Solid holography and massive gravity
JHE
Solid holography and massive gravity
Lasma Alberte 0 1 2 4 5
Matteo Baggioli 0 1 2 3 5
Andrei Khmelnitsky 0 1 2 4 5
Oriol Pujolas 0 1 2 5
0 Campus UAB , 08193 Bellaterra (Barcelona) , Spain
1 The Barcelona Institute of Science and Technology , BIST
2 Strada Costiera 11 , 34151, Trieste , Italy
3 Department of Physics, Institute for Condensed Matter Theory, University of Illinois
4 Abdus Salam International Centre for Theoretical Physics
5 1110 W. Green Street, Urbana, IL 61801 , U.S.A
Momentum dissipation is an important ingredient in condensed matter physics that requires a translation breaking sector. In the bottomup gauge/gravity duality, this implies that the gravity dual is massive. We start here a systematic analysis of holographic massive gravity (HMG) theories, which admit eld theory dual interpretations and which, therefore, might store interesting condensed matter applications. We show that there are many phases of HMG that are fully consistent e ective eld theories and which have been left overlooked in the literature. The most important distinction between the di erent HMG phases is that they can be clearly separated into solids and uids. This can be done both at the level of the unbroken spacetime symmetries as well as concerning the elastic properties of the dual materials. We extract the modulus of rigidity of the solid HMG black brane solutions and show how it relates to the graviton mass term. We also consider the implications of the di erent HMGs on the electric response. We show that the types of response that can be consistently described within this framework is much wider than what is captured by the narrow class of models mostly considered so far.
Holography and condensed matter physics (AdS/CMT); Spontaneous Sym

2.1
2.2
2.3
2.4
3.1
3.2
5.1
5.2
5.3
5.4
1 Introduction and motivation 2
Phases of Holographic Massive Gravity
General mass terms
Symmetries
Probe limit Lagrangian with T
Stressenergy tensor and the background solution
Quadratic mass terms
Transverse phonons
Linear and nonlinear consistency
6
Conclusions
A Second order action  linear consistency
A.1 Vector modes
A.2 Scalar modes
B Stability analysis in eikonal approximation
Introduction and motivation
Gravity (MG) has been unveiled: MG theories are especially relevant for CM applications
by way of the AdS/CFT correspondence [1{4]. This is quite appealing because CM contains
a large variety of poorly understood systems, whereas both MG and AdS/CFT have enjoyed
a vast development in the last decade.
The fact that CM and MG must be connected somehow is actually very natural: any
distribution of matter gives rise to a plasma mass for the graviton, much in the same way
the photon gets a mass in a medium with a nite density of freely moving charges. Since a
the UV region in comparison to the LI massive gravity theories.1 Thus, it seems to be
possible to match the great diversity of types of condensed matter materials with the great
diversity of consistent phases of LVMG.
This connection, surely quite obvious to most MG experts, has always been clear but
seemed of little consequence because the graviton plasma mass inside a material with an
energy density
can be estimated as mg2
MPl2, which is much smaller than the inverse
with the open problems in cosmology (for a recent review see, e.g. [8]).
In AdS/CFT correspondence [9], though, this connection opens an entirely new
dimension. MGs in AdSlike spaces are dual to strongly coupled materials that incorporate
a crucial aspect of CM: a sector that breaks spatial translation invariance. From the CM
perspective, including this sector in the lowenergy description is of central importance to
capture e.g. the e ects from phonons and from disorder (see below).
In our view, however, the way how MG plays this central role has not been clearly
stated yet in the AdS/CMT literature. To make contact with the AdS/CMT applications,
we shall stick to the common terminology that refers to the MG theories in AdS (or more
general spaces that allow a holographic interpretation) as Holographic Massive Gravity
(HMG). The main aim of this paper is to clarify the role, consistency, and classi cation of
HMG theories. The key messages will be:
1. HMGs can be broadly and clearly separated into solids and uids.2 This follows both
from the analysis of the residual di eomorphisms that are preserved as well as from
the presence/absence of an elastic shear response.
energy scales [7].
2. The large family of HMGs considered below are very healthy: they are fully
consistent as E ective Field Theories (EFTs) in that they are free of ghosts and other
pathologies at linear and nonlinear level, even if they are not necessarily of dRGT
type [10]. In addition, and in contrast to the application to cosmology, in holography
the graviton mass does not need to be so much smaller than the Planck mass: it
su ces that it is around the AdS curvature scale 1=L wich can be taken to be, say,
one to two orders of magnitude below MPl. In this case, the strong coupling scale
(which is usually a geometric mean of the form
n
(mgn 1MPl)1=n for n = 2; 3; 5)
associated with the graviton mass sector does not di er signi cantly from the Planck
scale, and in any case is not below 1=L. This is quite important for the holographic
application because it allows the gravitational description to continue to be weakly
1Recently it has been shown that some forms of LVMG can even be UVcompleted to almost Planckian
2A more thorough classi cation of the possible phases of HMG including super uid relatives as well as
other distinctions along the holographic direction is deferred for future work.
{ 2 {
coupled (at least at low frequencies and long wavelengths) even in the presence of a
fully nonlinear mass sector.
3. The class of HMGs relevant for CM is much more general than the dRGT models [10]
that have been considered mostly so far (see e.g. [1, 2, 4]), as was already argued
in [11]. The family of theories that are free from various pathologies (as discussed
e.g. in [11, 12]) and have the same symmetries is parameterized by free functions
that are related to the physical properties of the materials. This has some practical
implications because there are many properties of the dRGTlike massive gravities
that are not generic in the AdS/CMT context, and therefore can lead to misguided
All in all, this suggests that holographic massive gravities might be very relevant for
condensed matter applications.
In this paper we shall use both the presentation of massive gravity in terms of broken
di eomorphisms and in its covariantized form relying on the Stuckelberg elds
A. In the
latter language MG can be seen as a theory of general relativity coupled to a number of
scalar elds. The application of the scalar elds formalism to holographic massive gravity
was initiated in [13, 14] and was instrumental in identifying the additional degrees of
freedom as phonons [11].
The most meaningful way to characterize the various di erent phases of massive gravity
is phrased in terms of the symmetries that are left unbroken in each phase. The situation
is very similar to the organization of the EFTs for various materials according to the
spontaneous symmetry breaking pattern of the time and/or space translations, [15{18].
The classi cation of the phases of MG proceeds in very similar terms because on physical
grounds the LV mass terms are the possible forms of the plasmamass that is generated
in di erent types of materials. In other words, the EFTs of solids/ uids/etc. are related
to the phases of LVMG by gauging the spacetime symmetries, that is, by introducing the
coupling to the dynamical metric by the usual covariantization prescription. Thus it is not
at all surprising that one can speak of, e.g., solid and
uid phases of MG. For the sake of
simplicity, we shall restrict ourselves to these two cases and defer for future work a more
thorough analysis of other phases of MG.
As is nowadays relatively wellunderstood, the EFT for uids and solids in at space
involves a set of phonon scalars I (in 2+1 dimensions, I = 1; 2) that enjoy internal shift
and rotation symmetries for homogeneous and isotropic materials [15{18]. The internal
symmetry group for solids is the twodimensional Euclidean group of translations and
rotations. For uids, the internal group is much bigger and includes also volume preserving
di eomorphisms (VPDi s). The scalars acquire an expectation value
break the product of the (space transformations)
(internal transformations) to the
diagonal subgroup. For uids, the preserved symmetry includes a volume preserving diagonal
I
=
iI xi and
subgroup.
{ 3 {
(1.1)
(1.2)
The e ective Lagrangian at the lowest order in derivatives in the two cases can be
written as [18{21]
L(solids) = Vs(X; Z)
and
L( uids) = Vf (Z) ;
where X = tr IIJ and Z = det IIJ with IIJ
J .3 The functions Vs;f encode
the linear and nonlinear properties of the solid and uid, and they are free functions subject
to mild consistency constraints. It is easy to realize that gauging these theories leads to
graviton mass terms around the solution with
I
= iI xi. The simplest way to see this is
to replace
with g
and to go to the unitary gauge where the scalar elds are xed to be
equal to their background con guration. The above solid/ uid Lagrangians then become
nonlinear potential terms for the metric
Vs tr gij ; det gij
and
Vf det gij ;
where gij denotes the spatial part of the inverse metric. We note that exactly the same
procedure was followed in the socalled `solid in ation', ref. [21] (see also [22] for a generic
EFT description of broken spatial di eomorphisms on cosmological backgrounds).
At this point, it is quite clear that there must exist a fully equivalent nonlinear
formulation of solid/ uid MGs that is phrased entirely in terms of a unitarygauge metric variable.
The form of this action is dictated by requiring it to be invariant under certain subset of the
di eomorphisms, that do not include the spatial di eomorphisms xi 7! x~i(t; xj ). The
preserved di eomorphisms are the ones enjoyed by the potential terms in (1.2). Both for solid
and
and rotations that force the potential not to depend explicitly on xi and to contract the
uid MGs, these include the timereparametrizations t 7! f (t) plus global translations
spatial indices with Kronecker delta ij .4 For uid MG the potential is also invariant under
the spatial VPDi s, that forces it to be a function of det gij only. Importantly, as we shall
see below, the VPDi symmetry protects the vanishing of the physical mass parameter of
the metric tensor modes.
In order to link the de ning symmetries of solid/ uid MGs to the structure of mass
parameters, it is important to identify the relevant notion of the mass terms, which is not
entirely obvious since we need to work in curved backgrounds. The most practical de nition
is to expand the action around the solutions of interest and look at the nonderivative
quadratic terms once the kinetic terms have a canonical form. Limiting the discussion
here to homogeneous and isotropic backgrounds, one can follow [5, 6] and parameterize the
possible mass terms by ve constant mass parameters,5
m02h200 + 2m12h20i
m22hi2j + m32hi2i
2m42h00hii :
(1.3)
For uids/solids, the preserved symmetries at the level of the Lagrangian in the unitary
gauge formulation, and the allowed mass terms are as follows:
3In higher dimensions there are more invariants. For instance, in 3+1 dimensions tr IJI IKJ gives an
independent invariant.
4For simplicity, we shall assume that he kinetic part of the action is given by the standard
EinsteinHilbert term, which is not the most general one compatible with these symmetries.
5The precise link between these mass terms and the nonlinear Lagrangian (1.2) will be done in full detail
in sections 2 and 5 in a class of models that covers uids and solids.
{ 4 {
Solids Preserved symmetries: time reparametrizations and the diagonal subgroup of space
translations and rotations. Expressed as in nitesimal di eomorphisms (x
(x )), these are:
t(t), i = ci + ij xi
where ci = const; T =
. The
! x +
label
denotes that it is a combination of a spatial translation/rotation and a
corresponding internal transformation that leaves the background con guration invariant. The
allowed mass terms are m1;2;3 6= 0
Fluids Preserved symmetries: time reparametrizations and the diagonal subgroup of
translations, rotations, and volumepreserving di eomorphisms. In in nitesimal form,
these look like
t(t), Vi P (xj ) (VP = volume preserving, @i i = 0). The allowed
mass terms are m1;3 6= 0. Importantly, for uids m2 = 0.
Let us insist that in both cases the spatial translations are broken (or nonlinearly
realized), which we emphasize with the
label that should remind that this is not a
standard translation. In the realization of these phases as HMGs presented in the main
body of the paper, both cases lead to a
nite DC conductivity in the electric response.
Also, both the uid/solid EFTs and the uid/solid MGs are very healthy in the sense that
the Lagrangian can be always chosen in order to avoid ghosts or other pathologies at linear
and nonlinear levels. We show that explicitly in section 5.
