2011

Theoretical Aspects of Massive Gravity

Hinterbichler, Kurt

Understand

Massive gravity has seen a resurgence of interest due to recent progress which has overcome its traditional problems, yielding an avenue for addressing important open questions such as the cosmological constant naturalness problem.

The possibility of a massive graviton has been studied on and off for the past 70 years.
During this time, curiosities such as the vDVZ discontinuity and the Boulware-Deser ghost were uncovered.
We re-derive these results in a pedagogical manner, and develop the St\"ukelberg formalism to discuss them from the modern effective field theory viewpoint.

Built on

G. Nordstrom Ann. Phys
1913
Earlier work this paper cites.
G. Nordstrom Ann. Phys
1913
Earlier work this paper cites.
A. Einstein and A. Fokker Ann. Phys
1914
Earlier work this paper cites.
A. Einstein, “The foundation of the general theory of relativity,” Annalen Phys
1916
Earlier work this paper cites.
T. Kaluza, “On the Problem of Unity in Physics,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. )
1921
Earlier work this paper cites.
O. Klein, “Quantum theory and five-dimensional theory of relativity,” Z. Phys
1926
Earlier work this paper cites.
G. Gabadadze, “Cargese lectures on brane induced gravity,” Nucl. Phys. Proc. Suppl
Original
1929
Earlier work this paper cites.
M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,” Proc. Roy. Soc. Lond
1939
Earlier work this paper cites.
S. N. Gupta, “Gravitation and Electromagnetism,” Phys. Rev
1954
Earlier work this paper cites.
R. H. Kraichnan, “Special-Relativistic Derivation of Generally Covariant Gravitation Theory,” Phys. Rev
1955
Earlier work this paper cites.
E. C. G. Stukelberg Helv. Phys. Act
1957
Earlier work this paper cites.
R. L. Arnowitt, S. Deser, and C. W. Misner, “Canonical variables for general relativity,” Phys. Rev
1960
Earlier work this paper cites.
S. V. Babak and L. P. Grishchuk, “Finite-range gravity and its role in gravitational waves, black holes and cosmology,” Int. J. Mod. Phys
1960
Earlier work this paper cites.
S. Weinberg, “Photons and gravitons in perturbation theory: Derivation of Maxwell’s and Einstein’s equations,” Phys. Rev
1965
Earlier work this paper cites.
N. T. Evans, “Discrete Series for the Universal Covering Group of the 3+2 de Sitter Group,” Journal of Mathematical Physics
1967
Earlier work this paper cites.
C. Deffayet, S. Deser, and G. Esposito-Farese, “Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors,” Phys. Rev
Original
1967
Earlier work this paper cites.
P. G. O. Freund and Y. Nambu, “Scalar field coupled to the trace of the energy-momentum tensor,” Phys. Rev
1968
Earlier work this paper cites.
S. Deser, “Self-interaction and gauge invariance,” Gen. Rel. Grav
1970
Earlier work this paper cites.
S. Deser and L. Halpern Gen. Relativ. Gravit
1970
Earlier work this paper cites.
H. van Dam and M. J. G. Veltman, “Massive and massless Yang-Mills and gravitational fields,” Nucl. Phys
1970
Earlier work this paper cites.
V. I. Zakharov JETP Letters (Sov. Phys.)
1970
Earlier work this paper cites.
Y. Iwasaki, “Consistency condition for propagators,” Phys. Rev. D
1970
Earlier work this paper cites.
A. I. Vainshtein, “To the problem of nonvanishing gravitation mass,” Phys. Lett
1972
Earlier work this paper cites.
D. G. Boulware and S. Deser, “Can gravitation have a finite range?,” Phys. Rev
1972
Earlier work this paper cites.
P. Van Nieuwenhuizen, “On ghost-free tensor lagrangians and linearized gravitation,” Nucl. Phys
1973
Earlier work this paper cites.
P. Van Nieuwenhuizen, “Radiation of massive gravitation,” Phys.Rev
1973
Earlier work this paper cites.
L. P. S. Singh and C. R. Hagen, “Lagrangian formulation for arbitrary spin. 1. The boson case,” Phys. Rev
1974
Earlier work this paper cites.
D. G. Boulware and S. Deser, “Classical General Relativity Derived from Quantum Gravity,” Ann. Phys
1975
Earlier work this paper cites.
C. Fronsdal, “Massless Fields with Integer Spin,” Phys. Rev
1978
Earlier work this paper cites.
