"Massive Gravity"
Claudia de Rham 
1 Introduction
2 Massive and Interacting Fields
2.1 Proca field
2.2 Spin-2 field
2.3 From linearized diffeomorphism to full diffeomorphism invariance
2.4 Non-linear Stückelberg decomposition
2.5 Boulware–Deser ghost
I Massive Gravity from Extra Dimensions
3 Higher-Dimensional Scenarios
4 The Dvali–Gabadadze–Porrati Model
4.1 Gravity induced on a brane
4.2 Brane-bending mode
4.3 Phenomenology of DGP
4.4 Self-acceleration branch
4.5 Degravitation
5 Deconstruction
5.1 Formalism
5.2 Ghost-free massive gravity
5.3 Multi-gravity
5.4 Bi-gravity
5.5 Coupling to matter
5.6 No new kinetic interactions
II Ghost-free Massive Gravity
6 Massive, Bi- and Multi-Gravity Formulation: A Summary
7 Evading the BD Ghost in Massive Gravity
7.1 ADM formulation
7.2 Absence of ghost in the Stückelberg language
7.3 Absence of ghost in the vielbein formulation
7.4 Absence of ghosts in multi-gravity
8 Decoupling Limits
8.1 Scaling versus decoupling
8.2 Massive gravity as a decoupling limit of bi-gravity
8.3 Decoupling limit of massive gravity
8.4 Λ3-decoupling limit of bi-gravity
9 Extensions of Ghost-free Massive Gravity
9.1 Mass-varying
9.2 Quasi-dilaton
9.3 Partially massless
10 Massive Gravity Field Theory
10.1 Vainshtein mechanism
10.2 Validity of the EFT
10.3 Non-renormalization
10.4 Quantum corrections beyond the decoupling limit
10.5 Strong coupling scale vs cutoff
10.6 Superluminalities and (a)causality
10.7 Galileon duality
III Phenomenological Aspects of Ghost-free Massive Gravity
11 Phenomenology
11.1 Gravitational waves
11.2 Solar system
11.3 Lensing
11.4 Pulsars
11.5 Black holes
12 Cosmology
12.1 Cosmology in the decoupling limit
12.2 FLRW solutions in the full theory
12.3 Inhomogenous/anisotropic cosmological solutions
12.4 Massive gravity on FLRW and bi-gravity
12.5 Other proposals for cosmological solutions
IV Other Theories of Massive Gravity
13 New Massive Gravity
13.1 Formulation
13.2 Absence of Boulware–Deser ghost
13.3 Decoupling limit of new massive gravity
13.4 Connection with bi-gravity
13.5 3D massive gravity extensions
13.6 Other 3D theories
13.7 Black holes and other exact solutions
13.8 New massive gravity holography
13.9 Zwei-dreibein gravity
14 Lorentz-Violating Massive Gravity
14.1 SO(3)-invariant mass terms
14.2 Phase m1 = 0
14.3 General massive gravity (m0 = 0)
15 Non-local massive gravity
16 Outlook

13 New Massive Gravity

13.1 Formulation

Independently of the formal development of massive gravity in four dimensions described above, there has been interest in constructing a purely three dimensional theory of massive gravity. Three dimensions are special for the following reason: for a massless graviton in three dimensions there are no propagating degrees of freedom. This follows simply by counting, a symmetric tensor in three dimensions has six components. A massless graviton must admit a diffeomorphism symmetry which renders three of the degrees of freedom pure gauge, and the remaining three are non-dynamical due to the associated first class constraints. On the contrary, a massive graviton in three dimensions has the same number of degrees of freedom as a massless graviton in four dimensions, namely two. Combining these two facts together, in three dimensions it should be possible to construct a diffeomorphism invariant theory of massive gravity. The usual massless graviton implied by diffeomorphism invariance is absent and only the massive degree of freedom remains.

