, LWL, H.H. Tseng, 2014)

Ling-Wei Luo TGWG Group Seminar@NTNU 24/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Minimal and Non-Minimal Coupling Minimal coupled case T ∼ −R (TEGR).

TEGR in 5D KK scenario with the metric given by

2 gµν − k AµAν k Aµ 2 g¯MN = with k = κ4, k Aν −1 The eﬀective Lagrangian is 1 1 µν Leﬀ = e T − Fµν F (coincides with the form of GR). 2κ4 4

(de Andrade, Guillen, Pereira, 2000) Non-minimal coupled case φ T −φ R Remark: µ The curvature-torsion relation in TEGR: −R˜(e)= T − 2 ∇˜ µT .

Ling-Wei Luo TGWG Group Seminar@NTNU 25/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary 5D GR vs. 5D TEGR in KK Scenario The dimensional reduction of 5D GR

Brans-Dicke theory (Brans & Dicke, 1961) with ωBD = 0 √ p (5) (5) ˜ ˜ − − g R −→ − −g φR −φ . | {z } surface term The dimensional reduction of 5D TEGR (5) (5) µ e T −→ e φ T +2 T ∂µφ . | {z } no analogue

µ Substituting the relation −R˜(e) = T − 2 ∇˜ µT into the 4D eﬀective Lagrangian Equivalent to Brans-Dicke Theory Z −1 4 µ d x e φR˜(e) −2 ∇˜ µ(φ T ) . 2κ4 | {z } surface term

Ling-Wei Luo TGWG Group Seminar@NTNU 26/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Conformal Transformation

2 By conformal transformation (geµν = Ω (x) gµν ):

2 µν µν T = Ω Te − 4 Ω ge Teµ∂ν Ω + 6 ge ∂µΩ ∂ν Ω , −1 Tµ = Teµ + 3 Ω ∂µΩ .

Choosing φ = Ω2, the action reads Z 4 1 1 µν Seﬀ = d x e Te + ge ∂µψ ∂ν ψ , 2 κ4 2

(C.Q. Geng, Chang Lai, LWL, H.H. Tseng, 2014) 1/2 where ψ = (6/κ4) ln Ω is the dilaton ﬁeld. There exists an Einstein frame for such a non-minimal coupled eﬀective Lagrangian in teleparallel gravity.

Ling-Wei Luo TGWG Group Seminar@NTNU 27/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Adding the Vector and Scalar Fields 5D metric is given by 2 4 2 gµν − κ φ AµAν κ φ Aµ g¯MN = 2 2 , κ φ Aν −φ

The coframes in T5 are  i i µ i  θ = e µ dx + e 5 dy ,  |{z} µ i 1 5ˆ 0 =⇒ dy = κ A e θ + θ . µ i φ  5ˆ 5ˆ µ 5ˆ 5 µ  θ = e µ dx + e 5 dx = −κ φ Aµ dx + φ dy ,

The torsion components in T5

i i i µ ν µ ν T¯ jk = T jk + κ Aµ(∂5e ν )(ej ek − ek ej ) ,

i 1 i µ T¯ ˆ = (∂5e µ)ej , 5j φ ˆ κ T¯5 = − φ e µe ν F + κ2 φ A (∂ A )(e ν e µ − e ν e µ), ij 2 i j µν µ 5 ν i j j i 5ˆ 1 µ 1 µ µ T¯ ˆ = ei (∂µφ) + κ Aµei (∂5φ) + κ ei (∂5Aµ) . i5 φ φ

Ling-Wei Luo TGWG Group Seminar@NTNU 28/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Extending to 5D NGR torsion scalar in 4D is given by 1 T = a T T ijk + b T T kji + c T j T k i := T i Σ jk , NGR ijk ijk ji k 2 jk i where

jk jk kj jk k lj j lk Σi = 2a Ti + b (T i − T i) + c (δi T l − δi T l) . Extending to a 5D torsion scalar theory Extended torsion scalar

(5) (ext) LMN NML L N M T = a T¯LMN T¯ + b T¯LMN T¯ + c T¯ LM T¯ N .

5ˆ5ˆ Decomposition (η5ˆ5ˆ = η = −1)

(5) (ext) ¯ ¯ ¯i5ˆj ¯ ¯5îj ¯ ¯j5î ¯ ¯ji5ˆ T = TNGR + 2a Ti5ˆjT + a T5îjT + b Ti5ˆjT + 2b T5îjT ¯ ¯5î5ˆ ¯j i ¯5ˆ i ¯i ¯j 5ˆ + (2a + b + c) T5î5ˆT + 2c T j T 5ˆ + c T i5ˆT j .