The main di erences between the two types of theories are i) that the solid phases
exhibit propagating transverse phonons  the Goldstone modes of the broken space
translations, inhomogenenous in spatial coordinates  whereas the
uids do not; and ii) that
the tensor modes are massive/massless for solid/ uid phases respectively. In the HMG
constructions below, both statements will hold at the level of local propagating elds in AdS.
However, since local degrees of freedom in the dual eld theory (such as the transverse
phonons) can be overdamped in black brane backgrounds, the di erence in this respect
between the solid/ uid HMGs becomes slightly blurred. Instead, a clear distinction
survives at the level of the tensor modes being massive/massless. In turn, as we shall show
below, this allows to characterize the solid types of HMGs as the ones with rigidity, a
nonzero static elastic shear response, encoded in the nonvanishing m2.
After this quite generic review of the physical distinctions between MG phases, let us
introduce a bit more the speci cs of the holographic application (for some recent
developments concerning the importance of the translation breaking sectors see [1{4, 23{28] and
references therein). As already mentioned, we will focus on asymptotically AdS4
backgrounds, since they admit simple and wellunderstood 2+1 CFT interpretations. We shall
denote by t and xi the coordinates along the boundary directions and r the holographic
coordinate dual to the renormalization group scale. Since the holographic map already gives
r a special role, we consider gravity theories with an anisotropic mass term. The crucial
ingredient to make contact with realistic CM is to break translations in the xi directions.
Thus, in the nonunitary gauge description we are going to need at least two scalars I for
I = 1; 2 with vacuum expectation values
I
= iI xi.6 One can also consider additional
6There are other ways to accomplish the breaking of xitranslations like the socalled holographic lattices,
i.e., explicitly xidependent source terms. For this purpose, these constructions are equivalent to massive
gravity [3] and considerably less convenient to work with.
{ 5 {
radial and temporal elds r and t
. The role of a t
uid/solid versions of the dual material, and the role of r is basically the same as of the
commonlyused dilaton
eld. This identi cation especially holds when
r does not enjoy
an internal translational symmetry, and thus provides a rich variety of radial dependence
in black brane as well as in vacuum solutions (zero temperature and density). For the sake
of simplicity, we shall restrict here mostly to the simplest case that provides momentum
dissipation in the xi directions  a two eld model with the I scalars only.
These two eld HMG models are already quite rich (as they include solid and
uid
/ t scalar is to introduce super
behaviours), simple and interesting. Because of their dual eld theory interpretation, it
seems appropriate to call them solid/ uid CFTs. Indeed, the asymptotically AdS black
brane solutions in the solid/ uid MG admit an interpretation as CFTs with solid/ uid
behaviour that are able to dissipate momentum, and which are deformed only by a
nite
temperature and
nite density. The fact that momentum is not conserved can be seen
as a coupling of a standard momentumconserving CFT to a translationbreaking sector.
The latter is realized in the black brane solutions via the I hair. The magnitude of the
translationbreaking is proportional to the local energy density in the
I sector and is
position (i.e., scale) dependent. Importantly, depending on the form of the potential
V (X; Z) this energy density can be located closer or farther from the black brane horizon.
In the CFT, this characterizes the typical energy scale of the translation breaking. At
least for the solids it seems suggestive and consistent to identify the explicit breaking (that
is localized near the AdS boundary) as some type of lattice disorder in the sense that it
mimics a random distribution of lattice defects in an otherwise perfect crystal. We shall
not deepen much more into the nature of this disorder in this work (for the role that
holographic version of disorder can play in various contexts, see [29{46] and references
therein). Still, having identi ed a clear distinction between solid and
uid materials might
help identifying the nature of disorder, simply because the notion of disorder itself seems
much better de ned for solids than
uids in the rst place.
The focus of our work is to analyze the full set of mass terms that are available in HMG
and to clarify how the various mass terms control di erent transport properties of the dual
materials. Obviously, there are many more mass terms if one restricts to homogeneity
and isotropy of the CFT in the spatial directions, xi, but not in the holographic direction.
Hence, the full classi cation of the phases of HMG is rather complicated. We shall therefore
initiate the analysis of the e ects from the full range of possible mass terms, but will also
restrict to the cases that are simpler to interpret. Along the way, we will perform various
checks such as that the DC conductivity is indeed
nite in both solids and uids, and that
the solids, in addition, enjoy a nontrivial elastic response.
The rest of this paper is organized as follows. In section 2 we set up our notations
and introduce a classi cation of the massive gravity theories according to the preserved
subgroup of the di eomorphism symmetry. We describe two particular cases, on which we
focus in the present paper: the general theory with negligible background stress energy
tensor and the theory written in terms of two Stuckelberg
elds  the simplest theory
describing the solid phase of HMG. Section 3 is devoted to the electric response in the
dual eld theory. We derive the expression for the DC conductivity in terms of the mass
{ 6 {
parameters and discuss the phenomenology of the representative models. We proceed in
section 4 with discussing the elastic properties of the duals to the solid and
uid phases
of HMG. Section 5 contains the analysis of the massive gravity with two scalar
elds.
Section 6 is devoted to the conclusions and outlook. A detailed analysis of the HMGs with
four scalar elds and zero stressenergy tensor is presented in appendices A and B.
2
Phases of Holographic Massive Gravity
For holographic applications in condensed matter theory we are interested in massive
gravity theories that allow for asymptotically AdS charged black brane solutions. The action
that will be considered in this paper is the EinsteinMaxwell action with a negative
cosmological constant and a graviton mass term:7
Here the mass M is a constant of integration that can be determined by demanding that
f (r) vanishes on the horizon r = rh to be
S =
Z
d x
4 p
g
1
2
R +
6
L2
L2
4
F
F
+ L
:
The graviton mass term L can be written as a Lagrangian for the Stuckelberg scalar elds
A and will be speci ed in the following sections. We shall concentrate on the probe limit
and neglect the backreaction from the mass term on the background metric everywhere
apart from section 5. In this limit the action admits an asymptotically AdS
ReissnerNordstrom black brane solution:
ds2 = g^ dx dx = L
2
dr2
f (r)r2 +
f (r)dt2 + dx2 + dy2
r2
;
with the emblackening factor f (r) given by
f (r) = 1
M r3 +
r4 :
2
2rh2
M =
1
r
3 +
h
2
2rh
:
^A = x
A ;
A^t =
1
r
rh
:
;
{ 7 {
The solution for the Stuckelberg scalars and the Maxwell eld takes the form
In the following sections we shall discuss the di erent phenomenological consequences
in the dual theory induced by the presence of the graviton mass in the bulk and investigate
the bulk stability of the perturbations around the black brane background. To do so we
consider the perturbations of the metric and Maxwell elds de ned as
g
= g^
+ h
A
= A^ + a ;
7We work in the units where MP2l = (8 G) 1
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
and the perturbations of the Stuckelberg scalar elds
A de ned as
A = ^A +
A :
Since we are interested in homogenous and isotropic condensed matter systems, we
shall limit ourselves to the mass terms that preserve the constant shifts and rotations of
the transverse coordinates xi = fx; yg. It allows us to classify the perturbations according
to the scalar, vector, and tensor representations of the transverse O(2) rotation group.
The split depends on whether we consider homogeneous modes, i.e. independent on the
transverse coordinates xi, or inhomogeneous modes. For simplicity we shall consider only
the homogeneous case when the perturbations can be classi ed as
of massive gravity is to start with the noncovariant form of the massive gravity action
with the mass term written it terms of the metric perturbation h
and then restore the
di eomorphism invariance by the Stuckelberg trick. A similar analysis of the stability of
the Lorentz violating massive gravity around Minkowski background has been performed
previously in refs. [5, 6].
We de ne the inverse metric perturbations8 as
g
= g^
+ h
g
and write the most general quadratic mass term that preserves the rotations of the
transverse coordinates in the following form:
L (h ; r) =
1
2
m02(r)(htt)2 + 2m12(r)htihti
m22(r)hij hij
+ m32(r)hiihjj
2m42(r)htthii
+ m52(r)(hrr)2 + m62(r)htthrr
+ m72(r)hrihri + m82(r)htihri + m92(r)hrrhii
+ m120(r)hrthrt + m121(r)hrthii + m122(r)htthrt + m123(r)hrrhrt ;
(2.12)
where all the masses mi2 are functions of the radial coordinate. The repeated transverse
coordinate indices are contracted with ij . The numbering of the mass parameters is chosen
to be consistent with the notations of refs. [5, 6]. The novelty introduced by the planar
8We choose to de ne the mass term in terms of the inverse metric perturbations due to the usual Lorentz
invariant convention to write the graviton mass in terms of the matrix g f
where f
is an auxiliary
reference metric.
{ 8 {
AdS black brane background is the special role of the holographic coordinate r. For future
reference we note that the mass parameters de ned in (2.12) can be classi ed with respect
to the perturbations that they a ect as:
We note that this classi cation is valid only for the homogeneous modes.
The mass term (2.12) explicitly breaks the spacetime di eomorphisms since the metric
perturbation h
is not invariant under the coordinate transformations x 7! x~ (x ). In
fact, as we have discussed in the introduction, this can be considered as the de ning
property of massive gravity  a generic massive gravity theory is a theory that breaks
some subset of the di eomorphism invariance of the EinsteinHilbert gravity. In order
to reveal the symmetry breaking pattern of a given massive gravity theory it is useful
to restore the di eomorphism invariance by introducing four Stuckelberg scalar elds
which represent the physical coordinates.9 One can then write the mass term using two
A
ingredients: a gauge invariant version of the spacetime metric
and a reference metric in the con guration space of the scalar elds
I
AB
fAB( C ) ;
which can be a function of the Stuckelberg elds themselves. The metric fAB is used to
raise and lower the scalar elds space indices.10 Any mass term written in terms of I
AB
and fAB will be manifestly invariant under the general coordinate transformations. In
addition, depending on the exact form of the scalar elds Lagrangian, it can be invariant
under certain internal symmetries of the scalar elds A 7!
A( B). The eld con guration
A
= x
A
spontaneously breaks the internal symmetries and the spacetime di eomorphisms to a
diagonal subgroup  it is left invariant only by a combination of simultaneous internal
eld rede nitions and spacetime di eomorphisms. For example, under the transformations
A
7!
A
A ;
x 7! x +
have a covariant gravitational theory.
with the background metric of the spacetime.
the background transforms as
A 7!
A + A
A and it is left invariant if A =
A
.
This is the diagonal subgroup of the internal and spacetime symmetries.
9Also for various holographic applications (e.g. the holographic renormalization [47]) it is desirable to
10The reference metric fAB is usually taken to be the at Minkowski metric AB or is set to coincide
{ 9 {
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
HJEP02(16)4
In general, the scalar elds Lagrangian does not admit the full reparameterisation
invariance of the scalar
elds due to the fact that the reference metric fAB is
xed and,
hence, nondynamic. Thus, the scalar elds background spontaneously breaks the
spacetime di eomorphisms to the product of the residual internal symmetries and their
spacetime counterparts. For example, if the reference metric does not depend on the scalar
elds, the Lagrangian is invariant under constant shifts
A
7!
A + cA and the diagonal
subgroup contains the spacetime translations. An important case is when all the internal
scalar eld indices are contracted in the Lagrangian. Such theories are invariant under the
reparameterisations of the scalar elds that are isometries of the background metric fAB.
