Manifestly Gauge Invariant Relativistic Perturbation Theory

Transcription

1 Manifestly Gauge Invariant Relativistic Perturbation Theory Albert Einstein Institute ILQGS References: K.G., S. Hofmann, T. Thiemann, O.Winkler, arxiv: , arxiv: K.G., T. Thiemann, arxiv:

2 Plan of the Talk Content Application of Relational framework to General Relativity Special Case of Deparametrisation: Two examples Manifestly gauge-invariant framework for General Relativity (FRW and perturbation around FRW) : Reduced Phase Space

3 Problem of Time in General Relativity Observables in General Relativity Observables are by definition gauge invariant quantities The gauge group of GR is Diff(M) Canonical picture: Constraints c, c generate spatial and time gauge transformations O gauge invariant {c, O} = { c, O} = 0 Hamiltonian h can for Einstein Equations is linear combination of constraints and thus constrained to vanish Consequently: O gauge invariant {h can, O} = 0 Frozen picture, contradicts experiments, problem of time in GR

4 Relational Formalism Observables in General Relativity Basic Idea [Bergmann 60, Rovelli 90] Einstein Equations are no physical evolution equations Rather describe flow of unphysical quantities under gauge transf. : Take two gauge variant f, g and choose T := g as a clock Define gauge invariant extension of f denoted by F f,t in relation to values T takes F f,t : Values of f when clock T = g takes values 5, 17, 23, 42,... Solve α t (T ) = τ for t, then use solution t T (τ) for F f,t which becomes a function of τ

5 Relational Formalism: Idea Observables in General Relativity f, g move along gauge orbit PSfrag replacements f (t 3 ) g(t 4 ) f (t 1 ) gauge orbit f g(t 2 ) gauge orbit g

6 Relational Formalism Observables in General Relativity Basic Idea [Bergmann 60, Rovelli 90] Einstein Equations are no physical evolution equations Rather describe flow of unphysical quantities under gauge transf. : Take two gauge variant f, g and choose T := g as a clock Define gauge invariant extension of f denoted by F f,t in relation to values T takes F f,t : Values of f when clock T = g takes values 5, 17, 23, 42,... Solve α t (T ) = τ for t, then use solution t T (τ) for F f,t which becomes a function of τ

7 Relational Formalism Observables in General Relativity Explicit Form for F f,t [Dittrich 04] Take as many clocks T I as they are C I then F f,t (τ) can be expressed as powers series in T I with coefficients involving multiple Poisson brackets of C I and f. Explicit form in general quite complicated But: One has explicit strategy how to construct observables Analysed in several examples, application to cosmology and cosmological perturbations [Dittrich, Dittrich & Tambornino] Automorphism property {F f,t (τ), F f,t (τ)} = F {f,f },T (τ), If f (q, p) then F f,t = f (F q,t, F p,t )

8 Strategy of the Formalism Observables in General Relativity Steps to obtain EOM for observables Consider a physical System for instance gravity & some standard matter We would like to derive EOM for the observables associated to (q a, p a ) of gravity & matter Add additional action to the system which become clocks T We have c tot = c geo + c matter + c clock =: c + c clock = 0 ca tot = ca geo + ca matter + ca clock =: c a + ca clock = 0 Construct observables wrt to these constraints: F qa,t (τ) & F pa,t (τ) Construct so called physical Hamiltonian H phys which generates true evolution of F qa,t (τ), F pa,t (τ)

9 Special Case of Deparametrisation Steps technically simplify Deparametrisation: c tot and c tot a can be solved for p clock Expressions for F qa/p a,t (τ) and H phys simplify Note: H phys is in general different for each chosen clock system Evolution of observables is generated by H phys EOM for observables are clock dependent Consider two examples for clarification: scalar field without potential k-essence

