? Can we formulate General Relativity in a Lagrangian picture?

Transcription

1 6. General Relativity Astro Particle Physics description of the very early universe: curved space-time in the context of particle physics we need the particle physics description formulated in Hamiltonian mechanics Quantum Mechanics or formulated in Lagrangian mechanics using an action principle Quantum Mechanics through the Pathintegral formulation Special Relativity is the local symmetry group Lagrangian mechanics as the unifying framework? Can we formulate General Relativity in a Lagrangian picture?? what is the dynamic degree of freedom? the metric? what are the consequences? Thomas Gajdosik Concepts of Modern Cosmology

2 6. General Relativity Lagrangian formulation of gravity we can derive Einsteins equations also from an action: the starting point is the Einstein-Hilbert action S = d 4 x ( g L m R ) 16πG R = R(g µν ) is the Ricci scalar, G is Newtons gravitational constant g = det(g µν ) is the determinant of the metric why this g? using the differential calculus the volume element should be written as a 4-form d 4 x = dx 0 dx 1 dx 2 dx 3 = 1 4! ǫ µνρσdx µ dx ν dx ρ dx σ that transforms under coordinate transformations x x with xµ x α = Λ µ α d 4 x = 1 4! ǫ µνρσλ µ α Λν β Λρ γ Λσ δ dx α dx β dx γ dx δ = det[λ]d 4 x the metric transforms under these coordinate transformations x x g αβ = Λµ αλ ν β g µν g = det[g αβ ] = det[λµ αλ ν β g µν] = det[λ] 2 g since det[η µν ] = det[diag(1, 1, 1, 1)] = 1 g = det[g µν ] < 0 ( gd 4 x) is invariant: d 4 x g = det[λ] 1 d 4 x gdet[λ] 2 = d 4 x g Thomas Gajdosik Concepts of Modern Cosmology

3 6. General Relativity Lagrangian formulation of gravity we can derive Einsteins equations also from an action: the variation of the action gives the Euler-Lagrange equations varying the Einstein-Hilbert action we get δs = d 4 x ( δ( gl m ) δ( g) gαβ R αβ 16πG g (δgαβ )R αβ 16πG g gαβ (δr αβ ) 16πG the first term is the variation of the matter Lagrangian the second term can be calculated from the identity Tr[lnM] = ln(det[m]) Tr[M 1 δm] = det[m] 1 δdet[m] setting M = g µν we have M 1 = g µν and det[m] = det[g µν ] = det[(g µν ) 1 ] = 1/det[g µν ] = 1/g so Tr[M 1 δm] = g µν δg µν = gδ 1 g = gδg g 2 = δg g δ g = δg 2 g = 1 2 g ( g)g µνδg µν = 1 2 ggµν δg µν the third term has already the wanted differential δg µν the fourth term gives a total divergence (see next slide) it does not contribute to the equations of motion Thomas Gajdosik Concepts of Modern Cosmology )

4 6. General Relativity Lagrangian formulation of gravity we can derive Einsteins equations also from an action: the variation of the Ricci tensor is the contracted variation of the Riemann tensor: δr αβ = δr λ αλβ = δ(δκ λ Rλ ακβ ) = δκ λ δ[ κγ λ αβ +Γλ κργ ρ αβ (κ β)] the trick in the calculation is to realize, that δγ is a difference of two connections it is a tensor and we can calculate the covariant derivative: κ δγ λ αβ = κδγ λ αβ +Γλ κνδγ ν αβ Γµ καδγ λ µβ Γµ κβ δγλ αµ the antisymmetric part in (κ β) gives κ δγ λ αβ βδγ λ ακ = κ δγ λ αβ +Γλ κνδγ ν αβ Γµ καδγ λ µβ Γµ κβ δγλ αµ = δr λ ακβ so the term g αβ (δr αβ ) can be written as β δγ λ ακ Γ λ βν δγν ακ +Γ µ βα δγλ µκ +Γ µ βκ δγλ αµ = κ δγ λ αβ +Γλ κµδγ µ αβ +δγλ κµγ µ αβ (κ β) = δ[ κ Γ λ αβ +Γλ κµγ µ αβ ] (κ β) g αβ δr αβ = g αβ ( λ δγ λ αβ βδγ λ αλ ) = κ g κλ g αβ δγ λ αβ α δγ λ αλ = α [g αβ g µν δγ β µν δγ λ αλ ] = α V α and the integral over it gives only the boundary terms d 4 x g α [g αβ g µν δγ β µν δγ λ αλ] = [ ] g αβ g µν δγ β µν δγ λ αλ Ω 0 Ω Thomas Gajdosik Concepts of Modern Cosmology