This is the case in the Lorentz invariant massive gravity theories where the Minkowski
reference metric leads to the invariance under internal Poincare transformations. In the
context of e ective eld theories of solids and
uids (see [17, 18] and references therein),
the Lagrangian is invariant under rigid rotations and volume preserving di eomorphisms
of the scalar elds, respectively [19]. The background con guration (2.18) breaks these
symmetries down to the diagonal subgroup of internal and spacetime translations and
rotations, or volume preserving di eomorphisms. In the present work we are making the rst
steps towards using the e ective theories of solids and uids in the holographic context and
applying these theories to the condensed matter systems.
use I
Leaving some of the scalar
eld indices not contracted (e.g. when the Lagrangian
may contain a sole I
AB as opposed to I
ABfAB) breaks the internal scalar
eld metric
isometries to some residual subgroup. In general, this leads to Lorentz violating massive
gravity theories which are of the main interest of this paper. In particular, we choose to
preserve the internal O(2) rotations of the transverse scalar elds I
. This allows us to
AB as the building block of our e ective Lagrangian with the only condition that the
transverse indices have to be contracted with a reference metric proportional to IJ . As
described above, the full spacetime di eomorphisms are preserved by the action and are
broken spontaneously on the background con guration (2.18).
2.3
Probe limit Lagrangian with T = 0
A straightforward but not unique way of constructing a gauge invariant form of the most
generic quadratic mass term (2.12) is to set the scalar elds reference metric to be equal
to the background black brane metric g^ :
f AB( C
) = g^ ( C
) A B :
We can then introduce di eomorphism invariant metric perturbations as
HAB
I
AB
f AB( r) :
In the unitary gauge, when the scalar elds are set to be equal to their background
solution (2.18), the eld HAB equals to the inverse metric perturbations:
The covariant form of the action (2.12) is then obtained by replacing the di erent
components of the metric perturbations h
with the gauge invariant elds HAB and the explicit
(2.20)
(2.21)
(2.22)
rdependence of the masses with an explicit dependence on the radial Stuckelberg eld r
The resulting covariant mass term L (HAB; r) gives a stressenergy tensor that vanishes
on the background and, thus, does not contribute to the background equations for the
metric. Although the exact vanishing of the backreaction on the metric is a very particular
case it can be considered as a probe limit approximation for a generic massive gravity
theory. In particular, the number of propagating degrees of freedom as well as the stability
requirements of the theory should not be altered in this limit. The exact form of the
covariant mass term is given in eq. (A.1) in appendix A, where also the linear stability analysis
of the theory is performed. The phenomenological consequences of the mass term (2.12)
relevant for the electric response of the dual eld theory will be discussed in section 3.
We note here the obvious fact that the number of scalar elds used in our Lagrangian is
a matter of choice. However, it limits the available components of HAB and, consequently,
the subset of mass terms out of the ones presented in (2.12) that arise if not all four scalar
elds
A with A = t; r; x; y are used. Of particular interest for us is the case of only two
scalar elds, i.e. I with I = x; y, that is commonly used in the context of holographic
massive gravity and will be discussed in great detail in section 5. In this case, only the HIJ
components can be constructed which implies that all the quadratic mass terms in (2.12)
involving ft; rg components are absent:
m0 = 0 ; m1 = 0 ;
m4 = 0 ;
m5 = 0 ;
m6 = 0 ;
m7 = 0 ;
m8 = 0 ; m9 = 0 ; m10 = 0 ; m11 = 0 ; m12 = 0 ; m13 = 0 :
(2.23)
(2.24)
2
Moreover, since the black brane reference metric f AB( r) explicitly depends on the absent
eld
r then even HIJ can only be present in a particular combination that includes its
traceless part only, i.e. (HIJ )2
1 H2. This leads to a quadratic mass term of type (2.12)
with
m22 = 2 m32 :
The same conclusion about the available mass terms could have been reached by noting that
the absence of elds t; r in the mass term building blocks I
AB and fAB leads to a residual
symmetry that is left unbroken by the scalar elds background con guration
I
= xi.
These are the spacetime di eomorphisms xa 7! x~a(x ) with a = t; r. Hence, all mass terms
in (2.12) that transform nontrivially under these di eomorphisms are forbidden.
As a result, the above conditions prohibit the appearance of any components of the
metric perturbations that are scalars and vectors under the transverse rotations in quadratic
action. From the phenomenological point of view it is a case of no particular interest. We
note, however, that such strict constraints only arise if we consider generally covariant mass
terms with zero background stressenergy tensor, i.e. with zero backreaction on the metric.
In practice, the mass Lagrangian L (HAB; r) is not the only di eomorphism invariant
mass term that reduces to the quadratic action (2.12) once expanded up to the second
order in perturbation theory. The most general covariant mass terms can be obtained by
writing the action in terms of the matrix I
AB and arbitrary functions of the scalar eld
r
. The quadratic perturbative mass term obtained by these other actions will also be
of the form (2.12). However, a general mass Lagrangian of this sort will give rise to a
nonzero stressenergy tensor that will backreact on the spacetime metric. As a result,
some of the quadratic nonderivative terms arising in the perturbative expansion of L
will
vanish on the background equation of motion once combined with the EinsteinMaxwell
part in the full action (2.1). Hence, there is no onetoone correspondence of the quadratic
nonderivative term of metric perturbations in L
and the actual mass terms of the metric
perturbations in the case of generally covariant mass potentials with nonzero backreaction.
A particular example of a massive gravity action with nonzero stressenergy tensor we
would like to consider is a generic mass term that depends on only the two transverse
Stuckelberg elds I . Such model is a direct generalisation of the two elds dRGT theory,
and provides the simplest way to model momentum dissipation in the dual theory. Since
the action does not depend on the a scalar elds, the background solution
I
= xi iI
leaves the ft; rg spacetime di eomorphisms xa 7! x~a(x ) unbroken in this case. As before,
the most general graviton mass Lagrangian can be written in terms of the two building
blocks IIJ
J and some reference metric fIJ . Since we wish to preserve the
internal rotations in the (I; J ) plane we set the reference metric to be the identity matrix
fIJ = IJ . Hence, the mass Lagrangian is an arbitrary function of the powers of I with
all indices contracted with the identity matrix IJ . For a 2
2 matrix there are only two algebraically independent contractions. We choose them to be
I
I
I ;
I
I
I
J :
g L
=
1
r 1
2
Since any other contraction of I is a function of X and Z, the most general two elds mass
term takes the form:
S
Z
d x
4 p
Z
d x
4 p
g V (X; Z) :
The dRGT theory with two elds considered in ref. [1] is a particular case of the two elds
massive gravity with the Lagrangian given by
VdRGT =
X +
p
Z
p
2 Z :
The Lagrangian (2.27) of the scalar elds is similar to the e ective eld theories used
to describe perfect uids [19] and solids [20, 21]. In this context the scalar elds I denote
the comoving Lagrangian coordinates of the material. In order to describe perfect uids the
scalar elds action has to be invariant under eld space volume preserving di eomorphisms
For solids, the action is only invariant under constant shifts and rotations
I
7!
I ( J ) ;
det
= 1 :
I
7!
I = OJ
I J + cI :
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
We see that, in general, the scalar elds Lagrangian (2.27) describes solids. In the special
case when it depends only on the function Z, its symmetry is enhanced, and it describes
perfect uids. It is the presence of the eld X that \turns" the material into a solid. In this
sense, the two elds Lorentz violating massive gravity is an e ective theory that describes
gravity in the presence of a solid or a uid. We shall present a detailed analysis of the two
elds massive gravity in section 5.
We would like to emphasize that although the simplest massive gravity models written
in terms of the trace of the scalar elds kinetic terms, V = V (X), have been previously
studied in numerous works [11, 13, 14], the general massive gravities with V = V (X; Z)
are poorly explored so far. The only example that has been studied is the dRGT
potential (2.28). In particular, the distinction between the solid and
uid types of theories in
the context of holographic massive gravity is revealed here for the rst time. In the light of
this, a more careful study of the previously overlooked
uid type theories with V = V (Z)
would be of a particular interest. Let us restate that from the CFT perspective these
theories behave like momentum dissipating
uids. This is in contrast with Einstein gravity
black branes which are dual to momentum conserving
uids. A detailed study might thus
provide some deeper insights in the mechanism of the breaking of translational invariance
in the dual boundary theory. We leave this for future work.
3
Electric response
In this section we investigate the phenomenological consequences of allowing for the broad
class of the graviton mass terms as in (2.12). We shall concentrate on the dynamics of the
vector modes in particular and shall discuss the implications for momentum relaxation and
electric response in the dual CFT. Following the discussion of section 2.3 we shall consider
the di eomorphism invariant mass term relevant for vector perturbations
L
=
1
2
2m12( r)HtiHti + m72( r)HriHri + m82( r)HtiHri :
For simplicity we assume that the r dependence of the masses is given by
mi2( r) = M 2( r)f ( r) i mi2 ;
where i = 1; 7; 8, f ( r) is the emblackening factor written in a gauge invariant form, and
M 2( r) is a universal mass function of dimension mass squared that is regular and
nonvanishing at the horizon when r = rh. The masses mi2 and the powers i are dimensionless
constants. A detailed stability analysis of the quadratic Lagrangian in appendices A and B
leads to conditions
8 = 0 and
1 =
7 = 1. We also
nd in appendix B that for
arbitrary mass parameters mi the vector modes of the metric propagate on an e ective
acoustic background metric that is di erent from the ReissnerNordstrom background of
the Maxwell eld. The conditions on the mass parameters for the two e ective light cones
to coincide read
A priori these conditions do not need to be imposed as long as the acoustic metric describes
a causally stable spacetime [48]. In appendix B we nd no indications that this would not
be the case here. Importantly, the mass condition (3.3) is automatically satis ed in the two
elds massive gravity described in 2.4. This means that in these theories both dynamical
vector modes propagate on the same e ective metric. Since the two elds dRGT theory is
a subclass of these theories same conclusion applies. Nevertheless, in what follows we shall
leave the mass parameters mi2 unconstrained unless otherwise speci ed.
We also note that the mass term (3.1) gives rise to a vanishing stressenergy tensor
and as such can be considered as particular. As was already discussed in 2.3 both the
stability and phenomenology of mass terms with nonzero backreaction coincide with the
results presented below in the probe limit of vanishing graviton mass.
In appendix A we
nd that the dynamics of the vector sector of our model can be
described in terms of two gauge invariant vector
elds ai and
i
. The rst one is the
perturbations of the Maxwell eld while the second eld, i, was introduced as a Lagrange
multiplier and can be expressed in terms of the original elds as
i =
rh
ai
1
2r2
r
2
L2 hti
0
+
r
2
2L2 h_ri :
a(t; r) = a(r)e i!t ;
(t; r) = (r)e i!t
Henceforth we shall drop the index i. The resulting equations of motion for the Fourier
modes
then read
2m12 1
f M 2
!
2
i!m82 M1
M
0
m72
with det P = m48
a +
rh
a
= 0 ;
= 0 ;
In this subsection we make use of the analytic approach of ref. [2] for calculating the DC
conductivity (for an alternative approach, see [30]). The method relies on the existence of
a `massless mode' in the bulk that is a linear combination of the Maxwell eld a and the
graviton, which in our calculations is represented by the eld . In zero frequency limit,
the existence of this massless mode implies the conservation of a certain quantity
in the
radial direction. In [2] it was shown that the quantity
encodes the universal behaviour of
the DC conductivity and ensures that it remains constant as one moves from the horizon
to the boundary.
Here we shall repeat the analysis for the equations of motion (3.6), (3.7) with the mass
parameters set to m28 = 0; m27 =
2m21 so that to satisfy the condition (3.3). In this case
the equations of motion (3.6) and (3.7) can be written in the form
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
can be expressed in terms of our parameters m21 and M 2(r) as:
In particular, the 1 term corresponds to the mass function with
=
3 and the 2 term
is equivalent to
=
2.
