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 ∇ gµν = 0:
i ρ i i j d eµ − Γµ eρ + ω j eµ = 0 and ωij = − ωji .
Absolute parallelism for parallel vectors (Cartan 1922/Eisenhart 1925)
i i i ρ ∇ν eµ = ∂ν eµ − eρ Γµν = 0 .
w ρ ρ i =⇒ Weitzenb¨ockconnection Γµν = ei ∂ν eµ ←− ωijµ = 0. σ σ j i Curvature-free R ρµν (Γ) = ei eρR jµν (ω) = 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) = ∇uv − ∇vu − [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 = {µν } + 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 ∇˜ 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 .
The TEGR action 1 Z √ 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 {µν } = 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 (+ − − − ε). e
The tensor BMN := −∇˜ 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 ωMN = B[MN] −→ 0 (Hypersurface orthogonal) ,
where hMN =g ¯MN − ε nM nN the projection operator. Gauss’s equation
µ µ µ µ R¯ νρσ = R νρσ + ε(K σ Kνρ − K ρ Kνσ)
Extrinsic curvature ˜ 1 5 Kµν = B(µν) = −ε ∇µn · eν = −ε 2 Lngµν = ε {µν } 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 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 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 × , 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, induced torsion T¯ µν = T µν + C¯ µν , where
5ˆ C¯ µν z }| { C¯ρ =e ¯ρ(∂ e¯5ˆ − ∂ e¯5ˆ ) . µν 5ˆ µ ν ν µ
¯5ˆ 5ˆ 5ˆ M N 5ˆ C µν = Γνµ − Γµν = hµ hν T MN ∼ ωµν is related to the extrinsic torsion or twist ωµν .
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), ε φ (t, y) ,
ϑ¯0ˆ = dt , ϑ¯a = a(t, y) dxα , ϑ¯5ˆ = φ(t, y) dy . First Cartan structure equation
a˙ a0 φ˙ T¯0ˆ = d¯ϑ¯0ˆ = 0, T¯a = ϑ¯0ˆ ∧ ϑ¯a + ϑ¯5ˆ ∧ ϑ¯a, T¯5ˆ = ϑ¯0ˆ ∧ ϑ¯5ˆ , a aφ φ Torsion 5-form reads " !# 3 + 9 ε 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 ?¯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 ! a˙ 2 a˙ φ˙ 1 a00 a0 φ0 1 a02 κ + − − − = 5 T¯ . a2 a φ φ2 a a φ φ2 a2 3 0ˆ0ˆ
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) . 3ε 0 0 6ε 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. The 5D torsion scalar (5) σ µ 5 T = T +2 T σ T¯ µ5
The effective action of 5D TEGR Z 1 4 µ Seff = d x e (φ T +2 T ∂µφ) 2κ4
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 13/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory 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 1 µν e Leff = T − Fµν F (the form coincides with GR). 2κ4 4 (de Andrade et al. 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 RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 14/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary 5D GR vs. 5D TEGR in KK Scenario The Effective Lagrangian of 5D GR
ωBD = 0 case in Brans-Dicke theory. √ p (5) (5) ˜ ˜ − − g R −→ − −g φR−φ .
The Effective Lagrangian of 5D TEGR (5) (5) µ e T −→ e φ T +2 T ∂µφ . | {z } no analogy
µ Substituting the relation −R˜(e) = T − 2 ∇˜ µT into the 4D effective Lagrangian Equivalent to Scalar-Tensor Theory Z −1 4 µ d x e φR˜(e) −2 ∇˜ µ(φ T ) . 2κ4 | {z } surface term
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 15/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Conformal Transformation
2 Doing the conformal transformation (g˜µν = Ω (x) gµν ):
2 µν µν T = Ω T˜ − 4 Ωg ˜ T˜µ∂ν Ω + 6g ˜ ∂µΩ ∂ν Ω , −1 Tµ = T˜µ + 3 Ω ∂µΩ .
Choosing φ = Ω2, the action reads Z 4 1 1 µν Seff = d x e˜ T˜ + g˜ ∂µψ ∂ν ψ , 2 κ4 2
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 RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 16/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary Looking for the Topological Effect (current work) Including spinor i i L = e ψ¯ i γj eµ ∂ − ω (e)+K σjk ψ with σjk = [γj, γk] . D j µ 2 jkµ jkµ 4
Gravitational chiral anomaly inGR (Kimura 1969) 1 2 d ? j = R ∧ Rij = d Ω ∧ d Ω + Ω ∧ Ω ∧ Ω . A 384 π2 ij 3
The modified Chern-Simons gravity (Jackiw and Pi 2003) 1 θ L = R ∧ Rij −→ leptogenesis mCS 16 π G 4 ij (Alexander, Peskin and Sheikh-Jabbari 2004) Question
How about the anomaly induced gravity in W4? i i j i Nieh-Tan term Ti ∧ T − Rij ∧ϑ ∧ ϑ = d Ti ∧ ϑ |{z} =0
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 17/ 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 17/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory 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 object for TEGR in the braneworld scenario.
Braneworld theory of teleparallel gravity in the FLRW cosmology still provides an equivalent viewpoint as Einstein’s general relativity.