Ling-Wei Luo TGWG Group Seminar@NTNU 29/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary

ρσ l 2 µρ νσ i l T¯NGR = TNGR + 4aκ Tl Aρ(∂5e σ) − 2aκ (g AµAρ)(g ηil(∂5e ν )(∂5e σ))

2 νρ i µσ l σρ k − 2aκ ηil(g Aρ(∂5e ν ))(g Aµ(∂5e σ)) + 2bκT kAρ(∂5e σ) ρσ k 2 µρ i k ν σ − 2bκT kAρ(∂5e σ) + bκ (g AµAρ)(∂5e ν )(∂5e σ)ek ei 2 µσ k ρ i ν − 2bκ (g Aµ(∂5e σ))(Aρei )(∂5e ν )ek 2 µ ρ νσ i k + bκ (Aµek )(Aρei )(g (∂5e ν )(∂5e σ)) j σ ρ k j ρ σ k + 2cκT j (Aρek )(∂5e σ) − 2cκT j Aρek (∂5e σ) 2 µ ρ νσ j k + cκ (Aµej )(Aρek )(g (∂5e ν )(∂5e σ)) 2 µ k σ νρ j − 2cκ (Aµej )(∂5e σ)ek (g Aρ(∂5e ν )) 2 µρ j ν k σ + cκ (g AµAρ)(∂5e ν )ej (∂5e σ)ek ,

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i5ˆj 1 5ˆ5ˆ µν i k T¯ ˆ T¯ = η ηikg (∂5e µ)(∂5e ν ) , i5j φ2

2 ˆ κ T¯ T¯5ij = φ2η gµρgνσF F + 2κ3φ2η gµσgνρA (∂ A )F 5ˆij 4 5ˆ5ˆ µν ρσ 5ˆ5ˆ µ 5 ν σρ 4 2 µρ νσ + 2κ φ η5ˆ5ˆ(g AµAρ)(g (∂5Aν )(∂5Aσ)) 4 2 µσ νρ − 2κ φ η5ˆ5ˆ(g Aµ(∂5Aσ))(g (∂5Aν )Aρ) ,

j5ˆi 1 5ˆ5ˆ i j µ ν T¯ ˆ T¯ = η (∂5e µ)(∂5e ν )ej ei , i5j φ2

ˆ κ T¯ T¯ji5 = e ν gµρF (∂ ej ) − κ2(A e µ)(gνρ(∂ A )(∂ ej )) 5ˆij 2 j µν 5 ρ µ j 5 ν 5 ρ 2 µρ j ν + κ (Aµg )(∂5Aν )(∂5e ρ)ej ,

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ˆ ˆ 1 2κ ˆˆ T¯ T¯5i5 = η η5ˆ5ˆ(gµν (∂ φ)(∂ φ))+ η η55(∂ φ)(gµν (∂ φ)A ) 5ˆi5ˆ φ2 5ˆ5ˆ µ ν φ2 5ˆ5ˆ 5 µ ν

2 2κ ˆˆ κ ˆˆ + η η55(gµν (∂ φ)(∂ A )) + η η55(∂ φ)2(gµν A A ) φ 5ˆ5ˆ µ 5 ν φ2 5ˆ5ˆ 5 µ ν 2 2κ ˆˆ ˆˆ + η η55(∂ φ)(gµν A (∂ A )) + κ2η η55(gµν (∂ A )(∂ A )) , φ 5ˆ5ˆ 5 µ 5 ν 5ˆ5ˆ 5 µ 5 ν

ˆ 1 1 T¯j T¯5 i = − T ρ(∂ φ) − T ρA (∂ φ) − κ T ρ(∂ A ) ji 5ˆ φ ρ φ ρ 5 5 ρ κ κ − (A e µ)(gνρ(∂ ej )(∂ φ)) + (e ν (∂ ej ))(gµρA (∂ φ)) φ µ j 5 ν ρ φ j 5 ν µ ρ κ2 κ2 − (∂ φ)(A e µ)(gνρA (∂ ej )) + (∂ φ)(∂ ej )e ν (gµρA A ) φ 5 µ j ρ 5 ν φ2 5 5 ν j µ ρ 2 µ νρ j 2 j ν µρ − κ (Aµej )(g (∂5e ν )(∂5Aρ)) + κ (∂5e ν )ej (g Aµ(∂5Aρ)) , ˆ 1 ˆˆ T¯i T¯j 5 = η55(∂ ei )e µ(∂ ej )e ν . i5ˆ j φ2 5 µ i 5 ν j

Ling-Wei Luo TGWG Group Seminar@NTNU 32/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Kaluza-Klein Tower Harmonic expansions i X i(n) iny/2r X (n) iny/r e µ(x, y) = e µ(x) e =⇒ g(x, y)µν = gµν (x) e , n n X (n) iny/r Aµ(x, y) = Aµ (x) e , n X φ(x, y) = φ(n)(x) einy/r . n Zero KK mode  (0) T¯ = T ,  NGR NGR  2  ¯ ¯5îj κ (0) 2 µρ νσ (0) (0)  Tˆ T = φ ηˆˆg g Fµν Fρσ ,  5ij 4 55 ˆ ˆ 1 ˆˆ T¯ T¯5i5 = η η55(gµν (∂ φ(0))(∂ φ(0))) ,  5î5ˆ (0) 2 5ˆ5ˆ µ ν  φ   j i 5ˆ i 1 (0) ρ (0)  T¯ j T¯ ˆ = − T (∂ρφ ) .  5 φ(0) ¯ ¯i5ˆj ¯ ¯j5î ¯ ¯ji5ˆ ¯i ¯j 5ˆ Ti5ˆjT = Ti5ˆjT = T5îjT = T i5ˆT j = 0 Ling-Wei Luo TGWG Group Seminar@NTNU 33/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Specific Models Weak Field Approximation Summary Zero KK mode in 4D effective extended gravity in teleparallism Z 4 1 a 3 µρ νσ Seff = d x e φ TNGR − φ g g Fµν Fρσ 2κ4 8 2a + b + c 1 µν c µ + g ∂µφ∂ν φ − T ∂µφ . 2κ4 φ κ4