It is straightforward to check that the mass matrix M has a vanishing determinant
and, thus, one zero eigenvalue. The eigenvectors ~ 1; ~ 2 corresponding to the zero and
nonzero eigenvalues respectively can be taken to be
C ;
A
L2
1
The matrix M can then be diagonalized as M = U DU 1 where D = diag( 1 = 0; 2) and
U = (~ 1 ~ 2) is a matrix with its column vectors given by the eigenvectors ~ i. The massless
and massive modes,
1 and
2, are then:
M = BBB
0
2 2r2
r
2
h
M 2(r)r2 det P
4L2m21
mass matrix M de ned as
The equations (3.8) can be compared to the corresponding equations in [2] for the dRGT
massive gravity. We nd that the dRGT mass
HJEP02(16)4
The equations of motion (3.8) then take the form
1
2
!
a
!
U 1
= (det U ) 1 B
0
B
a +
L2
rh
a +
2rhM 2(r)m21 CC ;
1
A
U 1(r)
f
1
2
!
=
2
0
2
!
:
We emphasize that unless the mass function M 2(r) = const the matrix U(r) is a function
of r. In the zero frequency limit the massless equation can be written in the form of a
radial conservation law
0 = 0 ;
with
f a0 +
L2
f
2rhm12 M 2(r)
0 :
A few remarks are in order. First, we observe that the conserved quantity
can be
rewritten in terms of the massless and massive modes as
This form is very similar to the conserved
found in [2] for the dRGT massive gravity.
Second, we notice that in the unitary gauge the eld 0 is related to the metric
components as (see equations (A.12) in appendix A) relates to
0 =
2m12M 2(r)
r
2
f L4 hti :
= f a0
r
2
rhL2 hti :
i =
Sonshell :
Ai(r)
i(!) = lim
r!0 i! Ai
r!0 i! ai
1
i = lim
Hence, the conserved quantity , as de ned in (3.15), reduces to
This coincides exactly with the radial momentum conjugated to the Maxwell eld that can
be obtained by varying the onshell boundary action with respect to the boundary value
of the Maxwell eld Ai(r) [23, 25]:
The electric conductivity is then de ned as the functional derivative
The fact that in the zero frequency limit, the canonical momentum with respect to the
rfoliation of the Maxwell eld, , is conserved in the radial direction, was used by Iqbal
and Liu in [25] as a motivation to introduce a
ctitious membrane DC conductivity for
each constant r slice as a response to the massless mode
DC(r) = lim
1
(r; !)
!!0 i! (
1
)(r; !)
:
We note that this de nition of conductivity is equivalent to computing the linear response
of the boundary theory to the massless mode
1. In practice, we are interested in the linear
response to the Maxwell eld perturbations A. However, the membrane DC conductivity
has nice properties which we would like to exploit in order to extract information about
the actual DC conductivity de ned as
DC
lim
!!0
(!) = lim lim
!!0 r!0 i! a
1 a0
:
In the limit r ! 0 the two conductivities are related to each other as
r!0 DC = lim lim
lim
r!0 !!0 i!
= lim lim
r!0 !!0 i!
1
1
a0
a
A
+
(
1
) A
(
1
)
+
2L2
2rh2m21M 2(r)
0
:
(3.17)
HJEP02(16)4
(3.16)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
where we have used (3.13) to express the elds a; in terms of
; 2. For the two DC
conductivities to coincide we have to demand that the second term in the above expression
vanishes near the boundary, i.e. that
1
lim
r!0 M 2(r)
0
= 0 :
(3.24)
In the case of M 2(r) = const this means imposing the condition 0(0) = 0 on the boundary.
For other choices of mass function M 2(r) one has to similarly nd the appropriate condition
on the eld
at r = 0. We shall discuss this in more detail in the next section.
The claim of [25] is that the membrane DC conductivity does not evolve in the radial
direction and can be evaluated at arbitrary r. In particular, it can be evaluated at the
horizon, thus, emphasizing the fact that the DC conductivity of the boundary theory can be
entirely determined in terms of the horizon quantities. These ideas were used in the context
of massive holography by Blake and Tong in [2]. They have shown that also in the case
of massive gravity the membrane DC conductivity (3.21) is conserved. Thus, the electric
DC conductivity on the boundary can be evaluated as the membrane DC conductivity at
the horizon. Near the horizon the elds a and
are proportional to f (r) i!=(4 T ), and
the mass function M 2(rh) is regular. Hence f 01 = i! 1 while f 2 vanishes. Using the
equation (3.16) we nd the DC conductivity:
HJEP02(16)4
DC = DC(rh) = 1 +
2L2
2rh2M 2(rh)m21 = det U 1(rh) :
With the identi cation (3.11) this result coincides with the expression for the DC
conductivity in the dRGT massive gravity derived in [2].
Phenomenology.
A phenomenologically interesting question to investigate is the di
erent types of materials that can be described within the framework of holographic massive
gravity and, in particular, their ability to conduct an electric current. The distinction
between di erent classes of materials is well captured by the temperature dependence of their
DC electric conductivity. In the models of massive holography proposed in this paper, it
is controlled by the radial dependence of the mass function M 2(r). This determines the
dependence of the DC conductivity on the horizon temperature through T = jf 0(rh)j =(
4
).
For a mass function of the form
M 2(r) = L 2 r
DC = 1 +
2L2
2m21
2+
:
(3.26)
(3.27)
the DC conductivity becomes
This de nes the distiction between a metallic behaviour (d =dT < 0) and an insulating
behaviour (d =dT > 0). The parameter
determines the nature of the dual CFT as shown
in gure 1. For
<
2 the behaviour is metallic while for the opposite case,
>
2, we
for di erent values of the parameter .
are in the presence of an insulator.11 Since the mass function (3.10) in the dRGT theory
corresponds to the cases
= f 2; 3g, the dual materials exhibit a metallic behaviour
there. The possibility to mimic metallic and insulating behaviour (and a transition between
them) in the context of holographic massive gravity has been already pointed out in [11].
At last, we remark that the special temperature at which all the models with di erent
values of
have the same electric conductivity is given by
T = jf 0(L)j
4
(3.28)
and corresponds to the temperature of a black brane with the horizon radius rh = L.
We further exploit the freedom o ered by the generic power
to investigate the
possibility of having a linear T resistivity
= 1= DC / T 1. This is a special feature of strange
metals (see [26] and references therein) which evade the usual scaling predicted by the
Fermi liquid theory ( DC / T 2). Within our class of models we observe a linear scaling
in the resistivity at low temperature only for the power
=
3 as shown in
gure 2. As
mentioned above, the
=
3 mass function coincides with the 1 term of dRGT massive
gravity. The linear scaling of the DC resistivity for this particular case in the low
temperature regime12 has been already observed earlier in [14]. In [27], the same e ect has been
seen for a more complicated dilatonic model.
11We also note that within holographic models dual to EinsteinMaxwell theory it is impossible to get a
properly de ned insulator (
= 0 at T = 0), as pointed out in [32]. Our results suggest that the conclusion
reached in [32] also holds in
eld theories dual to massive gravity.
12At the best of our knowledge, the only holographic example showing linear T resistivity at high
temperature is [31] where a nontrivial dilatonic solution is exploited.
1.0
0.9
= 1= DC as a function of temperature for parameter values = 3,
= 1, m1 = 1. The dashed line is a linear t / T .
In order to nd the electric conductivity in the dual eld theory we need to numerically
solve the equations (3.6) and (3.7) with appropriate boundary conditions on the black
brane horizon r = rh and on the AdS boundary r = 0.
Boundary conditions on r = 0. In order to see what are the appropriate boundary
conditions for the eld
near the AdS boundary, we expand the equations (3.6) and (3.7)
in the limit r ! 0. We use the following ansatz for the radial dependence of the elds:
and nd that the leading order expansion of the elds a and
near the boundary is
a = r ;
= r ;
M 2(r) = r
a = a0 +
= 0 +
r
L a1 + : : :
+1
r
L
1 + : : : :
We see that the rst equation coincides with the usual near boundary behaviour of the
Maxwell eld perturbations in the EinsteinMaxwell theory. The second equation together
with (3.17) sets the near boundary behaviour of the metric perturbations. For an
everywhere constant mass function M 2 = L2 we recover the usual result for the metric
perturbations hti = L2=r2 ht(i0) (see e.g. [23]).
For M 2(r) = r with
+ 1 < 0 the second term in (3.31) diverges and and the 0 term
becomes subleading with respect to the 1 term. The correct near boundary expansion in
this case would be
For the regularity of the eld near the AdS boundary we shall thus demand that =
L j +1j
r
1 +
r j +1j
L
0 + : : : :
1 = 0(0) = 0
when
+ 1 < 0 :
= 1: left : mass dependence; center : temperature dependence with m2 = 1; right : temperature
dependence with m2 = 1 normalizing the quantity by the T dependent energy density .
5
Solid and uid CFTs
In this section we study in detail the model with two scalar elds, which provides the
simplest e ective description of holographic solids and perfect uids. The two elds mass
term (2.27) is the most general di eomorphism invariant Lagrangian that one can write
using two scalar elds only. All the stability requirements and holographic predictions can,
therefore, be translated directly to the form of the function V . This is a great advantage
of the two
elds massive gravity. However, we shall see below that the phenomenology
of (2.27) is less rich than that arising from the generic mass terms (A.1) that we consider
in appendix A.
5.1
Stressenergy tensor and the background solution
Let us start by asking when such a theory admits an asymptotically AdS black brane
solution. In distinction from the mass terms (A.1), the scalar
elds with action (2.27)
provide a nonvanishing contribution to the background stressenergy tensor even on the
solution (2.5). The scalar elds stressenergy tensor is given by
T
2
p
S
g g
=
i
i jj
I
i
j Iij VZ : (5.1)
Here and in what follows we drop the distinction between the indices I and i, use V
with subscripts to denote partial derivatives, e.g., VX
solution (2.2), (2.5) the quantities X and Z take values X^ = r2 and Z^ = r4, and the
1. On the
stressenergy tensor takes the form
^
Tab =
^
Tij =
g^ab V^ (r) ;
g^ij V^ (r)
r2 V^X (r)
2r4 V^Z (r) :
(5.2)
(5.3)
Here V^ (r)
V (X^ ; Z^) is the background value of the Lagrangian. The second line can be
rewritten in terms of the function V^ (r) only as
^
Tij =
g^ij
This structure exactly matches the structure of the Einstein tensor for the black brane
metric, i.e. Gii = Gaa
r2 ddr2 Gaa. It thus follows that the two elds massive gravity with
the action (2.27) admits an AdS black brane solution with the metric (2.2) for an
arbitrary choice of the function V (X; Z). The background solution of the total action (2.1) is
then completely determined by the rdependence of the background value V^ (r), with the
emblackening factor given by
HJEP02(16)4
f (r) = 1 +
2rh2 + r3 Z r
2r4
dr~r~4
1 V^ (r~) :
The integration constant parametrises the position of the black brane horizon as well as its
mass density and temperature. The fact that the resulting function f (r) is in uenced by the
graviton mass term only through the background value V^ (r) means that there are in nitely
many di erent actions that lead to the same background metric. From equation (5.5) it is
easy to see that, in principle, it is possible to design a two elds Lagrangian that does not
a ect the background metric. However, as we show below such theories do not contain any
propagating vector modes and cannot reproduce the generic mass term of the form (2.12).
This is di erent from the four elds case with zero backreaction studied in appendix A.
In order to nish the discussion of the background solution we also note that the con guration (2.5) is a solution to the equation of motion of the scalar elds for any choice of the function V (X; Z). Indeed, the equation of motion has the form:
On the con guration i = xi, g
= g^
the equation reads
= 0 :
and is satis ed since neither the background metric nor the function V^ (r) depend on xi.