5 6. General Relativity Lagrangian formulation of gravity we can derive Einsteins equations also from an action: varying the matter Lagrangian with respect to δg µν δs m ( glm ) ( glm ) g = δg µν g µν ρ =: 2 T µν g µν,ρ gives the definition of the symmetric Hilbert (stress energy) tensor! the canonical stress energy tensor ( using Φ,µ := µ Φ ) T µ ν := L m Φ,µ Φ,ν L m δ µ ν is not necessarily symmetric (when both indices are up or down) the Belinfante-Rosenfeld (stress-energy) tensor T µν B = Tµ λ gλν + λ (S µνλ +S νµλ S λνµ ) adds a divergence of the spin part S µνλ to make it symmetric it is equivalent to the Hilbert tensor Thomas Gajdosik Concepts of Modern Cosmology

6 6. General Relativity Lagrangian formulation of gravity we can derive Einsteins equations also from an action: putting the parts of the variation with respect to δg µν together δs g δg µν = 0 = 2 T µν ( 1 R 2 ggµν ) 16πG g R µν 16πG ( g = Rµν 1 2 g ) µνr 8πGT µν 16πG we get Einsteins equations we get also the field equations in the curved space time they are the normal Euler-Lagrange equations δs δφ = 0 = g L ( ) m g Φ L m ρ = ( Lm g Φ,ρ Φ L m ρ Φ,ρ the last equality only holds, if L m Φ,ρ = V ρ is a vector ) Thomas Gajdosik Concepts of Modern Cosmology

7 6. General Relativity examples of matter Lagrangians complex scalar Lagrangian: using only first derivatives of the complex scalar field φ the flat space Lagrangian L φ = ( µ φ )( µ φ) m 2 φ φ φ V(φ φ) can be written with covariant derivatives as L φ = g µν ( µ φ )( ν φ) m 2 φ φ φ V(φ φ) the canonical stress energy tensor T µ ν = L φ φ,ν + L φ φ,µ φ φ,ν L m δ µ ν,µ = g λµ ( λ φ )φ,ν +g λµ ( λ φ)φ,ν δ µ νg λκ ( λ φ )( κ φ) +[m 2 φ φ φ+v(φ φ)]δ µ ν is already symmetric (when both indices are up or down) and the same as the Hilbert tensor (since µ φ = µ φ = φ,µ ) Thomas Gajdosik Concepts of Modern Cosmology

8 6. General Relativity examples of matter Lagrangians Maxwell Lagrangian: using the vector potential A µ and its fieldstrength F µν = µ A ν ν A µ the flat space Lagrangian is L A = 1 4 Fµν F µν = 1 4 F2 the vector potential can be written as a one-form A = A µ dx µ the fieldstrength as a two-form F = 1 2 F µνdx µ dx ν = da there is no metric dependence in F µν the Lagrangian on a curved manifold is just L A = 1 4 gαµ g βν F αβ F µν = 1 2 Aβ,α (A β,α A α,β ) the canonical stress energy tensor T µ ν = L A A ρ,µ A ρ,ν L A δ µ ν = [ 1 2 gβρ g αµ (A β,α A α,β ) 1 2 Aβ,α (δ ρ β δµ α δ ρ αδ µ β )]A ρ,ν δµ νf 2 = F µρ A ρ,ν δµ νf 2 is not symmetric (when both indices are up or down) Thomas Gajdosik Concepts of Modern Cosmology

