Palatiniversus metricformalism inhigher-curvaturegravity
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- Μέλισσα Αναστασιάδης
- 7 χρόνια πριν
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1 Palatiniversus metricformalism inhigher-curvaturegravity Bert Janssen Universidad de Granada & CAFPE In collaboration with: M. Borunda and M. Bastero-Gil (UGR) References: JCAP 0811(2008) 008, arxiv: [hep-th. B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 1/29
2 Outlook 1. The Levi-Civita connection: what and why? B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
3 Outlook 1. The Levi-Civita connection: what and why? 2. Palatini formalism for Einstein-Hilbert B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
4 Outlook 1. The Levi-Civita connection: what and why? 2. Palatini formalism for Einstein-Hilbert 3. Higher-curvature corrections and Lovelock gravity B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
5 Outlook 1. The Levi-Civita connection: what and why? 2. Palatini formalism for Einstein-Hilbert 3. Higher-curvature corrections and Lovelock gravity 4. Palatini versus metric formalism for quadratic curvature gravity B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
6 Outlook 1. The Levi-Civita connection: what and why? 2. Palatini formalism for Einstein-Hilbert 3. Higher-curvature corrections and Lovelock gravity 4. Palatini versus metric formalism for quadratic curvature gravity 5. Palatini versus metric formalism for general Lagrangians B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
7 Outlook 1. The Levi-Civita connection: what and why? 2. Palatini formalism for Einstein-Hilbert 3. Higher-curvature corrections and Lovelock gravity 4. Palatini versus metric formalism for quadratic curvature gravity 5. Palatini versus metric formalism for general Lagrangians 6. Explicit example B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
8 Outlook 1. The Levi-Civita connection: what and why? 2. Palatini formalism for Einstein-Hilbert 3. Higher-curvature corrections and Lovelock gravity 4. Palatini versus metric formalism for quadratic curvature gravity 5. Palatini versus metric formalism for general Lagrangians 6. Explicit example 7. Conclusions B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 2/29
9 1 The Levi-Civita connection: what and why? Main lessons of General Relativity: Gravitational interaction is related to curvature of spacetime, determined by the energy and field content Spacetime is dynamical quantity, with physical degrees of freedom B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 3/29
10 1 The Levi-Civita connection: what and why? Main lessons of General Relativity: Gravitational interaction is related to curvature of spacetime, determined by the energy and field content Spacetime is dynamical quantity, with physical degrees of freedom Spacetime = D-dimLorentzian manifold with metric g µν and connection Γ ρ µν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 3/29
11 1 The Levi-Civita connection: what and why? Main lessons of General Relativity: Gravitational interaction is related to curvature of spacetime, determined by the energy and field content Spacetime is dynamical quantity, with physical degrees of freedom Spacetime = D-dimLorentzian manifold with metric g µν and connection Γ ρ µν Differential geometry: Metric: measures distances in manifold and angles in tangent space Connection: map between tangent spaces that defines parallel transport B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 3/29
12 1 The Levi-Civita connection: what and why? Main lessons of General Relativity: Gravitational interaction is related to curvature of spacetime, determined by the energy and field content Spacetime is dynamical quantity, with physical degrees of freedom Spacetime = D-dimLorentzian manifold with metric g µν and connection Γ ρ µν Differential geometry: Metric: measures distances in manifold and angles in tangent space Connection: map between tangent spaces that defines parallel transport Mathematically metric and connection are independent quantities B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 3/29
13 However if Γ ρ µν satisfies the properties symmetric: Γ ρ µν = Γ ρ νµ ( T ρ µν = 0) metric compatible: µ g νρ = 0 then Γ ρ µν isuniquely determined by the metric Γ ρ µν = 1 2 gρ ( µ g ν + µ g µ g µν ) (Levi-Civita connection) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 4/29
