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# Holographic stress tensor of colored Lifshitz spacetimes and hairy black
holes

Deniz O. Devecioğlu<EMAIL_ADDRESS>School of Physics, Huazhong University
of Science and Technology,
Wuhan, Hubei, 430074, China

###### Abstract

We compute the holographic stress tensor of colored Lifshitz spacetimes
following the proposal by Ross-Saremi for gravity duals of non-relativistic
theories. For a well-defined variational principle, we first construct a
finite on-shell action for the Einstein-Yang-Mills model in four dimensions
with Lifshitz spacetime as a solution. We then solve the linearised equations
of motion and identify the modes that preserve the asymptotically Lifshitz
condition. Employing these modes, we also show that the stress tensor is
finite, obeying the scaling and the diffeomorphism Ward identities, i.e.,
conservations laws. As a final application, we evaluate the energy density and
the spatial stress tensor of the previously found numerical black hole
solutions with various dynamical exponents $z$. The alternative Smarr relation
that has been used in Lifshitz black holes and the first law of thermodynamics
are shown to hold without a global Yang-Mills charge, indicating the black
holes in question are hairy.

###### Contents

  1. 1 Introduction
  2. 2 Boundary counterterms in EYM
  3. 3 Stress tensor for non-relativistic theories
  4. 4 Perturbation analysis
    1. 4.1 Perturbation setup
    2. 4.2 Constant perturbations and solutions
      1. 4.2.1 Scalar modes
      2. 4.2.2 Vector modes
      3. 4.2.3 Tensor modes
    3. 4.3 General perturbations and solutions
      1. 4.3.1 Scalar modes
      2. 4.3.2 Vector modes
  5. 5 Numerical black holes and thermodynamics
  6. 6 Conclusions
  7. A Linearized quantities
  8. B Field equations for generalised perturbations
    1. 2.1 Field equations for the scalar modes
    2. 2.2 Field equations for the vector modes

## 1 Introduction

A great deal of non-relativistic systems exhibit dynamical scaling at the
fixed points of their renormalisation group flow

$\displaystyle t\rightarrow\lambda^{z}t,\quad x^{i}\rightarrow\lambda
x^{i},\quad r\rightarrow\lambda^{-1}r,\quad z>1,$ (1.1)

which can be seen as a generalisation of the conformal scaling experienced in
relativistic theories. Following the logic of gauge/gravity duality, one can
then look for spacetimes that are weakly coupled dual descriptions of the
strongly coupled field theories enjoying this anisotropic scale invariance
Son:2008ye ; Balasubramanian:2008dm ; Kachru:2008yh . In addition to the scale
symmetry, demanding invariance under space and time translations, spatial
rotations will lead to the Lifshitz spacetimes in the bulk Kachru:2008yh

$\displaystyle
ds^{2}=-r^{2z}dt^{2}+r^{2}(dx^{2}+dy^{2})+\dfrac{dr^{2}}{r^{2}},$ (1.2)

which has curios geometric properties. For example, unlike AdS spacetimes,
where there is a well-defined boundary in the form of a conformal structure,
the Lifshitz spacetimes do not have a non-degenerate metric at the boundary
Horava:2010vho ; Copsey:2010ya . Therefore, the usual tools of holography like
the Fefferman-Graham (FG) expansion do not work, and others need a
modification to work in the non-relativistic case. There has been a flurry of
activity on developing notions and tools for non-relativistic holography (see
the review Taylor:2015glc  and references therein e.g., Chemissany:2014xsa ;
Mann:2011hg ; Ross:2011gu ; Ross:2009ar ; Christensen:2013rfa ;
Christensen:2013lma ; Hartong:2014oma ). In this regard, the non-relativistic
stress-energy tensor Ross:2009ar , inspired by Hollands:2005ya  will be our
primary tool in this work. To put it briefly, instead of working with the
boundary metric, the stress tensor will be defined by varying the boundary
frame fields and keeping the tangent space components of the of matter fields
fixed. While being more suited to a non-relativistic setting, this approach is
also crucial for defining correct conserved charges for gravity theories with
matter fields other than scalars. In Section 3 we will discuss the details of
this construction.

Another challenging feature of these spacetimes is that the cosmological
Einstein theory does not admit these as a solution; therefore, some type of
matter coupling or higher curvature terms should be considered Taylor:2008tg ;
Cai:2009ac ; Ayon-Beato:2009rgu ; Brenna:2011gp ; Lu:2012xu , making the
holographic analysis more intricate. An examplar is the massive vector model
Taylor:2008tg , which is simpler and includes all $z>1$ as a solution. This
system was the testing ground for the non-relativistic stress tensor, where it
was shown to generate finite results for linearised perturbations Ross:2009ar
. Later, building on this work, a definition of asymptotically locally
Lifshitz spacetimes is given, and the holographic renormalisation of one-point
functions is studied in Ross:2011gu . Another important model is the $z=2$
specific, four-dimensional system which is obtained from the Scherk-Schwarz
reduction of a five-dimensional axion-dilaton gravity living on an
asymptotically locally AdS spacetime Christensen:2013rfa ; Christensen:2013lma
. By computing the well-established higher dimensional AdS sources+VEVs, Ward
identities, the corresponding $z=2$ Lifshitz holographic dictionary was
identified. A major result of this work was the appearance of torsional
Newton-Cartan (TNC) geometry at the boundary Hartong:2015wxa ; Hartong:2014oma
.

In an attempt to further extend the Lifshitz holography, understand different
matter couplings better and search for dynamical curiosities, the main focus
of this work will be the four-dimensional Einstein-Yang-Mills (EYM) model. The
$SU(2)$ colored Lifshitz background solutions was first found in
Devecioglu:2014iia , later extended to five dimensions in Fan:2015yza . In
fact, the embedding of four-dimensional EYM in eleven-dimensional supergravity
was shown quite a long time ago Pope:1985bu , where the dynamical exponent for
the corresponding Lifshitz solutions should be fixed to an irrational number
$z=1+\sqrt{6}$ as a supersymmetry requirement Fan:2015yza .

