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Seminar 10 September 2019Institute of Physics
Tartu, Estonia
Are critical Lovelock Lagrangians topologicalin the metric-affine formulation?
Alejandro Jiménez Cano
Universidad de GranadaDpto. de Física Teórica y del Cosmos
www.ugr.es/~alejandrojc
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Structure of this presentation
1 Introduction (metric-affine formalism and geometry)
2 Metric-Affine Lovelock theory
3 The metric-affine Einstein Lagrangian in D = 2
4 The metric-affine Gauss-Bonnet Lagrangian in D = 4
5 Discussion of the general critical Lovelock term
6 Summary and conclusions
B. Janssen, A. Jiménez-Cano, J. A. Orejuela [Janssen, Jiménez, Orejuela 2019]
A non-trivial connection for the metric-affine Gauss-Bonnet theory in D = 4.Physics Letters B 795 (2019) 42 – 48
B. Janssen, A. Jiménez-Cano [Janssen, Jiménez 2019]
On the topological character of metric-affine Lovelock Lagrangians in critical dimensions.arXiv:1907.12100 [gr-qc]
A. Jiménez-Cano, [My PhD Thesis – Still in progress]
Metric-Affine Gauge theory of gravity. Foundations, perturbations and gravitational wave solutions.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 1
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1. Introduction (metric-affine formalism and geometry)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 2
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Geometric structures: metric
Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.
Geometric structures
p Metric structure: gµν (metric tensor)é Measuring (length, volume...)
s[γ](σ) =
ˆ σ
0
√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)
vol(U) =
ˆUωvol , ωvol :=
√|g| dx1 ∧ ... ∧ dxD . (1.2)
é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.
é Notion of scale (conformal transformations...)
gµν → e2Ωgµν . (1.3)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3
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Geometric structures: metric
Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.
Geometric structures
p Metric structure: gµν (metric tensor)é Measuring (length, volume...)
s[γ](σ) =
ˆ σ
0
√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)
vol(U) =
ˆUωvol , ωvol :=
√|g| dx1 ∧ ... ∧ dxD . (1.2)
é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.
é Notion of scale (conformal transformations...)
gµν → e2Ωgµν . (1.3)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3
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Geometric structures: metric
Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.
Geometric structures
p Metric structure: gµν (metric tensor)é Measuring (length, volume...)
s[γ](σ) =
ˆ σ
0
√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)
vol(U) =
ˆUωvol , ωvol :=
√|g| dx1 ∧ ... ∧ dxD . (1.2)
é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.
é Notion of scale (conformal transformations...)
gµν → e2Ωgµν . (1.3)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3
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Geometric structures: metric
Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.
Geometric structures
p Metric structure: gµν (metric tensor)é Measuring (length, volume...)
s[γ](σ) =
ˆ σ
0
√|gµν(σ′)xµ(σ′)xν(σ′)| dσ′ . (1.1)
vol(U) =
ˆUωvol , ωvol :=
√|g| dx1 ∧ ... ∧ dxD . (1.2)
é Module of a vector (not necessarily non-negative) ⇒ light cones ⇒ causality.
é Notion of scale (conformal transformations...)
gµν → e2Ωgµν . (1.3)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 3
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Geometric structures: affine connection
Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.
Geometric structures
p Affine structure: Γµνρ (affine connection)
é Notion of parallel inM ⇒ Covariant derivative∇µ
Different spaces
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 4
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Geometric structures: affine connection
Geometric gravity (Einstein 1915) The spacetime is modelled as a differentiable manifoldM.
Geometric structures
p Affine structure: Γµνρ (affine connection)
é Notion of parallel inM ⇒ Covariant derivative∇µ
é Geometrical objects:
Curvature: Rµνλρ := ∂µΓνλ
ρ − ∂νΓµλρ + Γµσ
ρΓνλσ − Γνσ
ρΓµλσ , (1.4)
Torsion: Tµνρ := Γµν
ρ − Γνµρ . (1.5)
Curvature Torsion
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 5
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Geometric structures
Def.: In the presence of metric and affine connection we define the non-metricity tensor:
Qµνρ := −∇µgνρ . (1.6)
Theorem. Given gµν , there is only one connection that satisfies
Tµνρ = 0 (torsionless condition), (1.7)
Qµνρ = 0 (compatibility condition), (1.8)
the Levi-Civita connection:Γµν
ρ =1
2gρσ [∂µgσν + ∂νgµσ − ∂σgµν ] . (1.9)
Notation. Objects associated to the Levi-Civita connection: Rµνλρ, Rµν , ∇µ...
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 6
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Geometric structures
Def.: In the presence of metric and affine connection we define the non-metricity tensor:
Qµνρ := −∇µgνρ . (1.6)
Theorem. Given gµν , there is only one connection that satisfies
Tµνρ = 0 (torsionless condition), (1.7)
Qµνρ = 0 (compatibility condition), (1.8)
the Levi-Civita connection:Γµν
ρ =1
2gρσ [∂µgσν + ∂νgµσ − ∂σgµν ] . (1.9)
Notation. Objects associated to the Levi-Civita connection: Rµνλρ, Rµν , ∇µ...
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 6
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Geometric structures
Def.: In the presence of metric and affine connection we define the non-metricity tensor:
Qµνρ := −∇µgνρ . (1.6)
Theorem. Given gµν , there is only one connection that satisfies
Tµνρ = 0 (torsionless condition), (1.7)
Qµνρ = 0 (compatibility condition), (1.8)
the Levi-Civita connection:Γµν
ρ =1
2gρσ [∂µgσν + ∂νgµσ − ∂σgµν ] . (1.9)
Notation. Objects associated to the Levi-Civita connection: Rµνλρ, Rµν , ∇µ...
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 6
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Metric-Affine formalism
p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):
S[g,Ψ] =
ˆL(g, Rµνρ
λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)
where Ψ are certain non-geometrical fields.
p We are assuming a particular affine structure, the one fixed by the metric.
Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...
p What if... Γµνρ(g) were fixed by the dynamics?
Metric-affine (or Palatini) formulationPromotion of Γµν
ρ to a general connection Γµνρ (independent field).
p Let us see what happens in the most simple case: Einstein gravity.The resulting action is
S[g, Γ, Ψ] =1
2κ
ˆgµνRµν(Γ)
√|g|dDx+ Smatter[g, Ψ] . (1.11)
(Hypothesis Smatter 6= Smatter[Γ]).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7
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Metric-Affine formalism
p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):
S[g,Ψ] =
ˆL(g, Rµνρ
λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)
where Ψ are certain non-geometrical fields.
p We are assuming a particular affine structure, the one fixed by the metric.
Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...
p What if... Γµνρ(g) were fixed by the dynamics?
Metric-affine (or Palatini) formulationPromotion of Γµν
ρ to a general connection Γµνρ (independent field).
p Let us see what happens in the most simple case: Einstein gravity.The resulting action is
S[g, Γ, Ψ] =1
2κ
ˆgµνRµν(Γ)
√|g|dDx+ Smatter[g, Ψ] . (1.11)
(Hypothesis Smatter 6= Smatter[Γ]).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7
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Metric-Affine formalism
p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):
S[g,Ψ] =
ˆL(g, Rµνρ
λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)
where Ψ are certain non-geometrical fields.
p We are assuming a particular affine structure, the one fixed by the metric.
Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...
p What if... Γµνρ(g) were fixed by the dynamics?
Metric-affine (or Palatini) formulationPromotion of Γµν
ρ to a general connection Γµνρ (independent field).
p Let us see what happens in the most simple case: Einstein gravity.The resulting action is
S[g, Γ, Ψ] =1
2κ
ˆgµνRµν(Γ)
√|g|dDx+ Smatter[g, Ψ] . (1.11)
(Hypothesis Smatter 6= Smatter[Γ]).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7
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Metric-Affine formalism
p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):
S[g,Ψ] =
ˆL(g, Rµνρ
λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)
where Ψ are certain non-geometrical fields.
p We are assuming a particular affine structure, the one fixed by the metric.
Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...
p What if... Γµνρ(g) were fixed by the dynamics?