14 However if Γ ρ µν satisfies the properties symmetric: Γ ρ µν = Γ ρ νµ ( T ρ µν = 0) metric compatible: µ g νρ = 0 then Γ ρ µν isuniquely determined by the metric Γ ρ µν = 1 2 gρ ( µ g ν + µ g µ g µν ) (Levi-Civita connection) One usually (tacitly) assumes that Levi-Civita describes correctly physics: uniqueness: see above simplicity: tensor identities simplify physical grounds: Equivalence principle & geodesics B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 4/29
15 Simplicity: many curvature tensor identities simplify for Levi-Civita Bianchi identity: 0 = R µνρ + R νρµ + R ρµν µ Tνρ ν Tρµ ρ Tµν + TµνT σ ρσ + TνρT σ µσ + TρµT σ νσ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 5/29
16 Simplicity: many curvature tensor identities simplify for Levi-Civita Bianchi identity: 0 = R µνρ + R νρµ + R ρµν µ Tνρ ν Tρµ ρ Tµν + TµνT σ ρσ + TνρT σ µσ + TρµT σ νσ Symmetry of Ricci tensor: R [µν = 1 2 R µν 1 2 T µν [µ T ν T ρ µνt ρ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 5/29
17 Simplicity: many curvature tensor identities simplify for Levi-Civita Bianchi identity: 0 = R µνρ + R νρµ + R ρµν µ Tνρ ν Tρµ ρ Tµν + TµνT σ ρσ + TνρT σ µσ + TρµT σ νσ Symmetry of Ricci tensor: R [µν = 1 2 R µν 1 2 T µν [µ T ν T ρ µνt ρ Divergence of Einstein tensor: [ µ G µ ν = TµνR ρ µ ρ 1T 2 µρr µρ ν ρg µ δµr ρ ν + δνr ρ ρ µ + R µν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 5/29
18 Simplicity: many curvature tensor identities simplify for Levi-Civita Bianchi identity: 0 = R µνρ + R νρµ + R ρµν µ Tνρ ν Tρµ ρ Tµν + TµνT σ ρσ + TνρT σ µσ + TρµT σ νσ Symmetry of Ricci tensor: R [µν = 1 2 R µν 1 2 T µν [µ T ν T ρ µνt ρ Divergence of Einstein tensor: [ µ G µ ν = TµνR ρ µ ρ 1T 2 µρr µρ ν ρg µ δµr ρ ν + δνr ρ ρ µ + R µν Partial integration: S = d D x g µ A ν B µν = d D x [ g µ B µν A ν +... ( 1 2 gστ µ g στ + Tµσ )A σ ν B µν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 5/29
19 Physical grounds: Equivalence Principle: gravitational force can locally be gauged away by an appropriate choice of coordinates B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 6/29
20 Physical grounds: Equivalence Principle: gravitational force can locally be gauged away by an appropriate choice of coordinates Mathematically: connection components can locally be set to zero if torsionless (NB: torsion is tensor) Γ ρ µν(p) = 0 T ρ µν = 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 6/29
21 Physical grounds: Equivalence Principle: gravitational force can locally be gauged away by an appropriate choice of coordinates Mathematically: connection components can locally be set to zero if torsionless (NB: torsion is tensor) Γ ρ µν(p) = 0 T ρ µν = 0 Affine and metric geodesics: metric geodesic = shortest curve between two points = curve obtained via Action Principle B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 6/29
22 Physical grounds: Equivalence Principle: gravitational force can locally be gauged away by an appropriate choice of coordinates Mathematically: connection components can locally be set to zero if torsionless (NB: torsion is tensor) Γ ρ µν(p) = 0 T ρ µν = 0 Affine and metric geodesics: metric geodesic = shortest curve between two points = curve obtained via Action Principle affine geodesic = curve with covariantly constant tangent vector = unaccelerated trajectory B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 6/29
23 Physical grounds: Equivalence Principle: gravitational force can locally be gauged away by an appropriate choice of coordinates Mathematically: connection components can locally be set to zero if torsionless (NB: torsion is tensor) Γ ρ µν(p) = 0 T ρ µν = 0 Affine and metric geodesics: metric geodesic = shortest curve between two points = curve obtained via Action Principle affine geodesic = curve with covariantly constant tangent vector = unaccelerated trajectory Affine geodesics and metric geodesics do not coincide, unless for Levi-Civita B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 6/29
24 However if Γ ρ µν satisfies the properties symmetric: Γ ρ µν = Γ ρ νµ ( T ρ µν = 0) metric compatible: µ g νρ = 0 then Γ ρ µν isuniquely determined by the metric Γ ρ µν = 1 2 gρ ( µ g ν + µ g µ g µν ) (Levi-Civita connection) One usually (tacitly) assumes that Levi-Civita describes correctly physics: uniqueness: see above simplicity: tensor identities simplify physical grounds: Equivalence principle & geodesics B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 7/29
