HAMILTONIAN FORMULATION OF f (Riemann) THEORIES OF GRAVITY

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1 ABSTRACT We present a canonical formulation of gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor, which, for example, arises in the low-energy limit of superstring theories. Our approach allows a unified treatment of various subcases and an easy identification of the degrees of freedom of the theory.

2 Spontaneous Workshop IV - Hot topics in Modern Cosmology Institut d Etudes Scientifiques de Cargèse, May 2010 HAMILTONIAN FORMULATION OF f (Riemann) THEORIES OF GRAVITY Yuuiti Sendouda (APC, Paris 7) with Nathalie Deruelle, Ahmed Youssef (APC, Paris 7) Misao Sasaki, Daisuke Yamauchi (YITP, Kyoto) Ref: Prog. Theor. Phys. 123, 169 (2010), Phys. Rev. D (2009)

3 PLAN Motivations for f (Riemann) gravity 1st-order action Hamiltonian formulation Phase space reduction Conclusion

4 Observational evidence The Universe is undergoing accelerated expansion : Not explained by known mechanisms. MOTIVATIONS What is responsible for the acceleration? Dark energy? Modified gravity? from NASA G µν = κ T µν

5 f (Riemann) GRAVITY Gravity theories with higher-curvature corrections S = 1 d D x gf(r abcd ) 2 Much more than currently fashionable f (R). M Various motivations from high-energy (quantum) physics : As counterterms to regularise Tab on curved spaces [Utiyama & DeWitt (1962)] Einstein-Hilbert action itself is not renormalisable [ t Hooft & Veltman (1974)] String theories predict this kind of modifications... String theories are still under development and their predictions are not secure. Cosmologists may have a chance to determine the true form of gravity from observations prior to particle physicists.

6 BASIC KNOWLEDGES Generally, the eom for the metric is 4th-order : R (a cde f R b)cde + 2 c d Because Riemann curvature tensor contains 2nd derivatives of metric. New dynamical dofs other than the metric will appear. Some landscape of f (Riemann) : f R (a c b)d 1 2 fgab = κ T ab f = f (R) : Metric+scalar on arbitrary background, any D Classical argument - Equivalent to scalar-tensor gravity [Teyssandier & Tourrenc (1983), Maeda (1989)] - Used to explain accelerating expansion [R+R 2 Starobinsky (1980); R+R -n Capozziello et al. (2003), Carroll et al. (2004); etc.] f = R 2 + Weyl 2 : Metric+scalar+traceless tensor on D = 4 Minkowski [Stelle (1978)] - Tensor has negative kinetic term : ghost Lovelock : 2nd-order eom, no extra dof [Lovelock (198?)]...

7 WHAT TO DO Questions to answer : How do we treat various sub-classes of f (Riemann) in a unified manner? How is the form of f reflected by the gravitational dynamics? How do we control the ghost?... As a first step, we construct Hamiltonian (1st-order) formulation of f (Riemann) gravity Dynamical (time-evolutional) properties become more transparent. Useful for stability analysis... We will keep f to be arbitrary as possible, but let me exclude Lovelock terms for a while...

8 1ST-ORDER f (Riemann) ACTION

9 BASIC IDEA We need an action consists of (at most) 1st derivatives. Remove 2nd derivatives L = f (φ, φ, φ) L = f (φ, φ, Ω) + ψ (Ω φ) L = f (φ, φ, Ω) + ψ Ω + ψ φ Ω is determined (implicitly) in terms of ψ via f Ω [φ, φ,ω(φ, φ,ψ)] + ψ = 0 Possible only when f is non-linear (non-degenerate) in the 2nd derivative. 2 dynamical dofs φ, ψ L = f [φ, φ,ω(φ, φ,ψ)] + ψ Ω(φ, φ,ψ)+ ψ φ Although the real story is a bit more complicated... Auxiliary field Integration by parts

10 AUXILIARY FIELDS We can first lower the order of derivative from 4 to 2. Eom contains higher-derivative (4th-order) due to nonlinearity of curvature (2nd derivative of metric) : S g [g ab ] = 1 d D x gf(r abcd ) 2 An equivalent action being linear in curvature S [g ab, abcd, ϕ abcd ] = 1 d D x g [ f ( abcd ) + ϕ abcd (R abcd abcd )] 2 M gives two 2nd-order eoms & one constraint equivalent to the 4th-order one : δs δg ab = δs δ abcd = δs δϕ abcd = 0 M δs g δg ab = 0

11 ADM DECOMPOSITION A geometrical way to define time [Arnowitt, Deser & Misner (1962)] t t a a t = 1 Metric is decomposed into dynamical/non-dynamical parts: g ab γ ab = g ab n a n b N = n a t a β a = γ a b t b (t a = Nn a + β a ) = n a n a n a Σ t M Spacetime tensors will be orthogonally decomposed using induced metric and normal vector.