5.2
Quadratic mass terms
In order to connect to the generic graviton mass term in (2.12) let us nd the quadratic mass
terms for the metric perturbations around the black brane. The expansion of the scalar
elds Lagrangian up to the second order in inverse metric perturbations in the unitary
(5.4)
(5.5)
(5.6)
(5.7)
gauge is given by
L
V^ (r) +
1 dV^
dr2 hii
1 d2V^
8 d (r2)2 (hii)2 +
1 d2V^
8 d (r2)2 (hii)2
dr2 (hij )2
1 4 dV^
2
dr2 haihbj g^ab + O(h3) ;
where we have used the relations
It is straightforward to check that the above Lagrangian transforms as a scalar under
the ft; rg di eomorphisms which is expected from the fact that the a elds are absent.
However, in situation when the scalar elds provide a nonvanishing background
stressenergy tensor, the quadratic expansion of the scalar elds Lagrangian L alone is insu cient
to describe the graviton mass term. Instead we shall consider the quadratic expansion of
the full action (2.1) including the EinsteinMaxwell part. As expected, the vector, tensor,
and scalar modes come in the quadratic expansion separately.
The action for the vector modes takes the same form as the vector mode action (A.10)
found in appendix A.1 with the condition (A.17) satis ed and with the mass parameters15
(hii)2 + O(h3)
(5.8)
2m12(r) = f (r)M 2(r) ;
m72(r) =
f (r) 1M 2(r) ;
The mass function M 2(r) introduced in section 3 is given in terms of the function V (X; Z):
M 2(r) =
1 dV^ (r)
r2 dr2 :
These masses automatically have the correct powers of f (r) and satisfy the conditions (3.3),
which guarantee that the gravitational vector mode propagates on the light cone de ned
by the background metric. The mass parameter of the vector modes, the analogue of the
m2(r) in the two elds dRGT theory speci ed in (3.10), is given by r2M 2(r) = dV^ (r) and
it is completely determined by the background value of the scalar elds Lagrangian. The
dr2
expression (3.25) for the DC conductivity in the two elds theory takes the form
DC = 1 + 2
dV^ (rh)
dr2
! 1
:
In a generic two elds massive gravity the rdependence of this mass parameter can be more
general than in the dRGT case (3.10) and is constrained only by the stability requirements.
In particular, from the quadratic action of the vector sector (A.18) one can see that for
15According to these de nitions, the dimensionless mass parameters mi2 take the values 2m12 =
m72 = 1.
(5.10)
(5.11)
(5.12)
to have a healthy kinetic term the vector modes mass parameter has to be
We also note that the functions V (X; Z) with constant background values V^ (r) are
special. In this case the scalar eld background contribution to the stressenergy tensor
takes the form of a cosmological constant and can be reabsorbed in the de nition of 3=L2.
Therefore, the scalar elds have vanishing stressenergy tensor and do not a ect the
background metric. However, it also leads to the vanishing of the vector mass parameter (5.11).
In the vector mode analysis of section A.1 it corresponds to the case when all the mass
parameters relevant for the vector sector are vanishing, m21(r) = m27(r) = m28(r) = 0.
The vector sector of such theory at quadratic level is identical to the one of pure
EinsteinMaxwell theory and does not contain any propagating gravitational degrees of freedom. On
the CFT side it implies the absence of momentum dissipation and in nite value of electric
DC conductivity. We conclude that the two elds action cannot provide a healthy graviton
vector mode without contributing nontrivially to the background stressenergy tensor. It
also implies, that in order to have a holographic massive gravity with propagating vectors
and
nite DC conductivity, which in the meantime does not contribute to the background
stressenergy tensor, one has to add more than two scalar elds.
Let us now turn to the helicitytwo tensor mode of the metric. In 3+1 dimensional
bulk this mode exists only for perturbations homogenous in the transverse directions and
encodes the viscoelastic response of the boundary theory in the holographic description.
The quadratic action for the traceless tensor mode takes the form
S =
Z
d4x
1
4r2
1
f (r)
( h_T )2
f (r)(h0T )2
2V^X (r2) h2T
;
where hT stands for any of the two helicitytwo components, which we parameterise as
hyy and h
Lr22 hxy. This action is equivalent to the transverse traceless
sector of the generic quadratic mass term (2.12) with the mass parameter m22(r) given by
m22(r) =
21r2 V^X (r) :
The mass of the tensor mode (5.15) is not completely determined by the background
behaviour of V^ (r), and is thus independent of the mass of the vector mode and of the value
of the DC conductivity. We would like to emphasize, that the tensor mass vanishes for
the two eld actions that only depend on Z, i.e., for the theories describing perfect uids.
This is to be expected, since such theories possess an additional symmetry of transverse
space di eomorphisms that forbids the appearance of the mass term for the helicity two
mode. This observation also agrees with an earlier nding in the two elds dRGT massive
gravity in [12] that the 2 term in the mass term (2.28) gives zero contribution to the mass
of the transverse graviton. Hence, in dRGT theory the 1 term describes solids, while the
2 term corresponds to a
uid.
In accordance to the symmetry considerations of the section 2.4, the scalar sector of
the theory does not contain any dynamical degrees of freedom. In particular, the mass
(5.14)
(5.15)
parameters m10 and m12 of the quadratic action vanish, and as shown in appendix A.2
all the scalar degrees of freedom can be eliminated by using constraints and are thus
nondynamical.
In order to analyse further the stability of the two elds theory we proceed with the study
of the quadratic action for the inhomogeneous vector modes in the decoupling limit and
derive their propagation speeds. For this we consider the quadratic action for the scalar
eld perturbations i = i
xi while keeping the metric to be xed, i.e. g
= g^ . In the
presence of xi dependence, the two component vector i can be split in the longitudinal
and transverse parts in the following way
i = p
L + p
T :
The quadratic Lagrangian for these components then reads
L
(2) =
1 2 dV^
1 2 dV^
r
r
2
2
2
2
r
2
VX + r2 d2V^ !
^
(dr2)2
where the repeated a indices are contracted with the metric g^abr 2. In accordance with
the full analysis of the homogeneous vector modes, the absence of ghost requires that ddrV^2
is positive. We can also read o the propagation speeds of both vector modes:
c2T = f (r)V^X
c2L = f (r) VX + r2 d2V^ !
^
(dr2)2
dV^ ! 1
dr2
= f (r)
2m22(r)
dr2
;
:
^
VX
0 ;
VX + r2 d2V^
^
(dr2)2
0 :
The absence of gradient instabilities puts additional constraints on the function V (X; Z)
One notices that the speed of propagation of the transverse phonons cT is proportional to
the m2 mass parameter, which is the one that is forced to vanish by the residual VPDi s in
the uid case. Hence, exactly as it happens for uids in at space, the VPDi s are
responsible for decoupling the transverse phonons, iT , by forcing them to be nonpropagating
(the dynamics of these degrees of freedom is related to vortices in a quite interesting way,
see [56]). That the vanishing of cT is equivalent to the vanishing of the tensor mode mass
m2 can be easily understood in terms of the Stuckelberg trick. The reason is that, in the
space part of the metric, hij is accompanied by @(i jT) to form a gauge invariant quantity,
and so the mass term for the transverse traceless mode and the spatial gradient term for
the transverse phonons iT are in fact the same thing.16 Notice, however, that the fact that
16See [21] for how this works in the cosmological solutions produced by solids/ uids. In the uid limit,
cT ! 0 and the tensor modes are massless.
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
the tensor mode mass must vanish for uids is not completely obvious from the nonlinear
form of the metric potential (1.2), which is a function of det gij . Naively, it seems that
it can still give rise to a mass term for tensor modes. However, this cannot happen and
the simplest physical reason is that the transverse phonons in a
uid must have vanishing
spatial gradient term. In turn, this further implies that black brane solutions acquire rigid
properties in solid HMGs but not in uid HMGs.
Linear and nonlinear consistency
Let us stress that the two elds massive gravity is free from the BoulwareDeser ghost [57],
i.e. that at full nonlinear level the number of unconstrained degrees of freedom is the same
as at the linear level. Showing this is almost trivial if one does not go to the unitary gauge,
where i are frozen to their vev
i = xi, but considers it as a theory of two scalar elds
minimally coupled to gravity, cf. [58]. For the most general Lagrangian that depends on
the rst derivatives of the scalar elds, V (X; Z), the equation of motion of the scalar elds
is given in (5.6) and schematically it looks like
i +
l = 0 :
Since this equation is of the second order in time and space derivatives even at full nonlinear
level, the full dynamics of the model is completely determined by specifying the same initial
data as for the linearized problem. This means that there are no additional degrees of
freedom showing up at nonlinear level. The other dynamical equations of the problem, the
Einstein equations, do not alter this conclusion, because the stress tensor from
order in derivatives. It is crucial that the scalar elds i are minimally coupled to gravity,
i is of rst
so that no additional degrees of freedom could arise in the gravity sector.
Note that the same conclusion is not so easy to reach if one starts directly in the
unitary gauge. In that gauge, the two elds theory looks like adding a potential term for
the metric V (tr(gij ) ; det(gij )) and the only dynamical equations are the Einstein equations.
Nevertheless, because of the above argument, this theory is completely free from the BD
problem for arbitrary choice of V (X; Z). The theory propagates four degrees of freedom at
non linear level in total. There are separate conditions to be imposed in order to insure that
the two additional degrees of freedom are healthy. But this is di erent from the BD ghost.
6
In this paper we have analyzed theoretical consistency and phenomenological implications
of a wide class of holographic massive gravity theories. These encompass all possible
Lorentz violating graviton mass terms that are compatible with symmetries of the AdS
black brane background solution and preserve the homogeneity and isotropy in the two
spatial directions of the dual eld theory.
Such theories are known to provide a holographic framework for momentum relaxation
and are used as a phenomenologically viable description of transport properties in various
condensed matter systems. However, only very particular massive gravity model, i.e. the
dRGT theory, has been mostly exploited for this purpose so far. Such restriction is justi ed
in the Lorentz invariant massive gravity where it is wellknown that a generic massive
gravity is plagued by a ghost instability. In the Lorentz violating case, however, there
are many more graviton mass terms that are allowed by theoretical consistency. For the
rst time, in this work we have spelled out all the mass terms that are permitted by the
symmetry of the black brane metric in the spirit of a similar analysis that has been carried
out earlier for the Minkowski background in [5, 6]. We have shown how generally covariant
form of these masses can be reconstructed by the usual Stuckelberg trick by using a set of
four scalar elds.
A large part of the paper is devoted to the discussion of massive gravity theories where
only two transverse Stuckelberg elds are employed. We argue that such theories are the
covariant version of the e ective eld theories that describe solids and
uids. As such,
also the two elds holographic massive gravity can be broadly split into theories describing
solids and
uids. Due to the enhanced symmetry in
uids, the set of allowed mass terms
is more constrained in this case. Of particular importance for this work is the observation
that the m2 mass term is absent for uids. We further show that this mass term implies
important phenomenological consequences for the response in the dual eld theory. We nd
that theories with m2 mass term enjoy a nonzero response to static shear deformations
thus signifying a sharp distinction between holographic solids and
uids.
We have also investigated the phenomenological consequences of the new classes of
the holographic massive gravities on the electric response in the dual theory.
We
nd
that it can be parametrised by a universal mass function M 2(r) that encodes the electric
properties of the dual system. We show that this function is proportional to the m2(r)
mass parameter in the dRGT theory. However, in our case the form of this function is
free. We generalize the universal expression for the DC conductivity found for the dRGT
massive gravity in [2] for this case. By assuming the particular dependence M 2(r) / r
we
nd that the parameter
determines whether the dual material has the properties
of an insulator or a metal. It also regulates the appearance of the phonons in the dual
theory. Moreover, we also show that for
scales linear with temperature. The same phenomenon has been previously observed in
the dRGT theory in [14]. Let us add here that the metric potential V (X; Z) that gives
rise to this behaviour scales like V
X (or equivalently V
Z1=4) which is similar to
p
the square root structure in the scalar DBI Lagrangian. Since the shape of the latter is
protected by a speci c reparametrizationlike symmetry it might be that also the universal
linear in T resistivity behaviour follows from a symmetry principle.