A diffeomorphism and parity invariant theory in three dimensions was given in [66] and referred to as ‘new massive gravity’ (NMG). In its original formulation the action is taken to be

[ ( ) ] 1 ∫ 3 √ --- 1 μν 3 2 SNMG = -2- d x − g σR + --2 RμνR − -R , (13.1 ) κ m 8
where κ2 = 1∕M3 defines the three dimensional Planck mass, σ = ±1 and m is the mass of the graviton. In this form the action is manifestly diffeomorphism invariant and constructed entirely out of the metric gμν. However, to see that it really describes a massive graviton, it is helpful to introduce an auxiliary field f μν which, as we will see below, also admits an interpretation as a metric, to give a quasi-bi-gravity formulation
[ ] ∫ 3 √ --- μν 1 2 μν 2 SNMG = M3 d x − g σR − q G μν − -m (qμνq − q ) . (13.2 ) 4
The kinetic term for qμν appears from the mixing with Gμν. Although this is not a true bi-gravity theory, since there is no direct Einstein–Hilbert term for qμν, we shall see below that it is a well-defined decoupling limit of a bi-gravity theory, and for this reason it makes sense to think of qμν as effectively a metric degree of freedom. In this form, we see that the special form of R2 − 3∕8R2 μν was designed so that q μν has the Fierz–Pauli mass term. It is now straightforward to see that this corresponds to a theory of massive gravity by perturbing around Minkowski spacetime. Defining
gμν = ημν + √-1---hμν, (13.3 ) M3
and perturbing to quadratic order in h μν and q μν we have
∫ [ ] 3 σ μν αβ μν αβ 1 2 μν 2 S2 = M3 d x − --h ˆℰμν hαβ − q ˆℰμν h αβ −-m (qμνq − q ) . (13.4 ) 2 4
Finally, diagonalizing as hμν = &tidle;hμν − σqμν we obtain
∫ [ ] 3 σ-&tidle;μν ˆα β&tidle; σ- μν ˆαβ 1- 2 μν 2 S2 = M3 d x − 2 h ℰμν hαβ + 2 q ℰμν qαβ − 4 m (qμνq − q ) , (13.5 )
which is manifestly a decoupled massless graviton and massive graviton. Crucially, however, we see that the kinetic terms of each have the opposite sign. Since only the degrees of freedom of the massive graviton q μν are propagating, unitarity when coupled to other sources forces us to choose σ = − 1. The apparently ghostly massless graviton does not lead to any unitarity violation, at least in perturbation theory, as there is no massless pole in the propagator. The stability of the vacua was further shown in different gauges in Ref. [252].

13.2 Absence of Boulware–Deser ghost

The auxiliary field formulation of new massive gravity is also useful for understanding the absence of the BD ghost [141*]. Setting σ = − 1 as imposed previously and working with the formulation (13.2*), we can introduce new vector and scalar degrees of freedom as follows

qμν = √1---¯qμν + ∇ μV ν + ∇ νV μ, (13.6 ) M3
1 ∇ π V μ = √------A μ + √--μ----, (13.7 ) M3m M3m2
where the factors of √M3--- and m are chosen for canonical normalization. A μ represents the helicity-1 mode which carries 1 degree of freedom and π the helicity-0 mode that carries 1 degree freedom. These two modes carries all the dynamical fields.

Introducing new fields in this way also introduced new symmetries. Specifically there is a U(1) symmetry

π → π + m χ, A → A − χ, (13.8 ) μ μ
and a linear diffeomorphism symmetry
√ -- ¯qμν → ¯qμν + ∇ μχ ν + ∇ νχ μ, A μ → A μ − m χμ. (13.9 )
Substituting in the action, integrating by parts and using the Bianchi identity ∇ μG μν = 0 we obtain
[ ∫ √--- ∘ ---- SNMG = d3x − g − M3R − M3 q¯μνG μν (13.10 ) 1 ( 2 − -- (m ¯qμν + ∇μA ν + ∇ νAμ + --∇ μ∇ νπ)2 4 ] m 2 2) − (m ¯q + 2∇A + m- □π ) .
Although this action contains apparently higher order terms due to its dependence on ∇ μ∇ νπ, this dependence is Galileon-like in that the equations of motion for all fields are second order. For instance the naively dangerous combination
(∇ μ∇ νπ)2 − (□ π)2 (13.11 )
is up to a boundary term equivalent to μ ν R μν∇ ∇ π. In [141*] it is shown that the resulting equations of motion of all fields are second order due to these special Fierz–Pauli combinations.