10 Scalar field as a Clock (LQC-Model) Deparametrisation for scalar field φ Constraints: c tot = c(q a, p a ) + 1 2λ ( π2 q + q ab φ,a φ,b ) c tot a = c a (q a, p a ) + πφ,a Using c tot a = 0 we get q ab φ,a φ,b = 1/π 2 q ab c a c b (more details later) Using c tot = 0 we get π = qλc q λ 2 c 2 q ab c a c b =: h φ (q a, p a ) Equivalent Hamiltonian constraint: c tot = π h φ (q a, p a ) Construct observables Q a (τ φ ) := F qa,φ(τ) and P a (τ φ ) := F pa,φ(τ) Evolution: Q a (τ φ ) = {H phys, Q a (τ φ )} and Ṗa (τ φ ) = {H phys, P a (τ φ )} H φ phys := d 3 σ QλC Q λ 2 C 2 Q ab C a C b

11 K-essence (Thiemann 06) Deparametrisation for k-essence field ϕ: Case I Constraints: c tot = c(q a, p a ) [1 + q ab ϕ,a ϕ,b ][π 2 + α 2 q], α > 0 c tot a = c a (q a, p a ) + πϕ,a Using c tot a = 0 we get again q ab ϕ,a ϕ,b = 1/π 2 q ab c a c b Using c tot = 0 we get π = h ϕ (q a, p a ) h ϕ (q a, p a ) := 1 2 (c2 q ab c a c b α 2 q) (c2 q ab c a c b α 2 q) 2 α 2 q ab c a c b q Equivalent Hamiltonian constraint: c tot = π + h ϕ (q a, p a ) Construct observables Q a (τ ϕ ) := F qa,ϕ(τ) and P a (τ ϕ ) := F pa,ϕ(τ) Q a (τ φ ) = {H phys, Q a (τ φ )} and Ṗa (τ ϕ ) = {H phys, P a (τ φ )} H ϕ phys := d 3 σh ϕ (Q a, P a )

12 K-essence (Thiemann 06) Deparametrisation for k-essence field ϕ: Case II Constraints: c tot = c (q a, p a ) [1 + q ab ϕ,a ϕ,b ][π 2 + α 2 q], α > 0 c tot a = c a(q a, p a ) + πϕ,a Using c tot a = 0 we get again q ab ϕ,a ϕ,b = 1/π 2 q ab c a c b Using c tot = 0 we get π = h ϕ(q a, p a ) h (q a, p a ) := 1 2 ((c ) 2 q ab c a c b α2 q) ((c ) 2 q ab c a c b α2 q) 2 α 2 q ab c a c b q Equivalent Hamiltonian constraint: c tot = π + h ϕ (q a, p a ) Construct observables Q a (τ ϕ ) := F qa,ϕ(τ) and P a (τ ϕ ) := F pa,ϕ(τ) Q a (τ φ ) = {H phys, Q a (τ φ )} and Ṗa (τ ϕ ) = {H phys, P a (τ φ )} H ϕ phys := d 3 σh ϕ (Q a, P a )

13 Comparison of both physical Hamiltonian Comparison of H φ phys and Hϕ phys s: (D 2 := Q ab C a C b ), (D ) 2 := Q ab C ac b ) H φ phys = d 3 σ QλC Q λ 2 C 2 D 2 H ϕ phys = d 3 1 σ 2 ((C ) 2 (D ) 2 α 2 1 Q) + 4 ((C ) 2 (D ) 2 α 2 Q) 2 α(d ) 2 Q Specialise both H phys to cosmology (FRW symmetry) Then D 2 = (D ) 2 = 0 and H φ phys = d 3 σ 2λ QC FRW and H ϕ phys = d 3 σc FRW Note that C FRW 0 here only C tot FRW = 0

14 Clocks for General Relativity Choose Clock and Ruler for GR Choose clock and ruler to give time & space physical meaning We need 1 clocks and 3 rulers: 4 scalar fields Chosen clocks & rulers such that good for cosmology: Free falling observer Standard cosmology C FRW as true Hamiltonian

