# Massive graviton on arbitrary background: derivation, syzygies, applications

## Abstract

We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a “reference metric' which is present in the non perturbative formulation. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaître-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reachmore »

- Authors:

- UPMC-CNRS, UMR7095, Institut d’Astrophysique de Paris, GReCO,98bis boulevard Arago, F-75014 Paris (France)
- (France)

- Publication Date:

- Sponsoring Org.:
- SCOAP3, CERN, Geneva (Switzerland)

- OSTI Identifier:
- 22458436

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2015; Journal Issue: 06; Other Information: PUBLISHER-ID: JCAP06(2015)038; OAI: oai:repo.scoap3.org:10783; cc-by Article funded by SCOAP3. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; EQUATIONS OF MOTION; FIELD EQUATIONS; GRAVITATION; LIMITING VALUES; METRICS; NONLUMINOUS MATTER; SPACE-TIME

### Citation Formats

```
Bernard, Laura, Deffayet, Cédric, IHES, Institut des Hautes Études Scientifiques,Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, and Strauss, Mikael von.
```*Massive graviton on arbitrary background: derivation, syzygies, applications*. United States: N. p., 2015.
Web. doi:10.1088/1475-7516/2015/06/038.

```
Bernard, Laura, Deffayet, Cédric, IHES, Institut des Hautes Études Scientifiques,Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, & Strauss, Mikael von.
```*Massive graviton on arbitrary background: derivation, syzygies, applications*. United States. doi:10.1088/1475-7516/2015/06/038.

```
Bernard, Laura, Deffayet, Cédric, IHES, Institut des Hautes Études Scientifiques,Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette, and Strauss, Mikael von. Tue .
"Massive graviton on arbitrary background: derivation, syzygies, applications". United States.
doi:10.1088/1475-7516/2015/06/038.
```

```
@article{osti_22458436,
```

title = {Massive graviton on arbitrary background: derivation, syzygies, applications},

author = {Bernard, Laura and Deffayet, Cédric and IHES, Institut des Hautes Études Scientifiques,Le Bois-Marie, 35 route de Chartres, F-91440 Bures-sur-Yvette and Strauss, Mikael von},

abstractNote = {We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a “reference metric' which is present in the non perturbative formulation. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaître-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reach our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.},

doi = {10.1088/1475-7516/2015/06/038},

journal = {Journal of Cosmology and Astroparticle Physics},

number = 06,

volume = 2015,

place = {United States},

year = {Tue Jun 23 00:00:00 EDT 2015},

month = {Tue Jun 23 00:00:00 EDT 2015}

}