- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

New formalism for perturbations of massive gravity theories around arbitrary background spacetimes

Kieran Wood*

  • *Contact author: kieran.wood@nottingham.ac.uk
PDF

Phys. Rev. D 113, 084054 – Published 27 April, 2026

DOI: https://doi.org/10.1103/vspq-pkfy

Abstract

We develop a new technique for studying the perturbations of dRGT-type massive gravity theories around arbitrary background spacetimes. Built initially from the vielbein formulation of the theory, but switching back to the metric formulation afterward, our approach bypasses many of the complications that arise in previous metric formulation approaches to linearizing massive gravity around generic backgrounds, naturally elucidates the ghost-free structure of the interactions, and readily generalizes to higher orders in perturbation theory, as well as to multiple interacting metric tensor fields. To demonstrate the power of our technique, we apply our formalism to a number of commonly occurring example backgrounds—proportional, cosmological, and black hole—recovering and extending many known results from the literature at linear order. Lastly, we provide, for the first time, the cubic order multigravity potential around a generic background spacetime.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (121)

  1. K. Hinterbichler, Theoretical aspects of massive gravity, Rev. Mod. Phys. 84, 671 (2012).
  2. C. de Rham, Massive gravity, Living Rev. Relativity 17, 1 (2014).
  3. A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory, J. Phys. A 49, 183001 (2016).
  4. K. Wood, Theoretical and phenomenological aspects of multi-gravity, Ph.D. thesis, University of Nottingham, 2025 (upcoming).
  5. M. Fierz and W. E. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. R. Soc. A 173, 211 (1939).
  6. A. Higuchi, Forbidden mass range for spin-2 field theory in de Sitter spacetime, Nucl. Phys. B282, 397 (1987).
  7. A. Higuchi, Massive symmetric tensor field in spacetimes with a positive cosmological constant, Nucl. Phys. B325, 745 (1989).
  8. D. G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6, 3368 (1972).
  9. P. Van Nieuwenhuizen, On ghost-free tensor lagrangians and linearized gravitation, Nucl. Phys. B60, 478 (1973).
  10. C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D 82, 044020 (2010).
  11. C. de Rham, G. Gabadadze, and A. J. Tolley, Resummation of massive gravity, Phys. Rev. Lett. 106, 231101 (2011).
  12. S. F. Hassan and R. A. Rosen, On non-linear actions for massive gravity, J. High Energy Phys. 07 (2011) 009.
  13. S. F. Hassan and R. A. Rosen, Resolving the ghost problem in nonlinear massive gravity, Phys. Rev. Lett. 108, 041101 (2012).
  14. S. F. Hassan, R. A. Rosen, and A. Schmidt-May, Ghost-free massive gravity with a general reference metric, J. High Energy Phys. 02 (2012) 026.
  15. S. Hassan, A. Schmidt-May, and M. von Strauss, Proof of consistency of nonlinear massive gravity in the Stückelberg formulation, Phys. Lett. B 715, 335 (2012).
  16. A. Golovnev, On the Hamiltonian analysis of non-linear massive gravity, Phys. Lett. B 707, 404 (2012).
  17. J. Klusoň, Nonlinear massive gravity with additional primary constraint and absence of ghosts, Phys. Rev. D 86, 044024 (2012).
  18. J. Klusoň, Note about Hamiltonian formalism for general nonlinear massive gravity action in Stückelberg formalism, Int. J. Mod. Phys. A 28, 1350160 (2013).