The case of holographic massive gravity with all four Stuckelberg elds employed has
never been considered elsewhere. Here we perform a thorough analysis of the homogeneous
scalar, vector and tensor perturbations of these theories. We
nd that the problematic
scalar sector can be healthy and propagate up to one helicity0 degree of freedom if the
mass parameters, as de ned in (2.12), satisfy
4m02m210
m142 = 0 :
(6.1)
The vector sector is in general stable and contains two dynamical vector elds with two
degrees of freedom each. However, we nd that the e ective light cones on which these
two elds propagate are, in general di erent from each other. While the Maxwell eld
perturbations propagate on the background ReissnerNordstrom geometry, the second vector
eld,
propagates on an e ective geometry determined by the graviton mass parameters.
The two light cones partially coincide only in the case when the mass parameters satisfy
m82 + m72 + 2m12 = 0 :
(6.2)
The light cone of the gravitational vector mode, although apparently causally stable, can
be wider than that of the Maxwell eld and thus allows superluminal propagation
velocities. Although bearing no apparent pathologies we
nd that the superluminal sector of
the holographic massive gravities is not suitable for describing momentum relaxation in
holographic framework. The reason for this is that the ingoing boundary conditions for the
eld
cannot be imposed in the usual manner in this case. In the theories, where the light
cones of both vector modes coincide, the electric response of the dual eld theory can also
be parametrized by a single mass function M 2(r) and is identical to the two elds case.
Acknowledgments
We thank Alessandro Braggio, Nicodemo Magnoli, Daniele Musso, Alex Pomarol and
Sebastien RenauxPetel for very useful discussions. MB would like to thank University
of Illinois, ICMT and Philip Phillips for the warm hospitality during the completion of
this work. We acknowledge support by the Spanish Ministry MEC under grants
FPA201455613P and FPA201125948, by the Generalitat de Catalunya grant 2014SGR1450 and
by the Severo Ochoa excellence program of MINECO (grant SO20120234).
A
Second order action  linear consistency
1
2
In this section we shall study the linear stability of the generally covariant graviton mass
term with four Stuckelberg elds described in section 2.3 and explicitly given by:
L (HAB; r) =
m02( r)(Htt)2 + 2m12( r)HtiHti
m22( r)Hij Hij
+ m32( r)HiiHjj
2m42( r)HttHii
+ m52( r)(Hrr)2 + m62( r)HttHrr
+ m72( r)HriHri + m82( r)HtiHri + m92( r)HrrHii
+ m120( r)HrtHrt + m121( r)HrtHii + m122( r)HttHrt + m123( r)HrrHrt :
For the analysis of quadratic Lagrangian it is enough to expand the gauge invariant elds
HAB de ned in (2.21) up to rst order in perturbations:
B + g^
where the last term arises from the perturbation of the reference metric f AB. We shall
classify the perturbations according to the scalar, vector, and tensor representations of the
transverse O(2) rotation group as in (2.8){(2.10). At quadratic level the Lagrangians of
the three sectors decouple from each other and will therefore be studied separately.
(A.1)
(A.2)
g L =
L2
2r2f
+
rh
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
(A.9)
Hir = g^rr i
r
Hit = g^tt i ;
t
i
i
t
( i)0
h~ri ;
h~ti :
hi
h~0ti
h~ri ;
hi = _ ri
( ti)0
1
2
f aihi +
f hi2
r2f 2a0i2 + r2 a_i2
The Maxwell perturbation ai is di eomorphism invariant by itself, and there is one more
useful gauge invariant quantity
that is, however, related to the other variables.
With these de nitions at hand, it is straightforward to rewrite the total action (2.1) up to the second order in perturbations in a di eomorphism invariant form
We consider the mass term relevant for vector perturbations
L
1
2
2m12( r)HtiHti + m72( r)HriHri + m82( r)HtiHri
and assume for simplicity that the r dependence of the masses is given by
mi2( r) = M 2( r)f ( r) i mi2 :
Here i = 1; 7; 8, f ( r) is the emblackening factor written in a gauge invariant form, and
M 2( r) is a universal mass function that is regular and nonvanishing at the horizon
r = rh. For concreteness, we also assume that the mass function M 2( r)
0 everywhere
and has the dimension of mass squared. The masses mi2 on the right hand side of (A.4)
and the powers i are dimensionles constants.
As was clari ed in section 2, vector perturbations are described by the components
HJEP02(16)4
hti
Lr22 h~ti;
hri
Lr22 h~ri;
ai;
i
:
Since the mass term above is di eomorphism invariant by construction it is possible to
rewrite it in terms of gauge invariant elds. Indeed, we nd that up to the rst order in
perturbations the elds Hir and Hit can be written as
where ri and ti are the gauge invariant elds introduced in [12] and are de ned as
2m12f 1 ( ti)2 + m72f 4+ 7 ( ri)2
m82f 2+ 8 ti ri :
(A.10)
The above action depends on the Maxwell eld ai and the three elds hi; ti; ri that are
related to each other via the constraint (A.9). The vector indices of the elds are always
contracted trivially, and in what follows we shall suppress them in order to simplify the
appearance of the expressions.
Since one expects at most two dynamical vector elds in the theory  the Maxwell
eld and the helicityone part of the massive graviton  it should be possible to integrate
out two of the four vector elds. In order to consistently impose the constraint (A.9) we
add it to the Lagrangian with a Lagrange multiplier :
p g L = L
2 h
_ r + ( t)0 :
The action (A.10) together with the above constraint contains ve vector elds t; r; h; a,
and
, all of which should be treated independently. The corresponding equations of
motion read
m28f 8 2m27f 2+ 7!
4m21f 1 2
m28f 8
r
M 2(r)P
r
a
2r2 ;
( t)0 ;
a
f 2a00 f f 0a0 +
h = 0 :
rh
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
The equations (A.12) and (A.13) can be used to eliminate the elds t; r and h from
the action (A.10) and rewrite it in terms of a and . We notice that the system of
equations (A.12) has a nondegenerate solution only in the case when M 2(r) 6= 0 and
det P = m84f 2 8
8m12m27f 1+ 7 6= 0 :
In order to satisfy the above condition at the horizon for a generic choice of the masses mi2
one has to impose
8 = 0 ;
1 =
7
:
Under these assumptions the det P is constant and nonvanishing everywhere. The case
when det P = 0 marks one particular phase of the Lorentz violating massive gravity and
has to be considered separately.
When det P 6= 0 we can solve for the elds t; r and h algebraically. After substituting
the solutions back in the Lagrangian (A.10) with the constraint term added we obtain:
p g L = L2
1 2
2f
2L4
+
M 2(r) det P
2
r
2
rh
2!
2m12f
2 _ 2 + m72f 2
0 2
m82 _ 0 :
(A.18)
When det P = 0 we nd that the Maxwell eld a and the eld
decouple from each
other. Moreover,
= 0 by its equation of motion and the Lagrangian for a reduces to
the case of pure EinsteinMaxwell theory. We shall not considered this case anymore in
what follows.
The Lagrangian (A.18) can be used to study the stability of the theory. The rst term
in the above expression coincides with the corresponding term found in the holographic
dRGT massive gravity in [12]. The equations of motion found by varying this action with
respect to the elds a and
can be easily recast in terms of the metric perturbations by
expressing
and its derivatives from the equations (A.12) and (A.13). The equations of
motion that follow from the action (A.18) are:
+ m82 M1(r)
M (r)
M 2(r)
0 +
0 r2 det P
2L2
In appendix B we analyse the stability of the vector modes in the highfrequency
approximation. We nd that the local causal structure for the eld
is determined by an
e ective metric geab given in (B.2). In general, this emergent light cone is di erent from the
light cone determined by the background metric g^ab. In particular, if the e ective light cone
is wider than that of the ReissnerNordstrom background, the perturbations
propagate
superluminally with respect to g^ab. A similar behaviour can be seen when discussing the
perturbations of the kessence [48, 59] and Galileons [60]. For the emergent geometry to
be causally stable it has to satisfy several consistency requirements investigated in detail
f a0 0
a
2 2r2
h
2 a +
2r2
a
= 0 ; (A.19)
= 0 : (A.20)
HJEP02(16)4
in appendix B.
A.2
Scalar modes
vant mass terms include
We further consider the stability of the homogeneous scalar sector modes (2.8). The
relem02( r)(Htt)2
m22( r)HijHij + m32( r)HiiHjj
2m42( r)HttHii + m52( r)(Hrr)2 + m62( r)HttHrr
+ m92( r)HrrHii + m120( r)HrtHrt
+ m121( r)HrtHii + m122( r)HttHrt + m123( r)HrrHrt ;
where we allow for an arbitrary r dependence of the masses.
There are in total eight components of elds which transform as scalars under the
transverse rotations: htt; htr; hrr; h; at; ar; t; r
. Two combinations of them can be set
to zero by an appropriate choice of the coordinate transformations t~ = t + t; r~ = r + r
Moreover, due to the U(
1
) symmetry of the Maxwell theory the components at; ar enter the
equations of motion only in the gauge invariant combination speci ed below. Hence, there
should be only
ve independent gauge invariant combinations of the scalar perturbations.
Three of them can be found to be [12]:
hr
ht
h~rr +
1
f
X
1
f
0
a0
t
a_r +
rh~0
2 Y_ ;
rf 0 h~ ;
2f
A^0
2f
t X + A^0t 2r h~ 0 ;
(A.21)
(A.22)
(A.23)
(A.24)
where we have introduced the short hand notations
X = h~tt +
and
LEM =
L2
4r4
3f 2hr2 +
f hra + 2r4a2
2rf 2hrht :
As we see, in the absence of the graviton mass term, there are no dynamical degrees of
freedom in the scalar sector of the EinsteinMaxwell theory.
For writing the mass term (A.21) in a di eomorphism invariant form, it is useful to
introduce other gauge invariant combinations:
These three gauge invariant elds allow us to rewrite the EinsteinMaxwell part of the action (2.1) up to second order in perturbations as
HJEP02(16)4
In terms of these elds the di erent components of HAB in the graviton mass term read:
t
tr
h~ +
2 r
h~rr +
;
2
~
htr + f ( t)0
2f _ t +
( r)0
2
f
1
1 r
_ :
f 0
2
f 0
r
Htt =
Htr =
2 1
L2 f 2 t ;
2
L2 tr ;
Hrr =
Hij =
2
L2
2 r ;
ij r
2
L2
h :
As explained above, there are in total only ve independent gauge invariant
combinations of the metric, Maxwell, and Stuckelberg eld perturbations. This means that there
are two relations between the seven gauge invariant elds h; ht; hr; a; t; r; tr. We nd
hr =
2
h 0
r ;
ht =
1
r f 0
2 f
h
0
r
h
f 2
t 0
f
2
f _ tr :
These expressions allow one to eliminate the
elds hr and ht from the action (A.27).
Moreover, from (A.34) we see directly how the presence of the Stuckelberg
elds
introduces dynamics in the scalar sector of the theory.
