On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism

Transcription

1 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: : Antonio N. Bernal, Bert Janssen, Alejandro Jimenez-Cano, J.A.O., Miguel Sanchez, Pablo Sanchez-Moreno November 17, 2016

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

3 Introduction General relativity: Gravity is a curvature effect. Free particles follow geodesics.

4 Introduction Spacetime: D-dimensional time-orientable Lorentzian manifold equipped with: Metric g µν. Levi-Civita connection: { } Γ ρ µν = ρµν = 1 2 gρλ ( µ g λν + ν g µλ λ g µν ).

5 Introduction Spacetime: D-dimensional time-orientable Lorentzian manifold equipped with: Metric g µν. Levi-Civita connection: Properties: { } Γ ρ µν = ρµν = 1 2 gρλ ( µ g λν + ν g µλ λ g µν ). T ρ µν = Γρ µν Γρ νµ = 0, µg νρ = 0.

6 Introduction Spacetime: D-dimensional time-orientable Lorentzian manifold equipped with: Metric g µν. Levi-Civita connection: Properties: { } Γ ρ µν = ρµν = 1 2 gρλ ( µ g λν + ν g µλ λ g µν ). T ρ µν = Γρ µν Γρ νµ = 0, µg νρ = 0. Geodesic curves (affine and metric): ẍ µ + Γ µ νρẋ ν ẋ ρ = 0.

7 Introduction Action: S = d D x [ ] 1 g 2κ gµν R µν + L M (φ, g).

8 Introduction Action: S = d D x [ ] 1 g 2κ gµν R µν + L M (φ, g). Equations of motion: R µν 1 2 g µνr = κt µν.

9 Introduction Action: S = d D x [ ] 1 g 2κ gµν R µν + L M (φ, g). Equations of motion: R µν 1 2 g µνr = κt µν. Geodesic curves: ẍ µ + Γ µ νρẋ ν ẋ ρ = 0.

10 Mathematical reasons: Absence of torsion. Metric compatibility. Introduction

11 Mathematical reasons: Absence of torsion. Metric compatibility. Uniqueness. Introduction

12 Mathematical reasons: Introduction Absence of torsion. Metric compatibility. Uniqueness. ( { } ) Physical reasons: Γ ρ µν(p) = ρµν + Sµν ρ + Tµν ρ Equivalence principle: Γ ρ µν(p) = 0 T ρ µν = 0.

13 Mathematical reasons: Introduction Absence of torsion. Metric compatibility. Uniqueness. ( { } ) Physical reasons: Γ ρ µν(p) = ρµν + Sµν ρ + Tµν ρ Equivalence principle: Γ ρ µν(p) = 0 T ρ µν = 0. We want metric geodesics = affine geodesics S ρ µν = 0.

14 Mathematical reasons: Introduction Absence of torsion. Metric compatibility. Uniqueness. ( { } ) Physical reasons: Γ ρ µν(p) = ρµν + Sµν ρ + Tµν ρ 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.

15 Mathematical reasons: Introduction Absence of torsion. Metric compatibility. Uniqueness. ( { } ) Physical reasons: Γ ρ µν(p) = ρµν + Sµν ρ + Tµν ρ 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?

16 Palatini formalism Metric g µν and connection Γ ρ µν independent, as in differential geometry. Action dependent on both: S = S(g, Γ) = d D x g [ ] 1 2κ gµν R µν (Γ) + L M (φ, g). δs Einstein equation. δg δs Connection equation. δγ

17 Palatini formalism Metric g µν and connection Γ ρ µν independent, as in differential geometry. Action dependent on both: S = S(g, Γ) = d D x g [ ] 1 2κ gµν R µν (Γ) + L M (φ, g). δs δg δs δγ Einstein equation. Connection equation. What do we expect? We hope to find Levi-Civita as the unique solution.

18 General solution S = d D x Equations of motion: [ ] 1 g 2κ gµν R µν (Γ) + L M (φ, g). R (µν) 1 2 g µνr = κt µν, R = g ρλ R ρλ, λ g µν T σ νλg σµ 1 D 1 T σ σλg µν 1 D 1 T σ σνg µλ = 0.

19 General solution S = d D x Equations of motion: [ ] 1 g 2κ gµν R µν (Γ) + L M (φ, g). R (µν) 1 2 g µνr = κt µν, R = g ρλ R ρλ, λ g µν T σ νλg σµ 1 D 1 T σ σλg µν 1 D 1 T σ σνg µλ = 0. General solution: Γ ρ µν = { ρµν } + A µ δ ρ ν.

20 Geometrical properties Palatini connections: Γ ρ µν = { ρµν } + A µ δ ρ ν.

21 Geometrical properties Palatini connections: { } Γ ρ µν = ρµν + A µ δν ρ. Torsion and metric derivative: T µν ρ = A µδν ρ A νδµ ρ, ρ g µν = 2A ρ g µν.

22 Geometrical properties Palatini connections: { } Γ ρ µν = ρµν + A µ δν ρ. Torsion and metric derivative: T µν ρ = A µδν ρ A νδµ ρ, ρ g µν = 2A ρ g µν. Curvature tensors: R λ µνρ = R λ µνρ + F µν δρ λ, Rµν = R µν + F µν, R = R, where F µν = µ A ν ν A µ = µ A ν ν A µ.

