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May 10, 2023 358 likes •824 views

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 Jos Alberto Orejuela

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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

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

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

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

Introduction

General relativity:

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

Introduction

Spacetime: D-dimensional time-orientable Lorentzian manifold equipped with:

  • Metric gµν.
  • Levi-Civita connection:

Γρ

µν =

  • ρ

µν

  • = 1

2gρλ (∂µgλν + ∂νgµλ − ∂λgµν) .

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

Introduction

Spacetime: D-dimensional time-orientable Lorentzian manifold equipped with:

  • Metric gµν.
  • Levi-Civita connection:

Γρ

µν =

  • ρ

µν

  • = 1

2gρλ (∂µgλν + ∂νgµλ − ∂λgµν) . Properties: T ρ

µν = Γρ µν − Γρ νµ = 0,

∇µgνρ = 0.

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

Introduction

Spacetime: D-dimensional time-orientable Lorentzian manifold equipped with:

  • Metric gµν.
  • Levi-Civita connection:

Γρ

µν =

  • ρ

µν

  • = 1

2gρλ (∂µgλν + ∂νgµλ − ∂λgµν) . Properties: T ρ

µν = Γρ µν − Γρ νµ = 0,

∇µgνρ = 0. Geodesic curves (affine and metric): ¨ xµ + Γµ

νρ ˙

xν ˙ xρ = 0.

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

Introduction

Action: S =

  • dDx
  • |g|

1

2κgµνRµν + LM(φ, g)

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

Introduction

Action: S =

  • dDx
  • |g|

1

2κgµνRµν + LM(φ, g)

  • .

Equations of motion: Rµν − 1 2gµνR = −κTµν.

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

Introduction

Action: S =

  • dDx
  • |g|

1

2κgµνRµν + LM(φ, g)

  • .

Equations of motion: Rµν − 1 2gµνR = −κTµν. Geodesic curves: ¨ xµ + Γµ

νρ ˙

xν ˙ xρ = 0.

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

Introduction

Mathematical reasons:

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

Introduction

Mathematical reasons:

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

Introduction

Mathematical reasons:

  • Absence of torsion.
  • Metric compatibility.
  • Uniqueness.

Physical reasons:

  • Γρ

µν(p) =

  • ρ

µν

  • + Sρ

µν + T ρ µν

  • Equivalence principle: Γρ

µν(p) = 0 ⇒ T ρ µν = 0.

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

Introduction

Mathematical reasons:

  • Absence of torsion.
  • Metric compatibility.
  • Uniqueness.

Physical reasons:

  • Γρ

µν(p) =

  • ρ

µν

  • + Sρ

µν + T ρ µν

  • Equivalence principle: Γρ

µν(p) = 0 ⇒ T ρ µν = 0.

  • We want metric geodesics = affine geodesics ⇒ Sρ

µν = 0.

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

Introduction

Mathematical reasons:

  • Absence of torsion.
  • Metric compatibility.
  • Uniqueness.

Physical reasons:

  • Γρ

µν(p) =

  • ρ

µν

  • + Sρ

µν + T ρ µν

  • Equivalence principle: Γρ

µν(p) = 0 ⇒ T ρ µν = 0.

  • We want metric geodesics = affine geodesics ⇒ Sρ

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.

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

Introduction

Mathematical reasons:

  • Absence of torsion.
  • Metric compatibility.
  • Uniqueness.

Physical reasons:

  • Γρ

µν(p) =

  • ρ

µν

  • + Sρ

µν + T ρ µν

  • Equivalence principle: Γρ

µν(p) = 0 ⇒ T ρ µν = 0.

  • We want metric geodesics = affine geodesics ⇒ Sρ

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?

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

Palatini formalism

Metric gµν and connection Γρ

µν independent, as in differential

  • geometry. Action dependent on both:

S = S(g, Γ) =

  • dDx
  • |g|

1

2κgµνRµν(Γ) + LM(φ, g)

  • .
  • δS

δg → Einstein equation.

  • δS

δΓ → Connection equation.

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

Palatini formalism

Metric gµν and connection Γρ

µν independent, as in differential

  • geometry. Action dependent on both:

S = S(g, Γ) =

  • dDx
  • |g|

1

2κgµνRµν(Γ) + LM(φ, g)

  • .
  • δS

δg → Einstein equation.

  • δS

δΓ → Connection equation. What do we expect? We hope to find Levi-Civita as the unique solution.

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

General solution

S =

  • dDx
  • |g|

1

2κgµνRµν(Γ) + LM(φ, g)

  • .

Equations of motion: R(µν) − 1 2gµνR = −κTµν, R = gρλRρλ, ∇λgµν − T σ

νλgσµ −

1 D − 1T σ

σλgµν −

1 D − 1T σ

σνgµλ = 0.

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

General solution

S =

  • dDx
  • |g|

1

2κgµνRµν(Γ) + LM(φ, g)

  • .

Equations of motion: R(µν) − 1 2gµνR = −κTµν, R = gρλRρλ, ∇λgµν − T σ

νλgσµ −

1 D − 1T σ

σλgµν −

1 D − 1T σ

σνgµλ = 0.

General solution: Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

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

Geometrical properties

Palatini connections: ¯ Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

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

Geometrical properties

Palatini connections: ¯ Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

Torsion and metric derivative: ¯ T ρ

µν = Aµδρ ν − Aνδρ µ,

¯ ∇ρgµν = −2Aρgµν.