Conformal 2 Weyl (5) (5) I Consider 5D matter Lagrangian Lm = e Lm(e M , Ψ, DM Ψ) The n-mode harmonic expansion of Ψ is given by µ X µ iny/r Ψ(x , y) = Ψ(x ) e , n For the massless zero mode, the matter field Ψ(0) is assumed to be localized on the T4 hypersurface, the resulting effective matter Lagrangian is i Lm, eff = eλφLm(e µ,Aµ, φ, Ψ, DµΨ) . Note: For 4D localized matter, the Lagrangian can be identified as

(4) i Lm = Lm, eﬀ = eLm(e µ, Ψ, DµΨ) . λ=1,Aµ=0, φ=1

Ling-Wei Luo TGWG Group Seminar@NTNU 34/ 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

i i 2 eµ = Ω eµ (λe = 1) =⇒ geµν = Ω (x) gµν (λg = 2) , Transformation of the torsion tensor and vector

i i i i Te µν = Ω T µν + (∂µΩ) eν − eµ (∂ν Ω) , −1 Teµ = Tµ − 3 Ω ∂µΩ , Torsion scalar of NGR, 2 µν TNGR = Ω TeNGR + 4 a + 2 b + 6 c ge Teµ(Ω ∂ν Ω) µν + 6 a + 3 b + 9 c ge ∂µΩ ∂ν Ω .

λ Scalar φ: φe = Ω φ φ with λφ = −2

Vector Aµ: Aeµ = Aµ and Feµν = Fµν

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Transformed eﬀective Lagrangian density 1 µν Lg = e φeTeNGR + 4 a + 2 b + 2 c φege Teµ ∂ν ω 2κ4 µν + 14 a + 7 b + c φege ∂µω ∂ν ω 2 6ω aκ 3 µρ νσ 1 µν − e φe ge ge Feµν Feρσ + 2a + b + c ge ∂µφe ∂ν φe 4 φe µν µν − 2 c ge Teµ ∂ν φe + 8 a + 4 b − 2 c ge ∂µω ∂ν φe , (∗)

with the conformal scalar ω := ln Ω. Conformal 1

Ling-Wei Luo TGWG Group Seminar@NTNU 36/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Einstein-Frame

In general, the Einstein-frame does not exist for the non-minimal torsion scalar φTNGR µν The non-minimal coupling ge Teµ(Ω ∂ν Ω) will be generated µν The Einstein-frame is obtained by eliminating the term ge Teµ∂ν ω in the transformed Lagrangian

the nessacery condition: 2 a + b + c = 0 .

The Lagrangian is reduced to

2 1 µν aκ 3 µρ νσ Lg = e φ TNGR − 2 c g Tµ∂ν φ − φ g g Fµν Fρσ . 2κ4 4

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The corresponding Einstein-frame

2 (E) 1 c µν aκ 6ω µρ νσ Lg = e TeNGR − ge ∂µϕ ∂ν ϕ − e ge ge Feµν Feρσ , 2κ4 2 8κ4

p where where ϕ := 6/κ4 ω and the ghost-free condition is c ≤ 0. Einstein-frame condition 2 a + b + c = 0 and c ≤ 0 .

For a simple choice of c = −1, one gets the minimal coupled one-parameter family model with 2a + b = 1 in teleparallelism.

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−1 µν µν Only keeping the terms of φe ge ∂µφe ∂ν φe and ge Teµ ∂ν φe Lagrangian 4a + 2b + 2c = 0 , 14a + 7b + c = 0 , 8a + 4b − 2c = 0 . Conformal Invariant condition 2a + b = 0 and c = 0 Corresponding to the simple one-parameter conformal invariant gravity in teleparallelism Z (c) 4 a ijk kji Sg = d x e φ Tijk T − 2 Tijk T . 2κ4

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Assuming that ω = ω(φ) under the conformal transformation 0 0 0 Due to dφ/dφe = ω exp(λφω)(1 + λφω φ) with ω := dω/dφ

φe = φe(φ) =⇒ ω(φe)

By setting ∂µω = ∂µ ln φe.