  19. T. Kugo and N. Ohta, Covariant approach to the no-ghost theorem in massive gravity, Prog. Theor. Exp. Phys. 2014, 43B04 (2014).
  20. N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Ann. Phys. (Amsterdam) 305, 96 (2003).
  21. P. Creminelli, A. Nicolis, M. Papucci, and E. Trincherini, Ghosts in massive gravity, J. High Energy Phys. 09 (2005) 003.
  22. C. Deffayet and J.-W. Rombouts, Ghosts, strong coupling, and accidental symmetries in massive gravity, Phys. Rev. D 72, 044003 (2005).
  23. K. Hinterbichler and R. A. Rosen, Interacting spin-2 fields, J. High Energy Phys. 07 (2012) 047.
  24. S. F. Hassan and R. A. Rosen, Bimetric gravity from ghost-free massive gravity, J. High Energy Phys. 02 (2012) 126.
  25. S. F. Hassan and R. A. Rosen, Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity, J. High Energy Phys. 04 (2012) 123.
  26. S. F. Hassan and A. Lundkvist, Analysis of constraints and their algebra in bimetric theory, J. High Energy Phys. 08 (2018) 182.
  27. N. Khosravi, N. Rahmanpour, H. R. Sepangi, and S. Shahidi, Multimetric gravity via massive gravity, Phys. Rev. D 85, 024049 (2012).
  28. H. R. Afshar, E. A. Bergshoeff, and W. Merbis, Interacting spin-2 fields in three dimensions, J. High Energy Phys. 01 (2015) 040.
  29. K. Nomura and J. Soda, When is multimetric gravity ghost-free?, Phys. Rev. D 86, 084052 (2012).
  30. J. H. Scargill, J. Noller, and P. G. Ferreira, Cycles of interactions in multi-gravity theories, J. High Energy Phys. 12 (2014) 160.
  31. S. F. Hassan and A. Schmidt-May, Interactions of multiple spin-2 fields beyond pairwise couplings, Phys. Rev. Lett. 122, 251101 (2019).
  32. C. de Rham, J. T. Deskins, A. J. Tolley, and S.-Y. Zhou, Graviton mass bounds, Rev. Mod. Phys. 89, 025004 (2017).
  33. M. Högås and E. Mörtsell, Constraints on bimetric gravity. Part I. Analytical constraints, J. Cosmol. Astropart. Phys. 05 (2021) 001.
  34. M. Högås and E. Mörtsell, Constraints on bimetric gravity. Part II. Observational constraints, J. Cosmol. Astropart. Phys. 05 (2021) 002.
  35. M. Högås and E. Mörtsell, Constraints on bimetric gravity from Big Bang nucleosynthesis, J. Cosmol. Astropart. Phys. 11 (2021) 001.
  36. K. Choi and S. H. Im, Realizing the relaxion from multiple axions and its UV completion with high scale supersymmetry, J. High Energy Phys. 01 (2016) 149.
  37. D. E. Kaplan and R. Rattazzi, Large field excursions and approximate discrete symmetries from a clockwork axion, Phys. Rev. D 93, 085007 (2016).
  38. G. F. Giudice and M. McCullough, A clockwork theory, J. High Energy Phys. 02 (2017) 036.
  39. F. Niedermann, A. Padilla, and P. M. Saffin, Higher order clockwork gravity, Phys. Rev. D 98, 104014 (2018).
  40. K. Wood, P. M. Saffin, and A. Avgoustidis, Clockwork cosmology, J. Cosmol. Astropart. Phys. 07 (2023) 062.
  41. E. Babichev, L. Marzola, M. Raidal, A. Schmidt-May, F. Urban, H. Veermäe, and M. von Strauss, Heavy spin-2 dark matter, J. Cosmol. Astropart. Phys. 09 (2016) 016.
  42. N. G. Albornoz, A. Schmidt-May, and M. von Strauss, Dark matter scenarios with multiple spin-2 fields, J. Cosmol. Astropart. Phys. 01 (2018) 014.
  43. J. Flinckman and S. F. Hassan, Mass spectrum & linear perturbations of ghost-free multi-spin-2 theory, J. High Energy Phys. 02 (2025) 176.
  44. J. Flinckman and S. F. Hassan, Existence of ghost-eliminating constraints in multivielbein theory, arXiv:2510.03014.
  45. N. Arkani-Hamed, A. G. Cohen, and H. Georgi, (De) constructing dimensions, Phys. Rev. Lett. 86, 4757 (2001).