J. Fang and C. Fronsdal, “Elementary particles in a curved space. 5. massive and massless spin-2 fields,” Lett.Math.Phys
1978
Earlier work this paper cites.
K. S. Stelle, “Classical Gravity with Higher Derivatives,” Gen. Rel. Grav
1978
Earlier work this paper cites.
J. Fang and C. Fronsdal, “Deformation of gauge groups: gravitation,” J. Math. Phys
1979
Earlier work this paper cites.
C. Fronsdal, “Smooth massless limit of field theories,” Nucl.Phys
1980
Earlier work this paper cites.
C. Aragone and S. Deser, “Consistency problems of spin-2 gravity couplings,” Nuovo Cim
1980
Earlier work this paper cites.
S. Deser, R. Jackiw, and S. Templeton, “Topologically massive gauge theories,” Ann. Phys
1982
Earlier work this paper cites.
D. Blas, C. Deffayet, and J. Garriga, “Bigravity and Lorentz-violating Massive Gravity,” Phys. Rev
Original
1982
Earlier work this paper cites.
F. A. Berends, G. J. H. Burgers, and H. Van Dam, “On spin-3 self interactions,” Z. Phys
1984
Earlier work this paper cites.
S. Deser and R. I. Nepomechie, “Gauge Invariance Versus Masslessness In De Sitter Space,” Ann. Phys
1984
Earlier work this paper cites.
C. S. Aulakh and D. Sahdev, “The infinite dimensional gauge structure of Kaluza Klein theories. 1. D = 1+4,” Phys. Lett
1985
Earlier work this paper cites.
R. M. Wald, “Spin-2 fields and general covariance,” Phys. Rev
1986
Earlier work this paper cites.
A. Maheshwari, “A study of the Higgs effect in the five dimensional Kaluza Klein theory,” Pramana
1986
Earlier work this paper cites.
A. Higuchi, “Forbidden mass range for spin-2 field theory in de sitter space time,” Nucl. Phys
1987
Earlier work this paper cites.
J. D. Barrow and F. J. Tipler, “The Anthropic Cosmological Principle,”. Oxford University Press, USA, 1988
1988
Earlier work this paper cites.
S. Weinberg, “The cosmological constant problem,” Rev. Mod. Phys
1989
Earlier work this paper cites.
M. B. Green and C. B. Thorn, “Continuing between closed and open strings,” Nucl. Phys
1991
Earlier work this paper cites.
M. Henneaux and C. Teitelboim, “Quantization of gauge systems,”. Princeton, USA: Univ. Pr. (1992) 520 p
1992
Earlier work this paper cites.
W. Siegel, “Hidden gravity in open string field theory,” Phys. Rev
1994
Earlier work this paper cites.
S. Weinberg, “The Quantum theory of fields. Vol. 1: Foundations,”. Cambridge, UK: Univ. Pr. (1995)
1995
Earlier work this paper cites.
I. Bengtsson, “Note on massive spin-2 in curved space,” J. Math. Phys
1995
Earlier work this paper cites.
A. Cucchieri, M. Porrati, and S. Deser, “Tree level unitarity constraints on the gravitational couplings of higher spin massive fields,” Phys. Rev
1995
Earlier work this paper cites.
J. M. Overduin and P. S. Wesson, “Kaluza-Klein gravity,” Phys. Rept
1997
Earlier work this paper cites.
M. Henneaux, “Consistent interactions between gauge fields: The cohomological approach,” Contemp. Math
1998
Earlier work this paper cites.
Supernova Search Team
1998
Earlier work this paper cites.
F. J. de Urries and J. Julve, “Ostrogradski formalism for higher-derivative scalar field theories,” J. Phys
1998
Earlier work this paper cites.
C.-K. Chow and S.-J. Rey, “Chiral perturbation theory for tensor mesons,” JHEP
1998
Earlier work this paper cites.
Supernova Cosmology Project
1999
Earlier work this paper cites.
A. Naqvi, “Propagators for massive symmetric tensor and p-forms in AdS(d+1),” JHEP
1999
Earlier work this paper cites.
A. Polishchuk, “Massive symmetric tensor field on AdS,” JHEP
1999
Earlier work this paper cites.
L. Randall and R. Sundrum, “A large mass hierarchy from a small extra dimension,” Phys. Rev. Lett
1999
Earlier work this paper cites.
G. R. Dvali, G. Gabadadze, and M. Porrati, “4D gravity on a brane in 5D Minkowski space,” Phys. Lett
2000
Earlier work this paper cites.
I. L. Buchbinder, D. M. Gitman, V. A. Krykhtin, and V. D. Pershin, “Equations of motion for massive spin 2 field coupled to gravity,” Nucl. Phys
2000
Earlier work this paper cites.
L. Brink, R. R. Metsaev, and M. A. Vasiliev, “How massless are massless fields in AdS(d),” Nucl. Phys
2000
Earlier work this paper cites.