23 Geometrical properties Affine geodesic equation: ẋ ρ ρ ẋ µ = 0 ẋ ρ ρ ẋ µ = A ρ ẋ ρ ẋ µ ) ( ṡ ẋ ρ ρ ẋ µ = ẋ µ, s(λ) = s λ 0 e λ ẋ ρ A ρ dλ 0 dλ

24 Geometrical properties Affine geodesic equation: ẋ ρ ρ ẋ µ = 0 ẋ ρ ρ ẋ µ = A ρ ẋ ρ ẋ µ ) ( ṡ ẋ ρ ρ ẋ µ = ẋ µ, s(λ) = s λ Same trajectories but with different parametrisation. 0 e λ ẋ ρ A ρ dλ 0 dλ

25 Parallel transport: Geometrical properties ẋ ρ ρ V µ ẋ ρ ρ V µ = ẋ ρ A ρ V µ

26 Parallel transport: Geometrical properties ẋ ρ ρ V µ ẋ ρ ρ V µ = ẋ ρ A ρ V µ V µ Γ (λ) = e G(λ) V µ (λ),

27 Parallel transport: Geometrical properties ẋ ρ ρ V µ ẋ ρ ρ V µ = ẋ ρ A ρ V µ V µ Γ (λ) = e G(λ) V µ (λ), Consequence of non metric compatibility. Similar to L-C transport composed with homothety. Uniqueness.

28 Physical observability Let s summarise: General solution Γ ρ µν = { ρµν } + A µ δ ρ ν. Curvature tensors are the same plus terms involving F µν. In particular: R µν = R µν + F µν.

29 Physical observability Let s summarise: General solution Γ ρ µν = { ρµν } + A µ δ ρ ν. Curvature tensors are the same plus terms involving F µν. In particular: R µν = R µν + F µν. Homothetic parallel transport.

30 Physical observability Same rough physics: Same solutions to Einstein equation: R (µν) = R µν R µν 1 2 g µνr = κt µν.

31 Physical observability Same rough physics: Same solutions to Einstein equation: R (µν) = R µν R µν 1 2 g µνr = κt µν. Same spacetime trajectories of free-falling test particles: ) ( s ẋ ρ ρ ẋ µ = A ρ ẋ ρ ẋ µ = ẋ µ ṡ Equivalence Principle preserved.

32 Physical observability Same rough physics: Same solutions to Einstein equation: R (µν) = R µν R µν 1 2 g µνr = κt µν. Same spacetime trajectories of free-falling test particles: ) ( s ẋ ρ ρ ẋ µ = A ρ ẋ ρ ẋ µ = ẋ µ ṡ Equivalence Principle preserved. Same tidal forces (geodesic deviation).

33 Physical observability Parallel transport with homothety Staticity: t r

34 Physical observability Parallel transport with homothety Staticity: t r There aren t any curvature effects except for the change of norm.

35 Physical observability Parallel transport with homothety Staticity: t r There aren t any curvature effects except for the change of norm. Could be interpreted as non-staticity, but that s because usually we work with metric-compatible connections.

36 Physical observability Parallel transport with homothety Staticity: t r There aren t any curvature effects except for the change of norm. Could be interpreted as non-staticity, but that s because usually we work with metric-compatible connections. The norm of a parallel transported vector cannot be physically measured.

37 Physical observability Parallel transport with homothety Staticity: t r There aren t any curvature effects except for the change of norm. Could be interpreted as non-staticity, but that s because usually we work with metric-compatible connections. The norm of a parallel transported vector cannot be physically measured. Usually we define a unit (say, a rod) and we transport it. Key point: the forces are not the geometrical ones.

38 Physical observability Parallel transport with homothety Staticity: t r There aren t any curvature effects except for the change of norm. Could be interpreted as non-staticity, but that s because usually we work with metric-compatible connections. The norm of a parallel transported vector cannot be physically measured. Usually we define a unit (say, a rod) and we transport it. Key point: the forces are not the geometrical ones. Moral: Compare directions with the connection and norms with the metric.

39 Future work Next step: Lovelock Gravities: Action of orden n in curvature but second order differential equations.

40 Future work Next step: Lovelock Gravities: Action of orden n in curvature but second order differential equations. In particular, Gauss-Bonnet, [ S = d D x g R µνρλ R µνρλ 4R µν R µν + R 2]. We have already obtained the variations of the action and have seen that Palatini connections are solutions.

41 Conclusions Summarising: Most general solution: Γ ρ µν = { ρµν } + A µ δ ρ ν.

42 Conclusions Summarising: Most general solution: Γ ρ µν = { ρµν } + A µ δ ρ ν. Same pregeodesics and same Einstein equation.

43 Conclusions Summarising: Most general solution: Γ ρ µν = { ρµν } + A µ δ ρ ν. Same pregeodesics and same Einstein equation. Unique with parallel transport homothetic to the L-C one.

44 Conclusions Summarising: Most general solution: Γ ρ µν = { ρµν } + A µ δ ρ ν. Same pregeodesics and same Einstein equation. Unique with parallel transport homothetic to the L-C one. No physical observable effects, so Palatini formalism yields an exact variational characterisation of such basic physics.

45 Conclusions Summarising: Most general solution: Γ ρ µν = { ρµν } + A µ δ ρ ν. Same pregeodesics and same Einstein equation. Unique with parallel transport homothetic to the L-C one. No physical observable effects, so Palatini formalism yields an exact variational characterisation of such basic physics. Relation between spacetimes with different geometry but same physics. Freedom of geodesic parametrisation.

46 Conclusions Summarising: Most general solution: Γ ρ µν = { ρµν } + A µ δ ρ ν. Same pregeodesics and same Einstein equation. Unique with parallel transport homothetic to the L-C one. No physical observable effects, so Palatini formalism yields an exact variational characterisation of such basic physics. Relation between spacetimes with different geometry but same physics. Freedom of geodesic parametrisation. Thanks for your attention!