25 However if Γ ρ µν satisfies the properties symmetric: Γ ρ µν = Γ ρ νµ ( T ρ µν = 0) metric compatible: µ g νρ = 0 then Γ ρ µν isuniquely determined by the metric Γ ρ µν = 1 2 gρ ( µ g ν + µ g µ g µν ) (Levi-Civita connection) One usually (tacitly) assumes that Levi-Civita describes correctly physics: uniqueness: see above simplicity: tensor identities simplify physical grounds: Equivalence principle & geodesics mathematically more rigorous reason? Palatini formalism! B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 7/29
26 2 Palatini formalism for Einstein-Hilbert 2.1 Metric formalism where S(g) = d D x g g µν R µν (g) R µν = µ Γ ν Γ µν + Γ µσγ σ ν Γ σ µνγ σ Γ ρ µν ( ) = 1 2 gρ µ g ν + µ g µ g µν S(g)non-linear andsecond orderin g µν expect fourth-order diff eqns B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 8/29
27 2 Palatini formalism for Einstein-Hilbert 2.1 Metric formalism where S(g) = d D x g g µν R µν (g) R µν = µ Γ ν Γ µν + Γ µσγ σ ν Γ σ µνγ σ Γ ρ µν ( ) = 1 2 gρ µ g ν + µ g µ g µν S(g)non-linear andsecond orderin g µν expect fourth-order diff eqns explicit calculation: only g g µν contributes to δs(g) R µν (g) 1 2 g µνr(g) = 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 8/29
28 2.2 Palatini formalism Consider metric and connection as independent: S(g, Γ) = d D x g g µν R µν (Γ) where R µν = µ Γ ν Γ µν + Γ µσγ σ ν Γ σ µνγ σ µ g νρ 0 and/or Γ ρ µν Γ ρ νµ δs δg µν R µν (Γ) 1 2 g µνr(γ) = 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 9/29
29 2.2 Palatini formalism Consider metric and connection as independent: S(g, Γ) = d D x g g µν R µν (Γ) where R µν = µ Γ ν Γ µν + Γ µσγ σ ν Γ σ µνγ σ µ g νρ 0 and/or Γ ρ µν Γ ρ νµ δs δg µν R µν (Γ) 1 2 g µνr(γ) = 0 Palatini identity: δr µνρ = µ (δγ νρ) ν (δγ µρ) + T σ µν(δγ σρ) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 9/29
30 2.2 Palatini formalism Consider metric and connection as independent: S(g, Γ) = d D x g g µν R µν (Γ) where R µν = µ Γ ν Γ µν + Γ µσγ σ ν Γ σ µνγ σ µ g νρ 0 and/or Γ ρ µν Γ ρ νµ δs δg µν R µν (Γ) 1 2 g µνr(γ) = 0 Palatini identity: δr µνρ = µ (δγ νρ) ν (δγ µρ) + Tµν(δΓ σ σρ) 0 δs(g, Γ) = d D x [ ( ) g (δγ µν) g µν gστ g στ + Tσ σ ρ g ρν δ µ (1 2 gστ ρ g στ + T σ ρσ g µν ) g ρν δ µ + gρν T µ ρ Γ µν = Levi Civita B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 9/29
31 Advantages of Palatini formalism: δs Practical: Einstein eqn δg µν S = d D x g easy tocalculate if R µνρ = R µνρ (Γ) [ g µν R µν (Γ) + L(φ, g µν ) R µν (Γ) 1 2 g µνr(γ) = T µν Γ µν = Levi Civita if matter isminimally coupled in L(φ, g µν ) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 10/29
32 Advantages of Palatini formalism: δs Practical: Einstein eqn δg µν S = d D x g easy tocalculate if R µνρ = R µνρ (Γ) [ g µν R µν (Γ) + L(φ, g µν ) R µν (Γ) 1 2 g µνr(γ) = T µν Γ µν = Levi Civita if matter isminimally coupled in L(φ, g µν ) Philosophical: Levi-Civita is not just a convenient choice, it is a minimum ofthe action, asolutionof theequations ofmotion Any other connection would not(necessarily) have this property. B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 10/29
33 Advantages of Palatini formalism: δs Practical: Einstein eqn δg µν S = d D x g easy tocalculate if R µνρ = R µνρ (Γ) [ g µν R µν (Γ) + L(φ, g µν ) R µν (Γ) 1 2 g µνr(γ) = T µν Γ µν = Levi Civita if matter isminimally coupled in L(φ, g µν ) Philosophical: Levi-Civita is not just a convenient choice, it is a minimum ofthe action, asolutionof theequations ofmotion Any other connection would not(necessarily) have this property. Question: how much remains valid in presence of curvature corrections? B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 10/29
34 3 Higher-curvature corrections& Lovelock gravity In D = 4, Einstein-Hilbert is most natural choice, but there is no reason to exclude higher-order corrections for D > 4. B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 11/29
35 3 Higher-curvature corrections& Lovelock gravity In D = 4, Einstein-Hilbert is most natural choice, but there is no reason to exclude higher-order corrections for D > 4. Lanczos (1938): S = d 5 x g [R 2 4R µν R µν + R µνρ R µνρ H µν = 2RR µν + 4R ρµν R ρ + 2R µρσ R ρσ ρ ν 4R µρ R ν 1g 2 µν [R 2 4R ρ R ρ + R ρστ R ρστ second order, divergence-free modified Einstein eqns B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 11/29
36 3 Higher-curvature corrections& Lovelock gravity In D = 4, Einstein-Hilbert is most natural choice, but there is no reason to exclude higher-order corrections for D > 4. Lanczos (1938): S = d 5 x g [R 2 4R µν R µν + R µνρ R µνρ H µν = 2RR µν + 4R ρµν R ρ + 2R µρσ R ρσ ρ ν 4R µρ R ν 1g 2 µν [R 2 4R ρ R ρ + R ρστ R ρστ second order, divergence-free modified Einstein eqns NB: R 2 4R µν R µν + R µνρ R µνρ topological invariant in D = 4 identically zeroin D < 4 dynamical in D > 4 isgauss-bonnet term B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 11/29