Our first step towards a holographic stress tensor for the EYM theory will be
to find the boundary terms to have a well-defined variational problem with an
extremized action on four-dimensional Lifshitz background solutions. This is
performed in Section 2 by considering the Hawking-Gibbons (HG) term and a
counter-term intrinsic to the boundary. Next, in order to find the
perturbative modes that contribute to the stress tensor in the dual field
theory, we solve the linearised field equations and identify the modes that
lead to the Lifshitz background (1.2). The identification of modes will be an
important stepping stone for further analysis, e.g., the holographic
renormalisation of the one-point functions. Through perturbation solutions, we
show that the stress tensor stays finite and obeys the Ward identities. This
is done for both constant and general perturbations in Section 4. For constant
perturbations, it is only possible to show that the scaling Ward identity
holds, as there is no dependence on time or boundary coordinates. On the other
hand, with general perturbations, we will show that the rest of the
conservation laws are satisfied by incorporating an expansion in wavenumber
$k$ and frequency $\omega$.

Leaving aside the possible applications in non-relativistic holography, the
EYM solutions (both particle-like and black holes) are certainly interesting
in their own rights. The asymptotically flat solutions by Bartnik-McKinnon
Bartnik:1988am  was quite a surprise for the community since Deser’s proof of
no static solutions to the Yang-Mills (YM) equations in four dimensions
Deser:1976wq , and Lichnerowicz theorem of no gravitational solitons Lichner
suggest that the EYM system do not admit particle-like solutions. However, the
non-linearities of both the gravity and the gauge system together make the
solutions possible. Besides particle-like solutions, the EYM model also admits
colored black holes in asymptotically flat Bizon:1990sr ; Volkov:1998cc  and
AdS cases Torii:1995wv ; Winstanley:1998sn , which are hairy and violates the
no-hair theorem.

On the Lifshitz side, the colored numerical black holes, which are
parametrised by the fine-tuned gauge field strength at the horizon, was
constructed for different horizon topologies Devecioglu:2014iia . The
behaviour of solutions differs for black holes that are large or small with
respect to the length scale $L$, which is fixed by the cosmological constant.
In some ways, the solutions are quite different from their conformal and
asymptotically flat cousins, e.g., for a given horizon radius $R_{0}$ and
dynamical exponent $z$, there is a unique value of the gauge field that leads
to desired Lifshitz asymptotics. This is in contrast to the solutions with AdS
asymptotics, where the solutions exist for a value of shooting parameter in
continuous open intervals Winstanley:2008ac . Apart from their functional
behaviour, the physics of these black holes are not studied. Therefore, the
second part of this work is devoted to the conserved quantities and the
thermodynamics of these black holes.

The generalised Smarr relation Kastor:2009wy  for Lifshitz black holes that is
valid for all horizon topologies reads

$\displaystyle(D-3)M=(D-2)TS-2PV,$ (1.3)

along with the first law

$\displaystyle dM=TdS+VdP,$ (1.4)

where $P,\,V$ are thermodynamic pressure and volume, respectively. As we
pointed out before, the Lifshitz asymptotics require matter, which in turn
makes the cosmological constant dependent on the couplings. This obscures the
definition of thermodynamic pressure, so instead, some of the works used
alternative Smarr relation paired with the first law as follows Liu:2014dva ;
Bertoldi:2009dt ; Dehghani:2010kd ; Berglund:2011cp ; Dehghani:2011hf ;
Dehghani:2013mba

$\displaystyle(D+z-2)M=(D-2)TS,\quad dM=TdS.$ (1.5)

Later, the alternative relation (1.5) was shown to be consistent with the
original one (1.3) when the horizon topology is planar Brenna:2015pqa . Thus,
we will make use of the reduced relation in the study of the thermodynamics of
the planar numerical solutions. One of the important results in our study of
thermodynamics will be the hairy nature of the solutions. Actually, the lack
of global YM charge was argued by linear analysis of field equations
Fan:2015yza . In Section 5 we will support this result by showing the first
law of thermodynamics and the Smarr relation (1.5) hold without any global YM
charge.

The outline of the paper is as follows: In Section 2 we set up our model and
find out the boundary terms for a well-defined variational principle. Then in
Section 3, we review the holographic stress-energy tensor for non-relativistic
theories following the recipe by Ross-Saremi Ross:2009ar . We then set the
stage for perturbation analysis by choosing a suitable basis for perturbation
of the frame fields and the gauge field components in Section 4.1. The
following two Sections 4.2, 4.3 will be reserved to the solutions and the
stress tensor computations of constant perturbations and general
perturbations, respectively. Section 5 deals with the energy and
thermodynamics of the numerical black hole solutions of Devecioglu:2014iia .
Finally we conclude in Section 6. Explicit results for the linearisation of
some expressions and the field equations of generalised perturbations are
presented in Appendix A-B.

Since the number and the type of indices are somewhat involved, let us
carefully display our conventions (along the way there will be several
addendeums). Here $F_{\mu\nu}$ is the gauge field strength $F_{\mu\nu}\equiv
F_{\mu\nu}^{\Lambda}\,T_{\Lambda}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]$
and we choose generators $T_{\Lambda}\equiv\tau_{\Lambda}/2,\,\,\Lambda=1,2,3$
with $\tau_{\Lambda}$ denoting Pauli matrices with upper case Greek indices
for the Yang-Mills algebra. The commutation relations and the normalization of
generators are given as
$[T_{\Lambda},T_{\Gamma}]=i\epsilon_{\Lambda\Gamma\Delta}T_{\Delta}$ and
$\text{Tr}(\,T_{\Lambda}T_{\Gamma})=\delta_{\Lambda\Gamma}/2$, respectively.
For the the signature of metric we use $(-,+,+,+)$, and the Riemann tensor is
taken as
$R^{\mu}\,_{\nu\alpha\beta}=\partial_{\alpha}\Gamma^{\mu}\,_{\beta\nu}-\cdots$
with $R_{\mu\nu}=R^{\alpha}\,_{\mu\alpha\nu}$. We also use
$x^{\mu}=(t,x_{i},r)$ for spacetime (curved) indices and $y^{a}=(t,x_{i})$ for
boundary coordinate indices. Finally we use $i,j=1,2$ indices for the planar
part. The integration constants of solutions will be displayed in fraktur font
e.g. $\mathfrak{b}_{1},\mathfrak{c}_{2},\mathfrak{s}_{3},\cdots$.

