#### Charge-hyperscaling violating Lifshitz hydrodynamics from black-holes

HJE
Charge-hyperscaling violating Lifshitz hydrodynamics
Elias Kiritsis 0 1 2 3 4 5 6 7 8
Yoshinori Matsuo 0 1 2 4 5 6 7 8
0 Department of Physics, University of Crete
1 71003 Heraklion , Greece
2 Crete Center for Quantum Complexity and Nanotechnology
3 APC, Astrparticule et Cosmologie, Universite Paris Diderot , CNRS/IN2P3, CEA/Irfu
4 Crete Center for Theoretical Physics
5 Open Access , c The Authors
6 There is also charge-
7 10, rue Alice Domon et Leonie Duquet , 75205 Paris Cedex 13 , France
8 Observatoire de Paris , Sorbonne Paris Cite
Non-equilibrium black hole horizons are considered in scaling theories with generic Lifshitz invariance and an unbroken U(1) symmetry. hyperscaling violation associated with a non-trivial conduction exponent. The boundary stress tensor is computed and renormalized and the associated hydrodynamic equations derived. Upon a non-trivial rede nition of boundary sources associated with the U(1) gauge eld, the equations are mapped to the standard non-relativistic hydrodynamics equations coupled to a mass current and an external Newton potential in accordance with the general theory of [43]. The shear viscosity to entropy ratio is the same as in the relativistic case. Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1508.02494 1http://hep.physics.uoc.gr/ kiritsis/.
1 Introduction
3 Hydrodynamic ansatz
4 The rst order solution
2 U(1)-invariant, charge-hyperscaling violating Lifshitz theory
5 Calculation and renormalization of the boundary stress tensor
5.1
Energy and momentum conservation
6 Newton Cartan theory and Milne-boost invariance
7 The entropy current
8 General background gauge eld
8.1
A gauge invariant and Milne-boost invariant stress-energy tensor
8.2 A gauge invariant stress-energy tensor that is not Milne-boost invariant
9 The case of general z
10 Results, interpretation and outlook
A Notations B Calculation of the equations of motion at rst order z = 2
B.1 The sound mode
B.2 Vector mode
B.3 Tensor mode
C Calculation of the stress-energy tensor
D More on counter terms
E Regularity conditions of the gauge eld at the horizon
F First order solution for general z G Counter terms for general z
{ 1 {
Introduction
The AdS/CFT correspondence [1{4] relates the anti-de Sitter (AdS) space-time to
conformal eld theory (CFT) on the boundary. It gives a semiclassical description of strong
coupling physics in the dual eld theory in terms of string theory or its low-energy limit:
(super) gravity. At nite temperature and in the long wavelength regime, the dual eld
theory can be e ectively described by uid mechanics and it can be related to black holes in
AdS space-time. The uid/gravity correspondence was rst studied using linear response
theory [5{8].
of motion.
Subsequently, fully dynamic descriptions were studied using boosted black holes in
asymptotically AdS geometries that led to relativistic uid dynamics in the dual CFT [9].1
In this formalism, the uid variables are encoded in the near-equilibrium black hole solution
and the uid equations appear as constraints on the solution imposed by the bulk equations
Recently, generalizations of the AdS/CFT correspondence to theories with
nonrelativistic scaling symmetry have been studied. In particular, many condensed matter
systems have critical points with non-relativistic scale invariance [10{14]. Some of these
systems have Lifshitz or Schrodinger symmetry [15{17]. Moreover, the hydrodynamics of
charge and energy in such systems may be interesting as has been argued recently for
the case of cold fermions at unitarity, [18], other strongly correlated systems, [19] and
graphene, [20{22].
Holographic techniques have been generalized to geometries with Lifshitz or
Schrodinger symmetry in connection with applications to condensed matter systems [23{
29]. In particular, in [28, 29, 31], all quantum critical holographic scaling theories with a
U(1) symmetry respecting translation invariance and spatial rotation invariance were
classi ed in terms of three scaling exponents. Two of them (z; ) appear in the metric while
another exponent,
appears in the pro le of the U(1) gauge eld (it is referred to as
in [28, 29, 31]).2 The exponent z is the Lifshitz (dynamical) scaling exponent, and
is
the hyperscaling-violation exponent, [29, 30]. Even though such theories have been studied
intensively many of the aspects are still unclear and, in particular, hydrodynamics with
Lifshitz scaling symmetry is not fully understood.
More recently, it was found that the boundary theory dual to space-times with Lifshitz
asymptotics can be described in terms of the torsional Newton-Cartan gravity theory,
which is a novel extension of the Newton-Cartan gravity with a speci c torsion tensor.
The application of the Newton-Cartan theory to non-relativistic condensed matter systems
(namely the Quantum Hall e ect) was rst discussed in [35]. Interactions between the
torsional Newton-Cartan gravity and matter were discussed in [36]. The correspondence
between the Lifshitz space-time and boundary torsional Newton-Cartan theory was rst
1A related work was presented in [8].
2This charge exponent control the anomalous scaling of the charge density, even if it is conserved, has
also been introduced independently in [56] and was studied in more detail in [32] and [34]. The reason for
the existence of anomalous charge exponent despite conservation is the RG running of the bulk coupling
for charged degrees of freedom.
{ 2 {
found in [38, 39] for speci c Lifshitz geometry and further studied in [40{43]. In these works,
the correspondence is studied by using the vielbein formalism, in which an appropriate
combination of the vielbeins and bulk gauge elds is considered. It turns out to be very
useful to use vielbeins to study the boundary theory. This is consistent with the holographic
renormalization in the asymptotically Lifshitz space-time, in which the scaling dimension
is calculated by using the vielbein [44, 45]. Counter terms in Lifshitz space-time were
discussed in generality in [46] by using the Hamilton-Jacobi formalism.
The uid/gravity correspondence for non-relativistic uids has been studied in [47, 48]
for a special case of the Schrodinger geometry which is related to ordinary AdS by the
TsT transformation. In these studies, the non-relativistic
uids are obtained by the
light
HJEP12(05)76
cone reduction of the relativistic
uids. The generalization to the charged
uid case is
studied in [49].
For the Lifshitz space-time, the correspondence to relativistic uids with Lifshitz
scaling, in which the velocity
eld is de ned by a normalized Lorentz vector, was studied
in [50{52]. In these works the anisotropic direction of the Lifshitz symmetry depends on
the frame. The
uid appears on the surface at
nite radius or on the horizon, contrary
to the Newton-Cartan theory which appears on the boundary. The hydrodynamics found
contains an antisymmetric part in the hydrodynamic stress tensor that contributes a new
transport coe cient to the dynamics.
In this paper, we consider the uid/gravity correspondence for Lifshitz geometries and
the relation to uids in boundary non-relativistic theories with Newton-Cartan symmetry.
We consider black holes in Lifshitz space-time with unbroken U(1) gauge symmetry that
are solutions of the Einstein-Maxwell-dilaton (EMD) theories. Although, the geometry
has Lifshitz scaling symmetry with dynamical exponent z, the bulk solution has
\chargehyperscaling violation"3 due to a nontrivial conduction exponent , associated with the
gauge eld and the non-trivial running of the dilaton.
We consider the black-hole solution of the theory, boost it using Galilean boosts and
then we make all parameters of the solution including the velocities, ~x-dependent. We then
proceed with the standard analysis introduced in [9]: we solve the bulk equations of motion
order by order in boundary derivatives and compute and renormalize the ( uid)
stressenergy tensor. We also calculate the entropy current and consider the thermodynamic
relations. What we nd is as follows:
The standard stress-energy tensor we obtain from the holographic calculation is
expressed in terms of the uid variables: velocity eld vi, energy density E and pressure
P , but also contains the (particle number) density n and external source Ai
associated to the U(1) symmetry current. It satis es the condition for Lifshitz invariant
theories zE = (d
1)P .
3This is distinct from what is called hyperscaling violation in condensed matter physics. Our de nition
of charge-hyperscaling violation is based on the existence of scaling but also the existence of anomalous
scaling dimensions in the charged sector. In particular, although the scaling dimensions of charge density
and conductivity are canonical, the scaling of the charge density and conductivity with temperature is
controlled by the conduction exponent , [31{34].
{ 3 {
By comparing the stress-energy tensor and the constraints in the bulk equations of
motion, we nd that the conservation law of the stress-energy tensor is di erent from
that of relativistic theories but agrees with that in the Newton-Cartan theory. In [50]
the stress-energy tensor required a modi cation (improvement) in order to satisfy
the trace Ward identity. Our stress-energy tensor satis es the Ward identity without
such a modi cation. It is Milne-boost invariant but is not gauge invariant.