As a result of the introduction of the new gauge symmetries, we straightforwardly count the number of non-perturbative degrees of freedom. The total number of fields are 16: six from gμν, six from qμν, three from A μ and one from π. The total number of gauge symmetries are 7: three from diffeomorphisms, three from linear diffeomorphisms and one from the U (1). Thus, the total number of degrees of freedom are 16 − 7(gauge ) − 7(constraint) = 2 which agrees with the linearized analysis. An independent argument leading to the same result is given in [317] where NMG including its topologically massive extension (see below) are presented in Hamiltonian form using Einstein–Cartan language (see also [176]).

13.3 Decoupling limit of new massive gravity

The formalism of Section 13.2 is also useful for deriving the decoupling limit of NMG which as in the higher dimensional case, determines the leading interactions for the helicity-0 mode. The decoupling limit [141] is defined as the limit

∘ ---- 2 2∕5 M3 → ∞, m → 0 Λ5∕2 = ( M3m ) = fixed. (13.12 )
As usual the metric is scaled as
gμν = ημν + √-1---hμν, (13.13 ) M3
and in the action
[ ∫ √--- ∘ ---- SNMG = d3x − g − M3R − M3 q¯μνG μν (13.14 ) 1 ( 2 − -- (m ¯qμν + ∇μA ν + ∇ νAμ + --∇ μ∇ νπ)2 4 ] m 2 2) − (m ¯q + 2∇A + m- □π ) ,
the normalizations have been chosen so that we keep Aμ and π fixed in the limit. We readily find
∫ [ S = d3x + 1h μν ˆℰαβh − q¯μν ˆℰαβh − ¯qμν(∂ ∂ π − η □ π) dec 2 μν αβ μν αβ μ ν μν ] 1 μν 1 μν αβ 2 − 4-FμνF + -5∕2ℰˆαβ h (∂μπ∂ νπ − ημν(∂π) ) , (13.15 ) Λ5∕2
where all raising and lowering is understood with respect to the 3 dimensional Minkowski metric. Performing the field redefinition &tidle; h μν = 2πημν + hμν + ¯qμν we finally obtain
[ ∫ 1 1 Sdec = d3x + --&tidle;hμν ˆℰαμβν &tidle;hαβ −-¯qμνℰˆαμβν ¯qαβ 2 2 ] − 1F F μν − 1(∂ π)2 − --1--(∂π )2□π . (13.16 ) 4 μν 2 2Λ5∕2 5∕2
Thus, we see that in the decoupling limit, NMG becomes equivalent to two massless gravitons which have no degrees of freedom, one massless spin-1 particle which has one degree of freedom, and one scalar π which has a cubic Galileon interaction. This confirms that the strong coupling scale for NMG is Λ5 ∕2.

The decoupling limit clarifies one crucial aspect of NMG. It has been suggested that NMG could be power counting renormalizable following previous arguments for topological massive gravity [196] due to the softer nature of divergences in three-dimensional and the existence of a dimensionless combination of the Planck mass and the graviton mass. This is in fact clearly not the case since the above cubic interaction is a non-renormalizable operator and dominates the Feynman diagrams leading to perturbative unitarity violation at the strong coupling scale Λ5∕2 (see Section 10.5 for further discussion on the distinction between the breakdown of perturbative unitarity and the breakdown of the theory).

13.4 Connection with bi-gravity

The existence of the NMG theory at first sight appears to be something of an anomaly that cannot be reproduced in higher dimensions. There also does not at first sight seem to be any obvious connection with the diffeomorphism breaking ghost-free massive gravity model (or dRGT) and multi-gravity extensions. However, in [425*] it was shown that NMG, and certain extensions to it, could all be obtained as scaling limits of the same 3-dimensional bi-gravity models that are consistent with ghost-free massive gravity in a different decoupling limit. As we already mentioned, the key to seeing this is the auxiliary formulation where the tensor f μν is related to the missing extra metric of the bi-gravity theory.