37 Lovelock (1970): generalise Lanczos to arbitrary D second order, divergence-free Einstein eqns for S = d D x g [Λ + R + L GB L [D/2 where L n = det g µ 1ν 1... g µ 2nν g µ 1ν 2n... g µ 2nν 2n R µ1 µ 2 ν 1 ν 2...R µ2n 1 µ 2n ν 2n 1 ν 2n B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 12/29
38 Lovelock (1970): generalise Lanczos to arbitrary D second order, divergence-free Einstein eqns for S = d D x g [Λ + R + L GB L [D/2 where L n = det g µ 1ν 1... g µ 2nν g µ 1ν 2n... g µ 2nν 2n R µ1 µ 2 ν 1 ν 2...R µ2n 1 µ 2n ν 2n 1 ν 2n String theory: higher-curvature terms appear as correction to sugra [Candelas, Horowitz, Strominger, Witten B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 12/29
39 Lovelock (1970): generalise Lanczos to arbitrary D second order, divergence-free Einstein eqns for S = d D x g [Λ + R + L GB L [D/2 where L n = det g µ 1ν 1... g µ 2nν g µ 1ν 2n... g µ 2nν 2n R µ1 µ 2 ν 1 ν 2...R µ2n 1 µ 2n ν 2n 1 ν 2n String theory: higher-curvature terms appear as correction to sugra [Candelas, Horowitz, Strominger, Witten In general ghosts appear due to higher-derivative terms, unless corrections areof the Lovelock type L n [Zwiebach[Zumino B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 12/29
40 Lovelock (1970): generalise Lanczos to arbitrary D second order, divergence-free Einstein eqns for S = d D x g [Λ + R + L GB L [D/2 where L n = det g µ 1ν 1... g µ 2nν g µ 1ν 2n... g µ 2nν 2n R µ1 µ 2 ν 1 ν 2...R µ2n 1 µ 2n ν 2n 1 ν 2n String theory: higher-curvature terms appear as correction to sugra [Candelas, Horowitz, Strominger, Witten In general ghosts appear due to higher-derivative terms, unless corrections areof the Lovelock type L n [Zwiebach[Zumino L n iseuler character in D = 2n identically zeroin D < 2n topological invariant in D = 2n dynamical in D > 2n B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 12/29
41 absence of ghosts make Lovelock gravities natural extensions to Einstein-Hilbert for D > 4 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 13/29
42 absence of ghosts make Lovelock gravities natural extensions to Einstein-Hilbert for D > 4 Not all stringy corrections are Lovelock, but dilaton might cure the problem. B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 13/29
43 absence of ghosts make Lovelock gravities natural extensions to Einstein-Hilbert for D > 4 Not all stringy corrections are Lovelock, but dilaton might cure the problem. Higher-curvature terms are popular in cosmology modify FRW dynamics that mimic dark matter/energy or cause late time acceleration? increasingly more difficult to derive eqns of motion Palatini come to help? B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 13/29
44 absence of ghosts make Lovelock gravities natural extensions to Einstein-Hilbert for D > 4 Not all stringy corrections are Lovelock, but dilaton might cure the problem. Higher-curvature terms are popular in cosmology modify FRW dynamics that mimic dark matter/energy or cause late time acceleration? increasingly more difficult to derive eqns of motion Palatini come to help? Palatini formalism has been studied in various contexts, though not systematically [Exirifard, Sheikh-Jabbari state that Palatini and metric are equivalent for Lovelock gravity B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 13/29
45 absence of ghosts make Lovelock gravities natural extensions to Einstein-Hilbert for D > 4 Not all stringy corrections are Lovelock, but dilaton might cure the problem. Higher-curvature terms are popular in cosmology modify FRW dynamics that mimic dark matter/energy or cause late time acceleration? increasingly more difficult to derive eqns of motion Palatini come to help? Palatini formalism has been studied in various contexts, though not systematically [Exirifard, Sheikh-Jabbari state that Palatini and metric are equivalent for Lovelock gravity We will demonstrate that metric includes Palatini, but not vice versa B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 13/29
46 4 Palatinivsmetricformalism for(riem) 2 gravity 4.1 Metric formalism S = d D x g [αr 2 + 4βR µν R µν + γr µνρ R µνρ = d D x g [α g µρ g σα δ νδ τβ + 4β g µσ g ρα δ νδ τβ + γ g µσ g ντ g ρα g β R µνρ R στα β Gravitational tensor via chain rule H µν 1 g δs(g) δg µν = 1 g δs(g) δg µν expl + 1 δs(g) g δr δ αβγ δr αβγ δ δγ σ ρ δγ σ ρ δg µν where δr µνρ = µ (δγ νρ) ν (δγ µρ), δγ ρ µν = [ 1 2 gρ µ (δg ν ) + ν (δg µ ) (δg µν ) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 14/29