The role of the (unbroken) U(1) symmetry in this class of theories is important. It
should be noted that this U(1) symmetry is responsible for the Lifshitz background
bulk solution. We
nd that it behaves very closely to the U(1) mass conservation
sis in [50, 51]. Even though the continuity equation and energy conservation equation
agree with those in the ordinary non-relativistic
uids, the Navier-Stokes equation
is di erent from that in the ordinary non-relativistic
uids. The e ects of pressure
become much larger than other contributions and some terms with velocity eld in
the Navier-Stokes equation are absent in our result. These absent terms are replaced
by external source-dependent terms associated to the U(1) gauge eld.
By rede ning the stress-energy tensor and allowing a (Milne-invariant) Newton
potential in our sources, [41]{[43] we can map the uid equations to the standard
nonrelativistic uid equations coupled to the torsional Newton-Cartan geometry in the
presence of a Newton potential. This is a universal result that we nd interesting
and far-reaching.
Moreover, there is a stress-energy tensor that is both gauge invariant and Milne-boost
invariant, but in this stress tensor the momentum density vanishes. There is also an
alternative gauge invariant but Milne-boost non-invariant stress-energy tensor which
agrees fully with the standard non-relativistic stress-energy tensor.
Our uid can be interpreted as a non-relativistic limit of a uid with Lifshitz scaling
symmetry.4 In the ordinary non-relativistic limit of uids, the relativistic energy is
separated into that from mass and the non-relativistic internal energy. The
nonrelativistic internal energy is much smaller than the mass energy, and hence than the
relativistic energy. In ordinary non-relativistic uids the pressure is at the same order
to the non-relativistic internal energy and hence is much smaller than the relativistic
energy density. However, in our case, pressure and the relativistic energy density are
at the same order due to the Lifshitz scaling symmetry, and the energy density is not
separated into the mass and the others. For non-relativistic
uids with Schrodinger
symmetry the uid equations are obtained by introducing the light-cone dimensional
reduction in [47{49]. Instead, our non-relativistic limit arises naturally as a rather
ordinary limit.
4The non-relativistic limit of the Lifshitz uid is also studied in [54, 55].
{ 4 {
We nd that the form of the uid equations is independent of the Lifshitz exponent
z as well as of the (non-trivial) conduction exponent, . It is only the constitutive
relations (equation of state) that depend on these scaling exponents.
The entropy satis es the local thermodynamic relation with the energy density and
pressure. The divergence of the entropy current is non-negative, compatible with the
second law.
This paper is organized as follows. In section 2, we introduce the model and its solution
of Lifshitz space-time. Then, we rst focus on the case with scaling exponent z = 2. In
section 3, we introduce the hydrodynamic ansatz. In section 4, we solve the equations of
HJEP12(05)76
motion by using the derivative expansion and obtain the solution to the rst order. In
section 5, we calculate the stress-energy tensor on the boundary and study its symmetries
and the conservation law. In section 6 we introduce the Newton Cartan geometry and
realize it in for the boundary
uid in question. In section 7, we investigate the entropy
and thermodynamic relation. In section 9, we consider the generalization to general z.
Section 10 is devoted to conclusion and discussions.
Appendix A contains a list of the variables used in this paper and their de nition. It
contains also comparison of variables with two other papers in the literature. More details
on the calculation for rst order solution and boundary stress-energy tensor are described
in appendix B and C, respectively. In appendix D, we consider the analysis of the general
counter terms. In appendix E, we discuss the regularity of the gauge eld at the horizon.
More details on the solution and counter terms for general z are discussed in appendices F
and G, respectively.
2
U(1)-invariant, charge-hyperscaling violating Lifshitz theory
We consider a holographic theory with Lifshitz scaling and an unbroken U(1) global
symmetry in d space-time dimensions. The dual (d + 1)-dimensional gravity theory will have
a massless U(1) gauge eld A and a dilaton . The bulk action is given by
S =
1
where F = dA is the eld strength of the gauge eld, and
is a dimensionless coupling
constant of the bulk theory. The equations of motion are given by
R
2F
F
1
d
1
e F 2g
0 = r (e F
=
=
d
1
4
2
1
g
e F 2 :
1
2
);
1
4
e
{ 5 {
This model has the Lifshitz geometry as a solution;
ds2 =
r2zdt2 +
dr2
r2 + X r2(dxi)2;
i
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
with the following gauge eld and dilaton;
are related to the parameters of the action (coupling
studied in holography in [25, 26] and was generalized in [28].
The metric (2.5) has the Lifshitz scaling symmetry
t ! czt ;
xi ! cxi ;
r ! c 1r ;
and no hyperscaling violation ( = 0). However, due to the running of the dilaton, the
scaling of the AC conductivity is anomalous and its scaling with temperature or frequency
is controlled by the conduction exponent , [31{34]. It is de ned from the solution for At,5
At
r
z
:
=
(d
1)
{ 6 {
We will call this charge-hyperscaling violation.
conduction exponent,6
The solution (2.6) is a solution with charge-hyperscaling violation coming from the
although the hyperscaling violation exponent
coming from the metric vanishes.
This model also has a black hole geometry as a solution [26, 28];
ds2 =
r2zf (r)dt2 +
dr2
f (r)r2 + X r2(dxi)2;
i
where
solution
The Hawking temperature of the black hole is given by
The gauge eld and dilaton take almost the same form as in the the zero temperature
At = a(rz+d 1
0
rz+d 1);
e
= r2(1 d) ;
but At must vanish at the horizon to be regular.
5Note that here we use a radial coordinate that is inverse to the one used in [31].
6Although the conduction exponent is usually referred to as , here we refer to it as in order to avoid
confusion with the bulk viscosity .
For a general solution the nite part of the conductivity scales as, [32]
0
r
and this is controlled by the conduction exponent, . We observe that the temperature
dependence although scaling, does not respect the natural dimension of conductivity. This
justi es the name charge-hyperscaling violation for the exponent .
In the particular case studied here, in view of (2.12) and (2.17) we obtain
In this paper, we focus on the case of d = 4. Extensions to other dimensions are
expected to be straightforward.
3
Hydrodynamic ansatz
of z.
Finkelstein coordinates;
In this section, we introduce an ansatz for the geometry which describes the physics of
uids in the boundary quantum
eld theory. We use the method proposed in [9]. We will
also x the Lifshitz exponent to be z = 2. In a later section, we will discuss other values
In order for the regularity at the horizon to become evident, we change to
Eddingtonwhere the null coordinate t+ is de ned by
The gauge eld becomes
ds2 =
r4f dt2+ + 2rdt+dr + r2(dxi)2 ;
dt+ = dt +
dr
r3f :
A = a r
5
r
05 dt+
ar2dr ;
(2.17)
(2.18)
HJEP12(05)76
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
where we have
xed the Ar = 0 gauge in the Fe erman-Graham coordinates.
Hereafter, we always take the Eddington-Finkelstein coordinates and t will stand for the null
coordinate t+.
To implement the hydrodynamic ansatz, we rst boost the black hole geometry. In
the case of the ordinary Schwarzschild-AdS5, the boundary
eld theory is a relativistic
conformal eld theory, and hence the Lorentz boost is employed. The Lifshitz geometry,
however, corresponds to the (torsional)-Newton-Cartan theory, [38]{[43]. Therefore, we
perform a Galilean boost on the black hole geometry. The metric becomes
ds2 =
(r4f
v2r2)dt2 + 2rdtdr
2r2vidt dxi + r2(dxi)2 :
The gauge eld and dilaton are not a ected by the Galilean boost;
A = a r
5
r
05 dt
ar2dr ;
e
= r 6:
Now, we replace the parameters r0 and vi by slowly varying function r0(x) and vi(x)
of the boundary coordinates x . Moreover, we promote a,
and the constant part of Ai
(which is usually gauged away) to space-time dependent functions.
ds2 =
(r4f
v2(x)r2)dt2 + 2rdtdr
2r2vi(x)dt dxi + r2(dxi)2
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(4.1)
(4.2)
(4.3)
(4.4)
where Ai(x) comes from the constant part of Ai but now is replaced by functions of x .
The above are no longer a solution of the equations of motion, and we must introduce
additional correction terms;
g
A
= g
+ h
;
= A + a ;
=
+ ' ;
where the background elds g , A and
are given by (3.6){(3.9).
4
The
rst order solution
In order to obtain the rst order solution for the hydrodynamic ansatz, we consider the
derivative expansion. Then, the equations of motion can be treated as ordinary di erential
equations with respect to r, and the correction terms, h , a and ', can be calculated
order by order.
The di erential equation can be solved at any given point, and we can take the point
to be x
= 0 without loss of generality. The parameters which are replaced by slowly
varying functions can be expanded around x = 0 as
)(0) +
;
:
;
;
The derivative expansion of the equations of motion gives the linear di erential equations
for the correction terms h , a and ' at the rst order. The next-to-leading terms in (4.1){
(4.4) are at the rst order and give the source terms in the di erential equations.