Similar

I. L. Buchbinder, D. M. Gitman, and V. D. Pershin, “Causality of massive spin 2 field in external gravity,” Phys. Lett
2000
Cited alongside, same era.
G. R. Dvali, G. Gabadadze, and M. Porrati, “Metastable gravitons and infinite volume extra dimensions,” Phys. Lett
2000
Cited alongside, same era.
C. Deffayet, “Cosmology on a brane in Minkowski bulk,” Phys. Lett
2001
Cited alongside, same era.
S. Deser and A. Waldron, “Stability of massive cosmological gravitons,” Phys. Lett
2001
Cited alongside, same era.
S. Deser and A. Waldron, “Partial masslessness of higher spins in (A)dS,” Nucl. Phys
2001
Cited alongside, same era.
S. Deser and A. Waldron, “Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations,” Phys. Lett
2001
Cited alongside, same era.
M. Porrati, “No van Dam-Veltman-Zakharov discontinuity in AdS space,” Phys. Lett
2001
Cited alongside, same era.
I. I. Kogan, S. Mouslopoulos, and A. Papazoglou, “The massless limit for massive graviton in dS(4) and AdS(4): How to circumvent the van Dam-Veltman-Zakharov discontinuity,” Phys. Lett
2001
Cited alongside, same era.
A. Karch, E. Katz, and L. Randall, “Absence of a VVDZ discontinuity in AdS(AdS),” JHEP
2001
Cited alongside, same era.
N. Arkani-Hamed, A. G. Cohen, and H. Georgi, “(De)constructing dimensions,” Phys. Rev. Lett
2001
Cited alongside, same era.
N. Arkani-Hamed, A. G. Cohen, and H. Georgi, “Electroweak symmetry breaking from dimensional deconstruction,” Phys. Lett
2001
Cited alongside, same era.
N. Boulanger, T. Damour, L. Gualtieri, and M. Henneaux, “Inconsistency of interacting, multigraviton theories,” Nucl. Phys
2001
Cited alongside, same era.

