Teleparallel Gravity in Five Dimensional Theories

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
Ling-Wei Luo
Pre-Workshop on Gravitation and Cosmology @ NTHU
1/ 15
Outline
Introduction
5-Dimensional Theories
Summary
Outline
1 Introduction
2 5-Dimensional Theories
3 Summary
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Pre-Workshop on Gravitation and Cosmology @ NTHU
1/ 15
Outline
Introduction
5-Dimensional Theories
Summary
Brief History of 5-Dimension Theories
KK theory: in order to unify electromagnetism and gravity by gauge
theory
Cylindrical condition (Kaluza 1921) ⇒ KK 0-mode
Compactification to small scale (Klein 1926)
As a KK generalization ⇒ induced matter theory (matter come from
the 5th-dimension) (Wesson 1998)
Large Extra dimension (ADD model) (Arkani-Hamed, Dimopoulos and Dvali
1998)
Solving hierarchy problem
SM 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) ⇒ solving
hierarchy problem
RS-II (only one brane) ⇒ compactification to generate 4-dimensional
gravity
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Outline
Introduction
5-Dimensional Theories
Summary
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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Outline
Introduction
5-Dimensional Theories
Summary
Teleparallelism
Introduce the orthonormal frame (veirbein) in Weitzenb¨ock
geometry W4
gµν = ηij eiµ ejν ,
ηij = diag(+1, −1, −1, −1)
where µ, ν, ρ, . . . = 0, 1, 2, 3 and i, j, k, . . . = ˆ
0, ˆ
1, ˆ2, ˆ3.
Metric compatible condition ∇ gµν = 0:
∇ eiν = 0 ,
Absolute parallelism
ωij = − ωji ,
(Teleparallelism, Einstein 1928)
for parallel vector
∇ν eiµ = ∂ν eiµ − eiρ Γρµν = 0
Weitzenb¨
ock connection ⇒ Γρµν = eρi ∂ν eiµ (ωijµ = 0)
Curvature-free Rσ ρµν (Γ) = eσi ejρ Ri jµν (ω) = 0
´ Cartan 1922)
Torsion tensor (Elie
T i µν ≡ ∂µ eiν − ∂ν eiµ
Contorsion tensor is defined as
1
K ρ µν = − (T ρ µν − Tµ ρ ν − Tν ρ µ )
2
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Introduction
5-Dimensional Theories
Summary
The connection can be decomposed as
ρ
Γρµν = {µν
} + K ρ µν ,
ρ
where {µν
}(e) is Levi-Civita connection
In W4 , Teleparallel Equivalent to GR (GRk or TEGR) based on the
the relation
˜ µT µ.
−R(e) = T − 2 ∇
The telaparallel Lagrangian is
Stele
1
=
2κ
Z
d4 x e T
Torsion Scalar
T =
1 ρ
1
1
T µν Tρ µν + T ρ µν T νµ ρ − T ν µν T σµ σ ≡ T i µν Si µν
4
2
2
Sρ µν ≡ K µν ρ + δρµ T σν σ − δρν T σµ σ = −Sρ νµ 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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Introduction
5-Dimensional Theories
Summary
Hypersurface of GR
The 5D metric can be decomposed as
ds2
= g¯M N dxM dxN
=
(¯
gµν + Aµ Aν ) dxµ dxν + 2 φAµ dxµ dx5 + ε φ2 dx5 dx5
where y = x5 with M, N = 0, 1, 2, 3, 5 and choose ε = −1.
Unit normal vector n and g¯55 = n · n
The tensor BM N = −∇M nN , hM N = g¯M N − ε nM nN
θ = hM N BM N ,
1
σM N = B(M N ) − θhM N ,
3
ωM N = B[M N ]
Gauss’s equation
¯ µ νρσ = Rµ νρσ + ε(K µ σ Kνρ − K µ ρ Kνσ )
R
5
Intrinsic curvature Kµν = −ε∇µ n · eν = −ε 12 Ln gµν = {µν
}
Later, we assume Aµ = 0
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5-Dimensional Theories
Summary
5-Dimension Setting
In normal coordinate, 5D metric is g¯M N = η¯IJ eIM eJN ,
η¯IJ = diag(+1, −1, −1, −1, ε) with ε = ±1,
M, N, O = 0, 1, 2, 3, 5 and I, J, K = ˆ
0, ˆ
1, ˆ
2, ˆ
3, ˆ
5.
The 5D torsion scalar can be decomposed as
(5)
1
T = |{z}
T¯ + T¯ρ5ν T¯ρ5ν + T¯ρ5ν T¯ν5ρ +2 T¯σ σ µ T¯5 µ5 −T¯ν 5ν T¯σ5 σ ,
2
induced torsion scalar
¯ ˆ5 µν
C
Induced torsion T¯ρ µν = T ρ µν + C¯ ρ µν with C¯ ρ µν
related to the extrinsic torsion or twist ωM N
z
}|
{
ˆ
ˆ
= eρˆ5 (∂µ e5ν − ∂ν e5µ )
ˆ
ˆ
ˆ
N ˆ
5
C¯ 5 µν = Γ5νµ − Γ5µν = hM
µ hν T M N ∼ ωM N
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Braneworld Theory
Metric is given by
g¯M N =
gµν (xµ , y)
0
,
0
εφ2 (xµ , y)
The tensor C¯ ρ µν = 0
⇒ induced torsion scalar T¯ = T
The Lagrangian
Sbulk =
1
2κ5
Z
1
dvol5 T + (Tij5 T ij5 + Ti5j T j5i )
2
2
+ ei (φ) T a − T5 T 5 ,
φ
where TA := T b bA
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The bulk metric g¯ is maximally symmetric 3-space with spatially flat
(k = 0) and has the form
g¯M N = diag −1, a2 (t, y), a2 (t, y), a2 (t, y), ε φ2 (t, y)
(1)
ˆ
ˆ
ˆ
ϑ¯0 = dt, ϑ¯i = a(t, y) dxi , ϑ¯5 = φ(t, y) dy.
