Canonical structure of Tetrad Bigravity

Sergei Alexandrov

Laboratoire Charles CoulombMontpellier

S.A. arXiv:1308.6586

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

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

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…

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 constraintremoving one d.o.f.)

Is it possible to find an interaction potential which is free from the

ghost pathology?

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?

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

=

antisymmetricityof the wedge product

linear in linear in

The model

The Hilbert-Palatini action:

The mass term:

Cartan equations Bimetric gravity in the tetrad formulation

Hilbert-Palatini action

3+1 decomposition:

the phase space the primary constraints

Non-covariant description

Solve constraints explicitly

primary constraints

secondary constraints

The kinetic term:

where

Covariant description

The symplectic structure is given by Dirac brackets

Don’t solve constraints explicitly

secondary constraints

We also need

Tetrad bigravity: phase spaceThe 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

Tetrad bigravity: primary constraints

Decomposition of the mass term:

where

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: 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

Tetrad bigravity: constraint algebra

secondary constraintfixing Lagrange multipliers

are second class

Tetrad bigravity: secondary constraint

where

One can compute explicitly

Stability condition for

fixes or are second class

secondary constraint stability condition for

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 graviton5 ― massive graviton

14 dim. phase space spaceor

7 degrees of freedom

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?