Metric-affine (or Palatini) formulationPromotion of Γµν
ρ to a general connection Γµνρ (independent field).
p Let us see what happens in the most simple case: Einstein gravity.The resulting action is
S[g, Γ, Ψ] =1
2κ
ˆgµνRµν(Γ)
√|g|dDx+ Smatter[g, Ψ] . (1.11)
(Hypothesis Smatter 6= Smatter[Γ]).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7
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Metric-Affine formalism
p Consider a theory depending on the metric structure and its associated curvature (Levi-Civita):
S[g,Ψ] =
ˆL(g, Rµνρ
λ(g),Ψ, ∇µΨ, ...)√|g|dDx , (1.10)
where Ψ are certain non-geometrical fields.
p We are assuming a particular affine structure, the one fixed by the metric.
Γµνρ(g) put by hand⇐ It is natural, it is the simplest one,...
p What if... Γµνρ(g) were fixed by the dynamics?
Metric-affine (or Palatini) formulationPromotion of Γµν
ρ to a general connection Γµνρ (independent field).
p Let us see what happens in the most simple case: Einstein gravity.The resulting action is
S[g, Γ, Ψ] =1
2κ
ˆgµνRµν(Γ)
√|g|dDx+ Smatter[g, Ψ] . (1.11)
(Hypothesis Smatter 6= Smatter[Γ]).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 7
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Metric-affine formalism for Einstein theory
p Action for the Einstein-Palatini theory
S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1
2κ
ˆgµνRµν(Γ)
√|g|dDx . (1.12)
p In D > 2 the equations of motion read
EoM g : 0 = R(µν) −1
2gµνR+ κTµν , (1.13)
EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1
D − 1Tσλ
σgµν −1
D − 1Tσν
σgµλ . (1.14)
p General solutions of the EoM of Γ
Γµνρ = Γµν
ρ +Aµδρν . (1.15)
Torsion, non-metricity and curvature tensors
Tµνρ = Aµδ
ρν −Aνδρµ , (1.16)
∇µgνρ = −2Aµgνρ , (1.17)
Rµνρλ = Rµνρ
λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)
Substituting this last condition into the metric equation one obtains the Einstein equations,
(EoM g)|Γon-shell : 0 = Rµν −1
2gµνR+ κTµν . (1.19)
p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:
proj : Γµνρ → Γµν
ρ + kµδρν (Rµνρ
λ → Rµνρλ + 2∂[µkν]δ
λρ ) ⇒ δprojLEP = 0 . (1.20)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8
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Metric-affine formalism for Einstein theory
p Action for the Einstein-Palatini theory
S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1
2κ
ˆgµνRµν(Γ)
√|g|dDx . (1.12)
p In D > 2 the equations of motion read
EoM g : 0 = R(µν) −1
2gµνR+ κTµν , (1.13)
EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1
D − 1Tσλ
σgµν −1
D − 1Tσν
σgµλ . (1.14)
p General solutions of the EoM of Γ
Γµνρ = Γµν
ρ +Aµδρν . (1.15)
Torsion, non-metricity and curvature tensors
Tµνρ = Aµδ
ρν −Aνδρµ , (1.16)
∇µgνρ = −2Aµgνρ , (1.17)
Rµνρλ = Rµνρ
λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)
Substituting this last condition into the metric equation one obtains the Einstein equations,
(EoM g)|Γon-shell : 0 = Rµν −1
2gµνR+ κTµν . (1.19)
p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:
proj : Γµνρ → Γµν
ρ + kµδρν (Rµνρ
λ → Rµνρλ + 2∂[µkν]δ
λρ ) ⇒ δprojLEP = 0 . (1.20)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8
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Metric-affine formalism for Einstein theory
p Action for the Einstein-Palatini theory
S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1
2κ
ˆgµνRµν(Γ)
√|g|dDx . (1.12)
p In D > 2 the equations of motion read
EoM g : 0 = R(µν) −1
2gµνR+ κTµν , (1.13)
EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1
D − 1Tσλ
σgµν −1
D − 1Tσν
σgµλ . (1.14)
p General solutions of the EoM of Γ
Γµνρ = Γµν
ρ +Aµδρν . (1.15)
Torsion, non-metricity and curvature tensors
Tµνρ = Aµδ
ρν −Aνδρµ , (1.16)
∇µgνρ = −2Aµgνρ , (1.17)
Rµνρλ = Rµνρ
λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)
Substituting this last condition into the metric equation one obtains the Einstein equations,
(EoM g)|Γon-shell : 0 = Rµν −1
2gµνR+ κTµν . (1.19)
p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:
proj : Γµνρ → Γµν
ρ + kµδρν (Rµνρ
λ → Rµνρλ + 2∂[µkν]δ
λρ ) ⇒ δprojLEP = 0 . (1.20)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8
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Metric-affine formalism for Einstein theory
p Action for the Einstein-Palatini theory
S[g, Γ, Ψ] = SEP[g, Γ] + Smatter[g, Ψ] , SEP[g, Γ] :=1
2κ
ˆgµνRµν(Γ)
√|g|dDx . (1.12)
p In D > 2 the equations of motion read
EoM g : 0 = R(µν) −1
2gµνR+ κTµν , (1.13)
EoM Γ : 0 = ∇λgµν − Tνλσgµσ −1
D − 1Tσλ
σgµν −1
D − 1Tσν
σgµλ . (1.14)
p General solutions of the EoM of Γ
Γµνρ = Γµν
ρ +Aµδρν . (1.15)
Torsion, non-metricity and curvature tensors
Tµνρ = Aµδ
ρν −Aνδρµ , (1.16)
∇µgνρ = −2Aµgνρ , (1.17)
Rµνρλ = Rµνρ
λ + 2∂[µAν]δλρ ⇒ R(µν) = Rµν . (1.18)
Substituting this last condition into the metric equation one obtains the Einstein equations,
(EoM g)|Γon-shell : 0 = Rµν −1
2gµνR+ κTµν . (1.19)
p Aµ is unphysical, since can be absorbed using the projective symmetry of the theory:
proj : Γµνρ → Γµν
ρ + kµδρν (Rµνρ
λ → Rµνρλ + 2∂[µkν]δ
λρ ) ⇒ δprojLEP = 0 . (1.20)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 8
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Metric Affine – Exterior notation
Three fundamental objects: coframe, metric and connection 1-form.
p Coframe. We can fix a general frame in the manifold and the corresponding dual basis (coframe):
ea = eµa∂µ , ϑa = eµadxµ [ϑa (eb) = δab ⇔ eµ
aeµb = δab ] . (1.21)
Notation:ϑa1...ak ≡ ϑa1 ∧ ... ∧ ϑak . (1.22)
p Metric. Components of the metric in the arbitrary basis:
gab = eµaeνbgµν . (1.23)
é Canonical volume form
ωvol :=1
D!Ea1...aDϑ
a1...aD =√|g|dx1 ∧ ... ∧ dxD , |g| ≡ |det(gµν)| . (1.24)
é Hodge star of an arbitrary k-form α = 1k!αa1...akϑ
a1...ak
? : Ωk(M) −→ ΩD−k(M)
α 7−→ ?α :=1
(D − k)!k!αb1...bkEb1...bkc1...cD−kϑ
c1...cD−k . (1.25)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 9
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Metric Affine – Exterior notation
Three fundamental objects: coframe, metric and connection 1-form.
p Coframe. We can fix a general frame in the manifold and the corresponding dual basis (coframe):
ea = eµa∂µ , ϑa = eµadxµ [ϑa (eb) = δab ⇔ eµ
aeµb = δab ] . (1.21)
Notation:ϑa1...ak ≡ ϑa1 ∧ ... ∧ ϑak . (1.22)
p Metric. Components of the metric in the arbitrary basis:
gab = eµaeνbgµν . (1.23)
é Canonical volume form
ωvol :=1
D!Ea1...aDϑ
a1...aD =√|g|dx1 ∧ ... ∧ dxD , |g| ≡ |det(gµν)| . (1.24)
é Hodge star of an arbitrary k-form α = 1k!αa1...akϑ
a1...ak
? : Ωk(M) −→ ΩD−k(M)
α 7−→ ?α :=1
(D − k)!k!αb1...bkEb1...bkc1...cD−kϑ
c1...cD−k . (1.25)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 9
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Metric Affine – Exterior notation
Three fundamental objects: coframe, metric and connection 1-form.
p Coframe. We can fix a general frame in the manifold and the corresponding dual basis (coframe):
ea = eµa∂µ , ϑa = eµadxµ [ϑa (eb) = δab ⇔ eµ
aeµb = δab ] . (1.21)
Notation:ϑa1...ak ≡ ϑa1 ∧ ... ∧ ϑak . (1.22)
p Metric. Components of the metric in the arbitrary basis:
gab = eµaeνbgµν . (1.23)
é Canonical volume form
ωvol :=1
D!Ea1...aDϑ
a1...aD =√|g|dx1 ∧ ... ∧ dxD , |g| ≡ |det(gµν)| . (1.24)
é Hodge star of an arbitrary k-form α = 1k!αa1...akϑ
a1...ak
? : Ωk(M) −→ ΩD−k(M)
α 7−→ ?α :=1
(D − k)!k!αb1...bkEb1...bkc1...cD−kϑ
c1...cD−k . (1.25)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 9
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Metric Affine – Exterior notation
Three fundamental objects: coframe, metric and connection 1-form.
p Connection 1-formωa
b = ωµabdxµ . (1.26)
where ωµab are the components of the affine connection in the anholonomic basis:
ωµab = eνaeλ
bΓµνλ + eσ
b∂µeσa . (1.27)
N.B. Γµνλ and ωµab contain the same information.