## 2 Boundary counterterms in EYM

We start with the four-dimensional Einstein gravity with a cosmological
constant minimally coupled to $SU(2)$ gauge fields described by the action

$S_{\mathcal{M}}=\int_{\mathcal{M}}d^{4}x\,\sqrt{-g}\Big{(}(R-2\Lambda)-\dfrac{1}{2g_{\text{\tiny
YM}}^{2}}\text{Tr}\left[\,F_{\mu\nu}F^{\mu\nu}\right]\Big{)},\,$ (2.1)

where $\Lambda$ is the cosmological and $g_{\text{\tiny YM}}^{2}$ is the gauge
coupling constant in dimensions of $1/\text{length}^{2}$. Unlike EYM solutions
in AdS, where the background is already a solution to the cosmological
Einstein theory; here, we use matter to support backgrounds with anisotropic
scaling symmetry.

The variation of (2.1) will amount to

$\displaystyle\delta
S_{\mathcal{M}}=\int_{\mathcal{M}}d^{4}x\,\sqrt{-g}(\mathcal{E}_{\mu\nu}\,\delta
g^{\mu\nu}+\mathcal{E}^{\Lambda}_{\mu}\,\delta
A_{\Lambda}^{\mu})+\text{boundary terms},$ (2.2)

where $\mathcal{E}_{\mu\nu}$ is the field equations for the metric

$\displaystyle\mathcal{E}_{\mu\nu}\equiv
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}-\frac{1}{g_{\text{\tiny
YM}}^{2}}T_{\mu\nu},$ (2.3)

with the traceless YM stress energy tensor defined as

$\displaystyle
T_{\mu\nu}\equiv\text{Tr}\,(F_{\mu}\,^{\alpha}F_{\nu\alpha}-\dfrac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}).$
(2.4)

For the matter sector, the field equations are

$\displaystyle\mathcal{E}_{\mu}^{\Lambda}\equiv
D^{\nu}F_{\nu\mu}^{\Lambda}=0,$ (2.5)

where the gauge covariant derivative is
$D_{\mu}\equiv\nabla_{\mu}-i[A_{\mu},\quad]$.

We assume a Lifshitz background solution of the form

$\displaystyle
ds^{2}=-r^{2z}dt^{2}+r^{2}(dx^{2}_{1}+dx^{2}_{2})+\dfrac{dr^{2}}{r^{2}},$
(2.6)

paired with the planar symmetric $SU(2)$ gauge field ansatz Basler:1986yr

$\displaystyle A_{\mu}dx^{\mu}=Q(r)T^{3}dt+R(r)T^{1}dx_{1}+R(r)T^{2}dx_{2}.$
(2.7)

Since we look for a background solution, we assume all functions depend on the
radial coordinate $r$, and for simplicity, we focus on the purely magnetic
case, i.e. $Q(r)=0$. Then through field equations (2.3), (2.5), the gauge
field solution is found to be

$\displaystyle R(r)=\pm\sigma r\,,\quad\text{for
all}\,\,z>1,\quad\sigma\equiv\sqrt{z+1},$ (2.8)

provided the cosmological constant and the gauge coupling are chosen as
follows

$\displaystyle\Lambda=-\dfrac{3+2z+z^{2}}{2},\quad g_{\text{\tiny
YM}}^{2}=\frac{1}{2}\dfrac{(z+1)}{(z-1)}.$ (2.9)

The generalization of this solution to five dimensions and the exact black
hole solutions with an extra Maxwell charge is found in Fan:2015yza . In the
conformal limit $z\rightarrow 1$, the YM part decouples from the gravity
action, and the decoupled gauge field is a solution to the pure YM. The sign
ambiguity of the gauge field corresponds to a gauge transformation. Hence in
what follows, we will continue with the positive sign gauge field.

Having reviewed the Lifshitz solution, let us construct the appropriate action
which satisfies $\delta S=0$ on-shell. The boundary terms in (2.2) consist of
the well known gravitational part $g^{\mu\nu}\delta
R_{\mu\nu}=\nabla_{\mu}f^{\mu}$ and the matter part

$\displaystyle\delta S_{\mathcal{M}}=\text{field
equations}+\int_{\mathcal{M}}d^{4}x\,\sqrt{-g}\nabla^{\mu}\left(f_{\mu}-\dfrac{1}{g_{\text{\tiny
YM}}^{2}}F^{\Lambda}_{\mu\nu}\delta A_{\Lambda}^{\nu}\right),$ (2.10)

where

$\displaystyle f^{\mu}\equiv
g^{\rho\nu}\delta\Gamma^{\mu}_{\rho\nu}-g^{\mu\nu}\delta\Gamma^{\rho}_{\rho\nu}.$
(2.11)

To cancel out the metric variations with derivatives and have a boundary value
problem with Dirichlet conditions at $\partial\mathcal{M}$, the addition of
the trace of extrinsic curvature known as the Hawking-Gibbons (HG) term is
necessary. However, this alone is not sufficient to pose a well-defined
variational problem in holography. We also require boundary counterterms to
extremize the action on solutions of the equations of motion. Taking these and
the gauge invariance into account, one is naturally led to choose

$\displaystyle S=S_{\mathcal{M}}+S_{\partial{\mathcal{M}}}$
$\displaystyle=\int_{\mathcal{M}}d^{4}x\,\sqrt{-g}\Big{(}(R-2\Lambda)-\dfrac{1}{2g_{\text{\tiny
YM}}^{2}}\text{Tr}[F_{\mu\nu}F^{\mu\nu}]\Big{)}$
$\displaystyle+\int_{\partial{\mathcal{M}}}d^{3}y\,\sqrt{-\gamma}\left(2K+k_{0}+\alpha\,\text{Tr}\left[F_{ab}F^{ab}\right]\right),\,$
(2.12)

where $y^{a}$ denotes the coordinates on the boundary at some constant $r$,
$\gamma_{ab}$ is the induced metric, $k_{0}$ and $\alpha$ are the constants to
be determined. The extrinsic curvature tensor is given as
$K_{ab}=\nabla_{(a}n_{b)}$ where $n^{\mu}=(0,0,0,r)$ is an outward-directed
unit vector orthogonal to the boundary. The antisymmetry of $F_{\mu\nu}$ makes
it possible to define the following tensor with a boundary index