Starting with the 3-dimensional version of bi-gravity [293] in the form

∫ [ ] 3 Mg √ --- Mf ∘ ---- 2 S = d x -2-- − gR [g ] +-2-- − fR [f ] − m 𝒰 [g,f ] , (13.17 )
where the bi-gravity potential takes the standard form in terms of characteristic polynomials similarly as in (6.4*)
3 𝒰 [g,f] = − Me-ff∑ α ℒ (𝒦 ), (13.18 ) 4 n n n=0
and 𝒦 is given in (6.7*) in terms of the two dynamical metrics g and f. The scale Me ff is defined as M −1= M −g1 + M −1 eff f. The idea is to define a scaling limit [425*] as follows
Mf → + ∞ (13.19 )
keeping M = − (M + M ) 3 g f fixed and keeping q μν fixed in the definition
M3-- f μν = gμν − M qμν. (13.20 ) f
Since 𝒦 μν → M2M3f-qμν, then we have in the limit
∫ [ 3 ( ) ] 3 M3-√ --- M3--μν m2Me--ff∑ n -M3- n S = d x − 2 − gR[g] − 2 q G μν(g) + 4 αn(− 1) M ℒn (q) n=0 f
which prompts the definition of a new set of coefficients
( )n (−-1)n ¯ M3-- cn = − 2M3 M αn Mf , (13.21 )
so that
[ ] ∫ √ --- ∑3 S = M3 d3x − − gR [g] − qμνG μν(g) − m2 cnℒn(q) . (13.22 ) n=0
Since this theory is obtained as a scaling limit of the ghost-free bi-gravity action, it is guaranteed to be free from the BD ghost. We see that in the case c2 = 1∕4, c3 = c4 = 0 we obtain the auxiliary field formulation of NMG, justifying the connection between the auxiliary field qμν and the bi-gravity metric fμν.

13.5 3D massive gravity extensions

The generic form of the auxiliary field formulation of NMG derived above [425]

∫ [ 3 ] 3 √ --- μν 2 ∑ S = M3 d x − − gR [g ] − q G μν(g) − m cnℒn (q) , (13.23 ) n=0
demonstrates that there exists a two parameter family extensions of NMG determined by nonzero coefficients for c3 and c4. The purely metric formulation for the generic case can be determined by integrating out the auxiliary field q μν. The equation of motion for q μν is given symbolically
∑3 − G − m2 ncn 𝜖𝜖qn−1g3−n = 0. (13.24 ) n=1
This is a quadratic equation for the tensor qμν. Together, these two additional degrees of freedom give the cubic curvature [447] and Born–Infeld extension NMG [279*]. Although additional higher derivative corrections have been proposed based on consistency with the holographic c-theorem [424], the above connection suggests that Eq. (13.23*) is the most general set of interactions allowed in NMG which are free from the BD ghost.

In the specific case of the Born–Infeld extension [279] the action is

∫ [ ∘ --------------------] 2 3 √ --- -1- SB.I = 4m M3 d x − g − − det [gμν − m2 G μν] . (13.25 )
It is straightforward to show that on expanding the square root to second order in 1∕m2 we recover the original NMG action. The specific case of the Born–Infeld extension of NMG, also has a surprising role as a counterterm in the AdS4 holographic renormalization group [329]. The significance of this relation is unclear at present.

13.6 Other 3D theories

13.6.1 Topological massive gravity

In four dimensions, the massive spin-2 representations of the Poincaré group must come in positive and negative helicity pairs. By contrast, in three dimensions the positive and negative helicity states are completely independent. Thus, while a parity preserving theory of massive gravity in three dimensions will contain two propagating degrees of freedom, it seems possible in principle for there to exist an interacting theory for one of the helicity modes alone. What is certainly possible is that one can give different interactions to the two helicity modes. Such a theory necessarily breaks parity, and was found in [180, 179]. This theory is known as ‘topologically massive gravity’ (TMG) and is described by the Einstein–Hilbert action, with cosmological constant, supplemented by a term constructed entirely out of the connection (hence the name topological)

∫ [ ] M3 3 √ --- 1 λμν ρ σ 2 σ τ S = ---- d x − g(R − 2Λ ) + --𝜖 Γ λσ ∂μ Γρν + -Γ μτΓ νρ . (13.26 ) 2 4μ 3
The new interaction is a gravitational Chern–Simons term and is responsible for the parity breaking. More generally, this action may be supplemented to the NMG Lagrangian interactions and so the TMG can be viewed as a special case of the full extended parity violating NMG.