Lagrangian density (∗) without Aµ becomes 1 µν Lg = e φeTeNGR + 4 a + 2 b ge Teµ ∂ν φe 2κ4 1 µν + 24a + 12b ge ∂µφe ∂ν φe . φe

Comparing with the eﬀective Lagrangian Lagrangian ) ( 2a + b = −c , 2a + b = 0 , =⇒ 24a + 12b = 2a + b + c , c = 0 .

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The gravitational equation of motion

1 φ e µ T j T ρν − 2 T νρ − 4 e ρ T j Kµν 2 i ρν j j i ρν j 2φ µν νµ µν νµ + ∂ν e Ti − 2 T i + 2 Ti − 2 T i ∂ν φ = 0 . e

The equation of motion of φ

a ijk kji e Tijk(T − 2 T ) = 0 . 2κ4

Tijk = 0 =⇒ no gravity =⇒ forbidden!! T ijk = 2 T kji =⇒ it implies T iik = 2T kii = 0, no torsion vector!!

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Writing the eﬀective Lagrangian into the new form 1 µν Lg = e φ TNGR − kc g TµTν 2κ4 c µν + g (∂µ − kTµ)φ(∂ν − kTν )φ , kφ c where k = is a ﬁxed ratio. Lagrangian 2a + b + c We need ghost-free 2a + b + c > 0 and c 6= 0 Conformal transformation

2ω −2ω gµν −→ e gµν ,Tµ −→ Tµ − 3 ∂µω , φ −→ e φ .

Ling-Wei Luo TGWG Group Seminar@NTNU 42/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Under the conformal transformation, the eﬀective Lagrangian becomes as 1 µν 1 µν Lg = e φeTeNGR − kc φege TeµTeν + + 2 − 3k 2c φege Teµ ∂ν ω |{z} 2κ4 k e e4ω 1 µν 6ω aκ2 3 µρ νσ + + 2 − 3k 3c φeg ∂µω ∂ν ω − e φe g g Feµν Feρσ k e 4 e e c µν + ge ∂µ − kTeµ + (2 − 3k)∂µω φe ∂ν − kTeν + (2 − 3k)∂ν ω φe . kφe | {z } c e−4ω gµν (∂ − kT )φ (∂ − kT )φ kφ µ µ ν ν 1 + 2 − 3k = 0= ⇒ conformal invariance!! k Conformal Invariant condition  2a + b + 4c = 0 for k = − 1  3 or with 2a + b + c > 0 and c 6= 0 2a + b = 0 for k = 1

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Weyl derivative for general ﬁeld ψ:

∗ (ψ) λψ k ∂µ = ∂µ + 2 Tµ

i Weyl derivative for φ, e ν and gνρ  ∗ (φ) = + λφk with = −2  ∂µ φ ∂µ 2 Tµ φ λφ   ∗ (e) i λek i ∂µ e ν = ∂µ + 2 Tµ e ν with λe = 1   ∗ (g) = + λg k with = 2  ∂µ gνρ ∂µ 2 Tµ gνρ λg

∗ ρ ρ ∗ (e) i ρ k ρ Deﬁne an invariant connection Γνµ = ei ∂µ e ν = Γνµ + 2 δν Tµ. λ k =⇒ ∗∂(g)g − Γσ g − Γσ g = g T g µ νρ νµ σρ ρµ νσ 2 µ νρ =⇒ Weyl-Cartan geometry!!!

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The torsion vector Tµ can be identified as the gauge field. The modified covariant derivative for ψ given by

∗∇(ψ) = ∗d(ψ) + ∗Γ .

The nonmetricity vanishes

∗ (g) ∗ (g) ∗ σ ∗ σ ∇µ gνρ = ∂µ gνρ − Γνµgσρ − Γρµgνσ = ∇µgνρ = 0 .

∗ ρ ∗ ρ We deﬁne Te µν := (gT ) µν , we have

 ∗ ρ ρ k ρ ρ  ∗ ρ ρ k ρ ρ  T µν = T µν + (δν Tµ − δµTν ) ,  Te µν = Te µν + (δν Teµ − δµTeν ) , 2 and 2 ∗ 3 ∗ 3  Tµ = (1 − k) Tµ ,  Teµ = (1 − k) Teµ . 2 2

∗ (ψ) ∗ ∗ (ψ) 2∗ ∂µ in terms of Tµ =⇒ ∂µ = ∂µ − λψk Tµ.

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∗ ρ ∗ ρ ρ ρ Due to Te µν = T µν − (1/2)(δν ∂µω − δµ∂ν ω), we have µν e φ TNGR − kc g TµTν ∗ µν ∗ ∗ = e φ TNGR − kc g Tµ Tν ∗ µν ∗ ∗ = e φe TeNGR − kc ge Teµ Teν

and

∗ (g) 2ω∗ (g) ∇e µ geνρ = e ∇µ gνρ = 0 , ∗ (φ) ∗ (φ) 3 2 −2ω∗ (φ) ∇e φe = ∂e + λφk − λφ ∂µω φe = e ∇ φ . µ µ 2 µ

Weyl gauge invariance 1 ∗ µν ∗ ∗ c µν ∗ (φ) ∗ (φ) Lg = e φe TeNGR − kc ge Teµ Teν + ge ∇e µ φe ∇e ν φe , 2κ4 kφe

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1 1 1 L = ke − φ2 T abcT + T abcT − T aT + k0 gµν D φD φ , 4 abc 2 cba 3 a µ ν

(Maluf and Faria, 2012) 0 where k = 1/(16πG), k = 6 and ηab = (−1, +1, +1, +1) as well as 1 Dµ := ∂µ − 3 Tµ, which gives the conformal invariant condition 2a + b + 3c = 0 .