Solving the (inhomogeneous) linear di erential equations for the correction terms h ,
a and ', we obtain the rst order solution in the derivative expansion. (See appendix B
for more details.) The integration constants generically modify the source terms near the
boundary. These contributions can be eliminated by setting the integration constants that
{ 8 {
modify the sources to zero. The rst order solution for the metric is given by
ds2 =
r4f dt2 + 2rdtdr + r2(dxi
vidt)2
3
+
2 r2@ividt2 + r2F ij (dxi
vidt)(dxj
vj dt) ;
where ij is the shear tensor
and the function F (r) is given by
F (r) =
dr
Z
r
3
r(r5
r
3
and integration constant is chosen such that F (r) ! 0 in r ! 1.
The rst order solution of the gauge eld is
A = a(x)
r
5
r05(x)
3
a(x)r2dr + Ai(x)(dxi
vi(x)dt) ;
(4.8)
and the dilaton has no correction term, ' = 0. Equations of motion implies that (2.9) and
its derivatives must be satis ed even after
and a is replaced by functions.
The solution above must satisfy the following constraints;
1
Cartan geometry. and
5
Calculation and renormalization of the boundary stress tensor
In this section, we consider the stress-energy tensor on the boundary. In order to study the
asymptotic behavior, we will use the vielbein formalism as this is adapted to the
Newton
The leading order term of the induced metric is expressed as
where
= g
n n , and g
is given by (3.6) and n is the normal vector to dr = 0
surface. On this surface, dr = 0, the vielbeins are given by
=
r2zf
+ r2 abe^a e^b
=
r 2zf 1
v^ v^ + r 2 abe^a e^b ;
dx = dt ;
v^ r
= rt + viri ;
e^a dx = dxa
vadt ;
e^a r
= ra :
{ 9 {
(4.5)
(4.6)
(4.7)
(4.9)
(4.10)
(4.11)
(5.1)
(5.2)
(5.3)
(5.4)
where
Sr = S + Sct :
Sct =
1
The variation of the renormalized action is expressed as
Sr =
Z
^0 v^ + S^a e^a + J^0 A^0 + J^a A^a + O
S
:
Then, the renormalized boundary theory stress-energy tensor and current are given by
, v^ and h
= e^a e^a will become the basic geometric data of Newton-Cartan geometry
that is discussed in the next section.
(1; vi) that depends on the velocity vi we introduced in the solution.
We also express the gauge eld in this frame;
The formulae above specify the vielbeins in a very speci c frame,
= (1; 0), v^
=
A^0 = v^ A ;
A^a = e^a A :
As will see in the next section that holographic data above will correspond to an in nite
set of Newton-Cartan data, all related by Milne boosts. Because of this the holographic
data above will be \Milne boost invariant".
In order to renormalize the expectation values, we have to add the counter terms to
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
(5.14)
T
b
J
T
r
J
r!1
r!1
= lim r5T
r ;
= lim r5Jr ;
= S^0v^
S^ae^a ;
r = J^0v^ + J^ae^a :
T(BY) =
1
) ;
It should be noted that the stress-energy tensor Tb
is not a symmetric tensor. It is also
not gauge invariant because we work in the vielbein formalism. As we will see later on,
when we introduce Milne-boosts, it will be Milne-boost invariant. In section 8 we will
de ne di erent stress-energy tensors with di erent properties under gauge transformations
and Milne boosts.
The stress-energy tensor (5.9) is related to the ordinary Brown-York tensor as
T
r
= T(BY)
+ J A + T(ct)
;
where T(ct)
is the counter term contribution and T(BY) is the Brown-York tensor, which
can be expressed in terms of the extrinsic curvature K
and J is calculated as
J
= p
= e n F
;
where n is the unit normal to the boundary.
The renormalized stress-energy tensor is obtained from (5.9), (5.10) as
a
The above expressions show that the stress-energy tensor Tb
contains the gauge eld
and hence is not gauge invariant. We will discuss several other de nitions of gauge
invariant stress-energy tensors in section 8.
5.1
Energy and momentum conservation
We now consider the conservation of the stress-energy tensor.
In the Newton-Cartan theory (that is described in more detail in the next section), the
conservation law takes a slightly di erent form from standard relativistic cases. It cannot
be expressed in a uni ed form in terms of the space-time stress-energy tensor, and we have
to introduce the energy vector Eb , momentum density Pb and stress tensor Tb
, which are
de ned by
(5.15)
(5.16)
(5.17)
(5.18)
(5.19)
(5.20)
(5.21)
(5.22)
(5.23)
(5.24)
(5.25)
(5.26)
(5.27)
Then, the conservation of energy and momentum is given by (see for example [36, 41, 43])
From the rst order solution, (5.16){(5.19), the energy vector Eb , momentum vector
(1form) Pb and stress tensor Tb
are given by
Eb0 =
Pb0 =
Tb ij =
3
1
The other components of Tb
condition becomes
, namely Tb 00 and Tb 0i vanish. The Lifshitz-scaling invariance
in terms of the stress-energy tensor Tb , or equivalently
z v^ Tb
+ e^a e^aTb
= 0 :
zEb0
Tb ii = 0 :
As was already noted, the momentum density contains a contribution from the external
source A , and hence is not gauge invariant. The external source A
dependence originates
from the non-standard de nition of the stress-energy tensor. Introducing an appropriate
rede nition of the stress-energy tensor, we will obtain a gauge invariant momentum density,
which is related to the velocity eld vi. We will discuss this in section 8.
1
2
Using (5.25){(5.27), the energy conservation (5.23) becomes
0 =
and its leading order terms give
5
2
which is the constraint (4.10). The momentum conservation (5.24) becomes
1
1 j
a
1
1
1
a
v
j
a
: (5.32)
The leading order terms give
1
a Aj @ivj +
1
1 j
a
1
a
v
j
a
(5.33)
which is a combination of (4.9) and (4.11).
The next-to-leading order terms of the conservation law provide the constraints at
second order. In order to calculate the solution to second order, we need to introduce the
derivative expansion of the correction terms, for example,
(5.28)
(5.29)
(5.30)
(5.31)
g
= g
+ h(1) + 2h(2) +
;
(5.34)
where g
is given by (3.6) and h(1) is the correction terms which we calculated in the
previous section. The expansion parameter is that of the derivative expansion, @ = O( ).
In order to calculate the second order solution, we further introduce the second order
correction terms h(2). However, as the correction terms do not contribute to the constraint
at rst order, these second order correction terms do not contribute to the constraint at
the second order. Therefore, we do not need to take h(2) into account to study the second
order constraints.
The background metric g
does not consist only of O( 0) contributions, but also
contains higher order corrections. The higher order corrections in g
are included in
x-dependent parameters r0, vi, a, and Ai. They can be expanded as
where r0(0), etc. are the leading order terms which we studied in the previous section, and
satis es the constraints at the rst order. The higher order terms, for example r0(1), do not
contribute the rst order terms, but must satisfy the second order constraints.
After some algebra, the second order constraint equation for the gauge eld gives
Together with the rst order constraint, it can be expressed as
where
is the heat conductivity and
is the shear viscosity whose values are
Eb0 = E ;
Ebi = E v
i
Pbi = nAi ;
Tb ij = P ij
ij ;
=
1
3
8 G r0 ;
=
1
3
16 G r0 :
(5.35)
(5.36)
(5.37)
(5.38)
(5.39)
(5.40)
(5.41)
(5.42)
(5.43)
(5.44)
(5.45)
(5.46)
(5.47)
(5.48)
This implies that there are no additional terms in this constraint at second order. In
a similar fashion, from the spatial component of the constraints in Einstein equation we
obtain
0 =
1
1 j
a
1
1
From the temporal component, we obtain
(5.42) agrees with (5.30), and an appropriate combination of (5.40) and (5.41) gives (5.32).
From the stress-energy tensor and the current, we can read o the energy density E ,
charge density n and pressure P as
E =
3
In terms of these quantities, the energy ow, momentum density and stress tensor are
expressed as
The Lifshitz invariance condition now becomes
the following form;
In terms of the above uid variables and transport coe cients, the uid equations take
ij ij
Eq. (5.58) can also be expressed as
where the current is given by
v^ is de ned in (5.4) and the eld strength is de ned as F = dA with
In terms of the temperature (note that here z = 2)
uid variables and transport coe cients can be expressed as
and
The scaling dimension under the Lifshitz scaling is given as
T =
4
= nv^ :
A = Ai(dxi
vidt) :
1
2
1
2
(5.49)
(5.50)
(5.51)
(5.52)
(5.53)
(5.54)
(5.55)
(5.56)
(5.57)
(5.58)
(5.59)
(5.60)
(5.61)
(5.62)
(5.63)
(5.64)
(5.65)
HJEP12(05)76
For comparison, the ordinary non-relativistic uid equations are given by
ij ij
where (5.63) gives conservation of energy, the Navier-Stokes equation (5.64) comes from
conservation of momentum, and the continuity equation (5.65) implies the conservation of
mass density. Eqs. (5.57) and (5.59) agree with the energy conservation (5.63) and
continuity equation (5.65), respectively. However, (5.58) is di erent from the Navier-Stokes
equation (5.64). Before we proceed further we must clarify the role of Newton Cartan theory.