The equations of motion for topologically massive gravity take the form

G + Λg + 1C = 0, (13.27 ) μν μν μ μν
where Cμν is the Cotton tensor which is given by
αβ 1- Cμν = 𝜖μ ∇ α(R βν − 4gβνR ). (13.28 )
Einstein metrics for which Gμν = − Λg μν remain as a subspace of general set of vacuum solutions. In the case where the cosmological constant is negative Λ = − 1 ∕ℓ2 we can use the correspondence of Brown and Henneaux [78] to map the theory of gravity on an asymptotically AdS3 space to a 2D CFT living at the boundary.

The AdS/CFT in the context of topological massive gravity was also studied in Ref. [449*].

13.6.2 Supergravity extensions

As with any gravitational theory, it is natural to ask whether extensions exist which exhibit local supersymmetry, i.e., supergravity. A supersymmetric extension to topologically massive gravity was given in [182]. An N = 1 supergravity extension of NMG including the topologically massive gravity terms was given in [21*] and further generalized in [67*]. The construction requires the introduction of an ‘auxiliary’ bosonic scalar field S so that the form of the action is

∫ [ ] 1 3 √ --- 1 1 1 1 S = -2- d x − g M ℒC + σℒE.H. + --2 ℒK + ---2ℒR2 + --2ℒS4 + --ℒS3 κ ∫ m 8&tidle;m ˆm ˆμ + d3x 1ℒ , (13.29 ) μ top
ℒC = S + fermions (13.30 ) ℒE.H.= R − 2S2 + fermions (13.31 ) 1 3 ℒK = K − --S2R − -S4 + fermions (13.32 ) 2[ 2 ] 2 9- 2 1- 2 ℒR2 = − 16 (∂S ) − 4(S + 6 R) + fermions (13.33 ) 3 ℒS4 = S4 + --RS2 + fermions (13.34 ) 10 ℒ 3 = S3 + 1RS + fermions (13.35 ) S 2 1 ρ [ 2 ] ℒtop = --𝜖λμνΓ λσ ∂ μΓ σρν +-Γ σμτΓ τνρ + fermions. (13.36 ) 4 3
The fermion terms complete each term in the Lagrangian into an independent supersymmetric invariant. In other words supersymmetry alone places no further restrictions on the parameters in the theory. It can be shown that the theory admits supersymmetric AdS vacua [21, 67*]. The extensions of this supergravity theory to larger numbers of supersymmetries is considered at the linearized level in [64].

Moreover, N = 2 supergravity extensions of TMG were recently constructed in Ref. [370] and its N = 3 and N = 4 supergravity extensions in Ref. [371].

13.6.3 Critical gravity

Finally, let us comment on a special case of three dimensional gravity known as log gravity [65] or critical gravity in analogy with the general dimension case [384, 183, 16]. For a special choice of parameters of the theory, there is a degeneracy in the equations of motion for the two degrees of freedom leading to the fact that one of the modes of the theory becomes a ‘logarithmic’ mode.

Indeed, at the special point μℓ = 1, (where ℓ is the AdS length scale, Λ = − 1∕ ℓ2), known as the ‘chiral point’ the left-moving (in the language of the boundary CFT) excitations of the theory become pure gauge and it has been argued that the theory then becomes purely an interacting theory for the right moving graviton [93*]. In Ref. [377] it was earlier argued that there was no massive graviton excitations at the critical point μℓ = 1, however Ref. [93] found one massive graviton excitation for every finite and non-zero value of μℓ, including at the critical point μ ℓ = 1.