Torsion 2-forms are
ˆ
ˆ
T¯0 = d¯ϑ¯0 = 0,
a˙ ˆ ˆ a0 ¯5 ¯ˆi
ˆ
ˆ
ϑ ∧ϑ ,
T¯i = d¯ϑ¯i = ϑ¯0 ∧ϑ¯i +
a
aφ
Torsion 5-form reads
"
¯
T = T+
3 − 9 ε a02
a˙ φ˙
+6
2
2
φ
a
a φ
φ˙ ˆ ˆ
ˆ
T¯5 = ϑ¯0 ∧ϑ¯5 ,
φ
(2)
!#
dvol5
The equations of motion
¯A
H
¯A
E
¯A
Σ
Ling-Wei Luo
1
= (−2)¯
?
T¯A − 2 (2) T¯A − (3) T¯A
2
B
¯
¯
¯
:= ie¯A (T ) + ie¯A (T ) ∧ HB ,
¯ mat
δL
:=
,
δ ϑ¯A
(1)
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Outline
Introduction
5-Dimensional Theories
Summary
The equation of motion of the bulk:
!
"
00
02 #
2
0 0
˙
a
˙
φ
ε
a
a
φ
1
+
ε
a
a
˙
¯H
¯ˆ − E
¯ˆ = 3
¯?ϑ¯ˆ0
+
− 2
−
−
D
0
0
a2
aφ
φ
a
a φ
2φ2
a2
!
a0 φ˙
3ε a˙ 0
¯ˆ
¯
−
?ϑ¯5ˆ = −κ5 Σ
+
0
φ
a
aφ
!
0 ˙
0
a
φ
a
˙
1 + ε a02 ¯
3
a
¨ 2a˙ 2
¯ˆ + 3
¯H
¯ˆ − E
¯ˆ =
¯
¯?ϑˆ5
D
−
?
ϑ
+
−
5
5
0
φ aφ
a
a
a2
2φ2
a2
¯ˆ .
= − κ5 Σ
5
¯
¯ A = T¯B ¯
The energy-momentum tensor is Σ
A ? ϑB ,
we have the Friedmann equation
!
a˙ 2
a˙ φ˙
1 a00
a0 φ0
1 a02
κ5 ¯
−
T00
+
−
−
=
a2
aφ
φ2 a
a φ
φ2 a2
3
The same as GR! See (Binetruy, Deffayet and Langlois 2000)
But the junction condition come from torsion itself!
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5-Dimensional Theories
Summary
KK Theory
Focus on low-energy effective gravitational theory
⇒ consider original KK theory
Cylindrical condition (no dependency x5 )
Compactify to S 1 and only consider zero KK mode
The metric reduce to
g¯M N
0
gµν (xµ )
=
,
0
−φ2 (xµ )
The residual components are T ρ µν , and T¯5 µ5 = ∂µ φ/φ
The torsion scalar is (5) T = T + 2 T σ σ µ T¯5 µ5
The effective Lagrangian is
Seff =
Ling-Wei Luo
1
2 κ4
Z
d4 x e (φ T + 2 T µ ∂µ φ)
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5-Dimensional Theories
Summary
Compare to GR
The effective Lagrangian of GR
p
−(5) g (5) R
→
√
−g φR
A specific case of Brans-Dicke theory (Brans-Dicke parameter ω = 0)
The effective Lagrangian of TEGR
(5) (5)
µ
e T → e φ T + 2 T ∂µ φ
˜ µT µ
Curvature-torsion −R(e) = T − 2∇
Z
−1
˜ µ (φ T µ )
⇒
d4 x e φR(e) − 2 ∇
2κ4
Equivalent to φR up to the total derivative term
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Conformal Transformation
By conformal transformation
T
Tµ
Ω2 T˜ − 4 Ω g˜µν T˜µ ∂ν Ω − 6 g˜µν ∂µ Ω ∂ν Ω
= T˜µ − 2 Ω−1 ∂µ Ω .
=
Choosing φ = Ω2 , the action reads
Z
1 ˜
4
µν
Seff = d x e˜
T − 14 g˜ ∂µ ψ ∂ν ψ ,
2 κ4
√
where ψ = (1/ 2 κ4 ) ln Ω.
There exist an Einstein frame for such non-minimal coupled effective
Lagrangian in teleparallel gravity.
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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
Summary
In GR, the extrinsic curvature plays an important role to give the
projected effect in the lower dimension
The effect on the lower dimensional manifold is totally determined
by a higher dimensional geometry for TEGR as our setting
braneworld theory of teleparallel gravity in the FLRW cosmology still
provides an equivalent viewpoint as Einstein’s general relativity.
The KK reduction of telaparallel gravity generate non-Brans-Dicke
type effective Lagrangian
The additional coupled term lead to an Einstein frame by conformal
transformation for the non-minimal coupled teleparallel gravity
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5-Dimensional Theories
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End
Tank You for Listening!!!
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