15 Observables in General Relativity Dust Lagrangian Add dust Lagrangian to Gravity & Standard Model S dust = 1 d 4 X det(g) ρ(g µν U µ U ν + 1) 2 where U µ = T,µ + W j S,µ j, ρ energy density M U µ = g µν U ν is a geodesic, fields W j, S j are constant along geodesics, T defines proper time along each geodesic T µν of a pressureless perfect fluid α t (T ) = τ becomes clock, α x (S j ) = σ j becomes ruler Dust serves as a physical reference system

16 A few Words on Notation Observables in General Relativity Canonical (3+1) split of Gravity + Standard Model + Dust Dust variables time α t (T ) = τ and space α x (S j ) = σ j : Conjugate momenta P and P j, j = 1, 2, 3 Remaining gravity & matter degrees of freedom q ab, p ab and φ, π are denoted by q a, p a Gauge variant quantities: Lower case letters q a, p a Gauge invariant quantities: Capital letters Q a, P a

17 Observables in General Relativity Deparametrisation of the Constraints in GR Canonical 3+1 split: (P,T ),(P j,s j ) & remaining non dust (p a, q a ) Detailed constraints analysis, then 1st class constraints c tot = c + c dust with c dust = P 2 + q ab (PT,b + P j S j,b )(PT,b + P j S j,b ) c tot a = c a + c dust a with c dust a = PT,a + P j S j,a : c dust = P 2 + q ab ca dust Use c tot a cb dust = 0 and replace c dust a Then solve c tot for P and c tot a by c a in c dust for P j Need to assume S j,a is invertible with inverse S a j

18 Deparametrisation of the Constraints in GR (Partial) Deparametrisation of the Constraints in GR Constraints in (partial) deparametrised form c tot = P + h with h(p a, q a ) := c 2 q ab c a c b c a tot = P j + h j with h j (T, S j, p a, q a ) = Sj a(c a ht,a ) c tot, c tot a mutually commute Here F f,t simplifies a lot Construction of F f,t in two steps 1.) Reduction wrt to c tot a : q ab (x, t) q ij (σ, t) 2.) Reduction wrt to c tot : q ij (σ, t) Q ij (σ, τ)

19 Observables with respect to Dust Clock & Rulers Space time points are labled by τ and σ j τ proper time on each geodesic Sfrag replacements x (σ 1 =1,σ 2 =4,σ 3 =35) x (σ 1 =8,σ 2 =0.3,σ 3 =44)

20 Construction of Observables Explicit Form of Observables 1.) Spatial diff -invariant quantities q ij (σ, t) = d 3 x det( S(x) x) δ(s(x), σ)q ab (x)s a i (x)s b j (x) local in σ but ultra non local in x 2.) Full Observables Q ij (σ, τ) = n=0 1 n! { h(τ), q ij (σ)} (n) where {f, g} (0) = g, {f, g} (n) := {f, {f, g} (n 1) }} and h(τ) := d 3 σ(τ T(σ)) h(σ) S S=range(σ) so called dust space

21 for GR H phys We have a strategy to construct gauge invariant extension for all p a, q a and get P i, Q i Due to automorphism property of F f,t, we can extend this to functions of p a, q a which just become functions of P i, Q i However, we would like to have so called physical Hamiltonian H phys for GR that generates evolution of observables Recall: We cannot use canonical Hamiltonian h can from Einstein equations because {h can, P i } = {h can, Q i } = 0 H phys should itself be gauge invariant H phys can be derived from deparametrised constraints

22 Reduced Phase Space & We have c tot = P + h(p a, q a ) with h = c 2 q ab c a c b H(σ, τ) := F h,t = C 2 (τ, σ) Q ij (τ, σ)c i (σ)c j (σ) is given by H phys = S d 3 σh(σ, τ) (S dust space) Physical Physical time evolution: df f,t (σ,τ) dτ = {H phys, F f,t (σ, τ)} Symmetries of H phys : {H phys, C j (σ)} = 0, {H phys, H(σ)} = 0 H phys no τ dependence: conservative system