é Exterior covariant derivative (of algebra-valued forms)
Dαa...b... = dαa...
b... + ωcb ∧αa...c... + ... − ωac ∧αc...b... − ... , (1.28)
é Curvature, torsion and non-metricity forms:
Rab := dωa
b + ωcb ∧ ωac =
1
2Rµνa
bdxµ ∧ dxν , (1.29)
T a := Dϑa =1
2Tµν
adxµ ∧ dxν , (1.30)
Qab := −Dgab = Qµabdxµ . (1.31)
é Notation for Levi-Civita: ωab, Rab.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 10
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Metric Affine – Exterior notation
Three fundamental objects: coframe, metric and connection 1-form.
p Connection 1-formωa
b = ωµabdxµ . (1.26)
where ωµab are the components of the affine connection in the anholonomic basis:
ωµab = eνaeλ
bΓµνλ + eσ
b∂µeσa . (1.27)
N.B. Γµνλ and ωµab contain the same information.
é Exterior covariant derivative (of algebra-valued forms)
Dαa...b... = dαa...
b... + ωcb ∧αa...c... + ... − ωac ∧αc...b... − ... , (1.28)
é Curvature, torsion and non-metricity forms:
Rab := dωa
b + ωcb ∧ ωac =
1
2Rµνa
bdxµ ∧ dxν , (1.29)
T a := Dϑa =1
2Tµν
adxµ ∧ dxν , (1.30)
Qab := −Dgab = Qµabdxµ . (1.31)
é Notation for Levi-Civita: ωab, Rab.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 10
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Metric Affine – Exterior notation
Three fundamental objects: coframe, metric and connection 1-form.
p Connection 1-formωa
b = ωµabdxµ . (1.26)
where ωµab are the components of the affine connection in the anholonomic basis:
ωµab = eνaeλ
bΓµνλ + eσ
b∂µeσa . (1.27)
N.B. Γµνλ and ωµab contain the same information.
é Exterior covariant derivative (of algebra-valued forms)
Dαa...b... = dαa...
b... + ωcb ∧αa...c... + ... − ωac ∧αc...b... − ... , (1.28)
é Curvature, torsion and non-metricity forms:
Rab := dωa
b + ωcb ∧ ωac =
1
2Rµνa
bdxµ ∧ dxν , (1.29)
T a := Dϑa =1
2Tµν
adxµ ∧ dxν , (1.30)
Qab := −Dgab = Qµabdxµ . (1.31)
é Notation for Levi-Civita: ωab, Rab.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 10
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2. Metric-Affine Lovelock theory
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Lovelock Theory
Def. (Metric) Lovelock term of order k in D dimensions:
S(D)k [g] =
ˆL(D)k
√|g|dDx , (2.1)
where
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)
Properties
p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]
p Total derivative in D = 2k dimensions (critical dimension).
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Lovelock Theory
Def. (Metric) Lovelock term of order k in D dimensions:
S(D)k [g] =
ˆL(D)k
√|g|dDx , (2.1)
where
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)
Properties
p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]
p Total derivative in D = 2k dimensions (critical dimension).
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Lovelock Theory
Def. (Metric) Lovelock term of order k in D dimensions:
S(D)k [g] =
ˆL(D)k
√|g|dDx , (2.1)
where
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)
Properties
p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]
p Total derivative in D = 2k dimensions (critical dimension).
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Lovelock Theory
Def. (Metric) Lovelock term of order k in D dimensions:
S(D)k [g] =
ˆL(D)k
√|g|dDx , (2.1)
where
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)
Properties
p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]
p Total derivative in D = 2k dimensions (critical dimension).
Example I. Case k = 1, Einstein(-Hilbert) lagrangian
sgn(g)L(D)1 = δ[ν1
µ1δν2]µ2Rν1ν2
µ1µ2 = R , (2.3)
⇒ [EoM gµν ] 0 = Rµν −1
2gµνR . (2.4)
In the critical dimension (D = 2):
p Conformal symmetry of the theory
p In D = 2 all the metrics are conformally flat
So the equation reduces to:0 = 0 No conditions. (2.5)
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Lovelock Theory
Def. (Metric) Lovelock term of order k in D dimensions:
S(D)k [g] =
ˆL(D)k
√|g|dDx , (2.1)
where
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.2)
Properties
p 2nd order differential equations for the metric (by constr.) [Lovelock 1971]
p Total derivative in D = 2k dimensions (critical dimension).
Example II. Case k = 2, Gauss-Bonnet lagrangian
sgn(g)L(D)2 = 3!δ[ν1
µ1δν2µ2
δν3µ3δν4]µ4Rν1ν2
µ1µ2Rν3ν4µ3µ4 = R2 − 4RµνR
µν + RµνρλRµνρλ . (2.3)
Equation of motion of the metric in critical dimension D = 4:
0 = RαβR+ 2RµαβνRµν − 2RµαR
µβ + Rµνα
λRµνβλ −1
4gαβ
(R2 − 4RµνR
µν + RµνρλRµνρλ
)= Cα
µνρCβµνρ −1
4gαβCµνρλC
µνρλ , Cµνρλ ≡ Weyl tensor (2.4)
And this is a known property of the Weyl tensor of ANY metric in D = 4⇒ no conditions.
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Lovelock theory: from metric to metric-affine
p The D-dimensional (metric) Lovelock lagrangian of order k,
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)
B.................. Jump to metric-affine .....................B
Def. D dimensional (metric-affine) Lovelock term of order k:
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)
B.................. Jump to exterior algebra notation .....................B
In the language of differential forms:
L(D)k ≡ L(D)
k
√|g|dDx ⇔ L
(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)
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Lovelock theory: from metric to metric-affine
p The D-dimensional (metric) Lovelock lagrangian of order k,
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)
B.................. Jump to metric-affine .....................B
Def. D dimensional (metric-affine) Lovelock term of order k:
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)
B.................. Jump to exterior algebra notation .....................B
In the language of differential forms:
L(D)k ≡ L(D)
k
√|g|dDx ⇔ L
(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13
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Lovelock theory: from metric to metric-affine
p The D-dimensional (metric) Lovelock lagrangian of order k,
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)
B.................. Jump to metric-affine .....................B
Def. D dimensional (metric-affine) Lovelock term of order k:
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)
B.................. Jump to exterior algebra notation .....................B
In the language of differential forms:
L(D)k ≡ L(D)
k
√|g|dDx ⇔ L
(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 13
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Lovelock theory: from metric to metric-affine
p The D-dimensional (metric) Lovelock lagrangian of order k,
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)
B.................. Jump to metric-affine .....................B
Def. D dimensional (metric-affine) Lovelock term of order k:
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)
B.................. Jump to exterior algebra notation .....................B
In the language of differential forms:
L(D)k ≡ L(D)
k
√|g|dDx ⇔ L
(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)
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Lovelock theory: from metric to metric-affine
p The D-dimensional (metric) Lovelock lagrangian of order k,
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.5)
B.................. Jump to metric-affine .....................B
Def. D dimensional (metric-affine) Lovelock term of order k:
L(D)k =
(2k)!