$\displaystyle n_{\mu}F^{\mu\nu}_{\Lambda}\rightarrow n_{\mu}F^{\mu
a}_{\Lambda}\equiv E^{a}_{\Lambda},$ (2.13)

which we use to write down (2.10) as

$\displaystyle\delta S_{\mathcal{M}}=\text{field
equations}+\int_{\partial\mathcal{M}}d^{3}y\,\sqrt{-\gamma}\left(n_{\mu}f^{\mu}-\dfrac{1}{g_{\text{\tiny
YM}}^{2}}E_{\Lambda}^{a}\delta A^{\Lambda}_{a}\right).$ (2.14)

The boundary contributions from the HG part of $S_{\partial{\mathcal{M}}}$ is

$\displaystyle\delta\left[\int_{\partial{\mathcal{M}}}d^{3}y\,\sqrt{-\gamma}\left(2K+k_{0}\right)\right]=$
$\displaystyle\int_{\partial{\mathcal{M}}}d^{3}y\,\sqrt{-\gamma}\left(-f^{\mu}n_{\mu}+\left[\Pi_{ab}-\frac{k_{0}}{2}\gamma_{ab}\right]\delta\gamma^{ab}\right)$
(2.15)

where we defined $\Pi_{ab}\equiv K_{ab}-K\gamma_{ab}$. The first term in
(2.15) handles the variations with derivatives and for the rest, we need the
components of the extrinsic curvature of the solution (2.6)

$\displaystyle K_{tt}=-zr^{2z},\quad K_{ij}=r^{2}\delta_{ij},\quad K=z+2.$
(2.16)

Likewise, the matter part of the variation reads

$\delta\left(\int_{\partial{\mathcal{M}}}d^{3}y\,\sqrt{-\gamma}\,\alpha\,\text{Tr}\left[F_{ab}F^{ab}\right]\right)=\alpha\int_{\partial{\mathcal{M}}}d^{3}y\,\sqrt{-\gamma}\Big{\\{}(F_{ac}^{\Lambda}F_{b}{}^{c}_{\Lambda}-\frac{1}{4}F_{cd}^{\Lambda}F_{\Lambda}^{cd}\gamma_{ab})\delta\gamma^{ab}\\\
-2\epsilon^{\Lambda\Gamma}{}_{\Delta}A_{\Gamma}^{b}F_{ba}^{\Delta}\delta
A^{a}_{\Lambda}\Big{\\}}.$ (2.17)

We also define the following general variation for future convenience

$\displaystyle\delta S=\int
d^{3}y(s_{ab}\delta\gamma^{ab}+s^{\Lambda}_{a}\delta A^{a}_{\Lambda}),$ (2.18)

where $s_{ab}$ and $s_{a}$ are

$\displaystyle s_{ab}\equiv$
$\displaystyle\sqrt{-\gamma}\left[\Pi_{ab}-\frac{k_{0}}{2}\gamma_{ab}+\alpha\left(F_{ac}^{\Lambda}F_{b}{}^{c}_{\Lambda}-\frac{1}{4}F_{cd}^{\Lambda}F_{\Lambda}^{cd}\gamma_{ab}\right)\right],$
(2.19) $\displaystyle s^{\Lambda}_{a}\equiv$
$\displaystyle-\sqrt{-\gamma}\left[\frac{1}{g_{\text{\tiny
YM}}^{2}}E^{\Lambda}_{a}+2\alpha\epsilon^{\Lambda\Gamma}{}_{\Delta}A^{b}_{\Gamma}F^{\Delta}_{ba}\right].$
(2.20)

Here the constants $k_{0}$ and $\alpha$ will be chosen such that $s_{ab}$ and
$s_{a}^{\Lambda}$ will be equal to zero on solutions making $\delta S=0$ for
arbitrary variations around (2.6), (2.8). With this in mind, evaluating the
solution on individual boundary terms we have

$\displaystyle\dfrac{1}{g_{\text{\tiny YM}}^{2}}E_{a}^{\Lambda}\,\delta
A_{\Lambda}^{a}$ $\displaystyle=\dfrac{1}{g_{\text{\tiny
YM}}^{2}}r\,\sqrt{z+1}(\delta A_{1}^{x_{1}}+\delta A_{2}^{x_{2}}),$ (2.21)
$\displaystyle\left[\Pi_{ab}-\frac{k_{0}}{2}\gamma_{ab}\right]\delta\gamma^{ab}$
$\displaystyle=\frac{1}{2}r^{2z}(4+k_{0})\delta\gamma^{tt}-\frac{1}{2}r^{2}(2(z+1)+k_{0})\delta_{ij}\delta\gamma^{ij},$
(2.22)
$\displaystyle\alpha(F_{ac}^{\Lambda}F_{b}{}^{c}_{\Lambda}-\frac{1}{4}F_{cd}^{\Lambda}F_{\Lambda}^{cd}\gamma_{ab})\delta\gamma^{ab}$
$\displaystyle=\frac{\alpha}{2}r^{2z}(z+1)^{2}\delta\gamma^{tt}+\frac{\alpha}{2}r^{2}(z+1)^{2}\delta_{ij}\delta\gamma^{ij},$
(2.23)
$\displaystyle-2\alpha\epsilon^{\Lambda\Gamma}{}_{\Delta}A^{b}_{\Gamma}F^{\Delta}_{ba}\delta
A_{\Lambda}^{a}$ $\displaystyle=-2\alpha r\,(z+1)^{3/2}(\delta
A_{1}^{x_{1}}+\delta A_{2}^{x_{2}}).$ (2.24)

Gathering all these pieces the general variation (2.18) reads

$\delta
S=\int_{\partial\mathcal{M}}d^{3}y\,\sqrt{-\gamma}\Bigg{\\{}\frac{1}{2}r^{2z}\left(4+k_{0}+\alpha(z+1)^{2}\right)\delta\gamma^{tt}+\frac{1}{2}r^{2}\Big{(}(z+1)(\alpha(z+1)-2)-k_{0}\Big{)}\delta^{ij}\delta\gamma_{ij}\\\
+\frac{r\sqrt{z+1}}{g_{\text{\tiny YM}}^{2}}\left[2\alpha g_{\text{\tiny
YM}}^{2}(z+1)-1\right](\delta A^{1}_{x_{1}}+\delta A^{2}_{x_{2}})\Bigg{\\}}.$
(2.25)