9 6. General Relativity examples of matter Lagrangians Maxwell Lagrangian: using the generators of Lorentz transformations acting on four-vectors the spin tensor (J µν ) α β = i(gαµ δ ν β gαν δ µ β ) S µνλ = i L A (J νλ ) α 2 A α β Aβ = 1,µ 2 g ραf ρµ (g αλ δ ν β gαν δ λ β )Aβ = 1 2 (Fµλ A ν F µν A λ ) = S µλν its derivative λ (S µνλ +S νµλ S λνµ ) = 1 2 λ(f λµ A ν +F µν A λ +F λν A µ +F νµ A λ F µλ A ν F λν A µ ) = λ (F λµ A ν ) = ( λ F λµ )A ν g αβ F αµ A ν,β = g αβ F αµ A ν,β we get the Belinfante-Rosenfeld tensor T µν B = T µ λ gλν + λ (S µνλ +S νµλ S λνµ ) = g αβ F αµ A β,ν gµν F 2 g αβ F αµ A ν,β = g αβ F αµ F βν gµν F 2 the Hilbert tensor T µν = 2 g µν [ 1 4 gαρ g βσ F αβ F ρσ ] g µν L A = 1 2 [δα µg ρ νg βσ +g αρ δ β µg σ ν]f αβ F ρσ g µν L A = 1 2 [F µβf νσ g βσ +F αµ F ρν g αρ ] g µν L A = F µβ F νσ g βσ g µνf 2 gives the same result Thomas Gajdosik Concepts of Modern Cosmology

10 6. General Relativity examples of matter Lagrangians Dirac Lagrangian: using the Dirac spinor ψ and its adjoint ψ = ψ γ 0 the flat space Lagrangian L ψ = ψ(i/ m)ψ can be written with a covariant derivative as L ψ = ψ(ig µν γ µ ν m)ψ the covariant derivative has to use the spin connection! the canonical stress energy tensor T µ ν = L ψ ψ,µ ψ,ν + ψ,ν L φ ψ,µ L ψ δ µ ν = ψiγ µ ψ,ν + ψ,ν 0 δ µ ν ψ(iγ λ λ m)ψ is not symmetric (when both indices are up or down) the Belinfante-Rosenfeld tensor is needed with the spintensor S µνλ = i 2 α is the spinor index of the Dirac spinor ψ (J µν ) α β = i 4 [γµ,γ ν ] α β for the Hilbert tensor T µν = 2 g δ( gl ψ ) δg µν L ψ (J µν ) α β ψβ ψ α,λ generates the Lorentz transformation on the spinors = 2 δl ψ δg µν g µνl ψ we have to include the dependence of the spin connections on the metric Thomas Gajdosik Concepts of Modern Cosmology

11 6. General Relativity paradigm of inflation most models of inflation assume one or more scalar fields taking real scalar fields φ k with the abbreviation φ 2 = k φ2 k S φ = d 4 x gl φ = d 4 x g 1 g µν ( 2 µ φ k )( ν φ k ) V(φ 2 ) we get the field equations δs φ δφ j (y) = 0 = d 4 x g g µν ( µ δ j k δ(x y))( νφ k ) V (φ 2 )2φ j (x)δ(x y) k = d 4 xδ(x y) µ [ gg µν ( ν φ j )]+ gv (φ 2 )2φ j (y) k = g ( µ [g µν ( ν φ j )]+2φ j V ) = g ( ν ( ν φ j )+2φ j V ) and the stress energy tensor T µν = 2 L φ g g µνl µν m = ( µ φ b )( ν φ b ) g µν [ 1 ( ρ φ 2 b )( ρ φ b ) V(φ 2 )] b b T 00 = 1 2 b [ φ 2 b +( φ b ) 2 ]+V(φ 2 ) = H(φ) T 0i = φ b b ( i φ b ) T ii = b [( iφ b ) 2 + a2 φ 2 2 b a2 2 ( φ b ) 2 ] a 2 V(φ 2 ) T jk = b ( jφ b )( k φ b ) for a homogeneous and isotropic field, we can set ( j φ b ) 0 so ( φ b ) 0, too) that gives us ρ = 1 2 b φ 2 b +V and p = 1 2 b φ 2 b V Thomas Gajdosik Concepts of Modern Cosmology