12 DECOMPOSITION OF ACTION : Projection by γa ADM decomposition of 2nd-order action : b S = 1 d D x n : Contraction with n a g [ f ( abcd ) + ϕ abcd (R abcd abcd )] 2 M ϕ abcd ( R abcd abcd ) + 4 ϕ abcn ( R abcn abcn ) + 2 Ψ ab ( R anbn Ω ab ) where Ψ ab 2 ϕ anbn, Ω ab anbn Two eoms immediately determine the redundant components of auxiliary field : Then we get δ ϕ abcd : abcd = R abcd, S = 1 2 M δ ϕ abcn : abcn = R abcn g [ f + 2 Ψ ab ( R anbn Ω ab )]

13 GEOMETRICAL RELATIONS Extrinsic curvature is the velocity of metric : K ab = γ c a c n b = 1 2 N ( γ ab + 2 D (a β b) ) We have Gauss Codazzi γ a e γ b f γ c g γ d h R e f gh = 2 K a[c K d]b + R abcd [γ] γ a d γ b e γ c f n g R defg = 2 D [a K b]c and Ricci relations γ a c n d γ b e n f R cde f = n K ab + K ac K b c D a D b N 2nd (time) derivative 1st (time) derivatives

14 1ST-ORDER ACTION Integrating by parts, Ψ appears to be dynamical : S = γ N [ Ψ ab (KK ab + K ac K c b N 1 D a D b N Ω ab ) M + N 1 K ab t β Ψ ab c (n c Ψ ab K ab ) f ] Divergence is canceled by the surface term : S = dσ a n a Ψ bc K bc M No 2nd derivatives in the total action S + S To be discussed : Eom for Ω determines # of DOFs. 2 Ψ ab = f Ω ab

15 HAMILTONIAN FORMULATION

16 Canonical momenta defined as HAMILTONIAN p ab δ(s + S ) δ γ ab, Π ab δ(s + S ) δ Ψ ab Canonical action found via Legendre transformation S + S = dtl= dt d D 1 x γ (p γ + Π Ψ) H Σ t where H = H[γ ab, p ab, Ψ ab, Π ab, Ω ab, N, β a ] = (N C + β a C a ) Σ t is the Hamiltonian, where Hamiltonian and momentum constraints are 2 C = (γ a(b Ψ c)d Π ab Π cd p Π) γ C a = 2 γ D c γ ab p bc Ψ bc Π ab γ γ γ 2 ( 2 Ψ Ω + f 2 D ad b Ψ ab ) + Π bc D a Ψ bc

17 EOMS AND CONSTRAINTS Constraints from variations wrt multipliers : δn : C = 0, δβ a : C a = 0 will (after second-class constraints are inserted into the action) turn to be first-class. Constraint from auxiliary field δω ab :2Ψ ab = f [γ ab, Π ab, Ω ab ] Ω ab will be used to reduce # of non-dynamical variables. Canonical eoms from variations wrt dynamical variables: γ ab = δh δp ab, ṗ ab = δh δγ ab, Ψ ab = δh δπ ab Π ab = δh δψ ab These 7 eqs recover the original 4th-order eom.

18 TRACE DECOMPOSITION Ψ can be decomposed into the trace and traceless parts : Φ γ Ψ D 1, ψab T Ψ ab Ψ ab γ Ψ D 1 γab Traceless There are (most generically) a scalar and (traceless) tensor degrees of freedom : L = (p γ + Π Ψ) H[γ ab, p ab, Ψ ab, Π ab, Ω ab, N, β a ] Σ t = ( p γ + Π Φ + π ψ) H[γ ab, p ab, Φ, Π, ψ ab, π ab, Ω ab, N, β a ] Σ t where p ab p ab 1 D 1 (γ ΠΨab + γ Ψ T (γ ac γ bd Π cd )), Π γ Π, π ab T Π ab

19 DONE We ve obtained canonical eoms and constraints for f (Riemann) in the most generic form. WHAT TO SEE BELOW Any symmetries of f give rise to additional constraints. They are usually 2nd-class and to be inserted into the action to eliminate unnecessary variables. One exceptional case is of conformal gravity where conformal (gauge) transformation is generated by a constraint.