Provided $k_{0}=-(z+3)$ and $\alpha=(z-1)/(z+1)^{2}$, the above expression is
zero and we have a well defined variational problem. Therefore our final
action is the following

$\displaystyle S=$
$\displaystyle\int_{\mathcal{M}}d^{4}x\sqrt{-g}\left((R-2\Lambda)-\frac{\text{Tr}[F_{\mu\nu}F^{\mu\nu}]}{2g_{\text{\tiny
YM}}^{2}}\right)$ $\displaystyle+$
$\displaystyle\int_{\partial\mathcal{M}}d^{3}y\sqrt{-\gamma}\left(2K-(z+3)+\frac{\text{Tr}[F_{ab}F^{ab}]}{2g_{\text{\tiny
YM}}^{2}(z+1)}\right).$ (2.26)

Note that in addition to the boundary terms we have considered, one can
include terms involving derivatives of the boundary fields, which we denote as
$S_{deriv}$. However, the scalars constructed from the boundary fields are
constant for the Lifshitz solution, and with a planar boundary metric, the
derivatives of these scalars do not contribute to the background (2.6). In the
case of general asymptotically Lifshitz spacetimes, the derivative terms can
be considered rendering the components of the stress tensor finite if needed
Ross:2009ar . Furthermore, inspired by the Einstein-Maxwell model, one can be
tempted to consider the following boundary term for the matter part
Braden:1990hw

$\displaystyle\int_{\partial{\mathcal{M}}}d^{3}y\,\sqrt{-\gamma}\,\text{Tr}\left[n^{\mu}\,F_{\mu\nu}A^{\nu}\right].$
(2.27)

However, unlike its abelian counterpart, this choice is not gauge invariant
on-shell, so our choice seems to be the only plausible one.

Having determined the boundary counterterms for EYM action with Lifshitz
asymptotics, we can now review the construction of stress tensor introduced in
Ross:2009ar  for non-relativistic spacetimes.

## 3 Stress tensor for non-relativistic theories

This section will recapitulate the procedure given in Ross:2009ar  for
calculating a boundary stress tensor from a bulk action principle in non-
relativistic theories. The construction is analogous to the relativistic case
Hollands:2005ya  and relies on the definition of a modified stress tensor in
the presence of non-scalar boundary fields. We will first work at a finite
cut-off $r_{c}$, then take $r_{c}\rightarrow\infty$ for the final result, as
there is no non-degenerate metric at the boundary.

The following definition is sufficient when the metric is the only non-trivial
boundary field in relativistic theory Balasubramanian:1999re

$\displaystyle T^{ab}\hat{\epsilon}=-2\frac{\delta S}{\delta\gamma_{ab}},$
(3.1)

where $\hat{\epsilon}$ is the volume form associated with $\gamma_{ab}$. The
covariant conservation of (3.1) can be shown by considering the action of
boundary diffeomorphisms on the variation. This definition was modified by
Hollands-Ishibashi-Marolf when the theory in question involves extra fields
other than scalars. Following the recipe in Hollands:2005ya , it is convenient
to consider the set of frame fields at the boundary

$\displaystyle\gamma_{ab}=\hat{e}^{M}_{a}\hat{e}^{N}_{b}\eta_{MN},$ (3.2)

where $\hat{e}^{M}_{a}$ are the boundary frame fields.111We choose
$I,J=0,1,2,3$ for bulk and $M,N,\cdots=0,1,2$ to denote the tangent space
directions on the boundary. The frame fields allows us to write any boundary
tensor as a collection of scalar fields, i.e.
$X_{ab\cdots}=X_{MN\cdots}\hat{e}^{M}_{a}\hat{e}^{N}_{b}$. Taking this into
account, the variation will only act on the boundary frame fields, while the
tangent space components of $X_{MN\cdots}$ will be held fixed. In the case of
asymptotically AdS spacetime, the choice of frame fields corresponds to the
boundary metric in Fefferman-Graham expansion.

This replacement of boundary metric with the boundary frame fields leads to
the modified boundary stress tensor

$\displaystyle T^{ab}\hat{\epsilon}=\frac{\delta
S}{\delta\hat{e}_{b}^{M}}\hat{e}^{aM}.$ (3.3)

The two definitions (3.1) and (3.3) agree when the nontrivial boundary fields
are metric and some scalars. However, if there are additional boundary fields,
then (3.3) will be the correct one containing the contributions from the
extras. In particular, consider the variation of a model supported by the non-
abelian gauge fields with a relativistic conformal field theory at the
boundary

$\displaystyle\delta
S=\int\hat{\epsilon}(T^{a}{}_{M}\delta\hat{e}^{M}_{a}+s^{\Lambda}_{M}\delta
A^{M}_{\Lambda}),$ (3.4)

where the covariant divergence of $T^{ab}$ can be shown to satisfy a modified
conservation law

$\displaystyle\nabla_{a}T^{ab}=s^{\Lambda}_{M}\nabla^{b}A_{\Lambda}^{M}+T^{a}_{M}\nabla^{b}\hat{e}^{M}_{a},$
(3.5)

which can also be written as

$\displaystyle
D_{a}T^{a}{}_{b}=s^{\Lambda}_{M}\partial_{b}A^{M}_{\Lambda}\,\,\text{with}\,\,D_{a}\hat{e}^{M}_{b}=0.$
(3.6)

This tensor allows the construction of the counter-term charges, which
generate the correct asymptotic symmetries of the theory Hollands:2005ya .