12 6. General Relativity paradigm of inflation using only a single scalar field with a sizeable, but slowly varying potential V(φ 2 ) and φ slowly varying, i.e. φ V we get the conditions like with the cosmological constant: ρ > 0 and p < 1 3 ρ the scalar field does not act like normal matter using the Friedmann equations for the Robertson-Walker metric 1 2 R ii R 00 = ȧ2 +k = 4πG a 2 3 ( φ V) 3 R 00 = ä a = 8πG 3 ( φ 2 V) together with the field equations, remembering ( j φ b ) 0, 0 = g µν µ ( ν φ)+2φv = g µν µ ( ν φ)+g µν Γ ρ µν ( ρφ)+2φv = φ 3ȧ a φ+2φv making the ansatz a = ce H inflt we get Hinfl 2 + k a = 4πG 2 3 (2V + φ 2 ) Hinfl 2 = 4πG 3 (2V 2 φ 2 ) k = 4πGa 2 φ 2 0 only flat or de Sitter... consistent with measurements and H infl 8π 3 V M 2 P for H infl t 60 V 45 t Pt M π 12 M 2 ( GeV) 2 P P Thomas Gajdosik Concepts of Modern Cosmology

13 6. General Relativity paradigm of inflation how does inflation stop? even with the slowly rolling inflaton field, there is a small change the field value approaches its minimum the inflaton field dominates, but there are the other fields, too it can decay into the other fields with the increasing scale factor, the temperature drops the effective potential can decrease to the minimum value assuming for example a form like the SM Higgs potential the value of the inflaton field stays large: φ is heavy but the value of the potential can go to zero: H infl 0 inflation stops including supersymmetry the value of the potential is bounded from below, mostly positive the heavy inflaton decays into the other fields: reheating Thomas Gajdosik Concepts of Modern Cosmology

14 6. General Relativity paradigm of inflation consequences of inflation the universe appears as flat, homogeneous, and isotropic as is seen in the CMB the seeds for structure formation can be understood as quantum fluctuations blown up to cosmic scales the primordial particle spectrum is thermal from the decay of the inflaton problems with inflation how natural are the conditions for inflation? how can we understand the ordered state after inflation coming from an unordered state before inflaton? it seems the initial conditions for inflation have to be more fine tuned than the conditions of the accelerating universe we see now Thomas Gajdosik Concepts of Modern Cosmology

15 6. General Relativity paradigm of inflation current research issues regarding inflation the simplest assumptions are too restrictive minimal coupling ( φ is only used as the source in T µν ) single field i φ 0 generalized G-inflation (Galileon inflation): more terms in the Lagrangian S = d 4 x g 5 i=1 L i R 16πG the first term, L 1, being a SM-like Higgs Lagrangian: L 1 = D µ φ 2 +λ( φ 2 v 2 ) 2 φ the inflaton field and D µ its covariant derivative λ the self-coupling of the inflaton v the vacuum expectation value of the inflaton field Thomas Gajdosik Concepts of Modern Cosmology

16 6. General Relativity paradigm of inflation current research issues regarding inflation other terms in generalized G-inflation: L 2 = K(φ,X) L 3 = G 3 φ [ L 4 = G 4 R+G 4X ( φ) 2 ( 2 φ) 2] L 5 = G 5 G µν ( µ ν φ) 1 6 G [ 5X ( φ) 3 3( φ)( 2 φ) 2 +2( 2 φ) 2] where the kinetic term of the inflaton is X = 1 2 gµν ( µ φ)( ν φ) K(φ,X) is the Kähler potential G i = G i (φ,x) is a paramterizing function with its derivative G ix = G i X and the abbreviations ( φ) = ( µ µ φ) ( 2 φ) 2 = ( µ ν φ)( µ ν φ) ( 2 φ) 3 = ( µ ν φ)( ν ρ φ)( ρ φ µ ) modifies the allowed potential for the Higgs field can easier accommodate initial conditions and end of inflation at the expense of several additional functions thereby being again less predictive Thomas Gajdosik Concepts of Modern Cosmology