47 H µν = α [2 µ ν R 2g µν 2 R + 2R µν R 12 g µνr 2 + 4β [ µ ν R 2 R µν 12 g µν 2 R 2R µνρ R ρ 12 g µνr ρ R ρ [ + γ 2 µ ν R 4 2 R µν 4R µρ R ρ ν 4R ρµν R ρ +2R µρσ R ν ρσ 1 2 g µνr ρστ R ρστ. B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 15/29
48 H µν = α [2 µ ν R 2g µν 2 R + 2R µν R 12 g µνr 2 + 4β [ µ ν R 2 R µν 12 g µν 2 R 2R µνρ R ρ 12 g µνr ρ R ρ [ + γ 2 µ ν R 4 2 R µν 4R µρ R ρ ν 4R ρµν R ρ +2R µρσ R ν ρσ 1 2 g µνr ρστ R ρστ. 2 (Riem) terms cancel for (α, β, γ) = (1, 1, 1) Gauss-Bonnet is2ndlovelock term L 2 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 15/29
49 H µν = α [2 µ ν R 2g µν 2 R + 2R µν R 12 g µνr 2 + 4β [ µ ν R 2 R µν 12 g µν 2 R 2R µνρ R ρ 12 g µνr ρ R ρ [ + γ 2 µ ν R 4 2 R µν 4R µρ R ρ ν 4R ρµν R ρ +2R µρσ R ν ρσ 1 2 g µνr ρστ R ρστ. 2 (Riem) terms cancel for (α, β, γ) = (1, 1, 1) Gauss-Bonnet is2ndlovelock term L 2 µ H µν = 0 for arbitrary (α, β, γ) energy conservation generally valid B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 15/29
50 H µν = α [2 µ ν R 2g µν 2 R + 2R µν R 12 g µνr 2 + 4β [ µ ν R 2 R µν 12 g µν 2 R 2R µνρ R ρ 12 g µνr ρ R ρ [ + γ 2 µ ν R 4 2 R µν 4R µρ R ρ ν 4R ρµν R ρ +2R µρσ R ν ρσ 1 2 g µνr ρστ R ρστ. 2 (Riem) terms cancel for (α, β, γ) = (1, 1, 1) Gauss-Bonnet is2ndlovelock term L 2 µ H µν = 0 for arbitrary (α, β, γ) energy conservation generally valid Compare H µν = T µν tocorresponding eqns ofpalatini formalism B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 15/29
51 4.2 Palatini formalism S(g, Γ) = d D x g [ αr 2 + β 1 R µν R νµ + 2β 2 R µν Rνµ + β 3 Rµν Rνµ + β 4 R µν R µν + 2β 5 R µν Rµν + β 6 Rµν Rµν + γ 1 R µνρ R ρµν + γ 2 R µνρ R µνρ + γ 3 R µνρ R µνρ with R µρ R µρ Rµ g νρ R µνρ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 16/29
52 4.2 Palatini formalism S(g, Γ) = d D x g [ αr 2 + β 1 R µν R νµ + 2β 2 R µν Rνµ + β 3 Rµν Rνµ + β 4 R µν R µν + 2β 5 R µν Rµν + β 6 Rµν Rµν + γ 1 R µνρ R ρµν + γ 2 R µνρ R µνρ + γ 3 R µνρ R µνρ with R µρ R µρ Rµ g νρ R µνρ Gravitational tensor: Hµν 1 δs(g,γ) g δg µν H µν = 2αR µν R + β 1 (R µ R ν + R µ R ν ) β 2 (2R (ν σ µ) R σ + R µr ν + R νr µ ) + 2β 3 R (ν ρ µ) R ρ + β 4 (R µ R ν + R µ R ν) β 5 (2R (ν σ µ) R σ + R µr ν + R νr µ ) + 2β 6 R (ν ρ µ) R ρ + γ 1 (R ρσµ R ν σρ + R ρµσ R σρ ν) + 2γ 3 R µρσ R ν ρσ + γ 2 (R ρσµ R ρσ ν + 2R µσρ R ν σρ R ρµσ R ρ ν σ ) 1 2 g µνl B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 16/29
53 Connection tensor: K µν 1 g δs(g, Γ) δγ µν [ = 2α g µν R δν µ R 2β 2 [ R νµ + δ ν σ R σµ + g µν σ R σ µ R ν +2β 4 [ R µν δ ν σ R µσ +2β 6 [ g µν σ Rσ + µ R ν + 4γ 3 σ R νσµ + O( g µν ) + O(T µν) + 2β 1 [ R νµ δ ν σ R σµ + 2β 3 [ µ R ν g µν σ R σ 2β 5 [ Rµν + δ ν σ Rµσ + g µν σ R σ µ R ν + 2γ 1 σ [R µνσ R µσν + 4γ 2 σ R νσ µ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 17/29
54 Connection tensor: K µν 1 g δs(g, Γ) δγ µν [ = 2α g µν R δν µ R 2β 2 [ R νµ + δ ν σ R σµ + g µν σ R σ µ R ν +2β 4 [ R µν δ ν σ R µσ +2β 6 [ g µν σ Rσ + µ R ν + 4γ 3 σ R νσµ + O( g µν ) + O(T µν) + 2β 1 [ R νµ δ ν σ R σµ + 2β 3 [ µ R ν g µν σ R σ 2β 5 [ Rµν + δ ν σ Rµσ + g µν σ R σ µ R ν + 2γ 1 σ [R µνσ R µσν + 4γ 2 σ R νσ µ H µν hasno 2 (Riem) terms, even forarbitrary (α, β, γ) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 17/29
55 Connection tensor: K µν 1 g δs(g, Γ) δγ µν [ = 2α g µν R δν µ R 2β 2 [ R νµ + δ ν σ R σµ + g µν σ R σ µ R ν +2β 4 [ R µν δ ν σ R µσ +2β 6 [ g µν σ Rσ + µ R ν + 4γ 3 σ R νσµ + O( g µν ) + O(T µν) + 2β 1 [ R νµ δ ν σ R σµ + 2β 3 [ µ R ν g µν σ R σ 2β 5 [ Rµν + δ ν σ Rµσ + g µν σ R σ µ R ν + 2γ 1 σ [R µνσ R µσν + 4γ 2 σ R νσ µ H µν hasno 2 (Riem) terms, even forarbitrary (α, β, γ) µ Hµν 0 inconsistency with H µν = T µν? B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 17/29
56 Comparison: takelevi-civita asansatz andsubstitute in H µν and K µν = R µν R µν R νµ R µν = H µν = H Levi Civita µν Kµν Kµν Levi Civita B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 18/29
57 Comparison: takelevi-civita asansatz andsubstitute in H µν and K µν = R µν R µν R νµ R µν = H µν = H Levi Civita µν Kµν Kµν Levi Civita H µν ) = 2αR µν R + 4β (R µρ R ρ ν + R µρν R ρ 1 2 g µν + 2γR µρσ R ν ρσ [αr 2 βr ρ R ρ + γr ρστ R ρστ K µν = 2(α + β)g µν R + 4(β + γ) R µν 2(α + β)δ ν µ R 4(β + γ) µ R ν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 18/29