In their discussion, a new arbitrary parameter k0 for the scalar kinetic term is introduced, resulting in a four-parameters model.

Note: In our model, the coeﬃcient of the kinetic term of φ is 2a + b + c so that the conformal invariant theory is totally determined by three parameters only.

Ling-Wei Luo TGWG Group Seminar@NTNU 47/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary Comparison Table 1 Minimal coupled models with TNGR 2κ

Additional Class Reference condition 2a + b + c = 0, - Einstein, 1929 1 1 (a, b, c) = ( 4 , 2 , −1) - Cho, 1976 - Hehl et al., 1978 2a + b + c = 0, - Nitsch and Hehl, 1980 c = −1 - Hayashi and Shirafuji, 1979 2a + b + c = 0, Static isotropic Hehl et al., 1978 1 (a, b, c) = ( 2 , 0, −1) metric by Scherrer Nitsch and Hehl, 1980 2a + b + c = 0, (a, b, c) = ( 1 , 1 , −1) Geng et al., 2014 4 2 Einstein-frame 2a + b + c = 0, c ≤ 0 X

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1 Non-minimal coupled models with φ TNGR (conformal invarience) 2κ

Additional Class Reference condition 0 µν k g DµφDν φ, 1 2a + b + 3c = 0 where Dµ := ∂µ − 3 Tµ Maluf and Faria, 2012 with arbitrary k0 2a + b + c = 0, - c = 0 X 2a + b + 4c = 0 2a + b + c > 0, 2a + b = 0 c 6= 0 X

Ling-Wei Luo TGWG Group Seminar@NTNU 49/ 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

S = Sg + Sm Z 1 aκ2κ2 = d4x e Φ2 T − 4 Φ6 gµρgνσF F 2 NGR 8 µν ρσ µν µν 2 + (4a + 2b + 2c) g ∂µΦ∂ν Φ − 2c g Tµ Φ∂ν Φ + κ4λΦ Lm .

i i i µ µ µ In the weak ﬁeld approximation e µ = δµ + h µ and ei = δi − hi with asymmetric ﬂuctuations

1 i hµν = γµν + aµν and h µ 1 . 2 |{z} | {z } anti-symmetric symmetric

i j The metric tensor gµν = ηije µe ν ≈ ηµν + γµν contains no anti-symmetric part of hµν .

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The torsion tensor and torsion vector are

( ρ ρ i i 2 T µν = δi (∂µh ν − ∂ν h µ) + O(hµν ) , µ 2 Tν = ∂µh ν − ∂ν h + O(hµν ) .

2 µ i µ e = 1 + h + O(hµν ) with h ≡ δi h µ = h µ = (1/2)γ. Lagrangian in the lowest order 1 aκ2κ2 L ≈ Φ2 T − 4 Φ6ηµρηνσF F g 2 NGR 8 µν ρσ µν µν + (4a + 2b + 2c) η ∂µΦ∂ν Φ − 2c η Tµ Φ∂ν Φ

µν The current-vector interaction η Tµ Φ∂ν Φ:

ρ ρ Tµ ≈ (1/2)∂ργ µ + ∂ρa µ − (1/2)∂µγ

Z Z Φ∂ Φ µ 4 ρν 4 2 ρν µ T =⇒ d x − 2c ∂ρa Φ∂ν Φ = d x c Φ ∂ν ∂ρ a −→ 0 symmetric

=⇒ No contribution from aµν

Ling-Wei Luo TGWG Group Seminar@NTNU 51/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary 1 µ νρ ρ µν TNGR = (2a + b)∂µγνρ ∂ γ − (2a + b)∂µγνρ ∂ γ 4 ρ σµ ρµ µ + c ∂ γρµ ∂σγ − 2c ∂µγ ∂ργ + c ∂µγ ∂ γ

ν µρ ρ σµ ρµ + (2a + b)∂µγνρ ∂ a + c ∂ γρµ ∂σa − c ∂µγ ∂ρa µ νρ ρ µν ρ σµ + (2a − b)∂µaνρ ∂ a + (2a − 3b)∂µaνρ ∂ a + c ∂ aρµ ∂σa

becomes the well-known Fierz-Pauli Lagrangian (Fierz and Pauli, 1939) by 1 1 setting (a, b, c) = ( 4 , 2 , −1) and aµν = 0. Gauge conditions  ∂ γµ = 0 transversed condition,  µ ν µ ∂µa ν = 0 transversed condition, =⇒ torsion vector Tµ vanishes. γ = 0 traceless condition.