The Galilei data ( ; h ) are constant under the covariant derivative;
HJEP12(05)76
r
= 0 ;
r h
= 0 :
Contrary to Einstein gravity, (6.2) does not uniquely x the Galilei connection. In order to
determine the connection, we must introduce a contravariant vector v (not to be confused
with velocities) and a two-form B . The vector v satis es the normalization condition
By using the vector v , we also de ne the spatial covariant (symmetric) metric h , which
satis es
h v = 0 ;
h h
= P
v :
Then, the Newton-Cartan connection is expressed as
In general, the Newton-Cartan connection (6.5) has torsion,
The curvature is de ned via the commutator of the covariant derivative and given by
Newton Cartan theory and Milne-boost invariance
We brie y review here the Newton-Cartan theory [64, 65]. We rst introduce the Galilei
metric of d-dimensional Galilei space-time, which consists of a 1-form
and a contravariant
symmetric tensor h
of rank (d
1). The 1-form
de nes the time direction of the
Galilei space-time, and h
gives the spatial inverse metric. They satisfy the orthogonality
condition,
It should be noted that v
has no relation to the uid velocity, and is in general,
di erent from the uid velocity vector v^ , although v is referred to as the velocity
eld
sometimes in the Newton-Cartan literature. Here, v is the inverse timelike vielbein to be
distinguished in general from the uid velocity eld.
1
2
h
(6.1)
(6.2)
(6.3)
(6.4)
(6.5)
(6.6)
(6.7)
(6.8)
1
2
R
If we impose the Newtonian condition
gauge eld; B = dB.
are (h ; ; v ; B ).
where [
] and (
) in the indices stand for the antisymmetric part and symmetric part,
respectively, we obtain the condition dB = 0. Then, B is (locally) the eld strength of a
To summarize the Newton-Cartan data that determine a given Newton-Cartan frame
h
However di erent Newton-Cartan data may describe the same physics. To see this we
introduce the concept of the Milne boost (see [66]), which is an internal symmetry of the
Newton-Cartan theory. Here, we focus on the torsion free cases, since for our solution on
the gravity side, (5.1), (5.2) we have
= (1; 0) and this give zero torsion in (6.6).7
We introduced v and B to de ne the Newton-Cartan connection. Two pairs (v ; B )
and (v0 ; B0 ) are physically the same if they give the same Newton-Cartan connection. The
torsionless part of the Newton-Cartan connection is invariant under the following (Milne
boost) transformation;
v
! v0 = v + h V ;
B ! B0 = B + P V dx
1
2
h V V
dx ;
h0 = h
( P +
P )V +
h V V
(6.9)
(6.10)
(6.11)
HJEP12(05)76
where V is a vector which parametrizes the Milne-boost transformation.
It should be noted that all non-trivial degrees of freedom of v can be absorbed into
B by using the Milne boost. The normal direction to the timeslice is xed by the
normalization condition
v
= 1 and the other directions are freely transformed by the Milne
boost (6.9). Therefore, we may choose an arbitrary but appropriately normalized inverse
timelike vielbein v .
Now, we will relate the Newton-Cartan data to the vielbeins (5.3) and (5.4) which we
introduced in the induced metric on the boundary (5.1) and (5.2). The timelike vielbein
is simply identi ed to that in the Newton-Cartan theory. The inverse spacelike vielbein
e^a should also be identi ed with that in the Newton-Cartan theory, which implies that the
inverse spatial metric in Newton-Cartan theory is expressed in terms of e^a as
h
= e^a e^a :
(6.12)
In the frame we use it is the unit matrix in the spatial directions. For the leftover
NewtonCartan data (v ; B ) there are many choices related by Milne boosts.
By using the Milne boost (6.9), we can choose a special \frame" in which the inverse
timelike vielbein equals the
uid velocity v0
= v^ .8
We will call this Newton-Cartan
frame the \holographic frame" from now on. It remains to identify the gauge eld B in
this frame. This is facilitated by comparing our equation in (5.60) with the one derived
in [36] in Newton Cartan theory (equation (5.17) of that paper). This gives the following
identi cation, B = A with
A = ( viAi; Ai) ;
Bt =
7The Milne boost in torsional cases is discussed in [36, 37].
8Here, \frame" is di erent from the coordinate frame but a special point in the internal space of the
Milne boost symmetry. No coordinate transformation is needed to take this \frame".
where Ai appears in (5.16){(5.19). We conclude, that in the \holographic frame" the
Newton-Cartan data are
A
0 0 0 1
v
! v^ = (1; ~v)
(6.14)
together with (6.13). In any other frame,
and h
remain invariant, but v and B
change by the Milne boosts, (6.9)
We would like now to change to a more canonical ( at) frame that is appropriate for
HJEP12(05)76
non-relativistic physics in particular the standard Navier-Stokes equation. We will call this
the \Newton frame" and it is determined by v
= (1; ~0). We will go from the holographic
frame to the Newton frame by a Milne boost with parameter V
= (0; ~v). In the Newton
frame we therefore obtain a new gauge eld that we call Ae using (6.9)
It should be noted that v^ does not transform under the Milne boost. The inverse vielbein
v transforms under the Milne boost, but the uid velocity v^ is Milne-boost invariant.
We will de ne also a class of gauge elds that are Milne-boost invariant. It is simple to
show that for any Milne-invariant vector X , that is normalized:
X
= 1, the following
gauge eld
Bb = B + h
X dx
h
X X
dx ;
is Milne-boost invariant as can be directly veri ed using the transformations in (6.9){(6.11).
We choose as such a vector the uid velocity vector, X
= v^ , which satis es
v^ = 1,
to de ne the invariant gauge eld as in (6.16)
Binv = B + h v^ dx
h v^ v^
dx :
1
2
1
2
We can evaluate Binv in the Newton frame (6.15) to nd that
1
2
Binv = Ae + h v^ dx
h v^ v^
dx = Ae + vidxi
2
1 v2dt = A :
As Binv is Milne-boost invariant its evaluation in the holographic frame, will also give the
same result. We conclude that the gauge eld A in the holographic frame, given in (6.13)
is Milne-boost invariant.
Next, we consider the Navier-Stokes equation. We have already found the
NavierStokes equation in the holographic frame in (5.58) to have the form
Fi J
where F = dA and where as we have shown above, all quantities that enter are
Milneboost invariant. However this does not look like the usual non-relativistic Navier-Stokes
(6.15)
(6.16)
(6.17)
(6.18)
(6.19)
equation (5.64) because we are not in the Newton frame. To do this we must rewrite it
using the gauge eld Ae in the Newton frame, (6.15). We directly compute
using (5.58), (5.61) and (6.15). Substituting this equation in (6.19) we obtain
Fei J
which is the conventional Navier Stokes equation albeit in the presence of an external force.
To bring this equation to an even more familiar form we follow [43], and choose the
gauge eld A such that
This implies
Then, (6.21) is expressed as
v^ = t + h
Aet =
A
This expression agrees with the Navier-Stokes equation with the external force Fi;
the force being the gravitational force from the Newton potential Fi = n@i e.
Therefore, (6.21) can be interpreted as the Navier-Stokes equation in the non-trivial Newton
potential;
e = Aet =
fected by the Newton potential. The conservation of total energy is obtained by
appropriate combination of (5.57) and (6.21), and can be expressed in terms of the Newton
potential (6.26) as
ne
E + P +
ne
v
i
ij v
j
This is consistent with the energy conservation of uids in the Newton potential; for
timeindependent Newton potential @t e = 0, the energy conservation is expressed as
ne
E + P +
ne
v
i
ij v
j
(6.28)
The stress-energy tensor (5.16){(5.19) is now rewritten as
1
2
nv2
1
2
nv2
Tb0i = nvi ;
Tbij = P ij
E
ne +
E + P
ne +
1
2
nv2
;
1
2
ij + nvivj :
1
2
nv2
1
2
nv2
(6.20)
(6.21)
(6.22)
(6.23)
(6.24)
(6.25)
(6.26)
(6.29)
(6.30)
(6.31)
(6.32)
This is the ordinary stress-energy tensor for non-relativistic uids with the Newton
potential terms. The conservation of total energy (6.27) and Navier-Stokes equation (6.24) can
be expressed in terms of this stress-energy tensor as
7
The entropy current
The holographic entropy current JS is de ned by the dual of the (d 1)-dimensional volume
form on the time slice on the horizon;
The entropy current JS can be expressed in terms of the normal vector n as
1 d JS 1 dx 2
^
^ dx d :
J
S =
p
h n
4G n0
;
T =
4
r0 ;
(6.33)
(7.1)
(7.2)
(7.3)
(7.4)
(7.5)
(7.6)
(7.7)
(7.8)
where the normal vector at the horizon is given by
to rst order in the derivative expansion.9
is obtained as
S = r
r0(x) ;
JS0 =
J Si =
Then, the holographic entropy current for the Lifshitz space-time with d = 4 and z = 2
Comparing this expression with (5.25) and (5.43), we nd that the entropy current satis es
T JS = Eb + P v^ =
T
b
v^
+ P v^ ;
where the Hawking temperature T is given by
to rst order in the derivative expansion.
of the entropy current is calculated as
We can easily check that the entropy current satis es the second law. The divergence
S =
1
4G
1
12G
9The horizon radius has corrections at higher order in the derivative expansion. Since these correction
terms appear from the second order, it does not contribute to the entropy current at the rst order.