In the FLRW universe, we found that the accelerated expansion of the universe can be achieved by the effective teleparallel gravity, which is the same as the effective KK scenario in general relativity.
The KK reduction of telaparallel gravity generates an additional coupling in the effective Lagrangian, which leads to an Einstein frame by the conformal transformation for the non-minimal coupled teleparallel gravity.
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 18/ 19 Outline Teleparallel Gravity Five-Dimensional Geometry Braneworld Scenario Kaluza-Klein Theory Summary End
Thank You!!!
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 19/ 19 Backup Slides Outline
6 Backup Slides
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 19/ 19 Backup Slides Alternative Gravitational Theory
Einstein’s unified field theory: Riemannian Geometry with Maintaining the Notion of Distant Parallelism (Teleparallelism, Einstein 1928)
Torsion scalar (Einstein 1929)
Teleparallel Lagrangian is equivalent to the Riemann scalar (Lanczos 1929)
Generalization: New General Relativity (NGR) (Hayashi & Shirafuji 1979)
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 20/ 19 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 Γµν = {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 spin connections in U4 is
ν ν i ν i j Γρµ = ei ∂µeρ + ei ω jµ eρ.
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 21/ 19 Backup Slides Poincar´eGauge Theory (PGT) 1 δ0φ = ( 2 ω(x) · M + ξ(x) · P )φ i i µ i i µ Gauge fields are ω j = ω jµdx and θ = eµdx Fields strength is
ij i ij i D ◦ D = R Mij + T Pi or [Dρ,Dσ] = R ρσMij−T ρσPi
where Mij and Pi are the rotational and the translational generators, respectively. i i i k i i Cartan equations: R j = dω j + ω k ∧ ω j and T = Dθ . Einstein-Cartan-Sciama-Kibble (ECSK) Theory The simplest Poincar´egauge theory:
Z 1 S = d4x e − R(e, ω) ECSK 2κ
ECSK extension: include supersymmetry and massless Rarita-Schwinger field (Rarita & Schwinger 1941) −→ N = 1 D = 4 Supergravity.
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 22/ 19 Backup Slides Notation in 5D
I J In orthonormal frame, 5D metric is g¯MN =η ¯IJ eM eN , 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.
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 23/ 19 Backup Slides Affine Connection and Spin 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(eρV ) + ω jµ V ] ν i ρ i ρ i j ρ = ei [(∂µeρ)V + eρ(∂µV ) + ω jµ eρ V ] ν i ρ ν ρ ν i j ρ = (ei ∂µeρ)V + δρ ∂µV +ei ω jµ eρ V ]
| {z ν } ∂µV ν ν i ν i j ρ = ∂µV + (ei ∂µeρ + ei ω jµ eρ)V ν ν ρ ν ≡ ∂µV +Γ ρµV = ∇µV .
The relation between affine connection and spin connection
ν ν i ν i j Γρµ ≡ ei ∂µeρ + ei ω jµ eρ
We have the definition of the total covariant derivative ∇µ i ν i i j =⇒ ∂µeρ − Γρµ eν + ω jµ eρ = 0 i =⇒ ∇µeρ = 0 (vielbein postulate). Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 24/ 19 Backup Slides
Different gravitational theories with geometry (arXiv:9602013[gr-qc]).
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 25/ 19 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 RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 26/ 19 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 RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 27/ 19 Backup Slides The 4-Form Equations of Motion
The 4-form equations of motion D¯H¯A − E¯A = κ5 Σ¯ A, where δT¯ 1 H¯ := = (−2)?¯ (1)T¯ − 2 (2)T¯ − (3)T¯ , A δT¯A A A 2 A ¯ ¯ ¯B ¯ EA := ie¯A (T ) + ie¯A (T ) ∧ HB , δL¯ Σ¯ := − mat . A δϑ¯A A = 0ˆ, 5ˆ components: " ! # a˙ 2 a˙ φ˙ ε a00 a0 φ0 1 − ε a02 D¯H¯ − E¯ = 3 + + − − ?¯ϑ¯ 0ˆ 0ˆ a2 a φ φ2 a a φ 2φ2 a2 0ˆ ! 3ε a˙ 0 a0 φ˙ + − ?¯ϑ¯ = κ Σ¯ , φ a a φ 5ˆ 5 0ˆ ! 3 a0 φ˙ a˙ 0 a¨ 2˙a2 1 − ε a02 D¯H¯ − E¯ = − ?¯ϑ¯ + 3 + − ?¯ϑ¯ 5ˆ 5ˆ φ a φ a 0ˆ a a2 2φ2 a2 5ˆ ¯ = κ5 Σ5ˆ .
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 28/ 19 Backup Slides Equation of Motion of The Effective Action
The gravitational equation of motion
1 eµ φ T + 2 T σ ∂ φ − eρ 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 + ∂ e (φ S µν + eµ ∂ν φ − eν ∂µφ) = κ Θµ e ν i i i 4 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 RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 29/ 19 Backup Slides
The equation of motion of scalar field φ in the
µ µ ν absence of matter 2 T − 2 ∂µT − 2T Γ + e Lm = 0 ======⇒ a a¨ +a ˙ = 0 . νµ ν α Γνµ=Γα0=3 (a/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.
Ling-Wei Luo RESCEU APCosPA Summer School 2015 @ Nikko City, Tochigi 30/ 19