20 PHASE SPACE REDUCTION

21 PROCEDURE Our generic Hamiltonian is still reducible in presence of 2nd-class constraints. The roles of the constraint eq : δω ab :2 Ψ ab = f Ω ab [γ ab, Π ab, Ω ab ] A) Ω determined (f is nonlinear in Ω) : Ψ dynamical... nothing happens B) Ω undetermined (f is at most linear in Ω) : Ψ non-dynamical Precisely, a scalar may arise from non-linearity of the trace while a tensor may arise from that of the traceless part of the second derivative. Ω ab = Ω D 1 γ ab + ω ab

22 1ST CLASS, 2ND CLASS Way to reduce action/hamiltonian : 1.Take time derivative of the above primary constraint to find a secondary constraint (Dirac) 2. If they are 2nd-class, insert them into the canonical action to reduce action/ Hamiltonian (Faddeev & Jackiw) 1st class constraints - commute with all the other constraints (modulo constraints), - generate gauge transformations ( Dirac conjecture ). - should be kept to make gauge symmetries of the system explicit. 2nd class constraints - do not commute with at least one other constraint, - are safely inserted into the action to eliminate non-dynamical dofs.

23 EXAMPLE 1: EINSTEIN Action Constraints Primary : Secondary : f = R Ψ ab = γ ab (2nd-class) Π ab = 2 p ab 2 γ p D δω ab : Ψ ab = γ ab γ ab (2nd-class) Action/Hamiltonian reduce into ADM s L = p γ H[γ ab, p ab, N, β a ] Σ t where No extra dof. p ab = p ab + 2 γ p D γab

24 Action Constraints Primary : Secondary : EXAMPLE 2: f (R) (2nd-class) (2nd-class) Reduced action/hamiltonian L = ( p γ + Π Φ) H[γ ab, p ab, Φ, Π, N, β a ] Σ t where f = f (R) Extra scalar dof. TΨ ab = 0 TΠ ab = 2 Φ γ ac γ bd T p cd p ab = p ab Φ γ ac γ bd Π cd δω ab : Ψ ab = f γ ab Agrees with the independent result [Deruelle, YS, Youssef (2009)]. Φ γ Ψ D 1 = f

25 Action Constraints Primary : EXAMPLE 3: C 2 f = C abcd C abcd = R abcd R abcd 4 D 2 R ab R ab + δω ab : Ψ ab = 4 D 2 [(D 3) TΩ ab + T ρ ab ] Secondary : γ Ψ = 0 (2nd-class) γ p D 2 T Ψ TΠ = 0 2 (D 1) (D 2) R2 ρ ab ΠΠ ab (Π Π) ab γ + R ab [γ] (depends on whether D = 4 or not) The secondary constraint can be 1st-class depending on # of dimensions. This moment we only use the primary constraint.

26 Action is reduced to be L = ( p γ + π ψ) H[γ ab, p ab, ψ ab, π ab, Π, N, β a ] Σ t where p ab p ab γ Π D 1 ψab, ψ ab T Ψ ab, π ab T Π ab, Π γ Π Extra traceless tensor dof (but see below). Hamiltonian H = (N C + β a C a + Π C Π ) Σ t where Π works as a Lagrange multiplier and C Π γ p D 2 ψ π In D = 4, CΠ is the generator of conformal transformation and commute with other constraints. [Boulware (1984)] If D > 4, more secondary constraints may arise. They might be used to further eliminate dofs (undone).

27 SUMMARY OF THIS PART Non-linearity of the second derivative determines what types of extra dofs arise : Tr part Tr-less part Extra dofs Extra gauge sym. R f (R) Linear NL - Scalar - C 2 - NL Tensor Conformal (D = 4)

28 CONCLUSION

29 Achievements CONCLUSION+ Hamiltonian formulation of f (Riemann) gravity has been established. Effective & simple way to reduce generic Hamiltonian to those of typical sub-cases (R, f (R), C 2 ) was shown. Plans for the future [all in progress] Properties of Ψ on various non-trivial backgrounds (e.g. FLRW, black holes) : Is there always ghost? If yes, what makes it harmless? Lovelock terms Energy in higher-derivative gravity theories Coupling to matter, Surface term: e.g. Junction conditions for braneworld Feedback to fundamental theories from phenomenological view point of gravity fin