This case was further analyzed in [273*], see also Ref. [274*] for a recent review. It was shown that the degeneration of the massive graviton mode with the left moving boundary graviton leads to logarithmic excitations.

To be more precise, starting with the auxiliary formulation of NMG with a cosmological constant 2 λm

∫ √ --- [ 1 ] SNMG = M3 d3x − g σR − 2λm2 − qμνG μν − -m2 (qμνqμν − q2) , (13.37 ) 4
we can look for AdS vacuum solutions for which the associated cosmological constant Λ = − 1∕ℓ2 in G μν = − Λgμν is not the same as 2 λm. The relation between the two is set by the vacuum equations to be
1 2 2 − ---2Λ − Λ σ + λm = 0, (13.38 ) 4m
which generically has two solutions. Perturbing the action to quadratic order around this vacuum solution we have
∫ [ ] 3 ¯σ μν μν 1 2 μν 2 S2 = M3 d x − 2h 𝒢μν − q 𝒢μν − 4m (qμνq − q ) . (13.39 )
𝒢 μν(h ) = ˆℰαμβν hαβ − 2Λh μν + Λ ¯gμνh (13.40 )
Λ ¯σ = σ − ---2 (13.41 ) 3m
where we raise and lower the indices with respect to the background AdS metric ¯gμν.

As usual, it is apparent that this theory describes one massless graviton (with no propagating degrees of freedom) and one massive one whose mass is given by M 2 = − m2σ¯. However, by choosing ¯σ = 0 the massive mode becomes degenerate with the existing massless one.

In this case, the action is

∫ [ ] 3 μν 1 2 μν 2 S2 = M3 d x − q 𝒢μν − 4-m (qμνq − q ) , (13.42 )
and varying with respect to hμν and qμν we obtain the equations of motion
𝒢μν(q) = 0, (13.43 ) 1- 𝒢μν(h) + 2(qμν − ¯gμνq) = 0. (13.44 )
Choosing the gauge μ ∇ hμν − ∇ νh = 0, the equations of motion imply h = 0 and the resulting equation of motion for hμν takes the form
[□ − 2 Λ]2hμν = 0. (13.45 )
It is this factorization of the equations of motion into a square of an operator that is characteristic of the critical/log gravity theories. Although the equation of motion is solved by the usual massless models for which [□ − 2Λ]h μν = 0, there are additional logarithmic modes which do not solve this equation but do solve Eq. (13.45*). These are so-called because they behave logarithmically in ρ asymptotically when the AdS metric is put in the form 2 2 2 2 2 2 2 d ρ = ℓ (− cosh(ρ) dτ + dρ + sinh (ρ ) d𝜃 ). The presence of these log modes was shown to remain beyond the linear regime, see Ref. [271].

Based on this result as well as on the finiteness and conservation of the stress tensor and on the emergence of a Jordan cell structure in the Hamiltonian, the correspondence to a logarithmic CFT was conjectured in Ref. [273], where the to be dual log CFTs representations have degeneracies in the spectrum of scaling dimensions.

Strong indications for this correspondence appeared in many different ways. First, consistent boundary conditions which allow the log modes were provided in Ref. [272], were it was shown that in addition to the Brown–Henneaux boundary conditions one could also consider more general ones. These boundary conditions were further explored in [305, 397], where it was shown that the stress-energy tensor for these boundary conditions are finite and not chiral, giving another indication that the theory could be dual to a logarithmic CFT.

Then specific correlator functions were computed and compared. Ref. [449] checked the 2-point correlators and Ref. [275] the 3-point ones. A similar analysis was also performed within the context of NMG in Ref. [270] where the 2-point correlators were computed at the chiral point and shown to behave as those of a logarithmic CFT.

Further checks for this AdS/log CFT include the 1-loop partition function as computed in Ref. [241]. See also Ref. [274] for a review of other checks.

It has been shown, however, that ultimately these theories are non-unitary due to the fact that there is a non-zero inner product between the log modes and the normal models and the inability to construct a positive definite norm on the Hilbert space [432].