  46. C. T. Hill, S. Pokorski, and J. Wang, Gauge invariant effective Lagrangian for Kaluza-Klein modes, Phys. Rev. D 64, 105005 (2001).
  47. C. Deffayet and J. Mourad, Multigravity from a discrete extra dimension, Phys. Lett. B 589, 48 (2004).
  48. C. Deffayet and J. Mourad, Deconstruction of gravity, Int. J. Theor. Phys. 44, 1743 (2005).
  49. C. De Rham, A. Matas, and A. J. Tolley, Deconstructing dimensions and massive gravity, Classical Quantum Gravity 31, 025004 (2013).
  50. A. Avgoustidis, F. Niedermann, A. Padilla, and P. M. Saffin, Deconstructing higher order clockwork gravity, Phys. Rev. D 103, 124007 (2021).
  51. K. Wood, A new look at multi-gravity and dimensional deconstruction, arXiv:2501.16442.
  52. R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70, 2837 (1993).
  53. R. Gregory, Black string instabilities in anti-de Sitter space, Classical Quantum Gravity 17, L125 (2000).
  54. R. Gregory and R. Laflamme, The instability of charged black strings and p-branes, Nucl. Phys. B428, 399 (1994).
  55. E. Babichev and A. Fabbri, Instability of black holes in massive gravity, Classical Quantum Gravity 30, 152001 (2013).
  56. E. Babichev and A. Fabbri, Stability analysis of black holes in massive gravity: A unified treatment, Phys. Rev. D 89, 081502 (2014).
  57. R. Brito, V. Cardoso, and P. Pani, Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass, Phys. Rev. D 88, 023514 (2013).
  58. K. Wood, P. M. Saffin, and A. Avgoustidis, Black holes in multimetric gravity, Phys. Rev. D 109, 124006 (2024).
  59. K. Wood, P. M. Saffin, and A. Avgoustidis, Black holes in multimetric gravity. II. Hairy solutions and linear stability of the non-and partially proportional branches, Phys. Rev. D 111, 024057 (2025).
  60. N. Boulanger, T. Damour, L. Gualtieri, and M. Henneaux, Inconsistency of interacting, multi-graviton theories, Nucl. Phys. B597, 127 (2001).
  61. S. F. Hassan, A. Schmidt-May, and M. von Strauss, On consistent theories of massive spin-2 fields coupled to gravity, J. High Energy Phys. 05 (2013) 086.
  62. O. Baldacchino and A. Schmidt-May, Structures in multiple spin-2 interactions, J. Phys. A 50, 175401 (2017).
  63. L. Bernard, C. Deffayet, and M. von Strauss, Consistent massive graviton on arbitrary backgrounds, Phys. Rev. D 91, 104013 (2015).
  64. L. Bernard, C. Deffayet, and M. von Strauss, Massive graviton on arbitrary background: derivation, syzygies, applications, J. Cosmol. Astropart. Phys. 06 (2015) 038.
  65. L. Bernard, C. Deffayet, A. Schmidt-May, and M. von Strauss, Linear spin-2 fields in most general backgrounds, Phys. Rev. D 93, 084020 (2016).
  66. C. Mazuet and M. S. Volkov, Massive gravitons in arbitrary spacetimes, Phys. Rev. D 96, 124023 (2017).
  67. C. Mazuet and M. S. Volkov, Massive spin-2 field in arbitrary spacetimes—the detailed derivation, J. Cosmol. Astropart. Phys. 07 (2018) 012.
  68. P. Guarato and R. Durrer, Perturbations for massive gravity theories, Phys. Rev. D 89, 084016 (2014).
  69. G. Cusin, R. Durrer, P. Guarato, and M. Motta, A general mass term for bigravity, J. Cosmol. Astropart. Phys. 04 (2016) 051.
  70. Q. Hu and D. Cheng, The polynomial solution to the Sylvester matrix equation, Appl. Math. Lett. 19, 859 (2006).
  71. N. J. Higham, Computing real square roots of a real matrix, Linear Algebra Appl. 88, 405 (1987).
  72. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, England, 1994).
  73. S. F. Hassan and M. Kocic, On the local structure of spacetime in ghost-free bimetric theory and massive gravity, J. High Energy Phys. 05 (2018) 099.