2ksgn(g)δ[ν1
µ1...δν2k]
µ2kRν1ν2
µ1µ2 ...Rν2k−1ν2kµ2k−1µ2k . (2.6)
B.................. Jump to exterior algebra notation .....................B
In the language of differential forms:
L(D)k ≡ L(D)
k
√|g|dDx ⇔ L
(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.7)
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Lovelock theory: from metric to metric-affine
Metric-affine Lovelock term of order k as the lagrangian D-form:
L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)
General properties
p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]
p Projective symmetry:
ωab → ωa
b +Aδba (⇔ Γµνρ → Γµν
ρ +Aµδρν) , (2.9)
⇒ Rab → Rab + dAgab(⇔ Rµνρ
λ → Rµνρλ + 2∂[µAν]δ
λρ
). (2.10)
Critical dimension D = 2k
p The Lagrangian becomes:
L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR
a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)
p Question: Is this a total derivative?
Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]
L(2)1 |Q=0 ∝ d
[Eabωab
], (2.12)
L(4)2 |Q=0 ∝ d
[Eabcd
(Ra
b ∧ ωcd + 13ωa
b ∧ ωce ∧ ωed)]
. (2.13)
(Exterior derivative of Chern-Simons like terms).
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Lovelock theory: from metric to metric-affine
Metric-affine Lovelock term of order k as the lagrangian D-form:
L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)
General properties
p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]
p Projective symmetry:
ωab → ωa
b +Aδba (⇔ Γµνρ → Γµν
ρ +Aµδρν) , (2.9)
⇒ Rab → Rab + dAgab(⇔ Rµνρ
λ → Rµνρλ + 2∂[µAν]δ
λρ
). (2.10)
Critical dimension D = 2k
p The Lagrangian becomes:
L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR
a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)
p Question: Is this a total derivative?
Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]
L(2)1 |Q=0 ∝ d
[Eabωab
], (2.12)
L(4)2 |Q=0 ∝ d
[Eabcd
(Ra
b ∧ ωcd + 13ωa
b ∧ ωce ∧ ωed)]
. (2.13)
(Exterior derivative of Chern-Simons like terms).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14
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Lovelock theory: from metric to metric-affine
Metric-affine Lovelock term of order k as the lagrangian D-form:
L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)
General properties
p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]
p Projective symmetry:
ωab → ωa
b +Aδba (⇔ Γµνρ → Γµν
ρ +Aµδρν) , (2.9)
⇒ Rab → Rab + dAgab(⇔ Rµνρ
λ → Rµνρλ + 2∂[µAν]δ
λρ
). (2.10)
Critical dimension D = 2k
p The Lagrangian becomes:
L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR
a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)
p Question: Is this a total derivative?
Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]
L(2)1 |Q=0 ∝ d
[Eabωab
], (2.12)
L(4)2 |Q=0 ∝ d
[Eabcd
(Ra
b ∧ ωcd + 13ωa
b ∧ ωce ∧ ωed)]
. (2.13)
(Exterior derivative of Chern-Simons like terms).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14
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Lovelock theory: from metric to metric-affine
Metric-affine Lovelock term of order k as the lagrangian D-form:
L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)
General properties
p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]
p Projective symmetry:
ωab → ωa
b +Aδba (⇔ Γµνρ → Γµν
ρ +Aµδρν) , (2.9)
⇒ Rab → Rab + dAgab(⇔ Rµνρ
λ → Rµνρλ + 2∂[µAν]δ
λρ
). (2.10)
Critical dimension D = 2k
p The Lagrangian becomes:
L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR
a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)
p Question: Is this a total derivative?
Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]
L(2)1 |Q=0 ∝ d
[Eabωab
], (2.12)
L(4)2 |Q=0 ∝ d
[Eabcd
(Ra
b ∧ ωcd + 13ωa
b ∧ ωce ∧ ωed)]
. (2.13)
(Exterior derivative of Chern-Simons like terms).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14
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Lovelock theory: from metric to metric-affine
Metric-affine Lovelock term of order k as the lagrangian D-form:
L(D)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k . (2.8)
General properties
p Levi-Civita is a solution of the palatini formalism EoM. [Borunda, Janssen, Bastero 2008]
p Projective symmetry:
ωab → ωa
b +Aδba (⇔ Γµνρ → Γµν
ρ +Aµδρν) , (2.9)
⇒ Rab → Rab + dAgab(⇔ Rµνρ
λ → Rµνρλ + 2∂[µAν]δ
λρ
). (2.10)
Critical dimension D = 2k
p The Lagrangian becomes:
L(2k)k = Ra1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑa1...a2k ≡ Ea1...a2kR
a1a2 ∧ ... ∧Ra2k−1a2k , (2.11)
p Question: Is this a total derivative?
Yes for the Riemann-Cartan case (metric-compatible)é Two examples (orthonormal frame chosen, i.e. gab ≡ ηab): [Hehl, McCrea, Mielke, Ne’eman 1995]
L(2)1 |Q=0 ∝ d
[Eabωab
], (2.12)
L(4)2 |Q=0 ∝ d
[Eabcd
(Ra
b ∧ ωcd + 13ωa
b ∧ ωce ∧ ωed)]
. (2.13)
(Exterior derivative of Chern-Simons like terms).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 14
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3. The metric-affine Einstein Lagrangian in D = 2
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 15
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The metric-affine Einstein Lagrangian
p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)
L(D)1 = gcbRa
b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa
b(ω)√|g|dDx , (3.1)
Reminder. In D > 2, the solution of the EoM of the connection is:
ωab = ωa
b +Aδba ⇔ Γµνρ = Γµν
ρ +Aµδρν . (3.2)
Unphysical projective mode← can be eliminated using a symmetry of the theory.
p Critical dimension D = 2.é Equation of motion of the connection
0 = DEab = −QcaEbc where Qab = Qab −
1
2gabQc
c . (3.3)
Therefore the general solution is one that verifies
Qab = 0 . (3.4)
é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?
Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM
Tµνρ 1
2D2(D − 1) 2 (pure trace) [None]
Qµλλ D 2 [None] (in any D due to proj. symmetry)
Qµνρ12D(D + 2)(D − 1) 4 They are zero
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16
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The metric-affine Einstein Lagrangian
p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)
L(D)1 = gcbRa
b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa
b(ω)√|g|dDx , (3.1)
Reminder. In D > 2, the solution of the EoM of the connection is:
ωab = ωa
b +Aδba ⇔ Γµνρ = Γµν
ρ +Aµδρν . (3.2)
Unphysical projective mode← can be eliminated using a symmetry of the theory.
p Critical dimension D = 2.é Equation of motion of the connection
0 = DEab = −QcaEbc where Qab = Qab −
1
2gabQc
c . (3.3)
Therefore the general solution is one that verifies
Qab = 0 . (3.4)
é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?
Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM
Tµνρ 1
2D2(D − 1) 2 (pure trace) [None]
Qµλλ D 2 [None] (in any D due to proj. symmetry)
Qµνρ12D(D + 2)(D − 1) 4 They are zero
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16
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The metric-affine Einstein Lagrangian
p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)
L(D)1 = gcbRa
b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa
b(ω)√|g|dDx , (3.1)
Reminder. In D > 2, the solution of the EoM of the connection is:
ωab = ωa
b +Aδba ⇔ Γµνρ = Γµν
ρ +Aµδρν . (3.2)
Unphysical projective mode← can be eliminated using a symmetry of the theory.
p Critical dimension D = 2.é Equation of motion of the connection
0 = DEab = −QcaEbc where Qab = Qab −
1
2gabQc
c . (3.3)
Therefore the general solution is one that verifies
Qab = 0 . (3.4)
é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?
Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM
Tµνρ 1
2D2(D − 1) 2 (pure trace) [None]
Qµλλ D 2 [None] (in any D due to proj. symmetry)
Qµνρ12D(D + 2)(D − 1) 4 They are zero
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16
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The metric-affine Einstein Lagrangian
p Einstein Lagrangian (arbitrary dimension) (We drop the factor (2κ)−1)
L(D)1 = gcbRa
b ∧ ?ϑac = sgn(g)eνbeµcgcaRµνa
b(ω)√|g|dDx , (3.1)
Reminder. In D > 2, the solution of the EoM of the connection is:
ωab = ωa
b +Aδba ⇔ Γµνρ = Γµν
ρ +Aµδρν . (3.2)
Unphysical projective mode← can be eliminated using a symmetry of the theory.
p Critical dimension D = 2.é Equation of motion of the connection
0 = DEab = −QcaEbc where Qab = Qab −
1
2gabQc
c . (3.3)
Therefore the general solution is one that verifies
Qab = 0 . (3.4)
é But, is this trivial? Or are there conditions over the D3 = 8 degrees of freedom of the connection?