We now turn our attention to the non-relativistic limit of this prescription.
In a non-relativistic field theory, instead of a covariant energy momentum
tensor, we have an energy density $\mathcal{E}$, an energy flux
$\mathcal{E}_{i}$, a momentum density $\mathcal{P}_{i}$ and a symmetric
spatial stress tensor $\Pi_{ij}$. This stress tensor complex satisfies the
following conservation equations, i.e. diffeomorphism Ward identities

$\displaystyle\partial_{t}\mathcal{E}+\partial_{i}\mathcal{E}^{i}=0,\quad\partial_{t}\mathcal{P}_{j}+\partial_{i}\Pi^{i}{}_{j}=0.$
(3.7)

In addition to these, we have the dilatation Ward identity for theories with
Lifshitz scaling symmetry

$\displaystyle z\mathcal{E}=\Pi^{i}{}_{i}.$ (3.8)

The main idea behind the procedure of Ross:2009ar  is to consider the non-
relativistic limit of the energy-momentum tensor (3.4) since, all of the above
identities can be derived as a non-relativistic limit of a relativistic
conservation equation. For a consistent limit, the boundary frame fields
should be related to the bulk frame fields by an appropriate power of $r$ such
that $\hat{e}^{M}_{b}$ stays finite as $r\rightarrow\infty$. With the
following rescaling

$\displaystyle e^{(0)}=r^{z}\hat{e}^{(0)},\quad e^{(i)}=r\hat{e}^{(i)},\quad
e^{(3)}=\frac{dr}{r},$ (3.9)

the Lifshitz background solution is then the condition
$\hat{e}^{(0)}\rightarrow dr$, $\hat{e}^{(i)}\rightarrow dx^{i}$,
$A^{\Lambda}_{M}\rightarrow\sigma\delta_{M}^{\Lambda}$ as
$r\rightarrow\infty$. Now, the scaled frame fields can be considered as the
boundary data as follows

$\displaystyle\delta S=\int
d^{3}y(-2s^{ab}\hat{e}_{bM}+s^{a}_{\Lambda}A_{M}^{\Lambda})\delta\hat{e}^{M}_{a}.$
(3.10)

Decomposing (3.10) into components of the stress energy complex gives

$\displaystyle\delta S=\int
d^{3}y\left[-\mathcal{E}\delta\hat{e}_{t}^{0}-\mathcal{E}^{i}\delta\hat{e}_{i}^{0}+\mathcal{P}_{i}\delta\hat{e}_{t}^{i}+\Pi_{i}{}^{j}\delta\hat{e}_{j}^{i}\right],$
(3.11)

where

$\displaystyle\mathcal{E}$
$\displaystyle=2s^{t}_{t}-s^{t}_{\Lambda}A_{t}^{\Lambda},$
$\displaystyle\mathcal{E}^{i}$
$\displaystyle=2s^{i}{}_{t}-s^{i}_{\Lambda}A_{t}^{\Lambda},$ (3.12)
$\displaystyle\mathcal{P}_{i}$
$\displaystyle=-2s^{t}{}_{i}+s^{t}_{\Lambda}A_{i}^{\Lambda},$
$\displaystyle\Pi_{i}{}^{j}$
$\displaystyle=-2s^{j}{}_{i}+s^{j}_{\Lambda}A_{i}^{\Lambda}.$ (3.13)

To be precise, it is clear from the variation (3.10) that the lower indices
for $\mathcal{P}_{i}$ and $\Pi_{i}{}^{j}$ are flat. However, for the sake of
simplicity, we converted all of them to the spacetime ones by multiplying the
expressions with their respective frame fields.

To sum up, starting from the relativistic counterpart, we have reviewed the
procedure of constructing a non-relativistic boundary stress tensor. Since the
theory we consider involves non-scalar boundary fields, special attention must
be given to their variation. This is handled by introducing vielbeins, which
makes it possible to keep the components of boundary fields fixed, resulting
in a stress tensor that has correct conserved charges. For the non-
relativistic limit, the frame fields in Lifshitz spacetime need to be scaled
appropriately to have finite values at the boundary. This approach is also
intrinsically natural for non-relativistic theories where space and time are
scaled differently.

There remains now to perform a perturbation analysis in order to check whether
the stress tensor and the action are finite for a more general class of
asymptotically Lifshitz spacetimes.

## 4 Perturbation analysis

We now focus on a more general class of asymptotically Lifshitz spacetimes and
show both the action and the stress tensor stays finite. As with most of the
EYM solutions, the Lifshitz black holes are numerical solutions and studied in
Devecioglu:2014iia . We will return to their analysis in section 5. On the
other hand, the exact solutions found in Fan:2015yza  are dressed with an
extra Maxwell field and admit the colored Lifshitz background as a vacuum. We
refrain ourselves studying these exact solutions with extra matter fields and
resort to perturbative analysis with the perturbations that are asymptotic to
the Lifshitz background.

In the following, we first introduce the appropriate choice of basis for
boundary frame fields, which allows the decomposition of the gauge field
perturbation into scalar and vector components that respect the symmetry of
the background solution. We then solve the linearised system for constant
perturbations and identify the modes that preserve asymptotically Lifshitz
condition. Employing these modes, we compute the on shell value of the action
and components of the stress tensor, showing they are finite and obey Ward
identities. In the final part of this section, we perform a similar analysis
for the generalised perturbations with an extra step of expanding the
linearised equations in powers of wave number and frequency in order to find
solutions.

### 4.1 Perturbation setup

Since we will perform the analysis in the linearised regime, let us carefully
put together all the ingredients. We consider a generic metric as a background
and its perturbation as follows

$\displaystyle g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu},$ (4.1)

where the deviation $h_{\mu\nu}$ should vanish sufficiently rapidly so that
the metric approaches Lifshitz vacuum $\bar{g}_{\mu\nu}$ as
$r\rightarrow\infty$. We will work in Gaussian gauge, i.e. $h_{\mu r}=0$ and
the perturbations are also scaled by an appropriate power of $r$ accordant
with the arguments in the previous section

$\displaystyle h_{tt}=-r^{2z}\hat{h}_{tt},\quad
h_{ti}=-r^{2z}\hat{v}_{1i}+r^{2}\hat{v}_{2i},\quad h_{ij}=r^{2}\hat{h}_{ij}.$
(4.2)

For this choice of perturbation, the orthonormal frames are222The background
metric $\bar{g}_{\mu\nu}$ is responsible for raising and lowering of the
spacetime indices $\mu,\nu\cdots$ and defining covariant derivative
$\bar{\nabla}_{\mu}$. For the sake of simplicity, we also use the following
shorthands:
$\hat{h}^{i}_{j}=\delta^{ik}\hat{h}_{kj},\,\hat{h}^{i}_{i}=\hat{h}_{ij}\delta^{ij}$.