The uid has the following properties;
The stress-energy tensor has a form similar to that of non-relativistic uids, and is
expressed in terms of the
uid variables: the velocity eld vi, the energy density E ,
the pressure P and the charge density n. It also contains the external gauge eld A.
At rst order, it has as transport coe cients the thermal conductivity
and shear
viscosity
while the bulk viscosity is zero due to the Lifshitz scaling symmetry. All
the above variables and transport coe cients are functions of temperature T .
The (particle number) density appears associated to the external gauge eld. The
stress-energy tensor calculated directly from the bulk solution agrees with that of
nonrelativistic
uids except for the terms where the density appears. The terms which
contain the density are di erent from those of a non-relativistic uid. In particular,
the momentum density T 0i is proportional to Ai and vanishes for Ai = 0.
The conservation law of the stress-energy tensor is not given in the form of its
covariant derivative, but is that appropriate to a boundary Newton-Cartan theory. It
is expressed in terms of the energy
ow E , momentum density Pi and stress
tensor T ij . The stress-energy tensor satis es the Ward identity of the scale invariance
1)P without any modi cation. In terms of the uid variables, the
conservation law takes the form of the uid equations: the energy conservation, continuity
equation and the Navier-Stokes equation.
The uid equations are similar to the non-relativistic
uid equations. The
continuity equation and energy conservation equation agree with that for standard
nonrelativistic
uids. However, the Navier-Stokes equation is di erent from that for
ordinary non-relativistic
uids. It does not contain the velocity
eld except the
terms appearing in the shear tensor. It contains the coupling to the external gauge
eld instead. A related property of the associated stress tensor is that it is gauge
non-invariant but Milne-boost invariant.
The absence of the velocity elds in our Navier-Stokes equation can be explained
as follows. For an ordinary
uid, the pressure is comparable to the non-relativistic
energy which does not include the mass energy, and hence it is much smaller than
the relativistic energy density. In our case, the pressure is of the same order as the
energy density because of the Lifshitz scaling symmetry. For this to happen the uid
equations must be di erent and this is what we nd, namely, the contribution from
pressure is much larger than that in ordinary uids, and then, the terms with velocity
elds becomes negligible compared to the pressure.
This is the reason that some terms with velocity elds in the Navier-Stokes equation
are absent. These terms are replaced by the external source which is identi ed with
the gauge eld in the Newton-Cartan theory.
We may do some rede nitions of the stress-energy tensor and by identifying the
gauge eld A to the gauge eld in Newton-Cartan theory, which takes the form of
A = Ae + vidxi
1 v2dt our Navier-Stokes equation agrees with the standard
nonrelativistic Navier-Stokes equation. In such a case the stress-energy tensor is gauge
invariant but Milne-boost non-invariant. Finally there is a de nition of the
stressenergy tensor that is both Milne-boost and gauge invariant.
Since the gauge eld in the Newton-Cartan theory is a generalization of the
Newtonian gravity theory, a general external gauge
eld gives a
uid in a non-trivial
gravitational potential. The Newton potential appears in the gauge eld as e = Aet.
The Navier-Stokes equation we obtain agrees with the ordinary Navier-Stokes
equation in the presence of an external (gravitational) force. The gauge
eld does not
contribute to the conservation of (internal) energy density E
. The conservation of
total energy can be obtained from the conservation of E and the Navier-Stokes
equation and agrees with that for ordinary non-relativistic uid in a non-trivial Newton
potential.
The entropy density is de ned in terms of the horizon area and satis es the local
thermodynamic relation with energy density and pressure. The divergence of the
entropy current is non-negative, which is consistent with the second law.
The form of the uid equations is independent of the Lifshitz exponent z as well as
of the conduction exponent
. This dependence appears at rst order, inside the
various state functions and therefore only in the constitutive relations.
There are several obvious interesting questions that remain unanswered by our work.
The rst is the extension of our results to hydrodynamics in the presence of hyperscaling
violation in the metric ( 6
= 0). This is under way. A naive guess would be that the
hydrodynamics would be a dimensional reduction of the one found here, along the lines
described in [29, 59]. In particular in [29] it was shown that Lifshitz solutions with
hyperscaling violation can be obtained as suitable dimensional reductions of higher-dimensional
Lifshitz invariant theories without hyperscaling violation. The associated reduction of the
hydrodynamics will provide equations similar to the ones here but with a non-zero bulk
viscosity. This needs to be veri ed.
actively pursued in [60].
A further extension involves Lifshitz geometries with broken U(1) symmetry. This is
An interesting question in relation to the above is: what is the appropriate
hydrodynamics for QFTs that are RG Flows that interpolate between relativistic and
nonrelativistic theories. To motivate the answer to this question, we consider rst non-Lorentz
invariant (but rotationally invariant) ows between Lorentz invariant xed points,11 [62],
but where the velocity of light in the IR is di erent for that in the UV. In such a case, the
hydrodynamics of this theory, is relativistic, but with a speed of light that is temperature
dependent.
This example suggests that in an (Lorentz-violating) RG
ow from a CFT (with an
unbroken U(1) symmetry that is used to drive the breaking of Lorentz invariance) to an
11The fact that the speed o light can vary on branes was pointed out rst in [61].
of the associated theories.
necessary.
Acknowledgments
IR non-relativistic scaling (rotational invariant) geometry at an arbitrary temperature, the
hydrodynamics will be again of the relativistic form (but with a general equation of state)
and with a speed of light c(T ) that is again temperature dependent. In the IR, c(T ! 0) =
1 and the hydrodynamics reduces to the one found here with the U(1) symmetry becoming
the mass-related symmetry. This is nothing else than the standard non-relativistic limit12
of the the relativistic hydrodynamics while all thermodynamic functions and transport
coe cients are smooth functions of T (if no phase transition exists at nite T ). Otherwise
they follow the standard behavior at phase transitions.
A more general breaking of Lorentz invariance during a RG
ow must involve higher
form
elds of tensors in the bulk, or a multitude of vector elds and the details of the RG
ow become complicated. It is important that such
ows are analyzed as they hold the
key to understanding general non-relativistic ows as well as generalized hydrodynamics
Finally a more detailed study of the above issues in the absence of U(1) symmetry is
We would like to thank A. Mukhopadhyay for discussions. We especially thank J. Hartong
and N. Obers for participating in early stages of this work and for extensive and illuminating
discussions on the topics presented here.
This work was supported in part by European Union's Seventh Framework
Programme under grant agreements (FP7-REGPOT-2012-2013-1) no. 316165, the EU program
\Thales" MIS 375734 and was also co nanced by the European Union (European Social
Fund, ESF) and Greek national funds through the Operational Program \Education and
Lifelong Learning" of the National Strategic Reference Framework (NSRF) under
\Funding of proposals that have received a positive evaluation in the 3rd and 4th Call of ERC
Grant Schemes."
A
Notations
Since in the topic treated in this paper there are a lots of variables involved and many
rede nitions, we present here a list of the variables and their de nitions with comments when
necessary. We also present a translation dictionary to the variables used by Jensen, [37]
and Hartong et al., [43]. In table 1, we present the relation between the variables in this
paper and those in [35] and [41]. In table 2, we present the correspondence of the uid
variables in this paper and those in [35].
Variables de ned on the gravity side.
d is the dimension of the space-time boundary, and the dimension of the boundary
12This is expected to happen along the lines presented in [63] although this needs to be veri ed.
vi: boost parameter introduced into the (static) black hole geometry in (3.4).
r0: the horizon radius which is de ned in (2.14).
a: the coe cient of the rz+d 1 term of the gauge eld, introduced in (2.6). Note that
a 6= a dx .