23 Reduced Phase space of Gravity + Scalar field and Dust Comparison with Unreduced Phase Space Standard unreduced framework: Gravity & scalar field Einstein Equations: EOM for q ab, p ab and matter dof Hamiltonian h can = Σ d 3 x (n(x)c(x) + n a (x)c a (x)) Constraints c := c geo + c matter = 0 and c a := c geo a + c matter a = 0 Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler] Manifestly gauge invariant EOM for Q ij, P ij and matter dof H phys = S d 3 σ C 2 Q ij C i C j (σ) Energy & momentum conservation H = ɛ, C j = ɛ j Lapse & Shift dynamical: N = C/H, N j = Q ij C i /H

24 Equation of Motion for Unreduced Case Second Order Time Derivative Equation of Motion for q ab q ab = [ṅ n ( det(q)) n ( det(q) ) ] ( + L n qab ( L n q ) det(q) det(q) n ab) +q cd( q ac ( L n q ) )( q ac bd ( L n q ) ) bd [ n 2 κ +q ab 2 det(q) C + n2( 2Λ + κ 2λ v(ξ))] + n 2[ κ ] λ ξ,aξ,b 2R ab +2n ( D a D b n ) + 2 ( L n q ) ab + ( L n q ) ab ( L n ( L n q )) ab

25 Reduced Phase space of Gravity + Scalar field and Dust Comparison with Unreduced Phase Space Standard unreduced framework: Gravity & scalar field Einstein Equations: EOM for q ab, p ab and matter dof Hamiltonian h can = Σ d 3 x (n(x)c(x) + n a (x)c a (x)) Constraints c := c geo + c matter = 0 and c a := c geo a + c matter a = 0 Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler] Manifestly gauge invariant EOM for Q ij, P ij and matter dof H phys = S d 3 σ C 2 Q ij C i C j (σ) Energy & momentum conservation H = ɛ, C j = ɛ j Lapse & Shift dynamical: N = C/H, N j = Q ij C i /H

26 Equation of Motion for Reduced Case Second Order Time Derivative Equation of Motion for Q jk Q jk = [ Ṅ N ( det Q) + N ( det Q ) ] ( L N Qjk ( L N Q ) det Q det Q N jk) +Q mn( Qmj ( L N Q ) mj)( Qnk ( ) L N Q) nk [ +Q jk N2 κ 2 det Q C + N2( 2Λ + κ 2λ v(ξ))] + N 2[ κ ] λ Ξ,jΞ,k 2R jk +2N ( D j D k N ) + 2 ( L N Q)jk + ( L Q ) N jk ( ( L N L N Q )) jk NH G jkmn N m N n det Q Q jk refers to derivative with respect to dust time τ here

27 Reduced Phase space of Gravity + Scalar field and Dust Comparison with Unreduced Phase Space Standard unreduced framework: Gravity & scalar field Einstein Equations: EOM for q ab, p ab and matter dof Hamiltonian h can = Σ d 3 x (n(x)c(x) + n a (x)c a (x)) Constraints c := c geo + c matter = 0 and c a := c geo a + c matter a = 0 Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler] Manifestly gauge invariant EOM for Q ij, P ij and matter dof H phys = S d 3 σ C 2 Q ij C i C j (σ) Energy & momentum conservation H = ɛ, C j = ɛ j Lapse & Shift dynamical: N = C/H, N j = Q ij C i /H

28 Equation of Motion for Reduced Case Second Order Time Derivative Equation of Motion for Q jk Q jk = [ Ṅ N ( det Q) + N ( det Q ) ] ( L N Qjk ( L N Q ) det Q det Q N jk) +Q mn( Qmj ( L N Q ) mj)( Qnk ( ) L N Q) nk [ +Q jk N2 κ 2 det Q C + N2( 2Λ + κ 2λ v(ξ))] + N 2[ κ ] λ Ξ,jΞ,k 2R jk +2N ( D j D k N ) + 2 ( L N Q)jk + ( L Q ) N jk ( ( L N L N Q )) jk NH G jkmn N m N n det Q

29 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Apply Manifestly Gauge Invariant Framework to FRW 1.) Specialise Q ij equations to FRW spacetime 2.) Consider linear perturbations around FRW spacetime 3.) Compare with standard results and check that dust clocks do not contradict current experiments