58 Compare: H µν = α [2 µ ν R 2g µν 2 R + 2R µν R 12 g µνr 2 +β [ µ ν R 2 R µν 12 g µν 2 R 2R µνρ R ρ 12 g µνr ρ R ρ +γ [2 µ ν R 4 2 R µν 4R µρ R ρ ν 4R ρµν R ρ + 2R µρσ R ρσ ν 12 g µνr ρστ R ρστ ) H µν = 2αR µν R + 4β (R µρ R ρ ν + R µρν R ρ ρσ + 2γR µρσ R ν 1g 2 µν [αr 2 βr ρ R ρ + γr ρστ R ρστ K µν = 2(α + β)g µν R + 4(β + γ) R µν 2(α + β)δ ν µ R 4(β + γ) µ R ν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 19/29
59 Compare: H µν = α [2 µ ν R 2g µν 2 R + 2R µν R 12 g µνr 2 +β [ µ ν R 2 R µν 12 g µν 2 R 2R µνρ R ρ 12 g µνr ρ R ρ +γ [2 µ ν R 4 2 R µν 4R µρ R ρ ν 4R ρµν R ρ + 2R µρσ R ρσ ν 12 g µνr ρστ R ρστ ) H µν = 2αR µν R + 4β (R µρ R ρ ν + R µρν R ρ ρσ + 2γR µρσ R ν 1g 2 µν [αr 2 βr ρ R ρ + γr ρστ R ρστ K µν = 2(α + β)g µν R + 4(β + γ) R µν 2(α + β)δ ν µ R 4(β + γ) µ R ν H µν = H µν 1 2 K (µν) g µ ρ K ρν g ν ρ K ρν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 19/29
60 K µν = 2(α + β)g µν R + 4(β + γ) R µν 2(α +β)δ ν µ R 4(β +γ) µ R ν 0 for (α, β, γ) = (1, 1, 1) equivalence for Gauss-Bonnet (= Lovelock) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 20/29
61 K µν = 2(α + β)g µν R + 4(β + γ) R µν 2(α +β)δ ν µ R 4(β +γ) µ R ν 0 for (α, β, γ) = (1, 1, 1) equivalence for Gauss-Bonnet (= Lovelock) All Lagrangianswith (α, β, γ) = (1, 1, 1)reduce togauss-bonnet, but notallhave K µν 0: K µν = 2(α β 2 + β 3 β 5 + β 6 )g µν R + 4(β 1 β 2 + β 4 β 5 + γ) R µν 2(α + β 1 β 2 + β 4 β 5 )δ ν µ R 4( β 2 + β 3 β 5 + β 6 + γ) µ R ν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 20/29
62 K µν = 2(α + β)g µν R + 4(β + γ) R µν 2(α +β)δ ν µ R 4(β +γ) µ R ν 0 for (α, β, γ) = (1, 1, 1) equivalence for Gauss-Bonnet (= Lovelock) All Lagrangianswith (α, β, γ) = (1, 1, 1)reduce togauss-bonnet, but notallhave K µν 0: K µν = 2(α β 2 + β 3 β 5 + β 6 )g µν R + 4(β 1 β 2 + β 4 β 5 + γ) R µν 2(α + β 1 β 2 + β 4 β 5 )δ ν µ R 4( β 2 + β 3 β 5 + β 6 + γ) µ R ν Conditions for K µν 0: α = γ 0, β 1 + β 4 = β 3 + β 6, α + β 1 + β 4 β 2 β 5 = 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 20/29
63 Lagrangiangswith K 0are S 1 (g, Γ) = d D x g [R 2 R µν R νµ 2R Rνµ µν R Rνµ µν + R µνρ R ρµν S 2 (g, Γ) = d D x g [R 2 R µν R νµ R µν R νµ + R µνρ R ρµν S 3 (g, Γ) = d D x g [R 2 2R Rνµ µν + R µνρ R ρµν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 21/29
64 Lagrangiangswith K 0are S 1 (g, Γ) = d D x g [R 2 R µν R νµ 2R Rνµ µν R Rνµ µν + R µνρ R ρµν S 2 (g, Γ) = d D x g [R 2 R µν R νµ R µν R νµ + R µνρ R ρµν S 3 (g, Γ) = d D x g [R 2 2R Rνµ µν + R µνρ R ρµν Lagrangians mustmimicthe symmetries of R µνρ inmetric formalism Determinant definition of Lovelock is too restrictive More (Palatini) inequivalent Lagrangians give Lovelock conditions H µν = H µν (g, g, g ) µ H µν = 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 21/29
65 5 More general Lagrangians S = d D x g L(g µν, R µνρ ) with L functional of (Riem), but not of (Riem) H µν = δl ( δl ) ( δg 1g µν 2 µνl + 1[ 2 α, β g δr ρ(µ δν) 1 2 ρ α αβρ ( δl ) ( ρ β g δr α(µ δν) α αβρ δl δr αβρ ) g β(µ g ν)ρ 1 2 β ( δl δr αβρ δl δr αβρ ) g β(µ δ ν) ) g α(µ g ν)ρ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 22/29
66 5 More general Lagrangians S = d D x g L(g µν, R µνρ ) with L functional of (Riem), but not of (Riem) H µν = δl ( δl ) ( δg 1g µν 2 µνl + 1[ 2 α, β g δr ρ(µ δν) 1 2 ρ α αβρ ( δl ) ( ρ β g δr α(µ δν) α αβρ δl δr αβρ ) g β(µ g ν)ρ 1 2 β ( δl δr αβρ δl δr αβρ ) g β(µ δ ν) ) g α(µ g ν)ρ H µν = δl δg µν K µρ = ν [( 1 2 g µνl δl δr ρνµ ) ( δl δr νρµ ) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 22/29
67 5 More general Lagrangians S = d D x g L(g µν, R µνρ ) with L functional of (Riem), but not of (Riem) H µν = δl ( δl ) ( δg 1g µν 2 µνl + 1[ 2 α, β g δr ρ(µ δν) 1 2 ρ α αβρ ( δl ) ( ρ β g δr α(µ δν) α αβρ δl δr αβρ ) g β(µ g ν)ρ 1 2 β ( δl δr αβρ δl δr αβρ ) g β(µ δ ν) ) g α(µ g ν)ρ H µν = δl δg µν K µρ = ν [( 1 2 g µνl δl δr ρνµ ) ( δl δr νρµ ) H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 22/29
68 Palatini formalism: H µν = T µν, K µν = 0 Metric formalism: H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) = T µν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 23/29
69 Palatini formalism: H µν = T µν, K µν = 0 Metric formalism: H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) = T µν µ H µν = 0 if Γ ρ µν is on-shell B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 23/29
70 Palatini formalism: H µν = T µν, K µν = 0 Metric formalism: H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) = T µν µ H µν = 0 if Γ ρ µν is on-shell Any solution ofpalatini issolutionof metric In general solutions of metric will not be solutions of Palatini Palatini is non-trivial subset of metric B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 23/29