Tensor d.o.f. in D dim. d.o.f. in 2 dim. Condition imposed by EoM
Tµνρ 1
2D2(D − 1) 2 (pure trace) [None]
Qµλλ D 2 [None] (in any D due to proj. symmetry)
Qµνρ12D(D + 2)(D − 1) 4 They are zero
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 16
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The critical metric-affine Einstein Lagrangian in D = 2. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(3.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (3.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(2)1 =
ˆEabRa
b(ω) =
ˆEab
[Ra
b(ω) − 14Qa
c ∧Qcb], (3.7)
p If we choose a orthonormal gauge, i.e. gab = ηab, one can use
Eab Rab(ω) = Eab dωab = d(Eab ωab) , (3.8)
to rewrite the Riemann-Cartan part of the action as a total derivative,
S(2)1 =
ˆd(Eab ωab) − 1
4
ˆEabQ
ac ∧Qcb . (3.9)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 17
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The critical metric-affine Einstein Lagrangian in D = 2. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(3.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (3.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(2)1 =
ˆEabRa
b(ω) =
ˆEab
[Ra
b(ω) − 14Qa
c ∧Qcb], (3.7)
p If we choose a orthonormal gauge, i.e. gab = ηab, one can use
Eab Rab(ω) = Eab dωab = d(Eab ωab) , (3.8)
to rewrite the Riemann-Cartan part of the action as a total derivative,
S(2)1 =
ˆd(Eab ωab) − 1
4
ˆEabQ
ac ∧Qcb . (3.9)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 17
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The critical metric-affine Einstein Lagrangian in D = 2. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(3.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (3.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(2)1 =
ˆEabRa
b(ω) =
ˆEab
[Ra
b(ω) − 14Qa
c ∧Qcb], (3.7)
p If we choose a orthonormal gauge, i.e. gab = ηab, one can use
Eab Rab(ω) = Eab dωab = d(Eab ωab) , (3.8)
to rewrite the Riemann-Cartan part of the action as a total derivative,
S(2)1 =
ˆd(Eab ωab) − 1
4
ˆEabQ
ac ∧Qcb . (3.9)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 17
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The critical Einstein-Palatini Lagrangian in D = 2. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(2)1 [gab,ϑ
a,ωab] =
ˆEabRa
b(ω) (3.10)
l equivalent
S(2)1 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term) − 14
ˆEabQ
ac ∧Qcb . (3.11)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (3.12)
EoM for ωab 0 = 0 , (3.13)
EoM forQab 0 = EabQcb ⇔ Qab = 0 . (3.14)
p Conclusion:There are conditions over the connection. So the Lagrangian CANNOT be a total derivative.As we have seen a term quadratic in the non-metricity survives.
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The critical Einstein-Palatini Lagrangian in D = 2. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(2)1 [gab,ϑ
a,ωab] =
ˆEabRa
b(ω) (3.10)
l equivalent
S(2)1 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term) − 14
ˆEabQ
ac ∧Qcb . (3.11)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (3.12)
EoM for ωab 0 = 0 , (3.13)
EoM forQab 0 = EabQcb ⇔ Qab = 0 . (3.14)
p Conclusion:There are conditions over the connection. So the Lagrangian CANNOT be a total derivative.As we have seen a term quadratic in the non-metricity survives.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 18
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The critical Einstein-Palatini Lagrangian in D = 2. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(2)1 [gab,ϑ
a,ωab] =
ˆEabRa
b(ω) (3.10)
l equivalent
S(2)1 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term) − 14
ˆEabQ
ac ∧Qcb . (3.11)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (3.12)
EoM for ωab 0 = 0 , (3.13)
EoM forQab 0 = EabQcb ⇔ Qab = 0 . (3.14)
p Conclusion:There are conditions over the connection. So the Lagrangian CANNOT be a total derivative.As we have seen a term quadratic in the non-metricity survives.
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4. The metric-affine Gauss-Bonnet Lagrangian in D = 4
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The metric-affine Gauss-Bonnet Lagrangian
p Gauss-Bonnet Lagrangian (arbitrary dimension)
L(D)2 = gmbgndRa
b ∧Rcd ∧ ?ϑamcn (4.1)
= sgn(g)[R2 −RµνRνµ + 2RµνR
νµ − RµνRνµ +RµνρλRρλµν
]√|g|dDx , (4.2)
whereRµν := Rµλν
λ , R := gµνRµν , Rµν := gλσRµλσ
ν . (4.3)
p N.B. In general D, the most general solution is not known. But we know there should be a free(unphysical) projective mode.
p Critical dimension D = 4.
D = 4 ⇒ ?ϑamcn = Eamcn ⇒ L(4)2 = Eabcd Ra
b ∧Rcd (4.4)
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 20
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The metric-affine Gauss-Bonnet Lagrangian
p Gauss-Bonnet Lagrangian (arbitrary dimension)
L(D)2 = gmbgndRa
b ∧Rcd ∧ ?ϑamcn (4.1)
= sgn(g)[R2 −RµνRνµ + 2RµνR
νµ − RµνRνµ +RµνρλRρλµν
]√|g|dDx , (4.2)
whereRµν := Rµλν
λ , R := gµνRµν , Rµν := gλσRµλσ
ν . (4.3)
p N.B. In general D, the most general solution is not known. But we know there should be a free(unphysical) projective mode.
p Critical dimension D = 4.
D = 4 ⇒ ?ϑamcn = Eamcn ⇒ L(4)2 = Eabcd Ra
b ∧Rcd (4.4)
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The metric-affine Gauss-Bonnet Lagrangian
p Gauss-Bonnet Lagrangian (arbitrary dimension)
L(D)2 = gmbgndRa
b ∧Rcd ∧ ?ϑamcn (4.1)
= sgn(g)[R2 −RµνRνµ + 2RµνR
νµ − RµνRνµ +RµνρλRρλµν
]√|g|dDx , (4.2)
whereRµν := Rµλν
λ , R := gµνRµν , Rµν := gλσRµλσ
ν . (4.3)
p N.B. In general D, the most general solution is not known. But we know there should be a free(unphysical) projective mode.
p Critical dimension D = 4.
D = 4 ⇒ ?ϑamcn = Eamcn ⇒ L(4)2 = Eabcd Ra
b ∧Rcd (4.4)
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(4.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (4.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(4)2 =
ˆEabcdRa
b(ω) ∧Rcd(ω)
=
ˆEabcd
[Ra
b(ω) ∧ Rcd(ω)− 1
2Rab(ω) ∧Qc
f ∧Qfd + 1
16Qae ∧Qe
b ∧Qcf ∧Qf
d]. (4.7)
p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,
S(4)2 =
ˆdC −
ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.8)
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(4.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (4.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(4)2 =
ˆEabcdRa
b(ω) ∧Rcd(ω)
=
ˆEabcd
[Ra
b(ω) ∧ Rcd(ω)− 1
2Rab(ω) ∧Qc
f ∧Qfd + 1
16Qae ∧Qe
b ∧Qcf ∧Qf
d]. (4.7)
p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,
S(4)2 =
ˆdC −
ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.8)
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(4.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (4.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(4)2 =
ˆEabcdRa
b(ω) ∧Rcd(ω)
=
ˆEabcd
[Ra
b(ω) ∧ Rcd(ω)− 1
2Rab(ω) ∧Qc
f ∧Qfd + 1
16Qae ∧Qe
b ∧Qcf ∧Qf
d]. (4.7)
p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,
S(4)2 =
ˆdC −
ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.8)
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
Appendix. Useful decomposition of the connection.In the presence of a metric, an arbitrary ωab can be split
ωab = ω[ab] + ω(ab) ⇒ω(ab) = 1
2Qab = 12
(Qab + 1
DgabQcc)
ω[ab] =: ωab(4.5)
It can be proved that ωab is also a connection with
Q = 0 , T = T (T ,Q) . (4.6)
p Instead of working in terms of ωab we can work with the independent fields ωab,Qcc andQab.
p Plugging this into the action we obtain
S(4)2 =
ˆEabcdRa
b(ω) ∧Rcd(ω)
=
ˆEabcd
[Ra
b(ω) ∧ Rcd(ω)− 1
2Rab(ω) ∧Qc
f ∧Qfd + 1
16Qae ∧Qe
b ∧Qcf ∧Qf
d]. (4.7)
p Choosing the orthonormal gauge, i.e. gab = ηab, the first term (the purely Riemann-Cartan one) canbe expressed as the Euler density and therefore,
S(4)2 =
ˆdC −
ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.8)
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(4)2 [gab,ϑ
a,ωab] =
ˆEabcdRa
b(ω) ∧Rcd(ω) (4.9)
l equivalent
S(4)2 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term)
−ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.10)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)
EoM for ωab 0 = D[Qc
a ∧Qbc], (4.12)
EoM forQab 0 = Eabcd[Rab − 1
4Qfa ∧Q
bf]∧Q
dm . (4.13)
p Counterexample. The following field configuration violates EoM of ωab:
gab = ηab , ωab = ωab + fα[aδb]t ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where
f is an arbitrary function,
αa = et(δay dy + δaz dz
).