$\displaystyle
e^{(0)}=r^{z}\hat{e}^{(0)}=r^{z}\left((1+\frac{1}{2}\hat{h}_{tt})dt+\hat{v}_{1i}dx^{i}\right),\,\,e^{(i)}=r\hat{e}^{(i)}=r\left(\hat{v}_{2i}dt+(\delta^{i}_{j}+\frac{1}{2}\hat{h}^{i}{}_{j})dx^{j}\right),\,\,e^{(3)}=\frac{dr}{r},$
(4.3)

where asymptotically vanishing perturbations will lead to the Lifshitz
background (2.6). The metric determinants for the bulk and the boundary read
respectively

$\displaystyle\sqrt{-g}=r^{z+1}\left[1+\frac{1}{2}\left(\hat{h}_{tt}+\hat{h}^{i}{}_{i}\right)\right],\quad\sqrt{-\gamma}=r^{z+2}\left[1+\frac{1}{2}\left(\hat{h}_{tt}+\hat{h}^{i}{}_{i}\right)\right].$
(4.4)

On the matter sector, we define the first order gauge field perturbations as

$\displaystyle A^{\Lambda}_{\mu}=\bar{A}^{\Lambda}_{\mu}+a^{\Lambda}_{\mu},$
(4.5)

where $\bar{A}_{\mu}$ is the background solution (2.8)

$\displaystyle\bar{A}=\sigma r(T^{1}dx_{1}+T^{2}dx_{2}).$ (4.6)

In order to decompose the field equations into vector and scalar parts, one
must choose the gauge field perturbation $a_{\mu}$ such that it respects the
symmetries of the background solution, i.e. a diagonal combination of the
spatial rotation and an internal flavour symmetry rotating
$A_{i}^{1},A_{i}^{2}$ into each other333We are grateful to Simon F. Ross for
the discussion of several issues in this section. We will also keep all the
components of the perturbation and let the field equations indicate the ones
that are pure gauge, and therefore, can be set to zero. With these in mind, we
make the following choice

$\displaystyle
a_{\mu}dx^{\mu}=(r\sigma\hat{v}_{2i}T^{i}-r^{z}\sigma\hat{a}_{3}T^{3})dt+\frac{1}{2}r\sigma\left(2\,\hat{b}_{i\Lambda}T^{\Lambda}+\hat{h}_{ij}\delta^{j}_{k}T^{k}\right)dx^{i}+\frac{\sigma}{r}\hat{c}_{\Lambda}T^{\Lambda}dr,$
(4.7)

where

$\displaystyle\hat{b}_{i\Lambda}=\hat{b}_{\Lambda
i},\quad\text{for}\quad\Lambda=1,2.$ (4.8)

or equivalently $\hat{b}_{ij}=\hat{b}_{ji}$ which corresponds to mixing of
spatial and group indices. The choice (4.7) might seem quite arbitrary, but
when written on flat indices

$\displaystyle
A^{I}_{\Lambda}T^{\Lambda}=\sigma\left(\hat{a}_{3}T^{3}\delta^{I}_{0}+\delta^{I}_{j}T^{j}+\hat{b}_{j\Lambda}T^{\Lambda}\delta^{Ij}+\hat{c}_{\Lambda}T^{\Lambda}\delta^{I}_{3}\right),$
(4.9)

the compatibility with the prescription we gave is evident, i.e. for vanishing
perturbations, the only non-zero component is the background solution which is
constant on tangent directions. This choice is also quite tractable in terms
of solving the coupled equations.

Finally, for completeness, we write down the linearized field equations for
both the metric and the gauge field

$\displaystyle\mathcal{E}_{\mu\nu}^{L}=$ $\displaystyle
R_{\mu\nu}^{L}-\frac{1}{2}\bar{R}h_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}R^{L}+\Lambda
h_{\mu\nu}-\dfrac{1}{g_{\text{\tiny
YM}}^{2}}\text{Tr}\Big{[}2f_{(\mu}{}^{\alpha}\bar{F}_{\nu)\alpha}-\bar{F}_{\mu\sigma}\bar{F}_{\nu\alpha}h^{\sigma\alpha}$
$\displaystyle-\frac{1}{2}f_{\alpha\beta}\bar{F}^{\alpha\beta}\bar{g}_{\mu\nu}+\frac{1}{2}\bar{F}_{\alpha}{}^{\gamma}\bar{F}^{\alpha\beta}h_{\gamma\beta}\bar{g}_{\mu\nu}-\frac{1}{4}\bar{F}_{\alpha\beta}\bar{F}^{\alpha\beta}h_{\mu\nu}\Big{]},$
(4.10) $\displaystyle\mathcal{E}_{\nu}^{L}=$
$\displaystyle-\bar{\nabla}_{\alpha}f_{\nu}{}^{\alpha}+\bar{\nabla}_{\beta}\bar{F}_{\nu\alpha}h^{\alpha\beta}-\frac{1}{2}\bar{F}_{\nu}{}^{\alpha}\bar{\nabla}_{\alpha}h+\bar{F}_{\nu}{}^{\alpha}\bar{\nabla}_{\beta}h_{\alpha}{}^{\beta}+\bar{F}_{\alpha\beta}\bar{\nabla}^{\beta}h_{\nu}{}^{\alpha}$
$\displaystyle-$ $\displaystyle
i\left([\bar{A}_{\alpha},f^{\alpha}_{\nu}]+[a_{\alpha},\bar{F}^{\alpha}_{\nu}]-[\bar{A}_{\alpha},\bar{F}^{\gamma}_{\nu}]h^{\alpha}{}_{\gamma}\right),$
(4.11)

where $f_{\mu\nu}\equiv\bar{\nabla}_{\mu}a_{\nu}-\bar{\nabla}_{\nu}a_{\mu}$
and the remaining expressions for the linearised objects are given in the
Appendix A.