71 Palatini formalism: H µν = T µν, K µν = 0 Metric formalism: H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) = T µν µ H µν = 0 if Γ ρ µν is on-shell Any solution ofpalatini issolutionof metric In general solutions of metric will not be solutions of Palatini Palatini is non-trivial subset of metric Solutions of metric are solution of Palatini K µρ ν B νµρ = 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 23/29
72 Palatini formalism: H µν = T µν, K µν = 0 Metric formalism: H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) = T µν µ H µν = 0 if Γ ρ µν is on-shell Any solution ofpalatini issolutionof metric In general solutions of metric will not be solutions of Palatini Palatini is non-trivial subset of metric Solutions of metric are solution of Palatini K µρ ν B νµρ = 0 B νµρ isconserved current B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 23/29
73 Palatini formalism: H µν = T µν, K µν = 0 Metric formalism: H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) = T µν µ H µν = 0 if Γ ρ µν is on-shell Any solution ofpalatini issolutionof metric In general solutions of metric will not be solutions of Palatini Palatini is non-trivial subset of metric Solutions of metric are solution of Palatini K µρ ν B νµρ = 0 B νµρ isconserved current g µν satisfies certain symmetry requirements B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 23/29
74 Einstein-Hilbert, L = R: K µρ = ν [ g µν δ ρ gρµ δ ν 0 no extra conditions! B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 24/29
75 Einstein-Hilbert, L = R: K µρ = ν [ g µν δ ρ gρµ δ ν 0 no extra conditions! L = R 2 : K µρ = 2 ν [ δ ρ gµν R δg ν µρ R = 0 E.g.: R = Λ (AdS m S n,...) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 24/29
76 Einstein-Hilbert, L = R: K µρ = ν [ g µν δ ρ gρµ δ ν 0 no extra conditions! L = R 2 : K µρ = 2 ν [ δ ρ gµν R δg ν µρ R = 0 E.g.: R = Λ (AdS m S n,...) L = R µν R µν + 2R Rµν µν + R Rµν µν : [ = 4 ν δr ν ρµ + g ρµ R ν δ ρ Rµν g µν R ρ K µρ = 0 E.g.: R µν = Λg µν (Einstein spaces) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 24/29
77 L = R µνρ R µνρ : K µρ = 4 νr µνρ = 0 E.g.: R µνρ = Λ(g µρ g ν g νρ g µ ) (max. symmetric spaces) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 25/29
78 L = R µνρ R µνρ : K µρ = 4 νr µνρ = 0 E.g.: R µνρ = Λ(g µρ g ν g νρ g µ ) (max. symmetric spaces) L = L(g µν, R µνρ l ): K µρ = ν [( δl ) δr ρνµ ( δl δr νρµ ) E.g.: δl δr µνρ = Λ (g µρ δ ν gνρ δ µ ) (diff eqn for g µν for given L) B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 25/29
79 6 Explicit example: FRW in Gauss-Bonnet gravity S = d D x g [R 2 4R µν R µν + R µνρ R µνρ NB: β 4 = 1, β 1 = β 3 = β 6 = 0 = NotPalatini-Lovelock! K µρ = 4 G µρ + 2 µ G ρ + 2 ρ G µ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 26/29
80 6 Explicit example: FRW in Gauss-Bonnet gravity S = d D x g [R 2 4R µν R µν + R µνρ R µνρ NB: β 4 = 1, β 1 = β 3 = β 6 = 0 = NotPalatini-Lovelock! K µρ = 4 G µρ + 2 µ G ρ + 2 ρ G µ Ansatz: ds 2 = dt 2 e 2A(t) ḡ mn dx m dx n with R mnp q = 0 T µν : perfect fluidwith P = ωρ B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 26/29
81 6 Explicit example: FRW in Gauss-Bonnet gravity S = d D x g [R 2 4R µν R µν + R µνρ R µνρ NB: β 4 = 1, β 1 = β 3 = β 6 = 0 = NotPalatini-Lovelock! K µρ = 4 G µρ + 2 µ G ρ + 2 ρ G µ Ansatz: ds 2 = dt 2 e 2A(t) ḡ mn dx m dx n with R mnp q = 0 T µν : perfect fluidwith P = ωρ Metric eqns: H tt = (D 1)! 2(D 5)! (A ) 4, H mn = (D 2)! (D 5)! e2a ḡ mn [ 2A (A ) (D 1)(A ) 4 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 26/29
82 6 Explicit example: FRW in Gauss-Bonnet gravity S = d D x g [R 2 4R µν R µν + R µνρ R µνρ NB: β 4 = 1, β 1 = β 3 = β 6 = 0 = NotPalatini-Lovelock! K µρ = 4 G µρ + 2 µ G ρ + 2 ρ G µ Ansatz: ds 2 = dt 2 e 2A(t) ḡ mn dx m dx n with R mnp q = 0 T µν : perfect fluidwith P = ωρ Metric eqns: H tt = (D 1)! 2(D 5)! (A ) 4, H mn = (D 2)! (D 5)! e2a ḡ mn [ 2A (A ) (D 1)(A ) 4 Solutions: ω = 1: desitter space with A(t) = 1 t ω 1: powerlaw solution with e A(t) = 2 ( (D 1)(1+ω) ) 4 t t 0 B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 26/29
83 Palatini eqns: H tt = (D 1)(D 2) [4(A ) 2 + 4A (A ) (D 3)(D 4)(A ) 4 [ H mn = e 2A ḡ mn 4(D 2)(A ) 2 + 2(D 2) 2 (D 5)A (A ) 2 + (D 1)! 2(D 5)! (A ) 4 Ktt t = 8(D 1)(D 2)A A K p tm = 2(D 2) δm [A p + (D 4)A A [ Kmn t = 4(D 2) e 2A ḡ mn A + (D 2)A A B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 27/29