(4.14)
since
D[Qc
a ∧Qbc]
= d[αa ∧αb
]= 2e2t
(δayδ
bz − δbyδ
az
)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)
p Conclusion: The Lagrangian CANNOT be a total derivative.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(4)2 [gab,ϑ
a,ωab] =
ˆEabcdRa
b(ω) ∧Rcd(ω) (4.9)
l equivalent
S(4)2 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term)
−ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.10)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)
EoM for ωab 0 = D[Qc
a ∧Qbc], (4.12)
EoM forQab 0 = Eabcd[Rab − 1
4Qfa ∧Q
bf]∧Q
dm . (4.13)
p Counterexample. The following field configuration violates EoM of ωab:
gab = ηab , ωab = ωab + fα[aδb]t ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where
f is an arbitrary function,
αa = et(δay dy + δaz dz
).
(4.14)
since
D[Qc
a ∧Qbc]
= d[αa ∧αb
]= 2e2t
(δayδ
bz − δbyδ
az
)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)
p Conclusion: The Lagrangian CANNOT be a total derivative.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(4)2 [gab,ϑ
a,ωab] =
ˆEabcdRa
b(ω) ∧Rcd(ω) (4.9)
l equivalent
S(4)2 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term)
−ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.10)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)
EoM for ωab 0 = D[Qc
a ∧Qbc], (4.12)
EoM forQab 0 = Eabcd[Rab − 1
4Qfa ∧Q
bf]∧Q
dm . (4.13)
p Counterexample. The following field configuration violates EoM of ωab:
gab = ηab , ωab = ωab + fα[aδb]t ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where
f is an arbitrary function,
αa = et(δay dy + δaz dz
).
(4.14)
since
D[Qc
a ∧Qbc]
= d[αa ∧αb
]= 2e2t
(δayδ
bz − δbyδ
az
)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)
p Conclusion: The Lagrangian CANNOT be a total derivative.
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 22
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The critical metric-affine Gauss-Bonnet Lagrangian in D = 4. Equivalent formulation
p The action can be rewritten in terms of the new fields as
S(4)2 [gab,ϑ
a,ωab] =
ˆEabcdRa
b(ω) ∧Rcd(ω) (4.9)
l equivalent
S(4)2 [gab,ϑ
a, ωab,Qab,Qc
c] = (boundary term)
−ˆEabcd
[12Ra
b(ω) ∧Qcf ∧Qf
d − 116Qa
e ∧Qeb ∧Qc
f ∧Qfd]. (4.10)
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (4.11)
EoM for ωab 0 = D[Qc
a ∧Qbc], (4.12)
EoM forQab 0 = Eabcd[Rab − 1
4Qfa ∧Q
bf]∧Q
dm . (4.13)
p Counterexample. The following field configuration violates EoM of ωab:
gab = ηab , ωab = ωab + fα[aδb]t ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where
f is an arbitrary function,
αa = et(δay dy + δaz dz
).
(4.14)
since
D[Qc
a ∧Qbc]
= d[αa ∧αb
]= 2e2t
(δayδ
bz − δbyδ
az
)dt ∧ dy ∧ dz 6= 0 in the entireM. (4.15)
p Conclusion: The Lagrangian CANNOT be a total derivative.
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5. Discussion of the general critical Lovelock term
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 23
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Critical dimension D = 2k.
L(2k)k = Ea1a2 ...a2k−1
a2kRa1a2 ∧ ... ∧Ra2k−1
a2k . (5.1)
p The action can be rewritten in terms of the new fields as
L(2k)k = Ea1...a2k
k∑m=0
1
4k−mk!
m!(k −m)!Ra1a2 ∧ ... ∧ Ra2m−1a2m∧ (5.2)
∧Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k (5.3)
basically,
L(2k)k = Ea1...a2k
[R ∧ R ∧ ... ∧ R ∧ R ← boundary term
+ R ∧ R ∧ ... ∧ R ∧Q2
+ R ∧ R ∧ ... ∧Q2 ∧Q
2
...
+ R ∧Q2 ∧ ... ∧Q
2 ∧Q2
+Q2 ∧Q
2 ∧ ... ∧Q2 ∧Q
2]
where Q2 ≡Q
aif ∧Qfai+1 (5.4)
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Critical dimension D = 2k.
L(2k)k = Ea1a2 ...a2k−1
a2kRa1a2 ∧ ... ∧Ra2k−1
a2k . (5.1)
p The action can be rewritten in terms of the new fields as
L(2k)k = Ea1...a2k
k∑m=0
1
4k−mk!
m!(k −m)!Ra1a2 ∧ ... ∧ Ra2m−1a2m∧ (5.2)
∧Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k (5.3)
basically,
L(2k)k = Ea1...a2k
[R ∧ R ∧ ... ∧ R ∧ R ← boundary term
+ R ∧ R ∧ ... ∧ R ∧Q2
+ R ∧ R ∧ ... ∧Q2 ∧Q
2
...
+ R ∧Q2 ∧ ... ∧Q
2 ∧Q2
+Q2 ∧Q
2 ∧ ... ∧Q2 ∧Q
2]
where Q2 ≡Q
aif ∧Qfai+1 (5.4)
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)
EoM for ωab 0 = Eaba3...a2kk−1∑m=1
1
4k−mk!
m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧
∧ D[Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k], (5.6)
EoM forQab 0 = ...omitted... . (5.7)
p Counterexample. Consider the following field configuration:
gab = ηab , ωab = ωab ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where αa = et(δa3dx3 + ... + δa2kdx2k
). (5.8)
An immediate consequence is Rab = 0, so
EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1
a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1
a2k], (5.9)
The ansatz (5.8) gives:
0 = d(αa3 ∧ ... ∧αa2k
)= 2(k − 1) e2(k−1)t (2k − 2)! δ
[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)
If k > 1 we get a contradiction:
0 = δ[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)
p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)
EoM for ωab 0 = Eaba3...a2kk−1∑m=1
1
4k−mk!
m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧
∧ D[Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k], (5.6)
EoM forQab 0 = ...omitted... . (5.7)
p Counterexample. Consider the following field configuration:
gab = ηab , ωab = ωab ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where αa = et(δa3dx3 + ... + δa2kdx2k
). (5.8)
An immediate consequence is Rab = 0,
so
EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1
a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1
a2k], (5.9)
The ansatz (5.8) gives:
0 = d(αa3 ∧ ... ∧αa2k
)= 2(k − 1) e2(k−1)t (2k − 2)! δ
[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)
If k > 1 we get a contradiction:
0 = δ[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)
p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)
EoM for ωab 0 = Eaba3...a2kk−1∑m=1
1
4k−mk!
m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧
∧ D[Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k], (5.6)
EoM forQab 0 = ...omitted... . (5.7)
p Counterexample. Consider the following field configuration:
gab = ηab , ωab = ωab ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where αa = et(δa3dx3 + ... + δa2kdx2k
). (5.8)
An immediate consequence is Rab = 0, so
EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1
a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1
a2k], (5.9)
The ansatz (5.8) gives:
0 = d(αa3 ∧ ... ∧αa2k
)= 2(k − 1) e2(k−1)t (2k − 2)! δ
[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)
If k > 1 we get a contradiction:
0 = δ[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)
p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)
EoM for ωab 0 = Eaba3...a2kk−1∑m=1
1
4k−mk!
