On the (non-)uniqueness of the Levi-Civita solution in the - PowerPoint PPT Presentation

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism José Alberto Orejuela Oviedo V Postgraduate Meeting On Theoretical Physics arXiv:1606.08756: Antonio N. Bernal, Bert Janssen, Alejandro Jimenez-Cano, J.A.O., Miguel Sanchez, Pablo Sanchez-Moreno November 17, 2016

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction General relativity: • Gravity is a curvature effect. • Free particles follow geodesics.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Spacetime: D -dimensional time-orientable Lorentzian manifold equipped with: • Metric g µν . • Levi-Civita connection: = 1 2 g ρλ ( ∂ µ g λν + ∂ ν g µλ − ∂ λ g µν ) . � � Γ ρ ρ µν = µν

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Spacetime: D -dimensional time-orientable Lorentzian manifold equipped with: • Metric g µν . • Levi-Civita connection: = 1 2 g ρλ ( ∂ µ g λν + ∂ ν g µλ − ∂ λ g µν ) . � � Γ ρ ρ µν = µν Properties: T ρ µν = Γ ρ µν − Γ ρ νµ = 0 , ∇ µ g νρ = 0 .

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Spacetime: D -dimensional time-orientable Lorentzian manifold equipped with: • Metric g µν . • Levi-Civita connection: = 1 2 g ρλ ( ∂ µ g λν + ∂ ν g µλ − ∂ λ g µν ) . � � Γ ρ ρ µν = µν Properties: T ρ µν = Γ ρ µν − Γ ρ νµ = 0 , ∇ µ g νρ = 0 . Geodesic curves (affine and metric): x µ + Γ µ x ν ˙ x ρ = 0 . ¨ νρ ˙

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Action: � 1 � � � d D x 2 κ g µν R µν + L M ( φ, g ) S = | g | .

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Action: � 1 � � � d D x 2 κ g µν R µν + L M ( φ, g ) S = | g | . Equations of motion: R µν − 1 2 g µν R = − κ T µν .

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Action: � 1 � � � d D x 2 κ g µν R µν + L M ( φ, g ) S = | g | . Equations of motion: R µν − 1 2 g µν R = − κ T µν . Geodesic curves: x µ + Γ µ x ν ˙ x ρ = 0 . ¨ νρ ˙

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Mathematical reasons: • Absence of torsion. • Metric compatibility.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Mathematical reasons: • Absence of torsion. • Metric compatibility. • Uniqueness.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Mathematical reasons: • Absence of torsion. • Metric compatibility. • Uniqueness. � � � � Γ ρ ρ + S ρ µν + T ρ Physical reasons: µν ( p ) = µν µν • Equivalence principle: Γ ρ µν ( p ) = 0 ⇒ T ρ µν = 0.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Mathematical reasons: • Absence of torsion. • Metric compatibility. • Uniqueness. � � � � Γ ρ ρ + S ρ µν + T ρ Physical reasons: µν ( p ) = µν µν • Equivalence principle: Γ ρ µν ( p ) = 0 ⇒ T ρ µν = 0. • We want metric geodesics = affine geodesics ⇒ S ρ µν = 0.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Mathematical reasons: • Absence of torsion. • Metric compatibility. • Uniqueness. � � � � Γ ρ ρ + S ρ µν + T ρ Physical reasons: µν ( p ) = µν µν • Equivalence principle: Γ ρ µν ( p ) = 0 ⇒ T ρ µν = 0. • We want metric geodesics = affine geodesics ⇒ S ρ µν = 0. Are they enough? • Although these are valid reasons, it seems that L-C is put by hand. • It would be perfect if there was a physical mechanism that selects Levi-Civita over other possibilities.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Introduction Mathematical reasons: • Absence of torsion. • Metric compatibility. • Uniqueness. � � � � Γ ρ ρ + S ρ µν + T ρ Physical reasons: µν ( p ) = µν µν • Equivalence principle: Γ ρ µν ( p ) = 0 ⇒ T ρ µν = 0. • We want metric geodesics = affine geodesics ⇒ S ρ µν = 0. Are they enough? • Although these are valid reasons, it seems that L-C is put by hand. • It would be perfect if there was a physical mechanism that selects Levi-Civita over other possibilities. • If I find a variational principle that have L-C as a solution, is it unique? Which one is the most general solution?

