Theoretical aspects of massive gravity
Kurt Hinterbichler
*
Center for Particle Cosmology, Department of Physics and Astronomy,
University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA
(published 7 May 2012)
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 van Dam, Veltman, and
Zakharov (vDVZ) discontinuity and the BoulwareDeser ghost were uncovered. These results are
rederived in a pedagogical manner and the Stu
¨ckelberg formalism to discuss them from the modern
effective Feld theory viewpoint is developed. Recent progress of the last decade is reviewed,
including the dissolution of the vDVZ discontinuity via the Vainshtein screening mechanism, the
existence of a consistent effective Feld theory with a stable hierarchy between the graviton mass and
the cutoff, and the existence of particular interactions which raise the maximal effective Feld theory
cutoff and remove the ghosts. In addition, some peculiarities of massive gravitons on curved space,
novel theories in three dimensions, and examples of the emergence of a massive graviton from extra
dimensions and brane worlds are reviewed.
DOI:
10.1103/RevModPhys.84.671
PACS numbers: 04.50.Kd, 11.10.
±
z
CONTENTS
I. Introduction
671
A. General relativity is massless spin 2
671
B. Modifying general relativity
673
C. History and outline
674
II. The ±ree ±ierzPauli Action
675
A. Hamiltonian and degree of freedom count
675
B. ±ree solutions and graviton mode functions
676
C. Propagator
678
III. Linear Response to Sources
679
A. General solution to the sourced equations
680
B. Solution for a point source
680
C. The vDVZ discontinuity
681
IV. The St
€u
ckelberg Trick
682
A. Vector example
682
B. Graviton Stu
¨ckelberg and origin of the
vDVZ discontinuity
683
V. Nonlinear Interactions
685
A. Massive general relativity
685
B. Spherical solutions and the Vainshtein radius
687
C. Nonlinear Hamiltonian and the BoulwareDeser mode 688
VI. The Nonlinear St
ckelberg ±ormalism
690
A. Stu
¨ckelberg for gravity and the restoration of
diffeomorphism invariance
690
B. Another way to Stu
¨ckelberg
692
VII. St
ckelberg Analysis of Interacting Massive Gravity
693
A. Decoupling limit and breakdown of linearity
693
B. Ghosts
694
C. Resolution of the vDVZ discontinuity and
the Vainshtein mechanism
694
D. Quantum corrections and the effective theory
695
VIII. The
±
3
Theory
697
A. Tuning interactions to raise the cutoff
697
B. The appearance of Galileons and the absence of ghosts
698
C. The
±
3
Vainshtein radius
699
D. The Vainshtein mechanism in the
±
3
theory
700
E. Quantum corrections in the
±
3
theory
700
IX. Brane worlds and the Resonance Graviton
701
A. The DGP action
701
B. Linear expansion
703
C. Resonance gravitons
705
X. Conclusions and ±uture Directions
706
Appendix: Total Derivative Combinations
707
I. INTRODUCTION
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 Fall '13
 Gravity, Mass, Quantum Field Theory, String Theory, General Relativity, The Land, Gauge theory, massive gravity

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