13.7 Black holes and other exact solutions

A great deal of physics can be learned from studying exact solutions, in particular those corresponding to black hole geometries. Black holes are also important probes of the non-perturbative aspects of gravitational theories. We briefly review here the types of exact solutions obtained in the literature.

In the case of topologically massive gravity, a one-parameter family of extensions to the BTZ black hole have been obtained in [245]. In the case of NMG as well as the usual BTZ black holes obtained in the presence of a negative cosmological constant there are in a addition a class of warped AdS3 black holes [102] whose metric takes the form

2 2 ( 2 )2 2 2 2 ρ-−-ρ-0 2 2 ρ +-(1 −-β-)ω- --1----dρ--- ds = − β r2 dt + r dϕ − r2 dt + β2 ζ2ρ2 − ρ20, (13.46 )
where the radial coordinate r is given by
2 2 2 2 -β2-ρ20- r = ρ + 2 ωρ + ω (1 − β ) + 1 − β2 . (13.47 )
and the parameters β and ζ are determined in terms of the graviton mass m and the cosmological constant Λ by
∘ ------------- 2 9-−-21Λ-∕m2-∓-2---3(5 +-7Λ∕m2-- −2 21-−-4-β2 β = 4(1 − Λ∕m2 ) , ζ = 8m2 . (13.48 )
This metric exhibits two horizons at ρ = ± ρ0, if β2 ≥ 0 and ρ0 is real. Absence of closed timelike curves requires that β2 ≤ 1. This puts the allowed range on the values of Λ to be
35m2 m2 − 289---≤ Λ ≤ 21-. (13.49 )
AdS waves, extensions of plane (pp) waves anti-de Sitter spacetime have been considered in [33]. Further work on extensions to black hole solutions, including charged black hole solutions can be found in [418, 101, 253, 5, 6, 372, 427, 250]. We note in particular the existence of a class of Lifshitz black holes [32*] that exhibit the Lifshitz anisotropic scale symmetry
z t → λ t, ⃗x → λ⃗x, (13.50 )
where z is the dynamical critical exponent. As an example for z = 3 the following Lifshitz black hole can be found [32]
6 ( 2) 2 ds2 = − r- 1 − M-l-- dt2 + (-2dr---) + r2dϕ2. (13.51 ) ℓ6 r2 rℓ2-− M
This metric has a curvature singularity at r = 0 and a horizon at √ --- r+ = ℓ M. The Lifshitz symmetry is preserved if we scale t → λ3t, x → λx, r → λ −1r and in addition we scale the black hole mass as M → λ −2M. The metric should be contrasted with the normal BTZ black hole which corresponds to z = 1
( ) 2 r2 M ℓ2 2 dr2 2 2 ds = − -2 1 − --2-- dt + (r2-----)-+ r dϕ . (13.52 ) ℓ r ℓ2 − M
Exact solutions for charged black holes were also derived in Ref. [249] and an exact, non-stationary solution of TMG and NMG with the asymptotic charges of a BTZ black hole was find in [227]. This exact solution was shown to admit a timelike singularity. Other exact asymptotically AdS-like solutions were found in Ref. [251].

13.8 New massive gravity holography

One of the most interesting avenues of exploration for NMG has been in the context of Maldacena’s AdS/CFT correspondence [396]. According to this correspondence, NMG with a cosmological constant chosen so that there are asymptotically anti-de Sitter solutions is dual to a conformal field theory (CFT). This has been considered in [67*, 381, 380] where it was found that the requirements of bulk unitarity actually lead to a negative central charge.

The argument for this proceeds from the identification of the central charge of the dual two dimensional field theory with the entropy of a black hole in the bulk using Cardy’s formula. The entropy of the black hole is given by [368]

ABTZ-- S = 4G3 Ω, (13.53 )
where G3 is the 3-dimensional Newton constant and 2G3 Ω = -3ℓ-c where l is the AdS radius and c is the central charge. This formula is such that c = 1 for pure Einstein–Hilbert gravity with a negative cosmological constant.