  74. J. H. Scargill and J. Noller, Strong-coupling scales and the graph structure of multi-gravity theories, J. High Energy Phys. 01 (2016) 029.
  75. S. F. Hassan, A. Schmidt-May, and M. von Strauss, Particular solutions in bimetric theory and their implications, Int. J. Mod. Phys. D 23, 1443002 (2014).
  76. S. F. Hassan, A. Schmidt-May, and M. von Strauss, Bimetric theory and partial masslessness with Lanczos–Lovelock terms in arbitrary dimensions, Classical Quantum Gravity 30, 184010 (2013).
  77. C. Deffayet, J. Mourad, and G. Zahariade, A note on “symmetric” vielbeins in bimetric, massive, perturbative and non perturbative gravities, J. High Energy Phys. 03 (2013) 086.
  78. J. Hoek, On the Deser-van Nieuwenhuizen algebraic vierbein gauge, Lett. Math. Phys. 6, 49 (1982).
  79. S. F. Hassan, A. Schmidt-May, and M. von Strauss, Metric formulation of ghost-free multivielbein theory, arXiv:1204.5202.
  80. C. de Rham and A. J. Tolley, Vielbein to the rescue? Breaking the symmetric vielbein condition in massive gravity and multigravity, Phys. Rev. D 92, 024024 (2015).
  81. H. Nersisyan, Y. Akrami, and L. Amendola, Consistent metric combinations in cosmology of massive bigravity, Phys. Rev. D 92, 104034 (2015).
  82. G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, and A. J. Tolley, Massive cosmologies, Phys. Rev. D 84, 124046 (2011).
  83. A. E. Gümrükçüoğlu, C. Lin, and S. Mukohyama, Open FRW universes and self-acceleration from nonlinear massive gravity, J. Cosmol. Astropart. Phys. 11 (2011) 030.
  84. B. Vakili and N. Khosravi, Classical and quantum massive cosmology for the open FRW universe, Phys. Rev. D 85, 083529 (2012).
  85. D. Comelli, M. Crisostomi, F. Nesti, and L. Pilo, FRW cosmology in ghost free massive gravity from bigravity, J. High Energy Phys. 03 (2012) 067.
  86. C. De Rham, G. Gabadadze, L. Heisenberg, and D. Pirtskhalava, Cosmic acceleration and the helicity-0 graviton, Phys. Rev. D 83, 103516 (2011).
  87. A. E. Gümrükçüoğlu, C. Lin, and S. Mukohyama, Anisotropic Friedmann–Robertson–Walker universe from nonlinear massive gravity, Phys. Lett. B 717, 295 (2012).
  88. M. von Strauss, A. Schmidt-May, J. Enander, E. Mörtsell, and S. F. Hassan, Cosmological solutions in bimetric gravity and their observational tests, J. Cosmol. Astropart. Phys. 03 (2012) 042.
  89. D. Comelli, M. Crisostomi, and L. Pilo, Perturbations in massive gravity cosmology, J. High Energy Phys. 06 (2012) 085.
  90. D. Comelli, M. Crisostomi, F. Nesti, and L. Pilo, FRW cosmology in ghost free massive gravity from bigravity, J. High Energy Phys. 03 (2012) 067.
  91. M. S. Volkov, Cosmological solutions with massive gravitons in the bigravity theory, J. High Energy Phys. 01 (2012) 035.
  92. A. De Felice, A. E. Gümrükçüoğlu, C. Lin, and S. Mukohyama, On the cosmology of massive gravity, Classical Quantum Gravity 30, 184004 (2013).
  93. M. Lagos and P. G. Ferreira, Cosmological perturbations in massive bigravity, J. Cosmol. Astropart. Phys. 12 (2014) 026.
  94. F. Könnig, A. Patil, and L. Amendola, Viable cosmological solutions in massive bimetric gravity, J. Cosmol. Astropart. Phys. 03 (2014) 029.
  95. A. De Felice, A. E. Gümrükçüoğlu, S. Mukohyama, N. Tanahashi, and T. Tanaka, Viable cosmology in bimetric theory, J. Cosmol. Astropart. Phys. 06 (2014) 037.
  96. Y. Akrami, S. F. Hassan, F. Könnig, A. Schmidt-May, and A. R. Solomon, Bimetric gravity is cosmologically viable, Phys. Lett. B 748, 37 (2015).