A : the constant part of the gauge eld, which is de ned in (3.8). This corresponds
to the Milne boost-invariant gauge eld in the Newton-Cartan theory Binv, or
equivalently, B in the holographic frame v = v^ . In this paper, it is sometimes expressed
as the 1-form A = A dx .
v^ : the timelike inverse vielbein on the boundary which is de ned up to the factor
r 2zf 1, or equivalently, de ned in (5.2) and (5.4) and given by v^
= (1; vi) in our
model. This corresponds to the velocity vector eld of the
uid. The holographic
frame of the boundary Newton-Cartan geometry is de ned by v = v^ .
: the timelike vielbein on the boundary which is de ned up to the factor of r2zf ,
or equivalently, de ned in (5.1) and (5.3) and given by
= (1; 0) in our model.
This corresponds to the timelike unit normal which de nes the time direction in the
Newton-Cartan theory. It is automatically invariant under the Milne boost.
e^a : the spacelike vielbein on the boundary which is de ned up to the factor of r2,
or equivalently, de ned in (5.1) and (5.3), and given by ea dx
= dxa
vadt in our
model. This is invariant under the Milne boost and equals to the spacelike vielbein
in the Newton-Cartan theory if we take the holographic frame v = v^ .
e^a : the spacelike inverse vielbein on the boundary which is de ned up to the factor of
r 2, or equivalently, de ned in (5.2) and (5.4), and given by ea @
This corresponds to the spacelike inverse vielbein. It is automatically invariant under
the Milne boost.
Variables in the (boundary) Newton-Cartan theory.
v : the timelike inverse vielbein in Newton-Cartan theory. This notation is
introduced above (6.3) to de ne the Galilei connection. The timelike inverse vielbein is not
invariant under the Milne boost but it is covariant. In the literature it is sometimes
called \velocity" but must be distinguished from the velocity of the uid.
h : the induced covariant metric on the time-slice. It is de ned by (6.4) and given by
h
= diag(0; 1; 1; 1) for a 4-dim space-time in the Newton frame. It is not invariant
under the Milne boost but it is covariant.
ea : the spacelike vielbein, which is introduced in (8.37){(8.38). It is given by ea =
diag(0; 1; 1; 1) for a 4-dim space-time. It satis es h
= ea ea.
ea : the spacelike vielbein, which is introduced in (8.37){(8.38). It is given by ea =
diag(0; 1; 1; 1) for a 4-dim space-time in the Newton frame. Since it is invariant under
a Milne boost, it is equal to e^a .
h
A
M
v^
u
v^
e^
a
{
e^
a
A
( e; 0)
n
T 0
e
At
Mt
v
u
i
i
Mi
from the relation of the metric and gauge
eld in gravity side. In [43], v^ is
de ned such that spacial components of the associated Milne invariant gauge
eld vanish. We can
also de ne such velocity
eld in our notation, v^
Ab . If we identify this combination to
v^
in [43], we obtain another correspondence. In this case, no variables in [43] correspond to the Milne
boost invariants in this paper.
B : the gauge eld in the Newton-Cartan theory. This notation is introduced
below (6.8). It is not invariant under the Milne boost but it is covariant.
Ae : the gauge eld in the Newton-Cartan theory in Newton frame v = (1; ~0). This
notation is introduced in (6.15). It is not invariant under the Milne boost but it is
covariant.
Bb: a Milne boost invariant combination for the gauge eld. It is de ned in (6.16),
with arbitrary but appropriately normalized Milne-invariant vector X .
Binv: the Milne boost invariant combination Bb for X
= v^ . It is de ned in (6.17).
vidxi
In terms of the gauge eld in the Newton frame Ae, it is expressed as Binv = Ae +
1 v2dt. It also equals to the gauge eld in the holographic frame A.
2
VEVs and uid variables.
T(nr) : the stress-energy tensor without counter terms on dr = 0 surface. To be
exact, the stress-energy tensor is the coe cient of O(r 5) term of this tensor (O(r z 3
for general z). It appears in (C.7){(C.10).
Tr : the renormalized stress-energy tensor on dr = 0 surface or its regular part in
the section in which we are discussing only on the boundary
uid. This is
introduced in (5.9) and the boundary stress-energy tensor is given by the coe cients of
O(r 5) terms.
T
b : the boundary stress-energy tensor, or its regular part in the section in which
we are discussing only the boundary uid. This is de ned in (5.9).
T
T
: it is de ned by (8.15), Tb
= T
+ J A
J A . This de nition is used to
de ne Milne-boost-invariants for a general background of A. It is both Milne-boost
invariant and gauge invariant.
: de ned by (8.30), T
= Tb
J Ae +
J Ae gives the standard
non-relativistic
uids' stress-energy tensor. It consists of the physical energy vector,
physical momentum density, and physical stress tensor. It is gauge invariant but not
Milne-boost invariant.
J : the current without counter terms de ned in (5.10). It is regular even without
the counter terms. It corresponds to the mass current in a non-relativistic theory.
Eb : de ned by
T
b
vector. It is not gauge invariant.
v^ in (5.20). It corresponds to the (Milne boost invariant) energy
Pb : de ned by Tb
e^ae^a in (5.21). It corresponds to the (Milne boost-invariant)
momentum density. It is di erent from the physical momentum density P , which is
not invariant under the Milne boost. It is not gauge invariant.
Tb
: de ned by Tb (e^ae^a )(e^b e^b ) in (5.22). It corresponds to the (Milne boost
invarivector. It is gauge invariant.
T
v^ in (8.17). It corresponds to the (Milne boost invariant) energy
P : de ned by T
e^ae^a in (8.18). It corresponds to the (Milne boost-invariant)
momentum density. It is di erent from the physical momentum density P , which is
not invariant under the Milne boost. It is gauge invariant.
: de ned by T (e^ae^a )(e^b e^b ) in (8.19). It corresponds to the (Milne boost
invariT
v in (8.36). It corresponds to the physical energy vector, which
T
T
E : de ned by
P : de ned by T
ant) stress tensor. It is gauge invariant.
contains a contribution from the mass density.
eaea in (8.37). It corresponds to the physical momentum density,
which contains a contribution from the mass density.
: de ned by T (eaea )(eb eb ) in (8.38). It corresponds to the physical stress tensor,
which contains a contribution from the mass density.
is introduced in (5.43) and equals E 0.
E : (Milne boost invariant) energy density, or equivalently, internal energy density. It
P : the pressure. It can be read o from the stress tensor and introduced in (5.43).
n: it is de ned by 1=a. It is introduced in (5.43). It corresponds to the particle
number density, or equivalently, the mass density.
T : the temperature. It can be calculated as the Hawking temperature of the black
hole (2.15).
S
case.
time-slice at the horizon.
J : the entropy current, which is de ned from the volume form in (7.2) on the
s: the entropy density, which is de ned in (7.10). It equals to JS0 in the non-relativistic
E
T
variables do not contain contributions from the external source.
energy tensor, or the uid equations.
stress-energy tensor, or uid equations.
e: the Newton potential, which is e = Aet. It is introduced around (6.24).
: the shear viscosity. It is introduced in (5.47) and can be read o from the
stress: the heat conductivity. It is introduced in (5.45), and can be read o from the
elds and constants in gravity side.
g : the metric. It is introduced in (2.1).
A: the gauge eld. It is introduced in (2.1).
: the dilaton. It is introduced in (2.1).
G: the Newton constant. It is introduced in (2.1).
A^: the gauge eld with local lorentz indices. It is de ned in (5.5).
J^: the current with local lorentz indices. It is de ned in (5.8).
h : the correction terms for the metric. It is introduced in (3.10).
a : the correction terms for the gauge eld. It is introduced in (3.11).
': the correction terms for the dilaton. It is introduced in (3.12).
R : the Ricci tensor. It appears rst in (2.1).
: the cosmological constant. It appears rst in (2.1).
T (bulk): the energy-momentum tensor in the bulk. It appears in (B.11).
T(BY): the Brown-York tensor which is de ned by (5.14), T(BY) = 8 1G (
K
K
).
n : the normal vector to the boundary or horizon. It appears rst in (5.15).
K : the extrinsic curvature on the boundary. It appears in (5.14). To be precise, it
is de ned on constant but nite r surface.
: the induced metric on the boundary. It is introduced in (5.1). To be precise, it
is de ned on constant but nite r surface.
Calculation of the equations of motion at rst order z = 2
In this section, we calculate the rst order solution in derivative expansion around (3.6){
(3.9). The coordinate can be chosen such that vi(x) = 0 at any given point, therefore
we may take vi(0) = 0 without loss of generality. Although we will work in vi(0) = 0
coordinates, the solution for vi(0) 6= 0 can be obtained by boosting the solution uniformly.
From now on we work at the point x
= 0, but we omit the subscript \(0)", hereafter.