A universal formula for this central charge has been obtained as is given by

ℓ ∂ℒ c = ----gμν-----. (13.54 ) 2G3 ∂R μν
This result essentially follows from using the Wald entropy formula [480] for a higher derivative gravity theory and identifying this with the central change through the Cardy formula. Applying this argument for new massive gravity we obtain [67]
( ) c = 3-ℓ- σ + --1--- . (13.55 ) 2G3 2m2 ℓ2
Since σ = − 1 is required for bulk unitarity, we must choose m2 > 0 to have a chance of getting c positive. Then we are led to conclude that the central charge is only positive if
Λ = − 1-< − 2m2. (13.56 ) ℓ2
However, unitarity in the bulk requires m2 > − Λ∕2 and this excludes this possibility. We are thus led to conclude that NMG cannot be unitary both in the bulk and in the dual CFT. This failure to maintain both bulk and boundary unitarity can be resolved by a modification of NMG to a full bi-gravity model, namely Zwei-Dreibein gravity to which we turn next.

13.9 Zwei-dreibein gravity

As we have seen, there is a conflict in NMG between unitarity in the bulk, i.e., the requirement that the massive gravitons are not ghosts, and unitarity in dual CFT as required by the positivity of the central charge. This conflict may be resolved, however, by replacing NMG with the 3-dimensional bi-gravity extension of ghost-free massive gravity that we have already discussed. In particular, if we work in the Einstein–Cartan formulation in three dimensions, then the metric is replaced by a ‘dreibein’ and since this is a bi-gravity model, we need two ‘dreibeins’. This gives us the Zwei-dreibein gravity [63*].

In the notation of [63*] the Lagrangian is given by

ℒ = − σM e Ra(e) − M f Ra (f) − 1m2M α 𝜖 eaebec 1 a 2 a 6 1 1abc 1- 2 a b c 1- 2 a b c a b c − 6m M2 α2𝜖abcf f f + 2m M12 𝜖abc(β1e e f + β2e f f ) (13.57 )
where we have suppressed the wedge products 3 e = e ∧ e ∧ e, a R (e) is Lorentz vector valued curvature two-form for the spin-connection associated with the dreibein e and Ra(f) that associated with the dreibein f. Since we are in three dimensions, the spin-connection can be written as a Lorentz vector dualizing with the Levi-Civita symbol ωa = 𝜖abcω bc. This is nothing other than the vierbein representation of bi-gravity with the usual ghost-free (dRGT) mass terms. As we have already discussed, NMG and its various extensions arise in appropriate scaling limits.

A computation of the central charge following the same procedure was given in [63*] with the result that

c = 12πℓ(σM1 + γM2 ). (13.58 )
Defining the parameter γ via the relation
(α2(σM1 + M2 ) + β2M2 )γ2 + 2(M2 β1 − σM1 β2)γ − σ (α1(σM1 + M2 ) + β1M1 ) = 0, (13.59 )
then bulk unitarity requires γ∕(σM1 + γM2 ) < 0. In order to have c > 0 we thus need γ < 0 which in turn implies σ > 1 (since M 1 and M 2 are defined as positive). The absence of tachyons in the AdS vacuum requires β1 + γβ2 > 0, and this assumes a real solution for γ for a negative Λ. There are an open set of such solutions to these conditions, which shows that the conditions for unitarity are not finely tuned. For example in [63] it is shown that there is an open set of solutions which are close to the special case M1 = M2, β1 = β2 = 1, γ = 1 and α = α = 3∕2 + --1- 1 2 ℓ2m2. This result is not in contradiction with the scaling limit that reproduces NMG, because this scaling limit requires the choice σ = − 1 which is in contradiction with positive central charge.

These results potentially have an impact on the higher dimensional case. We see that in three dimensions we potentially have a diffeomorphism invariant theory of massive gravity (i.e., bi-gravity) which at least for AdS solutions exhibits unitarity both in the bulk and in the boundary CFT for a finite range of parameters in the theory. However, these bi-gravity models are easily extended into all dimensions as we have already discussed and it is similarly easy to find AdS solutions which exhibit bulk unitarity. It would be extremely interesting to see if the associated dual CFTs are also unitary thus providing a potential holographic description of generalized theories of massive gravity.

  Go to previous page Scroll to top Go to next page