30 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Check Manifestly Gauge Invariant Equations for FRW Case Standard Framework: FRW Spacetime ds 2 = dt 2 + a(t) 2 δ ab dx a dx b = a(x 0 ) 2 η µν dx µ dx ν Metric q ab = a 2 (t)δ ab, Momenta p ab = 2ȧδ ab, c a = 0, FRW eqn from q ab = {h can, q ab } and ṗ ab = {h can, p ab }, c(q, p) = 0 FRW equation 3 ä a = Λ κ 4 (ρmatter + 3p matter ) Reduced Framework: FRW Spacetime FRW equation 3 Ä A = Λ κ 4 (ρmatter +ρ dust +3p matter )

31 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Standard Cosmological Perturbation Theory: Lagrange Formalism Einstein Equations G µν + Λg µν = R µν 1 2 g µν + Λg µν = κ 2 T µν Specialisation to FRW for (gravity + scalar field ξ) with k=0, (-,+,+,+) G 00 = 3H, H = a a, G ab = (2H + H 2 )δ ab T 00 = a 2 ρ, T ab = a 2 pδ ab FRW background quantities are indicated by a bar on the top

32 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Linear Perturbation around FRW Linear Perturbation [Mukhanov, Feldman, Brandenberger 1992] Consider perturbations δg µν := g µν g µν, δξ := ξ ξ Any F (g, ξ) is expanded up to linear order in δg and δξ δf denotes linear term in Taylor expansion F (g, ξ) F (g, ξ) One obtains equations for δg µν and δt µν One decomposes these equations into scalar, vector and tensor modes in order to extract physical dof 4 scalar fields φ, ψ, B, E, two transversal covector fields S a, F a and a traceless, symmetric,transversal tensor h ab and Z for scalar field contribution

33 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Linear Perturbation around FRW Perturbed metric δg 00 = 2a 2 φ, δg 0a = a 2 (S a +B,a ), δg ab = a 2 [2(ψδ ab +E,ab +F (a,b) )+h ab ] Metric is not invariant under gauge transformations x µ x µ + u µ One can construct seven invariants out of the 11 by using B E, F a in order to compensate gauge shift up to linear order The seven invariants Φ = φ 1 a [a(b E )], Ψ = ψ + H(B E ), V a = S a F a, h ab, Z = δξ + ξ (B E) Ten perturbed Einstein equation can be expressed in terms of these seven invariants

34 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Linear Perturbation around FRW Physical Degrees of Freedom Four of these equations do not contain second order time derivative of four of the seven fields These are constraints Four of the seven can be expressed in terms of the other three: V a = 0 and Φ, Z in terms of Ψ Finally: 3 physical dof: h ab, Ψ, for these evolution equations Usually in standard cosmological perturbation theory, gauge invariance is constructed order by order Repeat similar analysis in Hamiltonian framework [Langlois 1994] Additional aim: Use relational formalism to treat gauge invariance non perturbatively

35 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Reduced Phase space of GR with Dust Manifestly Gauge Invariant Cosmological Perturbation Theory [K.G., Hofmann, Thiemann, Winkler] We have EOM for Q ij and Ξ Specialise to FRW background: Equation formally agree (A a) Consider perturbation around FRW: δq ij = Q ij Q ij, δξ = Ξ Ξ δq ij and δξ are automatically gauge invariant Any power (δq ij ) n and (δξ) n will be also gauge invariant! Interesting for higher order perturbation theory

36 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Reduced Phase space of GR with Dust: Results Manifestly Gauge Invariant Cosmological Perturbation Theory [K.G., Hofmann, Thiemann, Winkler] Results for Linear Order Perturbation Theory Perturbed eqn for δ Q jk, δ Ξ agree up to one term which shows influence of the dust clock This has to be expected because we consider a gravitationally interacting observer Not an idealised observer as one has usually in cosmology Mukhanov et. al: Gravity + scalar field Here: Gravity + scalar field + dust Difference in physical degrees of freedom

