Canonical structure of Tetrad Bigravity Laboratoire Charles Coulomb Montpellier Sergei Alexandrov S.A. arXiv:1308.6586
Plan of the talk • Introduction: massiveandbimetricgravitty in the metric and tetrad formulations • Canonical structure of GR in the Hilbert-Palatini formulation • Canonical structure of tetrad bigravity • Conclusions
(In vacuum) is transverse and traceless and carries 5 d.o.f. 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 Massive gravity The idea: to give to the graviton a non-vanishing mass At the linearized level ― Fierz-Pauli theory breaks gauge invariance Diffeomorphism symmetry is broken
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…
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.) Boulware-Deser ghost Massive gravity describes a theory with 6 d.o.f. Boulware,Deser ’72 : Is it possible to find an interaction potential which is free from the ghost pathology?
Ghost-free potentials There is a three-parameter family of ghost-free potentials deRham,Gabadadze,Tolley ’10 : Symmetric polynomials: The ghost is absent because the lapse appears again as Lagrange multiplier and generates a (second class) constraint Hassan,Rosen ’11 How to deal with the awkward square root structure?
= Symmetricity condition = The mass term in the tetrad formalism (in 4d): • The symmetricity constraint follows from e.o.m. • The model should be absent from the BD ghost antisymmetricity of the wedge product linear in linear in Hinterbichler,Rosen ’12 Tetrad reformulation The idea: to reformulate the theory using tetrads The important property:
Cartan equations Bimetric gravity in the tetrad formulation The model The Hilbert-Palatini action: The mass term:
the phase space the primary constraints Hilbert-Palatini action 3+1 decomposition:
The kinetic term: where primary constraints secondary constraints Non-covariant description Solve constraints explicitly
We also need Covariant description Don’t solve constraints explicitly secondary constraints The symplectic structure is given byDirac brackets
does not depend on Covariant description Non-covariant description Phase space: Phase space: + s.c. constraints Symplectic structure: Symplectic structure: Dirac brackets canonical Poisson brackets + constraints affected by the mass term Tetrad bigravity: phase space The second class constraints of the two HP actions are not affected by the mass term
not expressible in terms of The total set of primary constraints: diagonal sector off-diagonal sector weakly commute with all primary constraints It remains to analyze the stability of Tetrad bigravity: primary constraints Decomposition of the mass term: where
secondary constraint condition on Lagrange multipliers Symmetricity condition Tetrad bigravity: symmetricity condition Stabilization of : Crucial property: where mixed metric Tetrad and metric formulations are indeed equivalent on-shell
fixing Lagrange multipliers secondary constraint are second class Tetrad bigravity: constraint algebra
Stability condition for are second class fixes or secondary constraint stability condition for Tetrad bigravity: secondary constraint One can compute explicitly where
14 dim. phase space space or 7 degrees of freedom 2 ― massless graviton 5 ― massive graviton Summary of the phase space structure • Phase space (in non-covariant description): + 2×(9+9+3+3) = 48 • Second class constraints: – (6+3+3+1+1) = –14 • First class constraints: – 2×(6+3+1) = –20 The BD ghost is absent!
Open problems • Superluminality, instabilities, tachyonic modes… • Partially massless theory • Degenerate sectors Additional gauge symmetry reducing the number of d.o.f. of the massive graviton from 5 to 4. Is it possible? What happens with the theory for configurations where some of the invertibility properties fail? Detailed study of the stability condition for