Outline Introduction 5-Dimensional Theories Summary

Pre-Workshop on Gravitation and Cosmology

Teleparallel Gravity in Five Dimensional Theories

Reference: arXiv:1403.3161 [gr-qc]

Ling-Wei Luo

Department of Physics, National Tsing Hua University (NTHU)

Collobrators: Chao-Qiang Geng (NCTS/NTHU), Huan Hsin Tseng (NTHU)

April 11, 2014 @ NTHU

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Outline Introduction 5-Dimensional Theories Summary

Outline

1 Introduction

2 5-Dimensional Theories

3 Summary

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Outline Introduction 5-Dimensional Theories Summary

Outline

1 Introduction

2 5-Dimensional Theories

3 Summary

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Outline Introduction 5-Dimensional Theories Summary

Brief History of 5-Dimension Theories

KK theory: in order to unify electromagnetism and gravity by gaugetheory

Cylindrical condition (Kaluza 1921) ⇒ KK 0-modeCompactification to small scale (Klein 1926)

As a KK generalization ⇒ induced matter theory (matter come fromthe 5th-dimension) (Wesson 1998)

Large Extra dimension (ADD model) (Arkani-Hamed, Dimopoulos and Dvali

1998)

Solving hierarchy problemSM particle confined on the 3-brane

Randall-Sundrum model in AdS5 spacetime (Randall and Sundrum 1999)

RS-I ( UV-brane and SM particle confined on IR-brane) ⇒ solvinghierarchy problemRS-II (only one brane) ⇒ compactification to generate 4-dimensionalgravity

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DGP model (Dvali, Gabadadze and Porrati 2000)

⇒ accelerating universe

Universal Extra dimension (Appelquist, Cheng and Dobrescu 2001)

Not only graviton but SM particle can propagate to extra dimension⇒ low compactification scale: reach to electroweak scale

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TeleparallelismIntroduce the orthonormal frame (veirbein) in Weitzenbockgeometry W4

gµν = ηij eiµ e

jν , ηij = diag(+1,−1,−1,−1)

where µ, ν, ρ, . . . = 0, 1, 2, 3 and i, j, k, . . . = 0, 1, 2, 3.Metric compatible condition ∇ gµν = 0:

∇ eiν = 0 , ωij = −ωji ,Absolute parallelism (Teleparallelism, Einstein 1928) for parallel vector

∇ν eiµ = ∂νeiµ − eiρ Γρµν = 0

Weitzenbock connection ⇒ Γρµν = eρi ∂νeiµ (ωijµ = 0)

Curvature-free Rσρµν(Γ) = eσi ejρ R

ijµν(ω) = 0

Torsion tensor (Elie Cartan 1922)

T iµν ≡ ∂µeiν − ∂νeiµContorsion tensor is defined as

Kρµν = −1

2(T ρµν − Tµρν − Tνρµ)

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The connection can be decomposed as

Γρµν = ρµν+Kρµν ,

where ρµν(e) is Levi-Civita connection

In W4, Teleparallel Equivalent to GR (GR‖ or TEGR) based on thethe relation

−R(e) = T − 2 ∇µTµ.

The telaparallel Lagrangian is

Stele =1

2κ

∫d4x e T

Torsion Scalar

T =1

4T ρµν Tρ

µν +1

2T ρµν T

νµρ − T νµν T

σµσ ≡

1

2T iµν Si

µν

Sρµν ≡ Kµν

ρ + δµρ Tσνσ − δνρ Tσµσ = −Sρνµ is superpotential

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Outline Introduction 5-Dimensional Theories Summary

Outline

1 Introduction

2 5-Dimensional Theories

3 Summary

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Outline Introduction 5-Dimensional Theories Summary

Hypersurface of GRThe 5D metric can be decomposed as

ds2 = gMN dxM dxN

= (gµν +AµAν) dxµ dxν + 2φAµ dxµ dx5 + ε φ2dx5 dx5

where y = x5 with M,N = 0, 1, 2, 3, 5 and choose ε = −1.

Unit normal vector n and g55 = n · nThe tensor BMN = −∇MnN , hMN = gMN − ε nMnN

θ = hMNBMN , σMN = B(MN) −1

3θhMN , ωMN = B[MN ]

Gauss’s equation

Rµνρσ = Rµνρσ + ε(KµσKνρ −Kµ

ρKνσ)

Intrinsic curvature Kµν = −ε∇µn · eν = −ε 12Lngµν = 5

µνLater, we assume Aµ = 0

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5-Dimension Setting

In normal coordinate, 5D metric is gMN = ηIJ eIM eJN ,

ηIJ = diag(+1,−1,−1,−1, ε) with ε = ±1,M,N,O = 0, 1, 2, 3, 5 and I, J,K = 0, 1, 2, 3, 5.