Deﬁne jµ := Φ∂µΦ and  µν 1 (µν) δLm  Tγ := T = −2 ,  2 δγµν  µν [µν] δLm  Ta := T = −2 . δaµν

Ling-Wei Luo TGWG Group Seminar@NTNU 52/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary EoM EoM of γµν : 2a + b ρ µν 2a + b ν ρµ µ ρν −jρ ∂ γ − ∂ γ + ∂ γ 2 4 c ρµ σν ρν σµ + η ∂σγ + η ∂σγ 4 c 1 ρµ ν 1 ρν µ µν σρ − η ∂ γ + η ∂ γ + η ∂σγ 2 2 2 c 2a + b + ηµν ∂ργ + ∂µaρν + ∂ν aρµ 2 2 c ρµ σν ρν σµ µν σρ + η ∂σα + η ∂σα − cη ∂σa 2 2 2a + b µν 2a + b ν µρ µ νρ −Φ 2γ − ∂ρ∂ γ + ∂ρ∂ γ 4 8 c µ σν ν σµ c µ ν σρ µν + ∂ ∂σγ + ∂ ∂σγ − ∂ ∂ γ + ∂ρ∂σγ η 8 4 c µν 2a + b µ ρν ν ρµ + η 2 γ + ∂ρ∂ a + ∂ρ∂ a 4 4 c µ σν ν σµ c µ ν c ν µ ρ µν 1 2 µν + ∂ ∂σa + ∂ ∂σa + ∂ j + ∂ j − c ∂ρj η = κ4λΦ T , 4 2 2 2 γ µν where 2 := η ∂µ∂ν .

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EoM of aµν : 2a + b µ ρν ν ρµ c ρµ σν ρν σµ −jρ ∂ γ − ∂ γ + η ∂σγ − η ∂σγ 2 2 c − ηρµ∂ν γ − ηρν ∂µγ + 2 2a − b ∂ρaµν 2 µ νρ ν ρµ ρµ σν ρν σµ + 2a − 3b ∂ a + ∂ a + c η ∂σa − η ∂σa 2 2a + b µ ρν ν ρµ c µ σν ν σµ −Φ ∂ρ∂ γ − ∂ρ∂ γ + ∂ ∂σγ − ∂ ∂σγ 4 4 µν 2a − 3b − c µ νρ ν ρµ 1 2 µν + 2a − b 2 a + ∂ρ∂ a + ∂ρ∂ a = κ4λΦ T . 2 2 a

EoM of Aµ:

2 2 2 δLm 2 2 5 µν aκ κ4 6 µν κ4λΦ + 3aκ κ4Φ (∂ν Φ)F + Φ ∂ν F = 0 . δAµ 2 EoM of Φ: 2 2 3aκ κ4 5 µν ΦT − Φ Fµν F + 2λκ4ΦLm NGR 4 2 δLm µ +κ4λΦ − 8a + 4b + 4c 2 Φ + 2c Φ∂µT = 0 . δΦ

Ling-Wei Luo TGWG Group Seminar@NTNU 54/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary

We consider the case of Weyl gauge invariance 2a + b + 4c = 0 or 2a + b = 0 µν µν along with the gauge conditions ∂µγ = 0, ∂µa = 0 and γ = 0

The EoM of γµν and aµν for 2a + b + 4c = 0  ρ µν µ ρν ν ρµ µ ρν ν ρµ  cjρ 2∂ γ − ∂ γ − ∂ γ + 2∂ a + 2∂ a    c c 1  +c Φ22 γµν + ∂µjν + ∂ν jµ − c ∂ jρηµν = κ λΦ2T µν ,  2 2 ρ 2 4 γ  µ ρν ν ρµ ρ µν  jρ 2c ∂ γ − ∂ γ + 4(b + 2c)∂ a    1  +4 b + c ∂µaνρ + ∂ν aρµ + 2 b + 2c Φ22 aµν = κ λΦ2T µν ,  2 4 a

Assuming that the scalar ﬁeld varies slowly

Φ ≈ Φc a constant ﬁeld, =⇒ jµ ≈ 0, The eq. reduce to  κ λ  2 γµν = 4 T µν ,  2c γ µν κ4λ µν  2 a = Ta .  4(b + 2c)

Ling-Wei Luo TGWG Group Seminar@NTNU 55/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary

The EoM of γµν and aµν for 2a + b = 0

 c µ ν c ν µ ρ µν 1 2 µν  ∂ j + ∂ j − c ∂ρj η = κ4λΦ T ,  2 2 2 γ 1  −8aj f ρµν − 4a Φ22 aµν = κ λΦ2T µν , ρ 2 4 a where f ρµν := ∂ρaµν + ∂µaνρ + ∂ν aρµ is the ﬁeld strength of aµν . For slowly varying scalar ﬁeld, the eq. reduce to

 µν  0 = Tγ , µν κ4λ µν  2 a = − Ta . 8a =⇒ only the anti-symmetric tensor aµν survives!!

