1. Canonical structure of Tetrad Bigravity Sergei Alexandrov Laboratoire Charles Coulomb Montpellier S.A. arXiv:1308.6586

2. 1. Introduction: massive and bimetric gravitty in the metric and tetrad formulations 2. Canonical structure of GR in the Hilbert- Palatini formulation 3. Canonical structure of tetrad bigravity 4. Conclusions Plan of the talk

3. Massive gravity The idea: to give to the graviton a non-vanishing mass At the linearized level ― Fierz-Pauli theory (In vacuum) is transverse and traceless and carries 5 d.o.f. breaks gauge invariance At the non-linear level ― Einstein-Hilbert action plus a potential an extra fixed metric has the flat space as a solution reduces to for and Diffeomorphism symmetry is broken

4. Bimetric gravity Two dynamical metrics coupled by a non-derivative interaction term Diffeomorphism symmetry: diagonal ― preserved by the mass term off-diagonal ― broken by the mass term The theory describes one massless and one massive gravitons In fact, not only…

5. Boulware-Deser ghost Massive gravity describes a theory with 6 d.o.f. Boulware,Deser ’72 : The trace becomes dynamical and describes a scalar ghost ― the Hamiltonian is unbounded from below In the canonical language: The lapse and shift do not enter linearly and therefore are not Lagrange multipliers anymore (In the FP Lagrangian generates a second class constraint removing one d.o.f.) Is it possible to find an interaction potential which is free from the ghost pathology?

6. Ghost-free potentials The ghost is absent because the lapse appears again as Lagrange multiplier and generates a (second class) constraint Symmetric polynomials: deRham,Gabadadze,Tolley ’10 : There is a three-parameter family of ghost-free potentials Hassan,Rosen ’11 How to deal with the awkward square root structure?

7. Tetrad reformulation The idea: to reformulate the theory using tetrads Hinterbichler,Rosen ’12 The important property: The symmetricity constraint follows from e.o.m. The model should be absent from the BD ghost The mass term in the tetrad formalism (in 4d): = Symmetricity condition = antisymmetricity of the wedge product linear in

8. The model The Hilbert-Palatini action: The mass term: Cartan equationsBimetric gravity in the tetrad formulation

9. Hilbert-Palatini action 3+1 decomposition: the phase spacethe primary constraints

10. Non-covariant description Solve constraints explicitly primary constraints secondary constraints The kinetic term: where

11. Covariant description The symplectic structure is given by Dirac brackets Don’t solve constraints explicitly secondary constraints We also need

12. Tetrad bigravity: phase space The second class constraints of the two HP actions are not affected by the mass term does not depend on Covariant description Phase space: + s.c. constraints Symplectic structure: Dirac brackets Non-covariant description Phase space: Symplectic structure: canonical Poisson brackets + constraints affected by the mass term

13. Tetrad bigravity: primary constraints Decomposition of the mass term: where not expressible in terms of The total set of primary constraints: diagonal sectoroff-diagonal sector weakly commute with all primary constraints It remains to analyze the stability of

14. Tetrad bigravity: symmetricity condition Stabilization of : Crucial property: mixed metric where secondary constraint condition on Lagrange multipliers Symmetricity condition Tetrad and metric formulations are indeed equivalent on-shell

15. Tetrad bigravity: constraint algebra secondary constraint fixing Lagrange multipliers are second class

16. Tetrad bigravity: secondary constraint where One can compute explicitly Stability condition for fixes orare second class secondary constraintstability condition for

17. Summary of the phase space structure + 2×(9+9+3+3) = 48 Phase space (in non-covariant description): Second class constraints: – (6+3+3+1+1) = –14 First class constraints: – 2×(6+3+1) = –20 The BD ghost is absent! 2 ― massless graviton 5 ― massive graviton 14 dim. phase space space or 7 degrees of freedom

18. Open problems Superluminality, instabilities, tachyonic modes… Partially massless theory Degenerate sectors What happens with the theory for configurations where some of the invertibility properties fail? Detailed study of the stability condition for Additional gauge symmetry reducing the number of d.o.f. of the massive graviton from 5 to 4. Is it possible?