Here, we take the following gauge conditions;
grr = 0 ;
gr / v^ ;
Tr[g 1h] = 0 ;
ar = 0 ;
(B.1)
The correction terms can be classi ed by using the SO(3) symmetry along the spatial
directions. The equations of motion are separated into that for scalar (sound mode), vector
mode and tensor mode. The sound mode consists of the following components
and vector mode
and the tensor mode is the traceless part of the metric
htt
htr
hii
at
' ;
hti
ai ;
hij :
To simplify the di erential equations, we rede ne the correction terms for the metric
as follows
htt
htr
hti
hxx
hij
gtt = r4( f + htt);
gtr =
gti = r2hti
2 htr;
gij = r2( ij + hxx ij + hij )
(B.2)
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
where hxx is the trace part and hij is the traceless part in xi-components. The gange
condition gives additional constraints;
and the (r; r)- and (r; i)-components of the correction terms must vanish. We de ne
The other correction terms a and ' are similar to the de nitions (3.11) and (3.12), but
ar is eliminated by the gauge condition.
2htr + 3hxx = 0
h1 =
1
2 hxx :
Some components of the equations of motion do not become di erential equations for the
correction terms, but give the constraints on the parameters. From the Einstein equation
we obtain
n
R
= 8 Gn
T (bulk)
where n is the normal vector and
is the induced metric on r =const. surfaces. In
fact the above equation contains no correction terms. For the sound mode, by contracting
the (B.11) with v^ = (1; vi), we obtain the following constraint;
0 =
p =
z + d
2(z
1
1)
1=2
:
Equation (B.12) must be satis ed for arbitrary r, and hence the rst and second terms
must vanish independently,
From (2.3), the r-component does not contain correction terms and gives another
where
constraint;
where the rst equation originates from (2.4) and the second from the t-component of (2.3).
The third to fth equations are the (t; t)-, (t; r)- and (r; r)-components of (2.2). The last
After substituting the above constraints to the equations of motions for the sound
mode, we obtain the following di erential equations;
0 =
2p6a0t(r)
a
0 = ra0t0(r)
0 =
6p6r5h01(r) + 6p6r05h01(r) + p6r5h0tt(r) + 5p6r4htt(r)
+ r6'00(r) rr05'00(r) + 6r5'0(r) r05'0(r) + 30r4'(r) 2p6r2@ivi
4a0t(r) + 30ar5h01(r) + 5p6a0r5'0(r)
+ 3ar6h0t0t(r) + 30ar5h0tt(r) + 60ar4htt(r) + 20p6ar4'(r)
12ar2@ivi + 8a0t(r)
72ar5h01(r)
18ar05h01(r)
240ar4h1(r)
0 = 3a
+ 8a0t(r) + 20p6ar4'(r) 240ar4h1(r) + 60ar4htt(r)
0 =
0 =
12h01 + p6'0 3rh010
2a0t(r)
ar2
3r4h010(r) +
3r05h010(r)
r
36r3h01(r) +
21r05h01(r)
r2
120r2h1(r)
(B.11)
(B.12)
(B.13)
(B.14)
(B.15)
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
(B.21)
(B.22)
HJEP12(05)76
(10), and at(0) are integration constants and h2 is given by
h2(r) =
5r04
0
q 37
2
5 1
2
0
; 2; r5 A
5r4
C2 G23;;13
r
5 4
p185 ; 110 13 + p185
equation is the trace part of the spatial component of (2.2). The above equations are not
independent but an appropriate combination gives the constraints, which we have already
imposed, and hence becomes trivial.
We rst impose the constraints to the parameters and then solve the di erential
equations. The solution for the sound modes is
htt =
p
2 6
5
5
1
3
h1 =
5
0
r5
1
6r05
r5
1
0
r5
1
r5
h02(r)
4 + 0
5
r5
1
a 15h(10) + p6r5 (0)
15r5h2(r)
p
r56 h(10) + 3
r 3
2 h2(r) +
r 3
2
rh02(r)
(B.23)
(B.24)
(B.25)
(B.26)
(B.27)
(B.28)
(B.29)
(B.30)
(B.32)
(B.33)
where pFq and Gpm;q;n are hypergeometric function and Meijer G-function, respectively [58],
and C1 and C2 are integration constants.
B.2
Vector mode
As for the sound modes, spatial component of (B.11) gives a constraint
a) :
Since this constraint must be satis ed at arbitrary r, the rst and second terms must vanish
independently,
Then, the equations of motion for the vector modes are
0 = r
0 = r2
5
5
0
r
5
r
05 a0i0(r) + 5ar7h0ti(r) + 7r05
+ ar2 r
5
0
r
5 h0t0i(r) + 4ar r
5
0
2r5 a0i(r) (B.31)
r
5 h0ti(r)
2 r
5
r
05 a0i(r)
0 = 2a0i + 4ar3h0ti + ar4h0t0i
where the rst equation is the xi-component of (2.3) and the others are the (t; xi)- and
(r; xi)-components of (2.2), respectively.
The solution for the vector modes is
hti =
Z dr
ht(i0)
3r05 Z
2
2
a ai +
2r3
where ht(i0) and ai(0) are integration constants. The function a1 is given by
and is expanded around r = r0 as
In order for the solution to be regular at r = r0, we we must take
and for the rest we obtain
a1 =
3C3
375r07(r
C3 =
hti =
ai =
ai(0) r5
r3
0
a2 ht(i0) + ai(0) r5 + r
2 0
3 5
Tensor mode
The equation of motion for the tensor mode is given by
0 = 2( 6r5 + r05)h0ij + 2( r6 + rr05)h0i0j
There are no constraints for the tensor mode.
The solution is where
hij =
ij
Z
r2dr
r5
r
0
5 + C4
Z
dr
r(r5
r05)
2
Regularity at r0 implies C4 = r03. We nally obtain
hij =
ij
Z (r3
r(r5
(B.34)
(B.35)
(B.36)
(B.37)
(B.38)
(B.39)
(B.40)
(B.41)
(B.42)
(B.43)
(B.44)
8 GT(BY)00 = 3
8 GT(BY)i0 =
8 GT(BY)0i =
8 GT(BY)ij = 4 ij +
3
2r7
vi +
1
1
2r5
4r05vi + @ir05 + O(r 7) ;
2r5 r0 ij + O(r 7) ;
3
16 GJ 0 =
16 GJ i =
2
1
ar5 +
ar10 r05 + O(r 11) ;
vi +
ar10 r05vi + O(r 11) :
to the rst order in the derivative expansion.
The current is given by
From (5.13) we can obtain the non-renormalized part of Tr
as
8 GTr(nr)00 = 4 +
2r05
8 GTr(nr)i0 =
8 GTr(nr)0i =
1
r5
1
8 GTr(nr)ij = 4 ij +
2 0
1
ar5 Ai + O(r 6) ;
1 viAi
+ O(r 6) ;
+ O(r 6) ;
1
r5
21 r05 ij
1 3
1 viAj
+ O(r 6) :
the terms of O(r 5) become
nite. Those of O(r0) diverge at the boundary, r ! 1,
but can be subtracted by introducing the boundary cosmological constant term. We can
further introduce a boundary counter term proportional to A2;
Sct =
1
16 G
Z
d x
4 p
This induces a counter term for the stress-energy tensor
8 + C +
Ce
Calculation of the stress-energy tensor
Here, we calculate the stress-energy tensor. For the solution (4.5), the Brown-York tensor
(C.1)
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
(C.7)
(C.8)
(C.9)
(C.10)
(C.11)
(C.12)
(C.13)
1
r5
Since the volume form on the boundary behaves as
r5 ;
1
1
5
2
1
2r5 0
r
5
8 GTr(ct)
(4
C)
+ C
The coe cient C in the counter term can be xed by the regularity condition for the
operator dual to the dilaton scalar . The vacuum expectation value of the operator O is
C 5
16 G r0 ;
P =
1 + C 5
16 G r0 :
O
= lim r5Or
r!1
Or = p
1
Sr = n r
:
Or
(nr) =
1
16 G
6
r 3 r05 !
2 r5
+ O(r 6) :
(C.14)
(C.15)
(C.16)
(C.17)
(C.18)
(C.19)
(C.20)
(C.21)
(C.22)
(C.23)
(C.24)
(C.25)
(C.26)
(C.27)
HJEP12(05)76
Subtracting the counterterm, the renormalized stress-energy tensor becomes
1
1
1
1 1
5
0
1 viAi ;
2 0
1 vivj Aj +
1 3
;
(1 + C)r05 ij
1 3
1 viAj t :
From the equations above we can read the energy density E and pressure P
Tb00 =
Tb0i =
Tbij =
r05 ij
3 5
2 0
r
1 viAi ;
2 0
1
a
1 vivj Aj +
1 3
;
1 viAj ;
For our rst order solution, the non-renormalized expectation value is calculated as
The regular term is at O(r 5) while the counter terms becomes
Or
(ct) =
1
5
16 G 2
C e
B B =
1
16 G
6 C + p
5
6 C r05
+ O(r 6) :
The renormalized expectation value is
Or =
1
16 G
6 (1
C)
r 3
2
(1
2C) r05
r5 !