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

Geometrical properties

Palatini connections: ¯ Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

Torsion and metric derivative: ¯ T ρ

µν = Aµδρ ν − Aνδρ µ,

¯ ∇ρgµν = −2Aρgµν. Curvature tensors: ¯ Rµνρλ = Rµνρλ + Fµνδλ

ρ,

¯ Rµν = Rµν + Fµν, ¯ R = R, where Fµν = ∂µAν − ∂νAµ = ∇µAν − ∇νAµ.

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

Geometrical properties

Affine geodesic equation: ˙ xρ ¯ ∇ρ ˙ xµ = 0 ⇔ ˙ xρ∇ρ ˙ xµ = −Aρ ˙ xρ ˙ xµ ⇔ ˙ xρ∇ρ ˙ xµ =

¨

s ˙ s

  • ˙

xµ, s(λ) =

λ

e− λ′

˙ xρAρ dλ′′ dλ′

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

Geometrical properties

Affine geodesic equation: ˙ xρ ¯ ∇ρ ˙ xµ = 0 ⇔ ˙ xρ∇ρ ˙ xµ = −Aρ ˙ xρ ˙ xµ ⇔ ˙ xρ∇ρ ˙ xµ =

¨

s ˙ s

  • ˙

xµ, s(λ) =

λ

e− λ′

˙ xρAρ dλ′′ dλ′

Same trajectories but with different parametrisation.

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

Geometrical properties

Parallel transport: ˙ xρ ¯ ∇ρV µ − ˙ xρ∇ρV µ = ˙ xρAρV µ

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

Geometrical properties

Parallel transport: ˙ xρ ¯ ∇ρV µ − ˙ xρ∇ρV µ = ˙ xρAρV µ ⇒ V µ

¯ Γ (λ) = e−G(λ)V µ(λ),

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

Geometrical properties

Parallel transport: ˙ xρ ¯ ∇ρV µ − ˙ xρ∇ρV µ = ˙ xρAρV 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.
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Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Physical observability

Let’s summarise:

  • General solution

¯ Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

  • Curvature tensors are the same plus terms involving Fµν. In

particular: ¯ Rµν = Rµν + Fµν.

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

Physical observability

Let’s summarise:

  • General solution

¯ Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

  • Curvature tensors are the same plus terms involving Fµν. In

particular: ¯ Rµν = Rµν + Fµν.

  • Homothetic parallel transport.
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Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Physical observability

Same rough physics:

  • Same solutions to Einstein equation:

¯ R(µν) = Rµν ⇒ Rµν − 1 2gµνR = −κTµν.

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

Physical observability

Same rough physics:

  • Same solutions to Einstein equation:

¯ R(µν) = Rµν ⇒ Rµν − 1 2gµνR = −κTµν.

  • Same spacetime trajectories of free-falling test particles:

˙ xρ∇ρ ˙ xµ = −Aρ ˙ xρ ˙ xµ =

¨

s ˙ s

  • ˙

xµ

  • Equivalence Principle preserved.
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Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Physical observability

Same rough physics:

  • Same solutions to Einstein equation:

¯ R(µν) = Rµν ⇒ Rµν − 1 2gµνR = −κTµν.

  • Same spacetime trajectories of free-falling test particles:

˙ xρ∇ρ ˙ xµ = −Aρ ˙ xρ ˙ xµ =

¨

s ˙ s

  • ˙

xµ

  • Equivalence Principle preserved.
  • Same tidal forces (geodesic deviation).
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Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Physical observability

  • Parallel transport with homothety ←

→ Staticity: r t

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

Physical observability

  • Parallel transport with homothety ←

→ Staticity: r t

  • There aren’t any curvature effects except for the change of

norm.

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

Physical observability

  • Parallel transport with homothety ←

→ Staticity: r t

  • 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.

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

Physical observability

  • Parallel transport with homothety ←

→ Staticity: r t

  • 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.

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

Physical observability

  • Parallel transport with homothety ←

→ Staticity: r t

  • 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.

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

Physical observability

  • Parallel transport with homothety ←

→ Staticity: r t

  • 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.

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

Future work

Next step: Lovelock Gravities:

  • Action of orden n in curvature but second order differential

equations.

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

Future work

Next step: Lovelock Gravities:

  • Action of orden n in curvature but second order differential

equations. In particular, Gauss-Bonnet, S =

  • dDx
  • |g|
  • RµνρλRµνρλ − 4RµνRµν + R2

. We have already obtained the variations of the action and have seen that Palatini connections are solutions.

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

Conclusions

Summarising:

  • Most general solution: ¯

Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

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

Conclusions

Summarising:

  • Most general solution: ¯

Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

  • Same pregeodesics and same Einstein equation.
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Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Conclusions

Summarising:

  • Most general solution: ¯

Γρ

µν =

  • ρ

µν

  • + Aµδρ

ν.

  • Same pregeodesics and same Einstein equation.
  • Unique with parallel transport homothetic to the L-C one.
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Introduction Palatini formalism General solution Geometrical properties Physical observability Future work Conclusions

Conclusions

Summarising:

  • Most general solution: ¯

Γρ

µν =

  • ρ

µν

  • + Aµδρ

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.

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

Conclusions

Summarising:

  • Most general solution: ¯

Γρ

µν =

  • ρ

µν

  • + Aµδρ

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.

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

Conclusions

Summarising:

  • Most general solution: ¯

Γρ

µν =

  • ρ

µν

  • + Aµδρ

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!