Ling-Wei Luo TGWG Group Seminar@NTNU 56/ 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

We have summarized the possible choices of the coefficients (a, b, c) on the torsion scalar shown by TABLE. The Einstein-frame can be achieved by taking 2a + b + c = 0 with c ≤ 0. We arrive at a new class of conformal invariant theories of gravity without the electromagnetic field Aµ. We provide a conformal invariant gravity in teleparallelism with the condition 2a + b = 0 with c = 0, which gives rise to the existence of the Einstein-frame. The Weyl gauge theory under the ghost-free constraints 2a + b + c > 0 and c 6= 0 can be obtained with the requirements either 2a + b + 4c = 0 or 2a + b = 0. For the conformal invariant models with 2a + b = 0, we find that only the anti-symmetric tensor field is allowed rather than the symmetric one.

Ling-Wei Luo TGWG Group Seminar@NTNU 57/ 58 Main References: Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Speciﬁc Models Weak Field Approximation Summary End

Thank You!!!

Ling-Wei Luo TGWG Group Seminar@NTNU 58/ 58 Backup Slides Outline

8 Backup Slides

Ling-Wei Luo TGWG Group Seminar@NTNU 58/ 58 Backup Slides

Riemann-Cartan Geometry U4

Einstein’s general relativity was established in 1915 and described on Riemannian geometry V4 with metric gµν and the metric-compatible ρ 1 ρλ Levi-Civita connection {µν } = 2 g (gνλ,µ + gλν,µ − gµν,λ). The metric-compatible aﬃne connection in U4 is

ρ ρ ρ Γ µν = {µν } + K µν ,

which can be decomposed into torsion and torsionless parts. Torsion: couple to spin angular momentum in gravity (Elie´ Cartan, 1922.) i j Introducing the orthonormal frame gµν = ηijeµeν , and the relation of aﬃne and Lorentz connections in U4 is

ν ν i ν i j Γ ρµ = ei ∂µeρ + ei ω jµ eρ.

Absolute parallelism

Ling-Wei Luo TGWG Group Seminar@NTNU 59/ 58 Backup Slides Notation in 5D

I J In orthonormal frame, 5D metric is g¯MN =η ¯IJ e M e N , η¯IJ = diag(+1, −1, −1, −1, ε) with ε = −1.

Coordinate frame M,N = 0, 1, 2, 3, 5, µ, ν = 0, 1, 2, 3, α, β = 1, 2, 3.

Orthonormal frame (anholonomic frame) A, B, I, J, K = 0ˆ, 1ˆ, 2ˆ, 3ˆ, 5ˆ, i, j, k = 0ˆ, 1ˆ, 2ˆ, 3ˆ, a, b = 1ˆ, 2ˆ, 3ˆ.

5D TEGR

Ling-Wei Luo TGWG Group Seminar@NTNU 60/ 58 Backup Slides Aﬃne Connection and Lorentz Connection Consider noncoordinate basis (orthonormal frame)

ν i ν i i j ei DµV = ei (∂µV + ω jµ V )

ν i ρ i j = ei ∂µ(e ρV ) + ω jµ V

ν i ρ i ρ i j ρ = ei (∂µe ρ)V + e ρ(∂µV ) + ω jµ e ρ V

ν i ρ ν ρ ν i j ρ = (ei ∂µe ρ)V + δρ ∂µV +ei ω jµ e ρ V

| {z ν } ∂µV ν ν i ν i j ρ = ∂µV + (ei ∂µe ρ + ei ω jµ e ρ)V ν ν ρ ν ≡ ∂µV +Γ ρµV = ∇µV . The relation between aﬃne connection and Lorentz connection

ν ν i ν i j Γ ρµ ≡ ei ∂µe ρ + ei ω jµ eρ

We can deﬁne the total covariant derivative ∇µ

i ν i i j ∂µe ρ − Γ ρµ e ν + ω jµ e ρ = 0 i =⇒∇µe ρ = 0 (vielbein postulate). Absolute parallelism

Ling-Wei Luo TGWG Group Seminar@NTNU 61/ 58 Backup Slides Brief History of 5-Dimensional Theories

Kaluza-Klein (KK) theory: to unify electromagnetism and gravity by gauge theory Cylindrical condition (Kaluza, 1921) Compactiﬁcation to a small scale (Klein, 1926)

Generalization of KK: induced-matter theory =⇒ matter from the 5th-dimension (Wesson, 1998)

Large Extra dimension (Arkani-Hamed, Dimopoulos and Dvali (ADD), 1998) Solving hierarchy problem SM particles conﬁned on the 3-brane

Ling-Wei Luo TGWG Group Seminar@NTNU 62/ 58 Backup Slides

Randall-Sundrum model in AdS5 spacetime (Randall and Sundrum, 1999) RS-I (UV-brane and SM particles conﬁned on IR-brane) =⇒ solving hierarchy problem RS-II (only one UV brane) =⇒ compactiﬁcation to generate 4-dimensional gravity