m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧
∧ D[Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k], (5.6)
EoM forQab 0 = ...omitted... . (5.7)
p Counterexample. Consider the following field configuration:
gab = ηab , ωab = ωab ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where αa = et(δa3dx3 + ... + δa2kdx2k
). (5.8)
An immediate consequence is Rab = 0, so
EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1
a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1
a2k], (5.9)
The ansatz (5.8) gives:
0 = d(αa3 ∧ ... ∧αa2k
)= 2(k − 1) e2(k−1)t (2k − 2)! δ
[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)
If k > 1 we get a contradiction:
0 = δ[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)
p Conclusion:
The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]
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General Lovelock in critical dimensions (k > 1). NOT a boundary term
p Instead of an equation of motion for ωab, now our splitting gives rise to three equations
EoM forQcc 0 = 0 ← (due to projective symmetry) , (5.5)
EoM for ωab 0 = Eaba3...a2kk−1∑m=1
1
4k−mk!
m!(k −m)!Ra3a4 ∧ ... ∧ Ra2m−1a2m∧
∧ D[Qa2m+1f1 ∧Qf1
a2m+2 ∧ ... ∧Qa2k−1fk−m ∧Qfk−m
a2k], (5.6)
EoM forQab 0 = ...omitted... . (5.7)
p Counterexample. Consider the following field configuration:
gab = ηab , ωab = ωab ,
ϑa = dxa , Qab = 2α(aδ
b)t ,
where αa = et(δa3dx3 + ... + δa2kdx2k
). (5.8)
An immediate consequence is Rab = 0, so
EoM for ωab 0 = Eaba3...a2kD[Qa3f1 ∧Qf1
a4 ∧ ... ∧Qa2k−1fk−1 ∧Qfk−1
a2k], (5.9)
The ansatz (5.8) gives:
0 = d(αa3 ∧ ... ∧αa2k
)= 2(k − 1) e2(k−1)t (2k − 2)! δ
[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k . (5.10)
If k > 1 we get a contradiction:
0 = δ[a33 ...δ
a2k]2k dt ∧ dx3 ∧ dx4 ∧ ... ∧ dx2k 6= 0 ∀p ∈M. (5.11)
p Conclusion: The Lagrangian (with k > 1) CANNOT be a total derivative. [Janssen, Jiménez 2019]
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General Lovelock in critical dimensions. Particular solutions
[Janssen, Jiménez 2019]
p The general equation of motion for the connection can be written:
0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3
aD−2 ⇔ (5.12)
⇔0 =
[Qca1Eca2...aD−2ab + ...+Q
caD−3Ea1...aD−4caD−2ab
+QcaEa1...aD−2cb
]∧Ra1a2 ∧ ... ∧RaD−3aD−2
(5.13)
é Particular cases:
(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)
(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q
cpEqabc] ∧Rpq ⇒ ? . (5.15)
p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).
p Families of solutions for arbitrary k > 1:é TeleparallelRc
d = 0.é Any connection such thatQab ∧Rc
d = 0.
p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).
Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]
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General Lovelock in critical dimensions. Particular solutions
[Janssen, Jiménez 2019]
p The general equation of motion for the connection can be written:
0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3
aD−2 ⇔ (5.12)
⇔0 =
[Qca1Eca2...aD−2ab + ...+Q
caD−3Ea1...aD−4caD−2ab
+QcaEa1...aD−2cb
]∧Ra1a2 ∧ ... ∧RaD−3aD−2
(5.13)
é Particular cases:
(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)
(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q
cpEqabc] ∧Rpq ⇒ ? . (5.15)
p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).
p Families of solutions for arbitrary k > 1:é TeleparallelRc
d = 0.é Any connection such thatQab ∧Rc
d = 0.
p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).
Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26
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General Lovelock in critical dimensions. Particular solutions
[Janssen, Jiménez 2019]
p The general equation of motion for the connection can be written:
0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3
aD−2 ⇔ (5.12)
⇔0 =
[Qca1Eca2...aD−2ab + ...+Q
caD−3Ea1...aD−4caD−2ab
+QcaEa1...aD−2cb
]∧Ra1a2 ∧ ... ∧RaD−3aD−2
(5.13)
é Particular cases:
(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)
(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q
cpEqabc] ∧Rpq ⇒ ? . (5.15)
p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).
p Families of solutions for arbitrary k > 1:é TeleparallelRc
d = 0.é Any connection such thatQab ∧Rc
d = 0.
p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).
Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26
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General Lovelock in critical dimensions. Particular solutions
[Janssen, Jiménez 2019]
p The general equation of motion for the connection can be written:
0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3
aD−2 ⇔ (5.12)
⇔0 =
[Qca1Eca2...aD−2ab + ...+Q
caD−3Ea1...aD−4caD−2ab
+QcaEa1...aD−2cb
]∧Ra1a2 ∧ ... ∧RaD−3aD−2
(5.13)
é Particular cases:
(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)
(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q
cpEqabc] ∧Rpq ⇒ ? . (5.15)
p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).
p Families of solutions for arbitrary k > 1:é TeleparallelRc
d = 0.é Any connection such thatQab ∧Rc
d = 0.
p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).
Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26
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General Lovelock in critical dimensions. Particular solutions
[Janssen, Jiménez 2019]
p The general equation of motion for the connection can be written:
0 = DEa1a2 ...ab ∧Ra1a2 ∧ ... ∧RaD−3
aD−2 ⇔ (5.12)
⇔0 =
[Qca1Eca2...aD−2ab + ...+Q
caD−3Ea1...aD−4caD−2ab
+QcaEa1...aD−2cb
]∧Ra1a2 ∧ ... ∧RaD−3aD−2
(5.13)
é Particular cases:
(k = 1) Einstein: 0 = QcaEbc ⇔ Qab = 0 (general sol.) . (5.14)
(k = 2) Gauss-Bonnet: 0 = [QcaEbcpq +Q
cpEqabc] ∧Rpq ⇒ ? . (5.15)
p Families of solutions for arbitrary k:é connection with Qµνρ = Vµgνρ (i.e.Qab = 0).
p Families of solutions for arbitrary k > 1:é TeleparallelRc
d = 0.é Any connection such thatQab ∧Rc
d = 0.
p Families of solutions for arbitrary k > 2:é Any connection such thatRab = αab ∧ k for certain 1-forms αab and k (due to k ∧ k ≡ 0).
Example. Ansatz of grav. wave: k is the dual form of the wave vector. [My PhD Thesis - still in progress]
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 26
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General Lovelock in critical dimensions. EoM of the coframe
p EoM in the general case
δS(D)k
δϑa= gabRa1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑ
a1...a2kb . (5.16)
p In D = 2k (critical) the red object becomes a (D + 1)-form, and the equation becomes trivial:
δS(2k)k
δϑa= 0 . (5.17)
p Another way of checking this is expanding the star in the Lagrangian to see the dependence on thecoframe,
L(D)k = Ra1
a2 ∧ ... ∧Ra2k−1
a2k ∧ ?ϑa1a2 ...a2k−1a2k (5.18)
= Ra1a2 ∧ ... ∧Ra2k−1a2k ∧(
1
(D − 2k)!Ea1...a2k b1...bD−2k
ϑb1...bD−2k
)(5.19)
In the critical case,L
(2k)k = Ea1a2 ...a2k−1
a2kRa1a2 ∧ ... ∧Ra2k−1
a2k , (5.20)
where
Ea1a2 ...a2k−1a2k depends only on gab (in the orthonormal case it is a constant tensor)
Rab is only connection-dependent .
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General Lovelock in critical dimensions. EoM of the coframe
p EoM in the general case
δS(D)k
δϑa= gabRa1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑ
a1...a2kb . (5.16)
p In D = 2k (critical) the red object becomes a (D + 1)-form, and the equation becomes trivial:
δS(2k)k
δϑa= 0 . (5.17)
p Another way of checking this is expanding the star in the Lagrangian to see the dependence on thecoframe,
L(D)k = Ra1
a2 ∧ ... ∧Ra2k−1
a2k ∧ ?ϑa1a2 ...a2k−1a2k (5.18)
= Ra1a2 ∧ ... ∧Ra2k−1a2k ∧(
1
(D − 2k)!Ea1...a2k b1...bD−2k
ϑb1...bD−2k
)(5.19)
In the critical case,L
(2k)k = Ea1a2 ...a2k−1
a2kRa1a2 ∧ ... ∧Ra2k−1
a2k , (5.20)
where
Ea1a2 ...a2k−1a2k depends only on gab (in the orthonormal case it is a constant tensor)
Rab is only connection-dependent .