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Palatini formalism Metric g µν and connection Γ ρ µν independent, as in differential geometry. Action dependent on both: � 1 � � � d D x 2 κ g µν R µν (Γ) + L M ( φ, g ) S = S ( g , Γ) = | g | . • δ S δ g → Einstein equation. • δ S δ Γ → Connection equation.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Palatini formalism Metric g µν and connection Γ ρ µν independent, as in differential geometry. Action dependent on both: � 1 � � � d D x 2 κ g µν R µν (Γ) + L M ( φ, g ) S = S ( g , Γ) = | g | . • δ S δ g → Einstein equation. • δ S δ Γ → Connection equation. What do we expect? We hope to find Levi-Civita as the unique solution.

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions General solution � 1 � � � d D x 2 κ g µν R µν (Γ) + L M ( φ, g ) S = | g | . Equations of motion: R ( µν ) − 1 R = g ρλ R ρλ , 2 g µν R = − κ T µν , 1 1 ∇ λ g µν − T σ D − 1 T σ D − 1 T σ νλ g σµ − σλ g µν − σν g µλ = 0 .

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions General solution � 1 � � � d D x 2 κ g µν R µν (Γ) + L M ( φ, g ) S = | g | . Equations of motion: R ( µν ) − 1 R = g ρλ R ρλ , 2 g µν R = − κ T µν , 1 1 ∇ λ g µν − T σ D − 1 T σ D − 1 T σ νλ g σµ − σλ g µν − σν g µλ = 0 . General solution: � � Γ ρ ρ + A µ δ ρ µν = ν . µν

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Geometrical properties Palatini connections: � � ¯ Γ ρ ρ + A µ δ ρ µν = ν . µν

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Geometrical properties Palatini connections: � � ¯ Γ ρ ρ + A µ δ ρ µν = ν . µν Torsion and metric derivative: ¯ ¯ T ρ µν = A µ δ ρ ν − A ν δ ρ µ , ∇ ρ g µν = − 2 A ρ g µν .

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Geometrical properties Palatini connections: � � ¯ Γ ρ ρ + A µ δ ρ µν = ν . µν Torsion and metric derivative: ¯ ¯ T ρ µν = A µ δ ρ ν − A ν δ ρ µ , ∇ ρ g µν = − 2 A ρ g µν . Curvature tensors: R µνρλ = R µνρλ + F µν δ λ ¯ ¯ ¯ ρ , R µν = R µν + F µν , R = R , where F µν = ∂ µ A ν − ∂ ν A µ = ∇ µ A ν − ∇ ν A µ .

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Geometrical properties Affine geodesic equation: x ρ ¯ x µ = 0 ⇔ ˙ x µ = −A ρ ˙ x ρ ˙ x ρ ∇ ρ ˙ x µ ˙ ∇ ρ ˙ � λ e − � λ ′ � ¨ s x ρ A ρ d λ ′′ d λ ′ � x µ = ˙ x ρ ∇ ρ ˙ x µ , s ( λ ) = ⇔ ˙ ˙ 0 s ˙ 0

Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions Geometrical properties Affine geodesic equation: x ρ ¯ x µ = 0 ⇔ ˙ x µ = −A ρ ˙ x ρ ˙ x ρ ∇ ρ ˙ x µ ˙ ∇ ρ ˙ � λ e − � λ ′ � ¨ s x ρ A ρ d λ ′′ d λ ′ � x µ = ˙ x ρ ∇ ρ ˙ x µ , s ( λ ) = ⇔ ˙ ˙ 0 s ˙ 0 Same trajectories but with different parametrisation.