DGP brane model (Dvali, Gabadadze and Porrati, 2000) =⇒ accelerating universe

Universal Extra Dimension (Appelquist, Cheng and Dobrescu, 2001) Not only graviton but SM particles can propagate to the extra dimension =⇒ low compactiﬁcation scale: reach to the electroweak scale

Ling-Wei Luo TGWG Group Seminar@NTNU 63/ 58 Backup Slides Equation of Motion of the TEGR Eﬀective Action The gravitational equation of motion

1 e µ φ T + 2 T σ ∂ φ − e ρ φ T j S µν 2 i σ i ρν j ν µ σ µ µ − ei ∂σφ T ν + ∂ν φ T + ∂ φ Tν 1 µν µ ν ν µ µ + ∂ν e (φ Si + ei ∂ φ − ei ∂ φ) = κ4 Θ e i

µ with Θν = diag(ρ, −P, −P, −P ) The modiﬁed Friedmann equations in ﬂat FLRW universe are

2 ˙ 3 φ H + 3 H φ = κ4 ρ , 2 ˙ ¨ 3 φ H + 2 φ H + 2 φ H˙ + φ = − κ4 P,

where H =a/a ˙ is the Hubble parameter (here ρ = P = 0 is assumed.)

Ling-Wei Luo TGWG Group Seminar@NTNU 64/ 58 Backup Slides

The equation of motion of scalar ﬁeld φ in the

µ µ ν absence of matter 2 T − 2 ∂µT − 2T Γνµ + e Lm = 0 ======⇒ a a¨ +a ˙ = 0 . ν α a˙ Γ νµ=Γ α0=3 a

Suppose the solution of a(t) is proportional to tm, the solution is √ a(t) = as + b t .

The constraint of the coeﬃcient: as b = 0 b = 0 case:

a(t) = as ⇒ the static universe.

as = 0 case: The Hubble parameter H = 1/(2 t) > 0 The the acceleration of scale factor a¨ = −b/(4 t2/3) > 0 for b < 0 =⇒ accelerated expanding universe. In general relativity, the equation of motion of φ is R˜(e) = 0 =⇒ the same solution for the scale factor in vacuum.

Ling-Wei Luo TGWG Group Seminar@NTNU 65/ 58 Backup Slides Equation of Motion of the NGR Eﬀective Action i Varying the full action S = Sg + Sm with respect to e µ, Aµ and φ 1 aκ2 2a + b + c e µ φ T − φ3 gλρgνσF F + gλν ∂ φ ∂ φ 2 i NGR 4 λν ρσ φ λ ν aκ2 −2 c gλν T ∂ φ − e ρ φ T j Σ µν − φ3 gµλgνσ F F λ ν i ρν j 2 λν ρσ 2a + b + c + gµλ∂ φ ∂ φ − c ∂ φ T µ σ + ∂ φ T µ + ∂µφ T φ λ ρ σ ρ ρ ρ 1 + ∂ e φ Σ µν − c e µ ∂ν φ + c e ν ∂µφ = κ λφ Θ µ , e ν i i i 4 i 2 2 λ δLm 1 aκ 2 µν 3aκ µν + φ ∂µ eF + φ ∂µφ F = 0 , e δAν e 2κ4 2κ4 3aκ2 φ δL T − φ2 gµρgνσF F + 2κ λ L + m NGR 4 µν ρσ 4 m e δφ 2a + b + c 2c + gµν ∂ φ∂ φ − 2φ 2ˆφ + ∂ egµν T = 0 , φ2 µ ν e ν µ

1 µν µ 1 δ Lm where 2ˆ := ∂ν (eg ∂µ) and Θi := − i is the dynamical e e δ e µ energy-momentum tensor with Lm := eLm.

Ling-Wei Luo TGWG Group Seminar@NTNU 66/ 58 Backup Slides Weyl Geometry Parallel transport of the vector V from point P(= xµ) to point P0(= xµ + dxµ) ν µ µ µ ∇V := dx (∇ν V )∂µ = (dV − δV )∂µ = 0 . µ ν µ Weyl Geometry (Weyl, 1918): Deﬁne the measure l := gµν V V of V , the variation of the measure l is proportional to l with the 1-form factor µ ϕ = ϕµdx ρ µ ν dl = −ϕl = −(ϕρdx )gµν V V . However the variation of l can be written as µ ν dl = d (gµν V V ) ρ µ ν ρ µ ρ σ ν ν ρ µ σ = ∂ρgµν dx V V dx − gµν Γ σρdx V V − gµν Γ σρdx V V σ σ ρ µ µ = (∂ρgµν − gσν Γ µρ − gµσΓ νρ)dx V V , µ µ µ σ ρ where dV = δV = −Γ σρV dx .   σ σ  ∂ρgµν − gσν Γ µρ − gµσΓ νρ = −ϕρgµν 6= 0 ,  Identity:  i ν i i j i  ∂µe ρ − Γ ρµ e ν + ω jµ e ρ = 0( ω iµ 6= 0) . WGT 

Ling-Wei Luo TGWG Group Seminar@NTNU 67/ 58