37 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Comparison MFB and Dust framework Counting physical degrees of freedom Start with 15 dof (gravity+scalar field + 4 dust fields) Lapse function and shift vector are pure gauge: reduction by 4 dof Hamiltonian & diffeomorphism constraint: reduction by 4 dof We end up with 7 physical dof Potentially dangerous, because 4 more than usual might contradict experiment Reason: We use dust fields to construct gauge invariant quantities, all components in three metric become physical Can we still match with the results obtained by Mukhanov, Feldman and Brandenberger? We need to show that these additional modes are zero or decay

38 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Comparison MFB and Dust framework Constants of motion in Dust framework Energy density H(σ) =: ɛ(σ) is constant of motion Momentum density C j (σ) =: ɛ j (σ) is constant of motion Perturbations δɛ, δɛ j are again constant of motion wrt perturbed Hamiltonian MFB: Constraints Additional modes decay: V a = 0, f mom(ψ, Φ, Z),a = 0 f energy (Ψ, Φ, Z) = 0 Dust: Energy & momentum conservation laws V j = κ δɛ j A, f mom(ψ, Φ, Z) 2,j = κ 4A ( 1 A δɛ j + ɛ[b E ],j ) f energy (Ψ, Φ, Z) = 1 A (δɛ ɛg(ψ, B, E)) In the limit of vanishing ɛ, δɛ and ɛ j, δɛ j exact agreement

39 Specialisation to FRW spacetimes Linear Cosmological Perturbation Theory Summary: Comparison with Standard Framework Background Equations agree formally Linear cosmological perturbation Theory: Results are in agreement with the one of Mukhanov et al. Dust seems to be appropriate clock for cosmological situations So far everything was purely classical.. Reduced phase space approach also of advantage when quantisation is considered

40 Reduced Phase Space in LQG in AQG Why is such a Framework Useful for? Advantages when is Considered Constraints have completely disappeared from the picture No Constraint Equations Constraints have been reduced classically Only algebra of observables of interest: Includes all physical degrees of freedom Direct access to physical Hilbert space

41 Reduced Phase Space in LQG in AQG Reduced Phase Space Reduced Phase Space for LQG [K.G., Thiemann] Algebra of observables simple {Q ij, P kl } {q ab (x), p cd (y)} = δ(a c δd b) δ3 (x, y) Easy to find representations of this algebra, even Fock possible However, apart from algebra representations need to support H phys Choose standard LQG representation used for H kin Physical Hilbert space where volume spectrum discrete! Problematic to preserve classical symmetries of H phys Recall: {H phys, H(σ)} = 0, {H phys, C j (σ)} = 0 This leads to infinitely number of conservation laws in LQG Moreover, physical Hilbert space is non-separable

42 Reduced Phase Space in LQG in AQG Reduced Phase Space for AQG Reduced Phase Space for AQG [K.G.,Thiemann] AQG:One fundamental algebraic graph, subgraphs are not preserved No additional infinitely many conservation laws can be performed using the techniques of AQG AQG is formulated as (background independent) Hamiltonian Lattice Gauge Theory Anomalies of H phys : Notion of Diff(S) is meaningless Idea: M-like functional S d 3 σ ah2 (σ)+bq jk C j C k (σ) det(q) H phys has no anomalies [H phys, [H phys, M]] M=0 = 0

43 Observables in General Relativity Problem of time in GR has been circumvented by dust clocks Results agree with standard cosmological perturbation theory Next Step: Second order and quantisation of perturbation Beyond linear order manifestly gauge invariant quantities should be of advantage compared to standard framework Improve (possible) anomaly issue of H phys Scattering Theory Relation of H phys with SM Hamiltonian on Minkowski space Vacuum problem in QFT on curved background

44 Conclusion & Outlook Observables in General Relativity Choosing dust as a clock... One could think of the dust as NIMP-particles (non interacting massless particles) It could be interpreted as the gravitational Higgs Hope for the future Extract some physics out of LQG such that working at an interface of a fundamental theory & (cosmological) observations becomes possible