+ O(r 6) :
To obtain a nite O , we must take C = 1, as the rst term is divergent at the boundary.
We obtain and Moreover the renormalized stress tensor (5.16){(5.19) satis es
O
1
16 G
1)P :
D
More on counter terms
We can also consider higher order terms of A for the counter terms. Due to the constraint
on a and , A always appears with the factor of e =2. Then, general counter terms is
d x
4 p
X c
5
2
e
A A
;
where the boundary cosmological constant term is xed such that the boundary
stressenergy tensor becomes nite. Then, the stress-energy tensor becomes
Tbij =
1
1
8 G a Ai ;
1
2 +
2
1 X c
2 0
1
0
1 viAi ;
1 vivj Aj +
1 3
j
1 + X c
1 3
The dual operator to the dilaton is calculated from
Or =
1
16 G
6
1
X c
1
X
c
! r5 #
0
r5
+ O(r 6) :
In order to regularize O
= limr!1 r5Or, the coe cient c must satisfy
(C.28)
(C.29)
(D.1)
(D.2)
(D.3)
(D.4)
(D.5)
(D.6)
(D.7)
Then, the stress-energy tensor is same to (5.16){(5.19).
E
Regularity conditions of the gauge eld at the horizon
If the guage eld A
has non-zero At at the horizon, it becomes singular at the horizon.
It can be easily seen by taking the wick rotation and by considering the Polyakov loop
wrapping on the time circle. Although the horizon is a point in the imaginary (euclidean)
time, it becomes two surfaces, future and past horizon in real time. In this section, we
show that the singularity appears only in the past horizon even for At 6= 0, if we take the
Eddington-Finkelstein coordinates. Here, we focus on the near horizon region and discuss
about the regularity of the gauge eld at the horizon.
In the near horizon region, the metric of the non-extremal black holes is universally
given by the Rindler space;
ds2 = rdt2 +
+ (dxi)2 ;
where r = 0 is the horizon of the black hole. Only outside of the horizon is covered by
this coordinates. In order to move to the Eddington-Finkelstein coordinates, we de ne null
and then, the metric is expressed as
t = t log r
ds2 = rdt2
2drdt + (dxi)2 :
The ingoing (outgoing) Eddington-Finkelstein coordinates (with t+ (t )) also cover inside
of the future (past) horizon. The Kruskal coordinate is de ned by
x = e t =2
ds2 = dx+dx + (dxi)2 :
and then, the metric becomes
This covers all region, and x+ = 0 and x = 0 are the past and future horizon, respectively.
If the gauge eld has non-vanishing At and regular Ar at the horizon, it is singular
there. It can be seen as follows. In the Kruskal coordinates, the gauge eld becomes
A = Atdt = At
:
Therefore, the gauge eld is singular at the future and past horizon. In the
EddingtonFinkelstein coordinates, it is expressed as
A = At dt
dx
x
:
(E.1)
(E.2)
(E.3)
(E.4)
(E.5)
(E.6)
(E.7)
(E.8)
However, if we take the ingoing Eddington-Finkelstein coordinates, and if Ar is not
singular at the horizon, the gauge eld is singular only at the past horizon and is regular
at the future horizon. In the Kruskal coordinates, the gauge eld is expressed as
A = A+dt+ + Ardr = A+ x+
dx+ + Ar (x+dx + x dx+) :
Therefore the gauge eld is singular at the past horizon x+ = 0 but regular at the future
horizon x
= 0.
First order solution for general z
The correction terms can be calculated in a similar fashion to the z = 2 case. We de ne
h , a and ' as
gtt = r2z( f + htt) ;
rz 1
2
gtr =
gti = r2hti ;
htr ;
gij = r2( ij + hxx ij + hij) ;
' =
log( r 6) :
A = a(x) r5
Ai(x)vi(x) dt a(x)r2dr + Ai(x)dxi + atdt + aidxi ;
and
In the vi(0) = 0 gauge, the constrains are expressed as
The rst order solution for the sound modes is
p(x) = a(x);
1
2(z
1) 0
1)r z 3h(0)
1
r z 3 rh3(r) 2(z
r 3 (z
2
5)pz
(z + 3)2
1 at(1)
1)
Z
dr h2(r)
(F.1)
(F.2)
(F.3)
(F.4)
(F.6)
(F.7)
(F.8)
(F.9)
(F.10)
(F.11)
(F.12)
(F.13)
(F.14)
z
rz+3
0
rz+3
a(1)
t
h2(r) r z 3 2z
(z
rz+3
3) r0z+3 h(0)
1
3) r0z+3
rz+3 Z
dr h2(r)
htt = 2 6
p
p
z
z + 3
+ r z 2 1
2
3
r 2 pz
3 z + 3
h1 =
at = at(0)
3
' = (0)
r 3
1 a(1)
t
+ r z 3h(10) + r z 3
Z
dr h2(r)
3
z
z + 3 ah(10)
1
1
z + 3
rz+3at(1) +
z + 3 Z
z
dr h2(r)
(10), a(0) and at(1) are integration constants. The function h2 is the solution
t
of the following di erential equation;
0 = r2 rz+3
rz+3 h020(r) r zrz+3 + (2z + 3)r0z+3 h02(r)
0
(z + 2) (2z + 7)rz+3 + (z + 2)r0z+3 h2(r)
2F1
+ C2r 21 z+1 p(z+3)(9z+19)
2F1
; 2 ; r0z+3
rz+3
+; +; 2 +; r0z+3
rz+3
(F.15)
(F.16)
(F.17)
(F.18)
(F.19)
(F.20)
(F.21)
(F.22)
(F.23)
The rst order solution for the vector modes is
=
2
1 pz + 3
p9z + 19
z + 3
:
hti =
Z
r6 z
ai = 2(z
ht(i0)
1)rz+3
2(z
(z
1)
ai
5)r0z+3 Z
where ht(i0) and ai(0) are integration constants. The function a1 is given by
r7a2(r)
a2(r) = C3
1)rz+3
(z
5)r0z+3]2
(z + 3)arz 5 rz+3
rz+3 ht(i0)
0
z + 3 rz+2
0
2(z
1) r5 a 10(z
1)rz+3
(z
5)(z
and a2(r) is expanded around r = r0 as
In order for the solution to be non-singular at r = r0, we have to take
a2(r) = C3 +
2(z
1)
z(z + 3)2ar02z@ir0 + O(r
r0):
C3 =
z(z + 3)2ar02z@ir0
2(z
1)
and then, a2(r) becomes
a2(r) = (z + 3)arz 5 rz+3
rz+3 ht(i0)
0
z +3 rz+2
2(z 1) r5 a 10(z 1)rz+3r02 +z(z +3)r5r0z (z j!5)(z 2)rz+3 @ir0 : (F.24)
0
hij =
Z
r2dr
rz+3
0
rz+3 + C4
Z
r(rz+3
The regularity at r0 implies C4 = r03. Then, we obtain
hij =
Z
Counter terms for general z
In order to obtain regular stress-energy tensor, we introduce the counter terms;
Sct =
1
Z
d x
4 p
(4 + 2z) + C +
A A
:
z + d
2(z
1
1)
C e
Since for general z, the volume form on the boundary behaves as
the regular contribution to the stress-energy tensor is given by O(r z 3) terms of Tr .
Then, the renormalized stress-energy tensor is obtained as
p
rz+3 ;
a
1 viAi ;
1 3
a
a
1 vivj Aj +
1 3
;
1
1
1 z
1
2
C
2
1
z + 2
2
0
rz+3
0
2
Ai ;
(1 + C)r0z+3 ij
(F.25)
(F.26)
(G.2)
(G.3)
(G.5)
(G.6)
(G.7)
(G.8)
(G.9)
(G.10)
and they satis es
becomes
O
Or =
= lim rz+3
1
Or
For C = z
1, the energy density and the pressure becomes
E =
rz+3 ;
P =
16 G 0
rz+3 ;
The constant C in the counter terms can be xed by the regularity of the dual operator
to the dilaton . By introducing the counter terms, the expectation value of the operator
r
1)
C
C
r
r 3(z
2
1) ! rz+3 #
0
rz+3
In order for the above expression to be regular, we have to take C = z
1, and then, we obtain and
1
1
1 z
2 0
1 viAi ;
Ai ;
z rz+3
2 0
1 vivj Aj +
1 3
j
;
1 3
1 viAj ;
O
1
16 G
r 3(z
2
1) rz+3 :
0
(G.11)
(G.13)
(G.14)
(G.15)
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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