Then

P. Batra, K. Hinterbichler, L. Hui, and D. N. Kabat, “Pseudo-redundant vacuum energy,” Phys. Rev
Original
2008
Later among the works it cites.
G. Gabadadze and A. Iglesias, “Special Massive Spin-2 on de Sitter Space,” JCAP
Original
2008
Later among the works it cites.
C. Deffayet, “Spherically symmetric solutions of massive gravity,” Class. Quant. Grav
2008
Later among the works it cites.
Z. Berezhiani, D. Comelli, F. Nesti, and L. Pilo, “Exact Spherically Symmetric Solutions in Massive Gravity,” JHEP
Original
2008
Later among the works it cites.
Z. Kakushadze, “Gravitational Higgs Mechanism and Massive Gravity,” Int. J. Mod. Phys
Original
2008
Later among the works it cites.
Z. Kakushadze, “Massless Limit of Gravitational Higgs Mechanism,” Int. J. Geom. Meth. Mod. Phys
Original
2008
Later among the works it cites.
G. Gabadadze and A. Iglesias, “Mass Screening in Modified Gravity,” Class. Quant. Grav
Original
2008
Later among the works it cites.
M. Osipov and V. Rubakov, “Superluminality in the Fierz–Pauli massive gravity,” Class. Quant. Grav
Original
2008
Later among the works it cites.
E. Babichev, V. Mukhanov, and A. Vikman, “k-Essence, superluminal propagation, causality and emergent geometry,” JHEP
Original
2008
Later among the works it cites.
M. Porrati and R. Rahman, “Intrinsic Cutoff and Acausality for Massive Spin 2 Fields Coupled to Electromagnetism,” Nucl. Phys
Original
2008
Later among the works it cites.
R. Gregory, “The three burials of Melquiades DGP,” Prog. Theor. Phys. Suppl
Original
2008
Later among the works it cites.
G. Dvali, O. Pujolas, and M. Redi, “Non Pauli-Fierz Massive Gravitons,” Phys. Rev. Lett
Original
2008
Later among the works it cites.
C. de Rham, “The Effective Field Theory of Codimension-two Branes,” JHEP
Original
2008
Later among the works it cites.
C. de Rham, S. Hofmann, J. Khoury, and A. J. Tolley, “Cascading Gravity and Degravitation,” JCAP
Original
2008
Later among the works it cites.
W. Li, W. Song, and A. Strominger, “Chiral Gravity in Three Dimensions,” JHEP
Original
2008
Later among the works it cites.
D. Grumiller and N. Johansson, “Instability in cosmological topologically massive gravity at the chiral point,” JHEP
Original
2008
Later among the works it cites.
V. A. Rubakov and P. G. Tinyakov, “Infrared-modified gravities and massive gravitons,” Phys. Usp
Original
2008
Later among the works it cites.
K. Izumi and T. Tanaka, “No de Sitter invariant vacuum in massive gravity theory with ghost,” Prog. Theor. Phys
Original
2009
Later among the works it cites.
E. Babichev, C. Deffayet, and R. Ziour, “Recovering General Relativity from massive gravity,” Phys. Rev. Lett
Original
2009
Later among the works it cites.
E. Dyer and K. Hinterbichler, “Boundary Terms, Variational Principles and Higher Derivative Modified Gravity,” Phys. Rev
Original
2009
Later among the works it cites.
D. A. Demir and N. K. Pak, “General Tensor Lagrangians from Gravitational Higgs Mechanism,” Class. Quant. Grav
Original
2009
Later among the works it cites.
C. Deffayet, G. Esposito-Farese, and A. Vikman, “Covariant Galileon,” Phys. Rev
Original
2009
Later among the works it cites.
C. de Rham, “An Introduction to Cascading Gravity and Degravitation,” Can. J. Phys
Original
2009
Later among the works it cites.
J. Khoury and M. Wyman, “N-Body Simulations of DGP and Degravitation Theories,” Phys. Rev
Original
2009
Later among the works it cites.
K. C. Chan and R. Scoccimarro, “Large-Scale Structure in Brane-Induced Gravity II. Numerical Simulations,” Phys. Rev
Original
2009
Later among the works it cites.
F. Schmidt, “Cosmological Simulations of Normal-Branch Braneworld Gravity,” Phys. Rev
Original
2009
Later among the works it cites.
G. Gabadadze, “General Relativity With An Auxiliary Dimension,” Phys. Lett
Original
2009
Later among the works it cites.
E. A. Bergshoeff, O. Hohm, and P. K. Townsend, “Massive Gravity in Three Dimensions,” Phys. Rev. Lett
Original
2009
Later among the works it cites.
I. Oda, “Renormalizability of Massive Gravity in Three Dimensions,” JHEP
Original
2009
Later among the works it cites.
K. Skenderis, M. Taylor, and B. C. van Rees, “Topologically Massive Gravity and the AdS/CFT Correspondence,” JHEP
Original
2009
Later among the works it cites.
E. Kiritsis and V. Niarchos, “Interacting String Multi-verses and Holographic Instabilities of Massive Gravity,” Nucl. Phys