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General Lovelock in critical dimensions. EoM of the coframe
p EoM in the general case
δS(D)k
δϑa= gabRa1a2 ∧ ... ∧Ra2k−1a2k ∧ ?ϑ
a1...a2kb . (5.16)
p In D = 2k (critical) the red object becomes a (D + 1)-form, and the equation becomes trivial:
δS(2k)k
δϑa= 0 . (5.17)
p Another way of checking this is expanding the star in the Lagrangian to see the dependence on thecoframe,
L(D)k = Ra1
a2 ∧ ... ∧Ra2k−1
a2k ∧ ?ϑa1a2 ...a2k−1a2k (5.18)
= Ra1a2 ∧ ... ∧Ra2k−1a2k ∧(
1
(D − 2k)!Ea1...a2k b1...bD−2k
ϑb1...bD−2k
)(5.19)
In the critical case,L
(2k)k = Ea1a2 ...a2k−1
a2kRa1a2 ∧ ... ∧Ra2k−1
a2k , (5.20)
where
Ea1a2 ...a2k−1a2k depends only on gab (in the orthonormal case it is a constant tensor)
Rab is only connection-dependent .
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6. Summary and conclusions
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Summary and conclusions
Ideas to remember. Consider metric-affine Lovelock in the critical dimension:
p The equation of the coframe (Vielbein) is a trivial identity (so, no information).
p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).
p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.
Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)
Open questions / further work
p Which is the role of the non-metricity in breaking the triviality in critical dimension?
p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]
Thanks for your attention!
Aitäh!
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29
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Summary and conclusions
Ideas to remember. Consider metric-affine Lovelock in the critical dimension:
p The equation of the coframe (Vielbein) is a trivial identity (so, no information).
p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).
p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.
Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)
Open questions / further work
p Which is the role of the non-metricity in breaking the triviality in critical dimension?
p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]
Thanks for your attention!
Aitäh!
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29
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Summary and conclusions
Ideas to remember. Consider metric-affine Lovelock in the critical dimension:
p The equation of the coframe (Vielbein) is a trivial identity (so, no information).
p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).
p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.
Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)
Open questions / further work
p Which is the role of the non-metricity in breaking the triviality in critical dimension?
p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]
Thanks for your attention!
Aitäh!
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29
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Summary and conclusions
Ideas to remember. Consider metric-affine Lovelock in the critical dimension:
p The equation of the coframe (Vielbein) is a trivial identity (so, no information).
p The Einstein-Palatini theory have been solved in all dimensions (even in the critical one).
p The big lesson:There are configurations that do not satisfy the EoM⇒ these theories are not boundary terms.
Consequently, we cannot use Lovelock terms to rewrite certain powers of R in terms of others!!(as we can in the metric or in the Riemann-Cartan case)
Open questions / further work
p Which is the role of the non-metricity in breaking the triviality in critical dimension?
p What about the complete Lovelock Lagrangian (all of the Lovelock terms allowed in that D)?The most interesting case is obviously D = 4, which includes EP + GBP. [Work in progress]
Thanks for your attention!
Aitäh!
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 29
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References
References for this presentation:p D. Lovelock, [Lovelock 1971]
The Einstein Tensor and Its Generalizations,
J. Math. Phys. 12 (1971), 498–501.p M. Borunda, B. Janssen, M. Bastero-Gil, , [Borunda, Janssen, Bastero 2008]
Palatini versus metric formulation in higher curvature gravity,
JCAP 0811 (2008), 008.p B. Janssen, A. Jiménez-Cano, J. A. Orejuela, [Janssen, Jiménez, Orejuela 2019]
A non-trivial connection for the metric-affine Gauss-Bonnet theory in D = 4,
Phys. Lett. B 795 (2019) 42 – 48.p F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman, [Hehl, McCrea, Mielke, Ne’eman 1995]
Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance,
Phys. Rep. 258 (1995) 1 – 171.p A. Jiménez-Cano, [My PhD Thesis – Still in progress]
Metric-Affine Gauge theory of gravity. Foundations, perturbations and gravitational wave solutions.p B. Janssen, A. Jiménez-Cano, [Janssen, Jiménez, 2019]
On the topological character of metric-affine Lovelock Lagrangians in critical dimensions.
arXiv:1907.12100 [gr-qc]
Other interesting references:p D. Iosifidis, and T. Koivisto,
Scale transformations in metric-affine geometry,
arXiv preprint: 1810.12276.p J. Beltrán Jiménez, L. Heisenberg, T. Koivisto,
Cosmology for quadratic gravity in generalized Weyl geometry,
JCAP 2016 (2016), no 04, 046.p J. Zanelli,
Chern-Simons Forms in Gravitation Theories,
Class. Quant. Grav. 29 (2012), 133001.
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Metric-affine lagrangians depending on curvature: EoM
Consider a gravitational lagrangian (vacuum) depending on the connection exclusively through thecurvature:
S[g,ϑ,ω] =
ˆL(gab,ϑ
a,Rab(ω)) ≡
ˆL(gab, eµ
a, Rµνab(ω))
√|g|dDx , (7.1)
with projective symmetry.
p Noether identities of Diff(M) and GL(D,R)⇒We only need the EoM of ϑa and ωab:
0 =δS
δϑa≡ ∂L
∂ϑa, (7.2)
0 =δS
δωab≡ D
(∂L
∂Rab
), (7.3)
or, in components,
0 =1√|g|
δS
δeµa≡ eµaL+
∂L∂eµa
, (7.4)
0 =−1
2√|g|
δS
δωµab≡(∇λ −
1
2Qλσ
σ + Tλσσ
)(∂L
∂Rλµab
)− 1
2Tλσ
µ ∂L∂Rλσab
. (7.5)
p Noether identity of projective symmetry⇒ the connection EoM is traceless (in a, b indices).
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Metric-affine lagrangians depending on curvature: EoM
Consider a gravitational lagrangian (vacuum) depending on the connection exclusively through thecurvature:
S[g,ϑ,ω] =
ˆL(gab,ϑ
a,Rab(ω)) ≡
ˆL(gab, eµ
a, Rµνab(ω))
√|g|dDx , (7.1)
with projective symmetry.
p Noether identities of Diff(M) and GL(D,R)⇒We only need the EoM of ϑa and ωab:
0 =δS
δϑa≡ ∂L
∂ϑa, (7.2)
0 =δS
δωab≡ D
(∂L
∂Rab
), (7.3)
or, in components,
0 =1√|g|
δS
δeµa≡ eµaL+
∂L∂eµa
, (7.4)
0 =−1
2√|g|
δS
δωµab≡(∇λ −
1
2Qλσ
σ + Tλσσ
)(∂L
∂Rλµab
)− 1
2Tλσ
µ ∂L∂Rλσab
. (7.5)
p Noether identity of projective symmetry⇒ the connection EoM is traceless (in a, b indices).
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Metric-affine lagrangians depending on curvature: EoM
Consider a gravitational lagrangian (vacuum) depending on the connection exclusively through thecurvature:
S[g,ϑ,ω] =
ˆL(gab,ϑ
a,Rab(ω)) ≡
ˆL(gab, eµ
a, Rµνab(ω))
√|g|dDx , (7.1)
with projective symmetry.
p Noether identities of Diff(M) and GL(D,R)⇒We only need the EoM of ϑa and ωab:
0 =δS
δϑa≡ ∂L
∂ϑa, (7.2)
0 =δS
δωab≡ D
(∂L
∂Rab
), (7.3)
or, in components,
0 =1√|g|
δS
δeµa≡ eµaL+
∂L∂eµa
, (7.4)
0 =−1
2√|g|
δS
δωµab≡(∇λ −
1
2Qλσ
σ + Tλσσ
)(∂L
∂Rλµab
)− 1
2Tλσ
µ ∂L∂Rλσab
. (7.5)
p Noether identity of projective symmetry⇒ the connection EoM is traceless (in a, b indices).
Dpto. Física Teórica y del Cosmos (UGR) Alejandro Jiménez Cano 31