84 Palatini eqns: H tt = (D 1)(D 2) [4(A ) 2 + 4A (A ) (D 3)(D 4)(A ) 4 [ H mn = e 2A ḡ mn 4(D 2)(A ) 2 + 2(D 2) 2 (D 5)A (A ) 2 + (D 1)! 2(D 5)! (A ) 4 Ktt t = 8(D 1)(D 2)A A K p tm = 2(D 2) δm [A p + (D 4)A A [ Kmn t = 4(D 2) e 2A ḡ mn A + (D 2)A A Solutions: K ρ µν = 0 forde Sitter solution H µν = T µν = H µν = T µν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 27/29
85 Palatini eqns: H tt = (D 1)(D 2) [4(A ) 2 + 4A (A ) (D 3)(D 4)(A ) 4 [ H mn = e 2A ḡ mn 4(D 2)(A ) 2 + 2(D 2) 2 (D 5)A (A ) 2 + (D 1)! 2(D 5)! (A ) 4 Ktt t = 8(D 1)(D 2)A A K p tm = 2(D 2) δm [A p + (D 4)A A [ Kmn t = 4(D 2) e 2A ḡ mn A + (D 2)A A Solutions: K ρ µν = 0 forde Sitter solution H µν = T µν = H µν = T µν K ρ µν 0 forpower law solution no solution for Palatini eqns! B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 27/29
86 Palatini eqns: H tt = (D 1)(D 2) [4(A ) 2 + 4A (A ) (D 3)(D 4)(A ) 4 [ H mn = e 2A ḡ mn 4(D 2)(A ) 2 + 2(D 2) 2 (D 5)A (A ) 2 + (D 1)! 2(D 5)! (A ) 4 Ktt t = 8(D 1)(D 2)A A K p tm = 2(D 2) δm [A p + (D 4)A A [ Kmn t = 4(D 2) e 2A ḡ mn A + (D 2)A A Solutions: K ρ µν = 0 forde Sitter solution H µν = T µν = H µν = T µν K ρ µν 0 forpower law solution no solution for Palatini eqns! Only desitter satisfies G µν = Λg µν Only ω = 1hasnecessary symmetries B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 27/29
87 7 Conclusions For an action S = d D x [ g L(g µν, R µνρ ) + L(g µν, φ) Metric formalism: H µν = T µν where H µν 1 g δs(g) δg µν B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 28/29
88 7 Conclusions For an action S = d D x [ g L(g µν, R µνρ ) + L(g µν, φ) Metric formalism: Palatini formalism: H µν = T µν where H µν 1 g δs(g) δg µν H µν = T µν, K µν = 0 where H µν 1 δs(g, Γ), K µν g δg µν 1 δs(g, Γ) g δγ µν Levi Civita B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 28/29
89 H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) Equivalence between both formalisms for K ρ µν 0 Lovelock in metric AND sufficient symmetries in Palatini Solutions of Palatini eqns are solutions of metric eqns Solutions of metric eqns are solutions of Palatini if required symmetry Palatini is non-trivial subset of metric B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 29/29
90 H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) Equivalence between both formalisms for K ρ µν 0 Lovelock in metric AND sufficient symmetries in Palatini Solutions of Palatini eqns are solutions of metric eqns Solutions of metric eqns are solutions of Palatini if required symmetry Palatini is non-trivial subset of metric Practical advantage? Still possible to calculate H µν frompalatini eqns butit isnotworthit... B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 29/29
91 H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) Equivalence between both formalisms for K ρ µν 0 Lovelock in metric AND sufficient symmetries in Palatini Solutions of Palatini eqns are solutions of metric eqns Solutions of metric eqns are solutions of Palatini if required symmetry Palatini is non-trivial subset of metric Practical advantage? Still possible to calculate H µν frompalatini eqns butit isnotworthit... Philosophical advantage? Levi-Civita is minimum of the action for Palatini solutions for other solutions, just convenient choice? B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 29/29
92 H µν = H µν 1 2 ρk ρ (µν) g µ ρ K (νρ) g ν ρ K (µρ) Equivalence between both formalisms for K ρ µν 0 Lovelock in metric AND sufficient symmetries in Palatini Solutions of Palatini eqns are solutions of metric eqns Solutions of metric eqns are solutions of Palatini if required symmetry Palatini is non-trivial subset of metric Practical advantage? Still possible to calculate H µν frompalatini eqns butit isnotworthit... Philosophical advantage? Levi-Civita is minimum of the action for Palatini solutions for other solutions, just convenient choice? Advantages are obvious when equivalence: Lovelock is very special! B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 29/29
93 Recall: H µν = 1 g δs(g) δg µν expl + 1 g δs(g) δr αβγ δ δr αβγ δ δγ σ ρ δγ σ ρ δg µν H µν 1 g δs(g, Γ) δg µν, K µν 1 g δs(g, Γ) δγ µν H µν = H µν 1 2 ρk ρ (µν) + 1g 2 µ ρ K(νρ) + 1g 2 ν ρ K(µρ) the chain rule reflects structure of Lovelock gravities no (Riem) terms K µν 0 equivalence between metric & Palatini B. Janssen (UGR) Palatini vs metric formalism in higher-curvature gravity 30/29
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