The previously nonderivative
terms ht and hr turn into terms with second order time derivatives in the action.
rf 0
2
~
h~ ;
rh
Y = h~tr +
r h~_ ;
2f
h
:
(A.25)
(A.26)
(A.27)
(A.28)
(A.29)
(A.30)
(A.31)
(A.32)
(A.33)
(A.34)
t
; r
We thus obtain the nal second order action for perturbations in two steps. We rst
replace the elds ht and hr in the EinsteinMaxwell action (A.27) by the corresponding
equations (A.34). We then add the mass term (A.21) where we replace the di erent
components of the matrix HAB by the corresponding expressions (A.32) and (A.33). The
resulting action depends on the ve gauge invariant elds a; h; r; t; tr. Three of them
are nondynamical and can be integrated out or act as Lagrange multipliers and impose
constraints among the remaining
elds. The nondynamical elds are the a; t; tr while
the two dynamical degrees of freedom are carried by h and r. Hence, there are at most
two propagating degrees of freedom in the scalar sector of the EinsteinMaxwell massive
gravity. It is a standard result in general theories of massive gravity. Around Minkowski
background it is known that one of the two scalar degrees of freedom is unhealthy and
propagates a ghost. This happens in both Lorentz invariant [57, 61] and Lorentz
violating [5, 6] massive gravity theories. In Lorentz violating massive gravity theories the number
of degrees of freedom in the scalar sector depends on the choice of the graviton masses: by
setting some of the mi's to zero, one can propagate either two, one or none of the helicity0
elds. Below we identify the mass terms that are responsible for the degree of freedom
count and present the main choices that lead to distinct number of propagating elds.
We shall not present the full Lagrangian here. The expression is quite long and becomes
even longer after we start integrating out the nondynamical elds. We list the di erent
steps we take in manipulating the action and discuss the di erent results depending on
the mass parameters mi. The actual calculations are straightforward. The rst step is to
integrate out the Maxwell eld a. It enters the action linearly and quadratically without
derivatives, which allows use to use its equation of motion to eliminate it from the action.
The next step is to integrate out the eld tr. It enters the action as:
1
2
L tr =
m120(r) t2r
tr F1 h_0; h_; _ r; h; t; r :
The result depends crucially on whether the mass parameter m10 in front of the term
quadratic in tr is vanishing or not.
In the case of a nonvanishing m10 the equation of motion of tr allows to express it
in terms of the other remaining elds h; t; r and eliminates it from the action. We then
proceed by integrating out the eld t which appears in the action as
1
L t =
8m210(r)f 4 4m02(r)m120(r)
2
tF2 h_0; h_; _ r; h00; h0; r0; h; r :
Further results depend on whether the combination 4m20m210
m412 is vanishing or not.
4m20m210
m412 6= 0
In this case we use the equation of motion for the eld t to eliminate it from the
action. The remaining action contains the two elds h and r. They both appear with
quadratic time derivatives and are kinetically mixed. We thus see two propagating
degrees of freedom. In analogy with the usual behaviour of the helicity0
elds in
massive gravity we expect that one of them is a ghost. We therefore conclude, that
the massive gravity theory (2.1) with the mass term (A.21) is unstable in this case.
In this case the quadratic term
t2 vanishes from the action and the eld t becomes
a Lagrange multiplier imposing the constraint F2 = 0 on the elds h and r. This
particular constraint allows to eliminate all the spatial derivatives of r from the
action. It means that in a high frequency limit one of the degrees of freedom has the
dispersion relation !2 = 0 and does not propagate. The other degree of freedom is a
linear combination of the elds h0; h; r and is dynamical. Hence, we conclude that
there is one propagating helicity0 degree of freedom in the high energy limit and this
phase can be stable.
When the mass parameter m10 vanishes, the eld tr turns into a Lagrange multiplier
and its equation of motion imposes a constraint F1 on the three
elds h; t; r. The
constraint has the form
F1 =
1
2f 2 m122(r) t + O
h_0; h_; _ r; h; r
= 0 :
(A.35)
Further results depend on the choice of m12.
m12 6= 0
For nonzero mass parameter m12 we can use the constraint F1 = 0 to eliminate t
from the action. The resulting action contains second order time derivatives of the
both remaining
elds h and
r. This shows that there are two propagating degrees
of freedom. As before, we conclude that this case is unstable since one of the degrees
of freedom is a ghost.
m12 = 0 and m0 6= 0
In this case we cannot use the constraint F1 to eliminate the eld t. Instead the
constraint provides a relation between the elds h and r. We use it in the action
in order to remove the term
trF1. The
eld t enters the remaining action only
linearly and quadratically without derivatives and can thus be integrated out by its
equation of motion. In the nal action there are no second order time derivatives.
We conclude that there are no propagating helicity0 degrees of freedom in this case.
m12 = 0 and m0 = 0
As before we use the constraint F1 to remove the term
tr from the action. We
As a result the only remaining terms that involve the eld r are: _ r h_; r2
further use it to express the spatial derivative r0 and eliminate it from the action.
; rh. The
highest derivative term of the h eld that arises in the quadratic action is ( h_)2; (h0)2.
In the high frequency limit this leads to the fact that both
elds have a dispersion
relation !2 = 0. We thus conclude that there are no propagating degrees of freedom
in this case.
Conclusion about the stability.
To summarise we conclude that the theory propagates
two degrees of freedom and is thus necessarily unstable if
4m02m210
m142 6= 0 :
The high energy limit allows us to further conclude that there is at most one dynamical
scalar degree of freedom in the parameter region:
4m02m210
m142 = 0 ;
and
m10 6= 0 :
m10 = 0 ;
and
m12 = 0 :
Finally, we nd that there are no dynamical helicity0 degrees of freedom if
We note that from the above cases, (A.37) and (A.38) were obtained in the high
frequency limit.
Our results presented here coincide with the earlier results for the Lorentz violating
massive gravity around the Minkowski background in [5, 6]. In particular, in the high
energy limit it was found that there is one propagating scalar degree of freedom when
m20 = 0; m210 6= 0 and no propagating scalars when m210 = 0, wherease the case with
m20 6= 0 and m210 6= 0 is unstable [6].17 These ndings coincide with our conclusions above.
B
Stability analysis in eikonal approximation
Here we investigate the short wavelength dynamics of the helicity1
elds a and
that
describe the dynamics of the vector perturbations of the Maxwell eld and the metric and
were introduced in appendix A.1. We analyze the action (A.18) and the corresponding
equations of motion (A.19) and (A.20) by using the eikonal approximation [62]. For this
we expand the elds as a(t; r) / exp(i!S^(t; r)) and
(t; r) / exp(i!S(t; r)) and take the
high frequency limit ! ! 1. In this limit only the terms quadratic in ! survive and the
equations read
and
new e ective metric de ning the geometry on which the perturbations
where the indices a; b = ft; rg and g^ab is the background metric (2.2) de ning the metric
on which the perturbations of the Maxwell eld propagate. In turn, the metric geab is a
propagate:
gab =
2m21f
12 m82
12 m28
m27f 2
(A.38)
(B.2)
The physical meaning of the equations (B.1) is that at the leading order in the eikonal limit
the elds a and
can be approximated as waves with a slowly changing phase S^ and S
respectively. The surface of constant phase is the corresponding wavefront and its normal
is given by the gradient @aS. The e ective acoustic cone of the wavefront propagation is
determined by the vectors tangential to the wavefront n
a
gab@bS. That the vector na
17We have identi ed the mass term m1 of [6] with the mass term m10 in our case.
is indeed tangential to the characteristic surface follows from equation (B.1). By further
rewriting of the equation (B.1) we nd another equation
geabnanb = 0 ;
in the spacetime geab.
where geab is the inverse of geab. Hence, the wavefronts of the eld
The e ective metric geab arises due to the scalar eld condensate that lls the
ReissnerNordstrom spacetime and de nes the local causal structure for the eld . In general, this
emergent light cone is di erent from the light cone determined by the background metric
g^ab. In particular, if the e ective light cone is wider than that of the ReissnerNordstrom
background, the perturbations
propagate superluminally with respect to g^ab. In such
situation, there are several conditions the emergent geometry has to satisfy in order for it
moves as if it were light
to be causally stable.
Hyperbolicity.
Obviously, the equations of motion (B.1) have to by hyperbollic, i.e. the
metric g~ab has to have a Lorentzian signature:
det ge 1 = 2m12m27
det P < 0 :
(B.3)
(B.4)
(B.5)
(B.6)
masses m2.
i
This means that det P = m48
8m21m27 > 0 and imposes the rst condition on the graviton
Closed timelike curves.
Due to the fact that, in general, superluminal propagation
might be possible in the metric gab one has to investigate whether closed timelike curves
can possibly form. To prove the causal stability of the emergent metric geab it is su cient
to
nd a di erentiable function
is a future directed timelike vector eld
(see [48, 63, section 8.2] and references therein). The function
then serves the role of a
global time function of the spacetime under consideration. We choose the function
to be
the time t of the original ReissnerNordstrom spacetime. Then the following condition has
to be satis ed
2m12f
2 < 0 :
Depending on the choice of
this expression can become in nite on the horizon when
f (rh) = 0 stressing the fact that the coordinate t is not a good choice for the time direction
near the horizon. This is the case also for the ReissnerNordstrom metric and thus we will
disregard this issue here. The above condition together with (B.4) implies the following
constraints on the masses:
m12 > 0 ;
and
m72 < m84=(8m12) :
In the special case when m28 = 0 this leads to m27 < 0.
Light cones. In order, to compare the light cone de ned by geab with the one de ned by
g^ab we consider a vector va that is null with respect to the metric ge, i.e. geabvavb = 0. We
then investigate whether this vector is spacelike or timelike with respect to the background
metric g^ab. By knowing that the vector va is a null vector with respect to ge we nd
vr =
1 f 2 v
t
2 2m21
pdet P ;
time axis. One can further check that
where the plus and minus indicate the two boundaries of the light cone. It is interesting
to notice, that only in the case m28 = 0 the light cone of ge is symmetric with respect to the
g^abva vb =
L2 f (vt)2
r2 (4m21)2 f 2(
1
)
pdet P
(B.8)
HJEP02(16)4
The two light cones coincide only if both g^abv+av+b = g^abva vb = 0. The sign of g^abva vb
determines whether the light cone of ge is wider or narrower than the light cone of g^. We
note that the mutual alignment of the two light cones for some xed values of the masses
mi2 can ip while the radial coordinate changes from 0 to rh due to the factor of f 2(
1
)
inside the square brackets. In order for this not to happen we demand that
(B.7)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
Under the above assumption the e ective metric simpli es to
= 1 :
gab =
2m21f 1 12mm2728f !
12 m82
:
is that they are related to each other by a conformal transformation
A general condition for the two metrics g^ab and geab to have the same causal structure
g^ab. Since the light cone of g^ab is symmetric with respect to the time axis, i.e. g^ab is a
diagonal metric, then this is possible only when m28 = 0. The condition g^abva vb = 0 then
(r) so that geab =
when
We also note that on the horizon f (rh) = 0 and hence both light cones coincide there
independently on the choice of the graviton masses m2.
i
Other particular choices of the graviton masses mi2 correspond to the cases when one
of the sides of the light cone of ge coincides with a side of the light cone of g^. This happens
when one of the conditions g^abv+av+b = 0 and g^abva vb = 0 is satis ed. For v+a this occurs
when
Similarly, for va we obtain
(m28 =
m27
m27
m28 = m27 + 2m21 ;
when
when
when
when
m27
m27
m27
m27
To analyse the mutual alignment of the two light cones for a particular choice of the
masses mi2 we have to check the sign of the norms gabva vb . If both gabva vb > 0, then the
null vectors va with respect to the metric geab are spacelike with respect to the
ReissnerNordstrom metric g^ab meaning that the light cone determined by the new e ective metric g~
is wider than the light cone of g^. Hence, the perturbations
can propagate superluminally.
Summary.
From the comparison of the two light cones de ned by the background metric
and the e ective metric, we rst conclude that for the mutual alignment of the two cones not
to ip while the radial coordinate changes from 0 to rh we need to demand that
= 1. We
further nd that a necessary condition for the two light cones to coincide is given in (B.11)
whereas the requirements for only one of the sides of the light cones to coincide read:
m27
m28 = m27 + 2m21 :
or
(B.14)
In general, whenever m28 6= 0 superluminal propagation of the perturbations
becomes
possible. This result has been obtained in the eikonal approxmation that corresponds to
taking the high frequency limit. In this limit, the phase velocity determined by the e ective
metric g approaches the front velocity of the propagation of
and describes the speed of
the signal propagation. Such a situation for perturbations propagating on a Minkowski
background would be pathological and should be avoided. However, as has been argued
previously in [48], having a superluminal propagation of perturbations on an emergent
e ective geometry due to a condensate of scalar elds does not lead to any causal paradoxes
as long as certain conditions are met. These include the requirement of the causal stability
of the emergent spacetime and of a wellposed Cauchy problem on an initial hypersurface
that is spacelike with respect to both the e ective metric g and the gravitational metric
g^ [48]. From our analysis above we nd that, the spacetime de ned by geab is causally stable
if m21 > 0 whereas the hyperbolicity condition is satis ed when m27 < m48=(8m21). The issue
of correctly posed Cauchy problem is more involved and has not be discussed here.
We also note that, strictly speaking, the eikonal approximation assumes that the
underlying theory is UV complete. The theory at hand is, however, an e ective theory and
is known to have a very low UV cuto
scale [64]. It might therefore be that the
superluminal propagation found in the eikonal limit is an artifact of the high frequency limit
and would not be present in the hypothethic UV complete massive gravity theory (for a
recent proposal of such a theory, see [7]). Whether or not such a completion is possible in
the presence of superluminalities in the low energy e ective eld theory is another subtle
question that has been studied in a number of earlier works [65, 66] and is beyond the
scope of this work.
We therefore conclude that restricting the allowed range of the graviton masses m2
i
basing on the fact that the theory permits superluminal propagation of eld perturbations
would be premature. Instead we shall accept the point of view that a superluminality in
an e ective eld theory is allowed and investigate whether this causes any inconsistencies
in the dual eld theory.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
[2] M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, Phys. Rev. D
88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].
Phys. Rev. Lett. 112 (2014) 071602 [arXiv:1310.3832] [INSPIRE].
086003 [arXiv:1306.5792] [INSPIRE].
scale and VDVZ discontinuity, hepth/0407104 [INSPIRE].
[5] V.A. Rubakov, Lorentzviolating graviton masses: Getting around ghosts, low strong coupling
[6] S.L. Dubovsky, Phases of massive gravity, JHEP 10 (2004) 076 [hepth/0409124] [INSPIRE].
[7] D. Blas and S. Sibiryakov, Completing Lorentz violating massive gravity at high energies, Zh.
Eksp. Teor. Fiz. 147 (2015) 578 [arXiv:1410.2408] [INSPIRE].
[8] C. de Rham, Massive Gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
[9] J.M. Maldacena, The LargeN limit of superconformal eld theories and supergravity, Int. J.
Theor. Phys. 38 (1999) 1113 [hepth/9711200] [INSPIRE].
Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
[10] C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev.
[11] M. Baggioli and O. Pujolas, ElectronPhonon Interactions, MetalInsulator Transitions and
Holographic Massive Gravity, Phys. Rev. Lett. 114 (2015) 251602 [arXiv:1411.1003]
[12] L. Alberte and A. Khmelnitsky, Stability of Massive Gravity Solutions for Holographic
Conductivity, Phys. Rev. D 91 (2015) 046006 [arXiv:1411.3027] [INSPIRE].
[13] T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05
(2014) 101 [arXiv:1311.5157] [INSPIRE].
[14] M. Taylor and W. Woodhead, Inhomogeneity simpli ed, Eur. Phys. J. C 74 (2014) 3176
[arXiv:1406.4870] [INSPIRE].
[hepph/9311264] [INSPIRE].
[hepph/9609466] [INSPIRE].
[15] H. Leutwyler, Nonrelativistic e ective Lagrangians, Phys. Rev. D 49 (1994) 3033
[16] H. Leutwyler, Phonons as goldstone bosons, Helv. Phys. Acta 70 (1997) 275
[17] A. Nicolis, R. Penco, F. Piazza and R. Rattazzi, Zoology of condensed matter: Framids,
ordinary stu , extraordinary stu , JHEP 06 (2015) 155 [arXiv:1501.03845] [INSPIRE].
[18] A. Nicolis, R. Penco and R.A. Rosen, Relativistic Fluids, Super uids, Solids and Supersolids
from a Coset Construction, Phys. Rev. D 89 (2014) 045002 [arXiv:1307.0517] [INSPIRE].
[19] S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, E ective eld theory for hydrodynamics:
thermodynamics and the derivative expansion, Phys. Rev. D 85 (2012) 085029
[arXiv:1107.0731] [INSPIRE].
[20] D.T. Son, E ective Lagrangian and topological interactions in supersolids, Phys. Rev. Lett.
94 (2005) 175301 [condmat/0501658] [INSPIRE].
arXiv:1501.07160 [INSPIRE].
[22] C. Lin and L.Z. Labun, E ective Field Theory of Broken Spatial Di eomorphisms,
[23] S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant.
Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
106009 [arXiv:0801.1693] [INSPIRE].
[24] S.A. Hartnoll and C.P. Herzog, Impure AdS/CFT correspondence, Phys. Rev. D 77 (2008)
[arXiv:1405.3651] [INSPIRE].
007 [arXiv:1401.5077] [INSPIRE].
[25] N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane
paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
[26] S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography,
JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].
[27] R.A. Davison, K. Schalm and J. Zaanen, Holographic duality and the resistivity of strange
metals, Phys. Rev. B 89 (2014) 245116 [arXiv:1311.2451] [INSPIRE].
[28] S.A. Hartnoll, Theory of universal incoherent metallic transport, Nature Phys. 11 (2015) 54
[29] A. Donos and J.P. Gauntlett, Novel metals and insulators from holography, JHEP 06 (2014)
[30] A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons,
JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].
[31] A. Amoretti, A. Braggio, N. Magnoli and D. Musso, Bounds on charge and heat di usivities
in momentum dissipating holography, JHEP 07 (2015) 102 [arXiv:1411.6631] [INSPIRE].
[32] S. Grozdanov, A. Lucas, S. Sachdev and K. Schalm, Absence of disorderdriven
metalinsulator transitions in simple holographic models, Phys. Rev. Lett. 115 (2015) 221601
[arXiv:1507.00003] [INSPIRE].
[33] R.A. Davison and B. Gouteraux, Momentum dissipation and e ective theories of coherent
and incoherent transport, JHEP 01 (2015) 039 [arXiv:1411.1062] [INSPIRE].
[34] S.A. Hartnoll and J.E. Santos, Disordered horizons: Holography of randomly disordered xed
points, Phys. Rev. Lett. 112 (2014) 231601 [arXiv:1402.0872] [INSPIRE].
[35] B. Gouteraux, Charge transport in holography with momentum dissipation, JHEP 04 (2014)
181 [arXiv:1401.5436] [INSPIRE].
critical point, arXiv:1508.04435 [INSPIRE].
[36] S.A. Hartnoll, D.M. Ramirez and J.E. Santos, Thermal conductivity at a disordered quantum
[37] A. Lucas, Hydrodynamic transport in strongly coupled disordered quantum eld theories, New
J. Phys. 17 (2015) 113007 [arXiv:1506.02662] [INSPIRE].
theories and the resistivity of strange metals with random eld disorder, Phys. Rev. D 89
(2014) 066018 [arXiv:1401.7993] [INSPIRE].
[40] D. Arean, L.A. Pando Zayas, I.S. Landea and A. Scardicchio, The Holographic
DisorderDriven SupeconductorMetal Transition, arXiv:1507.02280 [INSPIRE].
[41] D. Arean, A. Farahi, L.A. Pando Zayas, I.S. Landea and A. Scardicchio, Holographic
superconductor with disorder, Phys. Rev. D 89 (2014) 106003 [arXiv:1308.1920] [INSPIRE].
[42] A. Amoretti and D. Musso, Magnetotransport from momentum dissipating holography, JHEP
09 (2015) 094 [arXiv:1502.02631] [INSPIRE].
[43] M. Blake, A. Donos and N. Lohitsiri, Magnetothermoelectric Response from Holography,
JHEP 08 (2015) 124 [arXiv:1502.03789] [INSPIRE].
[44] M. Baggioli and M. Goykhman, Under The Dome: Doped holographic superconductors with
broken translational symmetry, JHEP 01 (2016) 011 [arXiv:1510.06363] [INSPIRE].
[45] M. Baggioli and M. Goykhman, Phases of holographic superconductors with broken
translational symmetry, JHEP 15 (2015) 35 [arXiv:1504.05561] [INSPIRE].
[46] M. Baggioli and D.K. Brattan, Drag Phenomena from Holographic Massive Gravity,
arXiv:1504.07635 [INSPIRE].
[hepth/0002230] [INSPIRE].
[47] S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of spacetime and
renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595
[48] E. Babichev, V. Mukhanov and A. Vikman, kEssence, superluminal propagation, causality
and emergent geometry, JHEP 02 (2008) 101 [arXiv:0708.0561] [INSPIRE].
[49] E. Megias and O. Pujolas, Naturally light dilatons from nearly marginal deformations, JHEP
08 (2014) 081 [arXiv:1401.4998] [INSPIRE].
[50] R. Argurio, D. Musso and D. Redigolo, Anatomy of new SUSY breaking holographic RG
ows, JHEP 03 (2015) 086 [arXiv:1411.2658] [INSPIRE].
[51] J.T. Devreese, Frohlich polaron concept. Lecture course including detailed theoretical
derivations, arXiv:1012.4576.
Pergamon Press (1970), pg. 1{12.
University Press (1995).
[52] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Course of Theoretical Physics, vol. 7,
[53] P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge
[54] L. Alberte, M. Baggioli and O. Pujolas, Viscosity bound violation in holographic solids and
the viscoelastic response, arXiv:1601.03384 [INSPIRE].
[55] G. Policastro, D.T. Son and A.O. Starinets, The Shear viscosity of strongly coupled N = 4
supersymmetric YangMills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hepth/0104066]
[INSPIRE].
JHEP 04 (2011) 102 [arXiv:1011.6396] [INSPIRE].
[56] S. Endlich, A. Nicolis, R. Rattazzi and J. Wang, The quantum mechanics of perfect uids,
[astroph/0505207] [INSPIRE].
HJEP02(16)4
electromagnetic eld, Proc. Roy. Soc. Lond. A 173 (1939) 211 [INSPIRE].
and gravity in theory space, Annals Phys. 305 (2003) 96 [hepth/0210184] [INSPIRE].
and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hepth/0602178] [INSPIRE].
arXiv:1502.05706 [INSPIRE].
[1] D. Vegh , Holography without translational symmetry , arXiv:1301 .0537 [INSPIRE].
[4] R.A. Davison , Momentum relaxation in holographic massive gravity , Phys. Rev. D 88 ( 2013 ) [21] S. Endlich , A. Nicolis and J. Wang , Solid In ation, JCAP 10 ( 2013 ) 011 [arXiv: 1210 .0569] to hyperscalingviolating regimes , Nucl. Phys. B 892 ( 2015 ) 239 [arXiv: 1411 .3331] [INSPIRE].
[39] A. Lucas , S. Sachdev and K. Schalm , Scaleinvariant hyperscalingviolating holographic [57] D.G. Boulware and S. Deser , Can gravitation have a nite range? , Phys. Rev. D 6 ( 1972 ) [58] L. Alberte and A. Khmelnitsky , Reduced massive gravity with two Stuckelberg elds , Phys. [59] C. ArmendarizPicon and E.A. Lim , Haloes of kessence , JCAP 08 ( 2005 ) 007 [60] A. Nicolis , R. Rattazzi and E. Trincherini, The Galileon as a local modi cation of gravity , Phys. Rev. D 79 ( 2009 ) 064036 [arXiv: 0811 .2197] [INSPIRE].