Original
2009
Later among the works it cites.
V. Niarchos, “Multi-String Theories, Massive Gravity and the AdS/CFT Correspondence,” Fortsch. Phys
Original
2009
Later among the works it cites.
M. Maeno and I. Oda, “Massive Gravity in Curved Cosmological Backgrounds,” Int. J. Mod. Phys
Original
2009
Later among the works it cites.
V. P. Nair, S. Randjbar-Daemi, and V. Rubakov, “Massive Spin-2 fields of Geometric Origin in Curved Spacetimes,” Phys. Rev
Original
2009
Later among the works it cites.
D. Bessada and O. D. Miranda, “CMB anisotropies induced by tensor modes in massive gravity,” JCAP
Original
2009
Later among the works it cites.
D. Bessada and O. D. Miranda, “CMB Polarization and Theories of Gravitation with Massive Gravitons,” Class. Quant. Grav
Original
2009
Later among the works it cites.
A. De Felice and S. Tsujikawa, “f(R) theories,” Living Rev. Rel
Original
2010
Later among the works it cites.
C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action,” Phys. Rev
Original
2010
Later among the works it cites.
E. Babichev, C. Deffayet, and R. Ziour, “The recovery of General Relativity in massive gravity via the Vainshtein mechanism,” Phys. Rev
Original
2010
Later among the works it cites.
A. H. Chamseddine and V. Mukhanov, “Higgs for Graviton: Simple and Elegant Solution,” JHEP
Original
2010
Later among the works it cites.
I. Oda, “Remarks on Higgs Mechanism for Gravitons,” Phys. Lett
Original
2010
Later among the works it cites.
I. Oda, “Higgs Mechanism for Gravitons,” Mod. Phys. Lett
Original
2010
Later among the works it cites.
K. Hinterbichler and J. Khoury, “Symmetron Fields: Screening Long-Range Forces Through Local Symmetry Restoration,” Phys. Rev. Lett
Original
2010
Later among the works it cites.
C. de Rham and A. J. Tolley, “DBI and the Galileon reunited,” JCAP
Original
2010
Later among the works it cites.
R. Maartens and K. Koyama, “Brane-World Gravity,” Living Rev. Rel
Original
2010
Later among the works it cites.
C. de Rham, J. Khoury, and A. J. Tolley, “Cascading Gravity is Ghost Free,” Phys. Rev
Original
2010
Later among the works it cites.
C. de Rham, “Massive gravity from Dirichlet boundary conditions,” Phys. Lett
Original
2010
Later among the works it cites.
H. Lu and Y. Pang, “Seven-Dimensional Gravity with Topological Terms,” Phys. Rev
Original
2010
Later among the works it cites.
D. Grumiller and I. Sachs, “AdS3/LCFT2 Correlators in Cosmological Topologically Massive Gravity,” JHEP
Original
2010
Later among the works it cites.
M. R. Gaberdiel, D. Grumiller, and D. Vassilevich, “Graviton 1-loop partition function for 3-dimensional massive gravity,” JHEP
Original
2010
Later among the works it cites.
A. Maloney, W. Song, and A. Strominger, “Chiral Gravity, Log Gravity and Extremal CFT,” Phys. Rev
Original
2010
Later among the works it cites.
A. Mironov, S. Mironov, A. Morozov, and A. Morozov, “Resolving Puzzles of Massive Gravity with and without violation of Lorentz symmetry,” Class. Quant. Grav
Original
2010
Later among the works it cites.
L. Grisa and L. Sorbo, “Pauli-Fierz Gravitons on Friedmann-Robertson-Walker Background,” Phys. Lett
Original
2010
Later among the works it cites.
F. Berkhahn, D. D. Dietrich, and S. Hofmann, “Self-Protection of Massive Cosmological Gravitons,” JCAP
Original
2010
Later among the works it cites.
S. Dubovsky, R. Flauger, A. Starobinsky, and I. Tkachev, “Signatures of a Graviton Mass in the Cosmic Microwave Background,” Phys. Rev
Original
2010
Later among the works it cites.
Particle Data Group
2010
Later among the works it cites.
A. S. Goldhaber and M. M. Nieto, “Photon and Graviton Mass Limits,” Rev. Mod. Phys
Original
2010
Later among the works it cites.
L. Alberte, A. H. Chamseddine, and V. Mukhanov, “Massive Gravity: Exorcising the Ghost,” JHEP
Original
2011
Closest in time.
S. F. Hassan, S. Hofmann, and M. von Strauss, “Brane Induced Gravity, its Ghost and the Cosmological Constant Problem,” JCAP
Original
2011
Closest in time.
F. Capela and P. G. Tinyakov, “Black Hole Thermodynamics and Massive Gravity,” JHEP
Original
2011
Closest in time.
M. Wyman, “Galilean-invariant scalar fields can strengthen gravitational lensing,” Phys. Rev. Lett
Original
2011
Closest in time.
C. de Rham et al
Original
2072
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…