  97. E. Mörtsell and J. Enander, Scalar instabilities in bimetric gravity: The Vainshtein mechanism and structure formation, J. Cosmol. Astropart. Phys. 10 (2015) 044.
  98. A. De Felice, A. E. Gümrükçüoğlu, and S. Mukohyama, Massive gravity: Nonlinear instability of a homogeneous and isotropic universe, Phys. Rev. Lett. 109, 171101 (2012).
  99. A. De Felice, A. E. Gümrükçüoğlu, C. Lin, and S. Mukohyama, Nonlinear stability of cosmological solutions in massive gravity, J. Cosmol. Astropart. Phys. 05 (2013) 035.
  100. A. Caravano, M. Lüben, and J. Weller, Combining cosmological and local bounds on bimetric theory, J. Cosmol. Astropart. Phys. 09 (2021) 035.
  101. S. Dwivedi and M. Högås, 2D BAO vs 3D BAO: Solving the Hubble tension with bimetric cosmology, Universe 10, 406 (2024).
  102. J. Smirnov, Dynamical dark energy emerges from massive gravity, arXiv:2505.03870.
  103. M. Högås and E. Mörtsell, Bimetric gravity improves the fit to DESI BAO and eases the Hubble tension, arXiv:2507.03743.
  104. D. Comelli, M. Crisostomi, and L. Pilo, FRW cosmological perturbations in massive bigravity, Phys. Rev. D 90, 084003 (2014).
  105. A. R. Solomon, Y. Akrami, and T. S. Koivisto, Linear growth of structure in massive bigravity, J. Cosmol. Astropart. Phys. 10 (2014) 066.
  106. S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008).
  107. M. Fasiello and A. J. Tolley, Cosmological stability bound in massive gravity and bigravity, J. Cosmol. Astropart. Phys. 12 (2013) 002.
  108. M. Fasiello and A. J. Tolley, Cosmological perturbations in massive gravity and the Higuchi bound, J. Cosmol. Astropart. Phys. 11 (2012) 035.
  109. F. Könnig, Higuchi ghosts and gradient instabilities in bimetric gravity, Phys. Rev. D 91, 104019 (2015).
  110. D. Comelli, M. Crisostomi, F. Nesti, and L. Pilo, Spherically symmetric solutions in ghost-free massive gravity, Phys. Rev. D 85, 024044 (2012).
  111. M. S. Volkov, Hairy black holes in the ghost-free bigravity theory, Phys. Rev. D 85, 124043 (2012).
  112. R. Brito, V. Cardoso, and P. Pani, Black holes with massive graviton hair, Phys. Rev. D 88, 064006 (2013).
  113. E. Babichev and A. Fabbri, Rotating black holes in massive gravity, Phys. Rev. D 90, 084019 (2014).
  114. E. Babichev and A. Fabbri, A class of charged black hole solutions in massive (bi) gravity, J. High Energy Phys. 07 (2014) 016.
  115. E. Ayón-Beato, D. Higuita-Borja, and J. A. Méndez-Zavaleta, Rotating (A) dS black holes in bigravity, Phys. Rev. D 93, 024049 (2016).
  116. E. Babichev and R. Brito, Black holes in massive gravity, Classical Quantum Gravity 32, 154001 (2015).
  117. R. Gervalle and M. S. Volkov, Asymptotically flat hairy black holes in massive bigravity, Phys. Rev. D 102, 124040 (2020).
  118. E. Babichev, R. Brito, and P. Pani, Linear stability of nonbidiagonal black holes in massive gravity, Phys. Rev. D 93, 044041 (2016).
  119. F. Könnig and L. Amendola, Instability in a minimal bimetric gravity model, Phys. Rev. D 90, 044030 (2014).
  120. M. Lüben, A. Schmidt-May, and J. Smirnov, Vainshtein screening in bimetric cosmology, Phys. Rev. D 102, 123529 (2020).
  121. J. M. Martin-Garcia, A. García-Parrado, A. Stecchina, B. Wardell, C. Pitrou, D. Brizuela et al., xact: Efficient tensor computer algebra for the Wolfram Language, http://www.xact.es.

Outline

Information

Phys. Rev. D 113, 084054– Published 27 April, 2026
  • Received 25 November 2025
  • Accepted 5 March 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation