#### Momentum and charge transport in non-relativistic holographic fluids from Hořava gravity

Received: July
and charge transport in non-relativistic holographic uids from Horava gravity
Richard A. Davison 0 1 3 6
Saso Grozdanov 0 1 3 4
Stefan Janiszewski 0 1 3 5
Matthias Kaminski 0 1 2 3
Tuscaloosa 0 1 3
AL 0 1 3
U.S.A. 0 1 3
Open Access 0 1 3
c The Authors. 0 1 3
0 Victoria , BC, V8W 3P6 , Canada
1 Niels Bohrweg 2 , Leiden 2333 CA , The Netherlands
2 Department of Physics and Astronomy, University of Alabama
3 Cambridge , MA 02138 , U.S.A
4 Instituut-Lorentz for Theoretical Physics, Leiden University
5 Department of Physics and Astronomy, University of Victoria
6 Department of Physics, Harvard University
We study the linearized transport of transverse momentum and charge in a conjectured eld theory dual to a black brane solution of Horava gravity with Lifshitz exponent z = 1. As expected from general hydrodynamic reasoning, we these quantities are di usive over distance and time scales larger than the inverse temperature. We compute the di usion constants and conductivities of transverse momentum and charge, as well the ratio of shear viscosity to entropy density, and nd that they di er from their relativistic counterparts. To derive these results, we propose how the holographic dictionary should be modi ed to deal with the multiple horizons and di ering propagation speeds of bulk excitations in Horava gravity. When possible, as a check on our methods and results, we use the covariant Einstein-Aether formulation of Horava gravity, along with eld rede nitions, to re-derive our results from a relativistic bulk theory.
holographic; AdS-CFT Correspondence; E ective eld theories; Gauge-gravity correspon-
1 Introduction 2 3 4
Introduction
Hydrodynamics and linear response
Momentum transport from a Horava black brane
Horava black brane solution
Black brane excitations
Holographic dictionary
Hydrodynamic solution of the equation of motion
Holographic renormalization and hydrodynamic Green's functions
Transport coe cients and susceptibilities
Charge transport from a Horava black brane
Covariant formalism and
eld rede nitions
Momentum transport
Charge transport
Sound waves when
= 0
The AdS/CFT correspondence [1] has proven to be an excellent tool with which to study
the properties of certain strongly interacting, relativistic quantum
eld theories. It has
taught us that these eld theories have a robust hydrodynamic limit with a large window
of applicability [2{5], and has enabled the calculation of various hydrodynamic properties
of these theories, most prominently the ratio of shear viscosity to entropy density [2, 6{20].
For this reason, holography has provided a fertile testing ground for ideas about
hydrodynamic descriptions of the quark-gluon plasma and of metals (see e.g. [21, 22]).
It is of both fundamental and practical interest to determine whether there are
classical gravitational descriptions of strongly interacting eld theories which are not relativistic
at zero temperature. See [23] for a recent review of this topic. One proposed class of
gravitational duals are non-relativistic solutions of relativistic theories of gravity (general
relativity (GR) coupled to appropriate matter content) [24{26]. A second approach [27, 28],
which we pursue here, is to work with an intrinsically non-relativistic theory of gravity, like
that proposed by Horava in [29]. This theory is not invariant under all spacetime di
eomorphisms, and arises as the dynamical theory of Newton-Cartan geometry [30] (to which
non-relativistic eld theories naturally couple [31{34]).
In this work, we study the linear response of a neutral black brane solution [35] of
(3+1)-dimensional Horava gravity. A manifestation of the non-relativistic nature of this
state is that the low energy, linearized excitations of di erent elds propagate at di
erent speeds, and each
eld has its own `sound horizon' (trapped surface) [36]. The causal
`universal' horizon of the solution traps modes of arbitrarily high speed, and is the
thermodynamic horizon of the solution [35{40]. The solution we study has an asymptotic Lifshitz
and space transform identically. It is invariant under spatial rotations and translations in
space and time. However, this solution does not have a Lorentz boost symmetry, as this
relativistic transformation is not a symmetry of Horava gravity. Moreover, the solution we
study has no Galilean boost symmetry.
Using the proposed holographic dictionary of [27, 28], with further re nement
following [30, 32, 41], we show that there is a simple hydrodynamic description of the linearized
transport of both charge density and transverse momentum density over long times and
distances in the conjectured dual eld theory. In particular, both of these quantities di use,
and have conductivities related to their di usion constants by Einstein relations. This is
an important consistency check of the existence of a holographically dual state of this black
brane. In terms of bulk quantities, we nd that the charge di usion constant D and the
transverse momentum di usion constant D can be neatly expressed as
(sound horizon radius);
(sound horizon radius);
where the relevant speed and sound horizon radius in each case is that of the corresponding
dual excitation in the gravitational theory. We note that these constants have the same form
as the analogous relativistic formulae, in which case di erent bulk excitations have the same
speed and sound horizon radius due to Lorentz invariance. In terms of the temperature of
the universal horizon, the di usion constants are given by (3.40) and (4.18).
Our results for momentum transport are complementary to those of [42] (which worked
with a related, covariant theory1), in which a non-linear hydrodynamic description of
transport was derived to leading order in perturbation theory in , one of the coupling constants
of Horava gravity.
parameterizes the di erence between the speed of one of the gravitons
and the null speed of the boundary metric.
While we study only linear (in amplitude)
perturbations, we work non-perturbatively in . Our non-perturbative result for the shear
agrees with that conjectured in [42]:
=s = 22=3=4 , where s is the entropy
universal horizon of the Horava solution does not coincide with the Killing horizon of the
GR solution in the limit
To obtain our results, we must modify the standard prescription for computing
twopoint retarded Green's functions in the relativistic case [43{46], due to the existence of
multiple horizons. We propose that the linear excitation of a
eld should obey ingoing
boundary conditions at its sound horizon. In some cases, we are able to check that this
1See section 5 for further comparison between these theories.
is a sensible prescription by rstly rewriting Horava gravity in a covariant form
(EinsteinAether theory), and then using a
eld rede nition invariance of this theory, as well as
di eomorphisms, to map the perturbation equations onto those of the Schwarzschild-AdS4
black brane solution of GR. This procedure maps the sound horizon radius of the original
Horava solution to the Killing horizon radius of the Schwarzschild-AdS4 black brane. It
also provides a natural explanation for the appearance of the speeds of the bulk excitations,
rather than the null speed of the boundary metric, in the di usion constants (1.1). We
expect that this general principle | that ingoing boundary conditions should be applied
at di erent values of r for bulk excitations which travel at di erent speeds | should be
valid in Horava gravity beyond these simple cases.
nite answers for the transverse momentum density correlators, we
performed holographic renormalization by including two counterterms which are invariant
under the symmetries of Horava gravity. Upon the mapping to a covariant Einstein-Aether
theory, these counterterms coincide with those of the GR calculation.
Finally, we exploit the eld rede nition invariance of the covariant form of Horava
gravity to identify a special point in the parameter space of the Horava theory (when the
to that of the Schwarzschild-AdS4 solution of GR. Therefore, at this special point, the
excitation spectrum of the dual eld theory contains a sound mode (5.21) with speed
proportional to the spin-2 graviton speed.
In the following section we provide a brief overview of linear response in hydrodynamics,
and a derivation of the expected forms of the retarded Green's functions for charge density
and transverse momentum density. In sections 3 and 4 we study linear perturbations
of the Horava black brane solution and derive from this the hydrodynamic forms of the
dual Green's functions. The relation between our Horava gravity calculations and those of
Einstein-Aether theory are described in section 5, before we conclude in section 6 with a
summary of our results and some open questions.
Hydrodynamics and linear response
In general, a system which is in local thermal equilibrium should have a coarse-grained,
hydrodynamic description over long lengths and times, with respect to the scales over
which the system is locally equilibrated (in our case, this is the inverse temperature). The
hydrodynamic variables are those which vary slowly over these long length and time scales.
These are typically the densities of the conserved charges of the system.
We are interested in the linear response properties of a U(1) charge density, and the
transverse momentum density, in a (2+1)-dimensional, rotationally and translationally
charges of the state are its energy, U(1) charge and momentum. The densities of these
conserved charges qa obey the following conservation equations
@tqa + r~ ~ja = 0;
where ~ja is the current density associated with the conserved charge density qa. We will
consider the response of states in which both the U(1) charge density and the momentum
density have vanishing expectation values.
The information about the linear response properties of the state are contained in its
two-point retarded Green's functions, which tell us how the expectation values of operators
respond to small external sources. The retarded Green's functions of the charge densities
and associated current densities can be computed within hydrodynamics using the
canonical method of Kadano
& Martin [47] (see [48] for a review). Heuristically, this method
proceeds in two steps.
When a small external source for a conserved charge density is
applied at an initial time, the response in the expectation value of the charge density at
that time is controlled by the susceptibility .2 This initial change in the expectation value
will then evolve in time via the equations of motion (2.1), and the variation of this response
at time t, with respect to the initial source, gives the retarded Green's function.
To determine the evolution in time of the charge densities, we must supplement the
equations (2.1) with constitutive relations for the current densities ~ja in terms of the
charges qb. Hydrodynamics is a universal e ective theory, and we therefore construct these
relations by writing down all terms containing the conserved charges and their derivatives
that are allowed by the symmetries of the system. The relations are written as a derivative
expansion, and are a good approximation at long distance and time scales. The microscopic
details of the system enter in the values of the coe cients of each term in these derivative
expansions. After a Fourier transform in the spatial directions, and using the constitutive
relations to replace ~ja with qb, the equations of motion for linear perturbations take the form
@tqa(t; k) + Mab(k)qb(t; k) = 0:
The two-point retarded Green's functions of the charges are then given by [48]
G(!; k) =
There has recently been a lot of progress in systematically constructing the full,
nonlinear constitutive relations of non-relativistic hydrodynamics [32, 41, 50, 51], and also for
Lifshitz hydrodynamics [52{56]. For our purposes this is overkill: by restricting to the
linear reponse of parity-invariant theories, simple Kadano -Martin arguments are valid. It
can be checked, for example, that imposing parity symmetry on the constitutive relations
of [32, 51] reduces the constitutive relations written in terms of Newton-Cartan data to the
usual Navier-Stokes equations.
Without loss of generality, we will align the y-axis with the direction along which linear
perturbations vary in space. The conserved charge densities of our system are the energy
density ", the U(1) charge density , and the momentum densities x and y. We also
assume that parity is unbroken, and that charge conjugation, under which only the U(1)
charge density and current ip sign, is a symmetry of the state. This last condition implies
we are studying a state which is not charged under this U(1).
2Such susceptibilities are also sometimes referred to as `thermodynamic transport coe cients', e.g. [49].
We begin with the constitutive relation for the longitudinal U(1) charge current density
jy. The goal is to write down, to linear order in perturbations, the most general derivative
expansion of the charges that is consistent with the symmetries above. To leading order in
the derivative expansion, only one term is allowed
jy =
momentum density x is equally simple
jyx =
The ellipsis denotes higher order terms in the derivative expansion. The constant D is a
transport coe cient that is not xed by this analysis but depends upon microscopic details
of the theory.
The linearized constitutive relation for the longitudinal current jyx of the transverse
In this case, it is parity symmetry (under which x !
x) that restricts the form of the
right hand side. D is a transport coe cient which, in general, is unrelated to D .
Combining the linearized constitutive relations with the conservation equations (2.1),
we nd that linearized perturbations of both the charge density and the transverse
momentum density obey a di usion equation
2 = 0;
D r2 x = 0;
and that the transport coe cients D and D are the di usion constants of charge density
and transverse momentum density, respectively. Di usion constants have dimensions of
Green's functions of the conserved charges
From the di usion equations (2.6), we can use (2.3) to compute the hydrodynamic
G (!; k) =
G x x(!; k) =
@vx vx=0
denote the static susceptibilities of the conserved charge densities
where the chemical potential
is the source for the charge density, and the velocity vx
is the source for the transverse momentum density.
has units of mass density. The
two-point retarded Green's functions of the associated current densities are xed by Ward
identities to be
Gjyjy (!; k) =
G (!; k);
G jy (!; k) = Gjy (!; k) =
G (!; k);
Gjyxjyx (!; k) =
G x x(!; k); G xjyx (!; k) = Gjyx x(!; k) =
up to contact terms.
In the long time (dc) limit, we de ne the linear response conductivities of U(1) charge,
and of transverse momentum as
Im kli!m0 Gjyjy (!; k) =
Im kli!m0 Gjyx jyx (!; k) =
respectively. The rst of these corresponds to the usual de nition of the conductivity via
Ohm's law, and the second corresponds to the usual de nition of the shear viscosity (see
e.g. [57]). The conductivities are xed in terms of the di usion constants by the Einstein
relations (2.10), which follow simply from the form of the Green's functions (2.7). From
now on we will refer to these conductivities by their conventional names of the electrical
conductivity and the shear viscosity, respectively.
We have refrained from a full discussion of non-relativistic [51] or Lifshitz [52{55]
hydrodynamics and have presented only the elements which are relevant for our holographic
computation. We have shown that transverse momentum and charge both di use,
regardless of whether the system is relativistic or not. We note that the presence of an additional
conserved particle number charge will not alter our conclusions, as it cannot enter the
linearized constitutive relations (2.4) and (2.5) due to symmetry reasons. In the following
sections, we will show that the Green's functions of the strongly interacting state
purportedly dual to a Horava gravity black brane are of the hydrodynamic form (2.7), and
will derive explicit expressions for the transport coe cients D
and D (or equivalently
and ) of this state.
Momentum transport from a Horava black brane
Horava gravity [29] is a non-relativistic quantum theory of gravity that breaks the local
Lorentz covariance between space and time enjoyed by GR. We are interested in the low
energy, classical regime of Horava gravity, whose degrees of freedom, GIJ , N I and N , are
the components of the ADM decomposition of a spacetime metric gXY
gXY dxX dxY =
N 2dt2 + GIJ dxI + N I dt
GIJ is the spatial metric on slices of constant global time t; N is the lapse function, which
encodes the normal distance between the leaves of the foliation by t; and N I is the shift
vector, which identi es events with the same spatial coordinates on di erent time slices.3
In (3+1)-dimensions, the two derivative bulk action of Horava gravity is
SH =
(1 + )K2 + (1 + )(R
3Our notation is that indices X; Y : : : are bulk spacetime indices with x0
t, while I; J : : : are bulk
spatial indices covering both the bulk radial direction x
r and the transverse directions shared with the
is the extrinsic curvature of slices of constant t, K is its trace, and R and G are the Ricci
scalar and the determinant of the spatial metric, respectively. Indices are raised and lowered
with GIJ and GIJ , and rI is the covariant derivative with respect to the spatial metric.
In addition to the cosmological constant
and the gravitational constant GH (which
has length dimension 2), there are three new coupling constants ( ; ; ) allowed by the
less restrictive symmetries of Horava gravity. These dimensionless constants must satisfy
2(1 + ), and
In comparison with the full spacetime di eomorphism invariance of GR, Horava gravity
is only invariant under the di eomorphisms that preserve the foliation by slices of constant
t. These are the spatial di eomorphisms xI ! x~I (t; xJ ) and reparametrizations of the
global time t ! t~(t). In particular, spatially dependent time di eomorphisms are not
symmetries of Horava gravity.
Horava black brane solution
For the case
= 0, and with cosmological constant
3, there is an asymptotically
AdS black brane solution to Horava gravity [35] with
GIJ = BB
N =
NI = @ r3
consistent with the
aforementioned constraints, and is smoothly connected to the numerical solutions of [35].
We have checked (to leading order in
) that, when
= 0, this solution has smooth
corrections. The corresponding spacetime metric (3.1) of this solution is asymptotically
AdS, and the boundary metric has a \null speed" of 1. This is a choice of units, and all
speeds in the formulae that follow are in units of this null speed.
normal distance between slices of constant t, vanishes [36{40]. In Horava gravity, causal
signals propagate only forward in global time t. The leaves of the asymptotic temporal
Therefore, events at r > rh can only signal to larger r, and can have no causal in uence
on those at r
rh. This causal event horizon traps modes of any speed, and is interpreted
as the thermodynamic horizon of the solution [35].
The Killing horizon of the solution is at rk
rh=(1 + p
1 + )1=3. Its physical
significance is that it is the trapped surface for modes of unit speed.4 While the null speed of
4The locations of the various trapped surfaces, or sound horizons, for di erent speeds can be determined
by examining the Killing horizon of an e ective metric, as will be explicitly demonstrated in section 5.
horizon of the spin-2 graviton rs, and the universal horizon rh, are trapped surfaces for waves with
speed s = 1, s2 = p
horizon rs can be inside or outside of the Killing horizon rk.
and s0 ! 1, respectively. Depending on the value of , the spin-2 sound
NI = 0, and N = 1, one
speeds squared of these modes are
the asymptotic metric at the boundary is 1, this is not the speed at which excitations of
generic elds travel in Horava gravity. In contrast to GR, Horava gravity has more than one
nds a spin-2 graviton and an additional spin-0 graviton. The
s22 = 1 + ;
s20 =
respectively [58]. Our background (3.4) also supports multiple gravitons, and we will see
shortly that the most important of these, for our purposes, travels at speed s2. This is
nite and therefore has a sound horizon (the trapped surface for modes of this
speed) at a radius rs, outside the universal horizon. When
= 0, s0 ! 1, and a mode
of this speed has a sound horizon which coincides with the universal horizon. A schematic
location of the various horizons is shown in gure 1.
Black brane excitations
To determine the linear response of the conserved momentum density
x of our purported
dual theory, we will study linearized excitations of the shift
Nx(t; r; y) around the black
brane solution. We will shortly outline in more detail the holographic dictionary that we use
to explicitly identify the sources.
Nx(t; r; y) couples to both
Gyx(t; r; y) and
Grx(t; r; y).
After choosing the gauge
motion for these linearized excitations are
kp1 + r3r3 (!g(r) + kn(r)) + i(r3
h
!rh3n0(r) = 0;
krh6r (!g(r) + kn(r)) + (r3
rh3)( 2n0(r) + rn00(r)) = 0;
p1 + (2r5 + rh3r2) + i!rh6 (!g(r) + kn(r))
ikp1 + r3r3n0(r) + (1 + )(2r6
h
3rh3r3 + rh6)g00(r) = 0;
where we have performed a Fourier transform with respect to the global time t and the
spatial direction y
Gyx(t; r; y)
Nx(t; r; y)
We leave it implicit that g and n both depend upon ! and k.
Only one of the two second order equations of motion is linearly independent (because
of the residual di eomorphism invariance after our gauge choice
Grx = 0), and we can
make this manifest by working directly with the variable
This eld is invariant under the gauge freedom and obeys the second order equation
i z4( 5 + z3) + ( 2 + z3)2(1 + 2z3)
3z3 + z6)2 + i z4( 10 + 6z3 + z6) i (z)
+ z( 2 + z3) 2q2( 1 + z3)(2 + z3( 3
00(z) = 0;
where we have de ned a rescaled radial coordinate z, frequency
and wavenumber q as
21=3r
2r1h=3 k;
21=3p1 +
This rescaling manifestly removes
from the equation of motion, as it entered only in
the combination with ! that we have de ned as . The rescaled radial coordinateordinate
response dynamics in this sector are completely independent of the coupling constant
Holographic dictionary
To extract the linear response correlators of the dual eld theory, we will follow a similar
procedure as in the relativistic case, e.g. [2, 43]. Firstly, we must solve the di erential
equation (3.9) subject to two boundary conditions. Our
rst boundary condition is to
To identify the sources in Horava gravity, we will use a re ned holographic dictionary
rst presented in [27, 28]. This was originally motivated by the understanding of
nonrelativistic symmetry groups of [24, 59], and now has a more rigorous formulation in terms
of the Newton-Cartan geometry of [30, 32]. The relation to Newton-Cartan geometry is
most clearly illustrated by comparing the large c ! 1 limits of [28] and [41]. In [41], the
most general spacetime metric is written as
which is satis ed by the ADM decomposition (3.1) for5
nX nY + hXY ;
nX = ( N; 0);
hXY =
In addition to this timelike vector nX and the degenerate symmetric \metric" hXY ,
Newton-Cartan geometry contains a \velocity" eld vX and an \inverse metric" hXY that
are de ned to obey
hXY vY = 0;
X = 1;
hXY nY = 0;
which implies that they can be expressed in terms of the ADM
X =
hXY =
We are now in position to express the sources of the eld theory in terms of the boundary
values of the Horava elds by using the de nition of Newton-Cartan sources found in [32]
(see also [33]): n0 is the source for energy density,6 h
is the source for the stress
tensor, and v is the source for momentum density. The barred notation is due to the fact
that the sources should be varied arbitrarily, while the elds v
must obey the
constraints (3.13). The explicit relation between variations of barred sources and unbarred
elds are given in [32], but for our background they are expressed in terms of the bulk
hij =
r2 Gij jr=0;
vi =
N ijr=0;
where the powers of r are needed to strip o the leading behavior of the bulk elds as we
approach the boundary at r ! 0. The sources of stress yx and momentum density x are
hyx =
r2 Gyxjr=0 = r2 Gyxjr=0
vx =
N xjr=0 = r2 Nxjr=0
respectively. This agrees with the discussion of boundary conditions at r ! 0 above. In the
notation of section 2, yx is equal jyx , the longitudinal component of the current associated
with the conserved charge density
the bulk elds in terms of eld theory sources, we need to apply another boundary condition
in order to solve the second order equation of motion (3.9). In the relativistic case, to
5To make this identi cation unambiguous, powers of c need to be reinstated in the ADM expansion,
6The lack of spatial components ni renders us unable to calculate the energy current.
determine the retarded Green's function of the dual eld theory, one must impose ingoing
boundary conditions at the black brane horizon [2, 43]. Heuristically, this is because the
retarded Green's function is the causal response function in the eld theory, and causality
in the bulk implies that nothing should come out of the black hole. The situation is
more subtle in Horava gravity; we must take care as there are multiple horizons. In
fact, the equation of motion (3.9) has singular points at both the spin-2 sound horizon
and the universal horizon. By studying the characteristic exponents near each singular
point, we nd that it is only possible to impose ingoing boundary conditions at the
spin2 horizon, which is the outermost singular point. We therefore choose this location to
impose ingoing (in global time) boundary conditions. These boundary conditions, as well
as the identi cation of sources, will be further justi ed in section 5 via a mapping to a
covariant calculation.
After imposing these boundary conditions, we will determine the dual Green's
functions from the on-shell action of Horava gravity, as in the relativistic case. This step
requires an appropriate holographic renormalization to obtain a
nite answer, as will be
explained shortly.
Hydrodynamic solution of the equation of motion
The equation of motion (3.9) cannot be solved analytically in general. It can be solved
analytically in a perturbative expansion at small frequencies and wavenumbers. This is
su cient for our purposes as we are ultimately interested in the dual Green's function
in this hydrodynamic limit. Anticipating the existence of a di usive excitation, we will
perform a perturbative expansion in small
As explained above, we
rst impose ingoing boundary conditions at the spin-2
horizon z = 1
(z) = (1
z ! 0. To nd the perturbative solutions, we expand
(z) =
and solve order by order in , imposing regularity at the horizon of
(z = 1) at each order.
At leading order in the small hydrodynamic expansion, the solution is
1(z) = C1 + C2z3:
We can identify these constants in terms of the near-boundary expansions of the
fundamental elds
using the de nition (3.8). Note that while the normalization of these elds is given by the
sources g0 and n0, and that g1, g2, n1, and n2 are determined in terms of these sources by
the near boundary equations of motion, g3 and n3 are un xed by near boundary analysis,
and are related to the expectation values of the
elds dual to g and n. Comparing the
expansion (3.20) to the leading order hydrodynamic solution yields
1(z) =
Moving to the rst subleading order in the perturbative expansion,
2 obeys the
same equation of motion as
1. Identifying the constants in terms of the near-boundary
expansions of the fundamental elds, we nd that
2(z) =
g3(0) + qn03(0) :
Recall that in addition to the second order equation we are solving for , there is also a
rst-order constraint equation (3.6) for g and n. This constraint equation places further
restrictions on the allowed near-boundary expansions (3.20). In particular it requires that
g3(0) = 0;
g30(0) =
To determine the Green's functions to leading order, we do not need to explicitly solve
the equations of motion at higher order in . However, we do need to identify the subleading
coe cients n3(0) and n03(0) in terms of the boundary values of the elds. To do this, it is
4(z) here. This results in
n3(0) =
n03(0) =
Holographic renormalization and hydrodynamic Green's functions
The nal step in our calculation of the retarded Green's functions in the hydrodynamic
limit is to evaluate the on-shell gravitational action at the boundary r ! 0. As in the usual
relativistic case, we must supplement the action (3.2) with two kinds of terms to obtain
the correct answer. Firstly, we have to add a Gibbons-Hawking-like boundary term such
that the variation problem is well-de ned. As shown in [35], the Horava-Gibbons-Hawking
the current case. It can be written as:
SHGH =
where H is the determinant of Hij (the induced spatial metric of the boundary), and 2
is the trace of its extrinsic curvature, as embedded in the bulk slices of constant t. Note
Secondly, we must renormalize the boundary action so that it is nite. To do this, we
supplement the action with counterterms of boundary geometric objects that are invariant
under foliation-preserving di eomorphisms (the symmetries of the theory). A nite answer
is obtained by including the following two counterterms
SHCT;1 =
SHCT;2 =
nite, and is of the form
where 2Kij is the extrinsic curvature of the slices of constant t, as embedded in the
boundThe result of including all of these terms is that the on-shell quadratic action is now
+ n0( !; k)Gng(!; k)g0(!; k) + n0( !; k)Gnn(!; k)n0(!; k) ; (3.29)
where we have explicitly reinstated the dependence upon ! and k. Equipped with the
onshell action, we now determine the retarded Green's functions of the operators x and jyx
using the standard relativistic prescription. This prescription is that the retarded Green's
functions GAB(!; k) are given by
GAB(!; k) = 2G'A'B (!; k);
where 'A is dual to the operator A. We identi ed the elds dual to x and jyx ( yx)
in (3.16) as n and g, respectively. One can roughly think of (3.30) as varying a generating
functional, provided by the on-shell gravity action, with respect to the sources n0 and g0.
Transport coe cients and susceptibilities
Using the solution of the linearized equation of motion in the small ! and k limit, and the
prescription (3.30), the hydrodynamic retarded Green's functions of the operators in the
dual eld theory are (up to contact terms independent of ! and k)
G x; x(!; k) =
Gjxy ; x(!; k) =
8 GH 21=3rh2 i!
8 GH 21=3rh2 i!
G x;jxy (!; k) = Gjxy ; x(!; k);
Gjxy ;jxy (!; k) =
8 GH 21=3rh2 i!
21=33
21=33
21=33
These Green's functions have the characteristic form associated with di usive
transport, in agreement with that expected based upon hydrodynamic considerations. The
hydrodynamic Green's functions (2.7) and (2.9) have two free parameters that are
determined by microscopic details of the speci c theory: the momentum susceptibility
the momentum di usion constant D . For the eld theory purportedly dual to our solution
of Horava gravity, these take the values
Our Green's functions obey the Einstein relation (2.10) and thus the eld theory viscosity
(or conductivity of transverse momentum) is
8 GH rh 3;
3 21=3 rh:
24=3 T
22=3
The di usion constant agrees with the expression (1.1) given in the introduction.
The Horava black brane solution (3.4) obeys a rst law-like relation [35]
d = T ds;
where is the ADM energy density of the solution, and the entropy density s and
temperature T are properties of the universal horizon
In terms of these thermodynamic quantities, the momentum susceptibility is
s =
T =
should have units of mass density. The expression (3.39) is telling us that the
rather than the null speed of the boundary metric.
of energy of the uid, we nd
where it is again apparent that s2 = p
is the natural speed of the system.
The ratio of viscosity to entropy density of the system takes the -independent value
This result was conjectured in [42], based upon a perturbative calculation to leading order
in , and we have provided an explicit veri cation of it. In light of the evidence above that
appears only when multiplying the null speed of the asymptotic metric, it is perhaps
~ = kB = 1).
Finally, we note that there is no continuity with the GR results in the
when we express quantities in terms of thermodynamic variables. In this limit, the universal
horizon of the Horava solution, which determines the thermodynamic quantities, does not
coincide with the spin-2 sound horizon, which determines, for example, . In GR, the
thermodynamic and sound horizons coincide.
Charge transport from a Horava black brane
We will now address the transport of a conserved U(1) charge in the eld theory.
Holographically, the situation is identical to the relativistic case, e.g. [2]. The source of a U(1)
charge and current is a U(1) background potential. These eld theory sources are simply
dual to the leading near-boundary term (which is
r0 for our black brane) of a bulk
U(1) gauge potential: the bulk gauge transformations that preserve the radial gauge choice
eld theory potential
source. The subleading near-boundary term (in this case
r1) of the bulk U(1) gauge
potential encodes the expectation values of the conserved U(1) charge and current.
A U(1) gauge potential is comprised of two parts: a spatial vector potential AI , and
a scalar potential . These transform as AI ! AI
gauge transformations, where
is the spacetime dependent gauge parameter. In GR
these two potentials combine to form a spacetime vector, but under the less restrictive
symmetries of Horava gravity they are separately well-de ned geometric objects.
minimal gauge invariant action of these U(1) potentials which is invariant under foliation
preserving di eomorphisms is
SHEM =
1 Z
F IJ FIJ
F JI NJ
where the magnetic and electric eld strengths are FIJ
@J AI and EI
@tAI , respectively; c is the speed of electromagnetic waves; and
0 is the vacuum
permeability, which gives the overall normalization of the action. From the point of view of the
action (4.1), the speed c is a coupling constant.
The combined action of (3.2) and (4.1) still has the black brane solution (3.4), with
AI = 0 and
this corresponds to a
eld theory state with zero density of the global U(1) charge. We
will concentrate on longitudinal perturbations of the gauge potential. These are dual to
charge density and longitudinal current density perturbations of the
eld theory. It is
these operators which should exhibit interesting physics in the hydrodynamic limit, as we
outlined in section 2.
The equations of motion for the longitudinal linear uctuations
Ay(r; t; y)
d!dke i!t+ikya(r);
d!dke i!t+iky (r);
k2 (r) + k!a(r)
00(r) = 0;
! 0(r) = 0;
r3)2 + (1 + )k2r6 e00(r) = 0:
where we have not explicitly written that both a(r) and (r) are also functions of ! and k,
only one is linearly independent. This is due to the residual U(1) gauge symmetry and can
be made manifest by working with the gauge invariant eld
!a(r) + k (r):
This is a multiple of the electric eld component Ey, and obeys the second order equation
3ip1 + r2) + k2(1 + )r6(!rh6 + 3ip1 + r2(rh3
r3)2(!(2rh6!
3ip1 + r2(rh3 + 2r3)) + k2(1 + )r6)i e(r)
r3)) + k2(1 + )r6)
a00(r) = 0;
To obtain the correlators of the dual eld theory operators, we will follow a similar
procedure as in the previous section. The equation of motion (4.5) has singular points both
r =
1=3 :
The latter of these is the sound horizon for excitations of speed c. Following our logic
in the previous section, we expect this to be the relevant horizon for the linear response
calculation. Indeed, this is the only singular point of the equation of motion at which
ingoing boundary conditions may be imposed, and is also the outermost singular point.
After imposing ingoing boundary conditions at r , we expand the remaining part of the
e(r) = 1
, as before.
In terms of the near-boundary expansions of the original elds
the lowest order solutions are
e1 = k 0(0) + rk 1(0);
e2 = !a0(0) + k 00(0) + !a1(0) + k 01(0) r:
is nite and has the form
Identifying 0 and a0 as the sources of
tion (3.30) to obtain the Green's functions
0( !; k)G (!; k) 0(!; k) + 0( !; k)G a(!; k)a0(!; k)
+ a0( !; k)Ga (!; k) 0(!; k)+a0( !; k)Gaa(!; k)a0(!; k) :
and jy, respectively, we use the
prescripSolving the constraint equation near the boundary imposes that
a1(0) = 0;
a01(0) =
And nally, demanding regularity of e3 and e4 at the sound horizon xes
1(0) =
01(0) =
k 00(0) + !a0(0) :
To determine the Green's functions of the dual operators
and jy, we again need to
evaluate the action on this solution. Unlike in the previous section, we do not need to add
any boundary terms or counterterms to the bulk action (4.1). This r ! 0 boundary action
GqR;q(!; k) =
GqR;jy (!; k) =
GjRy;jy (!; k) =
GjRy;q(!; k) = GqR;jy (!; k);
From the Einstein relation, we can then extract the electrical conductivity
obeys the relation (1.1).
bulk excitations
Note that the bulk electromagnetic wave speed c and the corresponding sound horizon
radius r appear naturally in these formulae. As previously advertised, the di usion constant
As a function of the temperature, the di usion constant depends on the speeds of both
D =
1=3 2 T
in the hydrodynamic limit. These Green's functions have the characteristic form (2.7)
and (2.9) associated with the di usive transport of a conserved charge density. In this
case, it is the U(1) charge density.
By comparing the hydrodynamic results to our holographic results (4.15), we can
extract the following expressions for the U(1) charge susceptibility
and di usion constant
D in the eld theory state dual to the black brane solution of Horava gravity
D = cr =
1=3 :
while the dimensionless ratio of momentum and charge di usion constants depends on
= 3 21=3
relativistic result [60].
Covariant formalism and eld rede nitions
Until this point, we have worked solely with the formalism of Horava gravity. To better
understand our results, and to give further justi cation for our choice of boundary
conditions and holographic renormalization counterterms, it is instructive to express Horava
gravity in a generally covariant way in terms of Einstein-Aether theory [61{63]. This is a
theory of a spacetime metric coupled to a dynamical, timelike `aether' vector uX of unit
norm. The two-derivative action of Einstein-Aether theory is [35]
SAE =
c3r~ X uY r~ Y uX + c4uX r~ X uY uZ r~ Z uY ;
where g is the determinant of the full spacetime metric gXY , r~ is its covariant derivative,
and R~ is its Ricci scalar. This action is invariant under the full coordinate di eomorphism
symmetry of general relativity. For a hypersurface orthogonal uX (like that arising in
Horava gravity), not all aether terms in the action are linearly independent, and one of
To map onto Horava gravity, one performs partial gauge xing by choosing the time
coordinate such that uX =
N Xt . This is possible when uX is hypersurface orthogonal.
The action (5.1) is then equal to the Horava action, as long as the spacetime metric is
decomposed in the ADM form (3.1). The coupling constants in each action are related in
the following way
= 1 +
We have been studying the
Momentum transport
We will now exploit this mapping to relate our Horava gravity calculations to a more
conventional holographic calculation: linear perturbations around the Schwarzschild-AdS4
black brane solution of the Einstein-Hilbert action with a cosmological constant. The
metric of this solution is
ds~2 =
r3=rs3 dt~2 +
r3=rs3)
and with aether vector given by
By a temporal di eomorphism, we may transform this Schwarzschild black brane into the
on the metric and the aether eld, where we have chosen the constant
After this eld rede nition, the solution takes the form
= s22 = 1 + .
ds2 =
ut =
where we have relabelled rh = 21=3rs.
unfamiliar form
ds^2 =
We now perform a non-trivial eld rede nition
u^r = 0:
This solution is identical to the Einstein-Aether formulation of our Horava gravity
solution (3.4). This is not a coincidence. Under the eld rede nitions (5.7), the
EinsteinAether action maps to itself, but with GAE ! GAE =
and with di erent values of the
c2 =
hypersurface orthogonal Einstein-Aether theory to Horava gravity using (5.2), we conclude
that the net e ect of these coordinate transformations and eld rede nitions is to convert
the GR action into the Horava gravity action with
6= 0,
= 0.
This explains why the series of coordinate transformations and eld rede nitions we
explicitly performed above turns the Schwarzschild-AdS solution into our Horava gravity
solutions, but since the Horava solution has
= 0 and is independent of , we can set
= 0 and utilize the mapping above.
The mapping is much more powerful than this. Not only is our background solution of
Horava gravity independent of , but the equations of motion for linearized perturbations of
the transverse gravitons around this state are also independent of . This implies that, after
suitable eld rede nitions and coordinate transformations, the GR equations of motion for
linearized transverse perturbations around the Schwarzschild-AdS solution are equivalent
to the linearized Horava equations of motion (3.6) around the solution (3.4). To be
explicit, they are identical after identifying hXY , the perturbations of the Schwarzschild-AdS
spacetime (5.3), with
and then replacing
As the Schwarzschild-AdS calculation is very well-understood, we can use this
equivalence to shed light on the results, and the calculational details, of our Horava gravity
computations in section 3. Firstly, we note that the black brane Killing horizon of the
Schwarzschild solution maps to the spin-2 sound horizon rs of the Horava gravity solution.
Therefore, our imposition of ingoing boundary conditions at this sound horizon in the
Horava formulation of the calculation is equivalent to imposing them at the Killing
horizon of the Schwarzschild metric. Secondly, the counter-terms (3.28) we added to obtain a
nite on-shell Horava action are equivalent to those of the relativistic case [65] after the
appropriate transformations. These give justi cation for our choice of boundary conditions
and counter-terms in the Horava formulation of the problem.
Under the mapping described above, the coe cients n0; n3; g0; g3 in the near-boundary
expansions (3.20) of the Horava gravity elds map in a trivial way to the corresponding
cohyx =
ht~x =
hrx =
e cients of the near-boundary expansions of the elds hxt; hxy in the Schwarzschild-AdS
calculation. There should therefore be a close relationship between the Green's functions
obtained from Horava gravity and those obtained from the Schwarzschild-AdS black brane
in [60]. The Green's functions obtained from the Horava calculation should be those of the
relativistic case, in which the black brane Killing horizon radius is replaced by the spin-2
sound horizon radius, and the relativistic speed of light is replaced by the spin-2 speed
s2 = p
null speed of the Schwarzschild-AdS solution (5.3) is given by p
1 + . This is precisely
1 + . The latter of these conditions is because, after the mapping, the asymptotic
what we found in section 3.6.
Finally, we note that we are not aware of any way of mapping thermodynamic
quantities using this procedure. Although the Schwarzschild-AdS black brane Killing horizon
maps to the spin-2 horizon of the Horava gravity solution, this is not the causal horizon of
the theory, as there may be excitations that travel at speeds greater than s2. This means
that there is no direct way of mapping thermodynamic quantities associated with the
horizon, such as the entropy and temperature, between the solutions. A covariant expression of
the location of the causal horizon in Einstein-Aether theory is given by uX
X = 0, where
gXY and g^XY above [38{40].
Charge transport
It is also possible to recast our calculation of charge di usion in section 4 in terms of a
covariant Einstein-Aether theory. The electromagnetic action (4.1) can be written in the
Einstein-Aether formalism as [66]
SEAM =
1 Z
where FXY = r~ X AY
r~ Y AX . In terms of the coupling constants used in (5.12), the speed
of electromagnetic waves is
The action (5.12) di ers from the usual covariant Maxwell action due to the second term
proportional to . We can again take advantage of eld rede nitions and coordinate
transformations to make this situation more intuitive. Under the eld rede nitions (where
g^AB = gAB
u^A = uA
F^AB = F AB;
the action (5.12) transforms to itself, but with di erent values of the coupling constants
and 0. This eld rede nition is the inverse of that in (5.7).
to be the speed of light squared
c2 =
= c2;
the new metric of the black brane solution is
ds^2 =
3
c2 1 2 rr3 + 1
h
which can be brought into the nicer diagonal form
ds~2 =
f (r) = 1
with a radially dependent temporal di eomorphism. This new metric is an \e ective
metric" for the electromagnetic modes, meaning that the sound horizon for modes of speed
c in the Horava theory is now a Killing horizon.7 The null speed in the asymptotic region
of the new metric is c. Although this is not the Schwarzschild-AdS solution, it gives an
intuitive reason for why it is the speed of light c and the sound horizon r for light, rather
than the null speed of the asymptotic metric and the universal horizon, which appear
and light both move at the same speed, this reduces to the Schwarzschild-AdS solution.
After the eld rede nition, the new coupling constants in the action are
^0 = c 0;
^ =
and so the action of the rede ned elds is still not the Maxwell action.
Sound waves when
= 0
Finally, we will comment brie y on longitudinal linearized perturbations around our Horava
This will tell us, amongst other things, about the longitudinal transport of
momentum in the dual eld theory. There is one limit in which these transport properties
are relatively simple to deduce.
In section 5.1, we described how to relate our
= 0 Horava equations of motion to
those of perturbations around the Schwarzschild-AdS solution of GR. For the transverse
linearized perturbations of Horava gravity, the equations of motion are independent of
and thus we could perform this mapping for any . The longitudinal perturbation equations
are -dependent and so generically we cannot use this mapping. However, in the special
of motion for perturbations hXY of the Schwarzschild-AdS metric (5.3) are transformed
7For c3 = (1
c2 + )=(1 +
) and adequate ut and ur, equation (5.17) is a solution. It would be
intersting to investigate this solution further.
ht~t~ =
hxx =
hyy =
hrr =
hrt~ =
hyt~ =
hyr =
into the Horava equations by identifying
2 (1 + ) N N
F (r) = p
followed by the replacement (5.11). We have not written either set of longitudinal equations
explicitly as they are very lengthy.
As the near-boundary expansions of each eld map in a trivial way, we can painlessly
determine the longitudinal, hydrodynamic quasi-normal modes of the Horava solution (3.4),
where we de ne a quasi-normal mode as a solution that is ingoing at the sound horizon
horizon radius with the spin-2 horizon radius, and the asymptotic null speed with p
and whose leading term vanishes at the asymptotic boundary. By replacing the Killing
in the Schwarzschild-AdS results [67], we nd the dispersion relations
! =
These are the dispersion relations of the sound waves in the eld theory dual of Horava
are most naturally expressed in terms of the speed of the spin-2 graviton, rather than
the null speed of the boundary metric. The attenuation coe cient may be rewritten as
to be more complicated, as there will be another graviton excitation.
In [42], the dual hydrodynamics of the solution (3.4) was studied (perturbatively in )
within Einstein-Aether theory and found to be consistent with relativistic hydrodynamics
to rst order in the derivative expansion. In our Horava theory calculation, this is only the
not solutions of Horava gravity in a global time.
We have used the non-relativistic holographic duality conjectured in [27, 28], and reinforced
in the language of [30, 32], to study the collective transport properties of a
dual to a black brane solution of Horava gravity. In agreement with the general principles
Grr + 2F (r) Nr + F (r)2
2 (1 + ) N N
Nr + F (r)
2 (1 + ) N N
Gyr + F (r) Ny;
of hydrodynamics outlined in section 2, we have shown that both charge and transverse
momentum di use, and calculated the di usion constants and conductivities associated
with these processes, in equations (3.35), (3.36), (4.16) and (4.17). The ratio (3.41) of
entropy density to viscosity is independent of
and is increased by a factor of 22=3 compared
to the relativistic case, in agreement with the conjecture of [42]. Geometrically, this factor
stems from the fact that the thermodynamic horizon does not coincide with the relevant
trapped surface, i.e. the spin-2 sound horizon. From the point of view of the gravitational
theory, we have derived new results for the hydrodynamic quasi-normal modes of the
solution (3.4) of Horava gravity.
We have outlined how the canonical method for determining linear response properties
from a dual theory of relativistic gravity should be modi ed for the non-relativistic case
of Horava gravity. The major di erence is that, in Horava gravity, di erent excitations
(e.g. di erent gravitons, the photon) have di erent speeds, and these speeds are di erent
from the null speed of the asymptotic metric. In GR, Lorentz invariance xes these speeds
to all be the same. In Horava gravity, these di erent excitations each have their own
independent sound horizon, which speci es the region from which the excitation cannot
escape. In principle, these are independent from the universal horizon, the causal horizon
of the spacetime itself. We propose that calculation of the two point retarded Green's
function of the dual eld theory requires one to impose ingoing boundary conditions at the
sound horizon of the dual excitation.
We have focused here on the most tractable Horava gravity calculations: the di usive
excitations around the analytic black brane solution (3.4). These examples make it easiest
to identify the fundamental di erences with the relativistic calculations of two-point
functions, and to identify how the canonical methods should be modi ed. Indeed, in some of
these cases we had the luxury of using the mapping to GR described in section 5 to check
that our proposed modi cations are sensible. In general, such a simple mapping is not
possible and one should then directly use the procedure we have outlined for Horava gravity.
There is one important feature of Horava gravity that does not a ect the examples
we have studied here: the existence of new graviton excitations due to the reduced di
eomorphism symmetry. These appear to be present in the
6= 0 calculation of longitudinal
transport, leading to a complicated set of coupled equations for linearized excitations, in
contrast to the single equation when
is therefore to complete our understanding of the dynamics in the longitudinal sector, and
to identify the nature and consequences of this new degree of freedom in the eld theory.
Due to the existence of this mode, we expect the hydrodynamics of the longitudinal sector
to be signi cantly di erent from that of the relativistic case.
A second important generalization of this work is to study Horava theories with
which is not unusual in the low energy limit of many-body systems. Given the recent
holography-inspired success of applying hydrodynamic techniques to explain the transport
properties of relativistic many-body systems, we are hopeful that the non-relativistic
generalization will be similarly useful. For example, in relativistic hydrodynamics, energy
cannot ow independently of momentum, and thus the thermoelectric conductivity matrix
6= 0.
depends only on one transport coe cient [68]. This will not generically be the case for
a system without Lorentz symmetry, and it is worthwhile verifying this directly from the
dual gravitational perspective. As mentioned in the dictionary discussion of section 3.2,
the Horava gravity theory (3.2) seems unable to encode the source of the energy current,
as there is no bulk eld corresponding to the spatial components ni of the Newton-Cartan
clock vector. To restore this source and allow calculation of correlators of the energy
current will require the study of a more general form of Horava gravity, a task explored in [30].
There is undoubtedly fruitful work to be pursued in this direction.
Although we have determined the counter-terms necessary to holographically
renormalize the Horava action for the di usive channel, a complete description of the process
is still lacking, although progress was made in [58]. A reasonable guiding principle is the
analog of the GR procedure: counter-terms should be constructed out of geometrically
invariant boundary data. For Horava gravity, this means terms invariant under foliation
vector Nr generically does not vanish. Therefore, in addition to the boundary data of the
induced spatial metric Gij , the induced shift vector Ni, and lapse N , the boundary value
of Nr should be included in possible counter-terms.
In the relativistic case, the calculation of the dual linear response conductivities can be
greatly simpli ed using `membrane paradigm' techniques, that relate them to properties
of the black brane horizon [68{71]. It may be possible to generalize these techniques to
Horava gravity, in which case we expect the properties of the sound horizon to play the
corresponding role. A more challenging task is to determine the full non-linear response of
the black brane and interpret it in terms of the non-linear hydrodynamics of a dual theory.
Some work in this direction was undertaken in [42].
Finally, it would be interesting to construct and study charged black brane solutions of
engineer a charged solution of Horava gravity by starting with the Reissner-Nordstrom-AdS
solution of GR and using the mappings described in section 5.
Acknowledgments
We are very grateful to Ste en Klug for his collaboration during the initial stages of this
project. We thank Christopher Eling and Jelle Hartong for useful discussions and remarks
on the draft of this paper. The work of R.A.D. is supported by the Gordon and Betty
Moore Foundation EPiQS Initiative through Grant GBMF#4306. S.G. is supported in
part by a VICI grant of the Netherlands Organisation for Scienti c Research (NWO), and
by the Netherlands Organisation for Scienti c Research/Ministry of Science and Education
(NWO/OCW). The work of S.J. is supported in part by NSERC of Canada. M.K. thanks
the Instituut-Lorentz and Koenraad Schaalm for hospitality during important phases of
Open Access.
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any medium, provided the original author(s) and source are credited.
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