The 5D torsion scalar can be decomposed as

(5)T = T︸︷︷︸induced torsion scalar

+1

2

(Tρ5ν T

ρ5ν + Tρ5ν Tν5ρ)+2 Tσσ

µ T 5µ5−T ν5ν T

σ5σ ,

Induced torsion T ρµν = T ρµν + Cρµν with Cρµν = eρ5(

C5µν︷ ︸︸ ︷

∂µe5ν − ∂νe5

µ)related to the extrinsic torsion or twist ωMN

C 5µν = Γ5

νµ − Γ5µν = hMµ hNν T

5MN ∼ ωMN

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Braneworld Theory

Metric is given by

gMN =

(gµν(xµ, y) 0

0 εφ2(xµ, y)

),

The tensor Cρµν = 0⇒ induced torsion scalar T = T

The Lagrangian

Sbulk =1

2κ5

∫dvol5

(T +

1

2(Tij5 T

ij5 + Ti5j Tj5i)

+2

φei(φ)T a − T5 T

5

),

where TA := T bbA

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The bulk metric g is maximally symmetric 3-space with spatially flat(k = 0) and has the form

gMN = diag(−1, a2(t, y), a2(t, y), a2(t, y), ε φ2(t, y)

)(1)

ϑ0 = dt, ϑi = a(t, y) dxi, ϑ5 = φ(t, y) dy.Torsion 2-forms are

T 0 = d ϑ0 = 0, T i = dϑi =a

aϑ0∧ϑi+ a′

aφϑ5∧ϑi, T 5 =

φ

φϑ0∧ϑ5 ,

(2)Torsion 5-form reads

T =

[T +

(3− 9 ε

φ2

a′2

a2+ 6

a

a

φ

φ

)]dvol5

The equations of motion

HA = (−2)?

((1)TA − 2 (2)TA −

1

2(3)TA

),

EA := ieA(T ) + ieA(TB) ∧ HB ,

ΣA :=δLmatδϑA

,

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The equation of motion of the bulk:

DH0 − E0 = 3

[(a2

a2+a

a

φ

φ

)− ε

φ2

(a′′

a− a′

a

φ′

φ

)−(

1 + ε

2φ2

)a′2

a2

]?ϑ0

+3ε

φ

(a′

a− a′

a

φ

φ

)?ϑ5 = −κ5 Σ0

DH5 − E5 =3

φ

(a′

a

φ

φ− a′

a

)?ϑ0 + 3

[(a

a+

2a2

a2

)−(

1 + ε

2φ2

)a′2

a2

]?ϑ5

=− κ5 Σ5 .

The energy-momentum tensor is ΣA = TBA ?ϑB ,

we have the Friedmann equation(a2

a2+a

a

φ

φ

)− 1

φ2

(a′′

a− a′

a

φ′

φ

)− 1

φ2

a′2

a2=κ5

3T00

The same as GR! See (Binetruy, Deffayet and Langlois 2000)

But the junction condition come from torsion itself!

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KK Theory

Focus on low-energy effective gravitational theory⇒ consider original KK theory

Cylindrical condition (no dependency x5)Compactify to S1 and only consider zero KK mode

The metric reduce to

gMN =

(gµν(xµ) 0

0 −φ2(xµ)

),

The residual components are T ρµν , and T 5µ5 = ∂µφ/φ

The torsion scalar is (5)T = T + 2Tσσµ T 5

µ5

The effective Lagrangian is

Seff =1

2κ4

∫d4x e (φT + 2Tµ ∂µφ)

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Compare to GR

The effective Lagrangian of GR√−(5)g (5)R →

√−g φR

A specific case of Brans-Dicke theory (Brans-Dicke parameter ω = 0)

The effective Lagrangian of TEGR

(5)e (5)T → e

(φT + 2Tµ ∂µφ

)Curvature-torsion −R(e) = T − 2∇µTµ

⇒ −1

2κ4

∫d4x e

(φR(e)− 2 ∇µ(φTµ)

)Equivalent to φR up to the total derivative term

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Conformal Transformation

By conformal transformation

T = Ω2 T − 4 Ω gµν Tµ∂νΩ− 6 gµν ∂µΩ ∂νΩ

Tµ = Tµ − 2 Ω−1 ∂µΩ .

Choosing φ = Ω2, the action reads

Seff =

∫d4x e

[1

2κ4T − 14 gµν∂µψ ∂νψ

],

where ψ = (1/√

2κ4) ln Ω.

There exist an Einstein frame for such non-minimal coupled effectiveLagrangian in teleparallel gravity.

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1 Introduction

2 5-Dimensional Theories

3 Summary

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Outline Introduction 5-Dimensional Theories Summary

Summary

In GR, the extrinsic curvature plays an important role to give theprojected effect in the lower dimension

The effect on the lower dimensional manifold is totally determinedby a higher dimensional geometry for TEGR as our setting

braneworld theory of teleparallel gravity in the FLRW cosmology stillprovides an equivalent viewpoint as Einstein’s general relativity.

The KK reduction of telaparallel gravity generate non-Brans-Dicketype effective Lagrangian

The additional coupled term lead to an Einstein frame by conformaltransformation for the non-minimal coupled teleparallel gravity

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Outline Introduction 5-Dimensional Theories Summary

End

Tank You for Listening!!!

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