Before discussing the solutions to the linearised equations, let us present
the action and stress-energy complex in terms of perturbations. As a result of
the background symmetry and the perturbation ansatz, the only Fourier modes
that will contribute to the action after the integration over the boundary
coordinates are the scalar ones

$S=-\int^{r_{c}}dr\,r^{z+1}\Big{\\{}2z(z+2)+z(z+2)(\hat{h}_{tt}+\hat{h}^{i}{}_{i})+2(z^{2}+z-2)\hat{b}^{i}{}_{i}+2(z-1)r\partial_{r}\hat{b}^{i}{}_{i}\\\
+(2z+3)r\partial_{r}[\hat{h}_{tt}+\hat{h}^{i}{}_{i}]+r^{2}\partial_{r}^{2}[\hat{h}_{tt}+\hat{h}^{i}{}_{i}]\Big{\\}}\\\
+r^{2+z}\left[2z+2(z-1)\hat{b}^{i}{}_{i}+z(\hat{h}_{tt}+\hat{h}^{i}{}_{i})+r\partial_{r}[\hat{h}_{tt}+\hat{h}^{i}{}_{i}]\right]\Big{|}_{r=r_{c}}.$
(4.12)

This result also implies that there will be no contribution from a possible
$S_{deriv}$ at linear order. In the following subsection, we will find the
explicit solutions for the perturbations and show (4.12) is finite.

On the other hand, in contrast to the action, all modes contribute to the
stress tensor complex

$\displaystyle\mathcal{E}$
$\displaystyle=-r^{z+2}(z-1)\left(2\hat{b}^{i}{}_{i}+\frac{r\partial_{r}\hat{h}^{i}{}_{i}}{(z-1)}-\frac{2\epsilon^{ij}\partial_{j}{\hat{b}_{i3}}}{r\sqrt{z+1}}\right)+\tilde{\mathcal{E}},$
(4.13) $\displaystyle\mathcal{E}^{i}$
$\displaystyle=-r^{z+2}(z-1)\delta^{ij}\left(\frac{r^{2z+1}\partial_{r}\hat{v}_{1j}}{(z-1)}-\frac{r\partial_{r}\hat{v}_{2j}}{(z-1)}-\frac{2}{\sqrt{z+1}}\epsilon_{j}{}^{k}(r^{z}\partial_{k}\hat{a}_{3}+r\partial_{t}\hat{b}_{k3})\right)+\tilde{\mathcal{E}}^{i},$
(4.14) $\displaystyle\mathcal{P}_{i}$
$\displaystyle=-r^{z+2}(z-1)\left(\frac{r\partial_{r}\hat{v}_{1i}}{(z-1)}-\frac{(2z-1)}{(z-1)}r^{-2z+3}\partial_{r}\hat{v}_{2i}+2r^{1-2z}\partial_{t}\hat{c}_{i}\right)+\tilde{\mathcal{P}}_{i},$
(4.15) $\displaystyle\Pi^{i}{}_{j}$
$\displaystyle=r^{z+2}(z-1)\Big{(}2(-2\hat{b}^{i}{}_{j}+\delta^{i}{}_{j}\hat{b}^{k}{}_{k})+2\sqrt{z+1}\epsilon^{i}{}_{j}\hat{c}_{3}+\frac{2}{r}\partial^{i}\hat{c}_{j}-2r\partial_{r}\hat{b}^{i}{}_{j}$
$\displaystyle+\frac{r}{(z-1)}\partial_{r}[\delta^{i}{}_{j}(\hat{h}_{tt}+\hat{h}^{k}{}_{k})-z\hat{h}^{i}{}_{j}]\Big{)}+\tilde{\Pi}^{i}{}_{j},$
(4.16)

where quantities with tilde denote the possible $S_{deriv}$ contributions.

Several observations are in order here. Firstly, all components of the matter
sector of the perturbation are multiplied by $(z-1)$, so in the limit $z=1$,
they are all decoupled, and only the metric perturbations survive. The
remaining piece should coincide with the relativistic definition (3.1).
Secondly, now it is possible to consider the $S_{deriv}$ contributions from
the action, which were quite important to make the components of
$\Pi^{i}{}_{j}$ finite for generalised perturbations in Ross:2009ar . Thirdly,
although we discussed the importance of the modification of conservation
equation (3.6), when linearised, the right-hand side of the equation reads

$\displaystyle
D_{a}T^{a}{}_{b}=\bar{s}_{M}^{\Lambda}(\partial_{b}A^{M}_{\Lambda})^{L}+(s_{M}^{\Lambda})^{L}\partial_{b}\bar{A}^{M}_{\Lambda}.$
(4.17)

The first term on the right-hand side vanishes on-shell and the second one
vanishes from the constancy of the background gauge field on flat indices.
Therefore, the linearisation is blind to this modification. Also, unlike its
relativistic counterpart, the spatial energy-momentum tensor (4.16) by
construction is not symmetric off-shell. However, we expect to have a
symmetric one on-shell, or it can always be improved to be symmetric. Finally,
it is worth repeating the argument of Ross:2009ar  on the applicability of the
linearisation. The equations (4.13)-(4.16) imply that the modes with a fall
off $r^{-(z+2)}$ will contribute to the components of stress-energy tensor at
linear order. Then, any possible solution with $r^{-(z+2)/2}$ fall off will
also contribute to the quadratic order, signalling the insufficiency of the
linear regime and a need for non-linear analysis.

We are now in a position to find the solutions to the perturbations and
explicitly compute the components of the stress-energy complex, and show it
satisfies the Ward identities.

### 4.2 Constant perturbations and solutions

Let us first consider the perturbations that are constant in the boundary
directions. With the help of the symmetry of mixing the spatial and the
flavour indices, we can decompose the constant perturbations into scalar
$f(r),j(r),k(r)$, vector $\hat{v}_{1i}(r),\hat{v}_{2i}(r)$ and tensor modes
$t_{d}(r),t_{o}(r),b_{d}(r),b_{o}(r)$ as follows

$\displaystyle h_{tt}=-r^{2z}f(r),\quad
h_{ti}=-r^{2z}\hat{v}_{1i}(r)+r^{2}\hat{v}_{2i}(r),\quad
h_{ij}=r^{2}k(r)\delta_{ij}+r^{2}k_{ij},\quad\hat{b}_{ij}=\delta_{ij}j(r)+J_{ij},$
(4.18)

where

$\displaystyle k_{ij}=\begin{pmatrix}t_{d}(r)&t_{o}(r)\\\
t_{o}(r)&-t_{d}(r)\end{pmatrix},\quad
J_{ij}=\begin{pmatrix}b_{d}(r)&b_{o}(r)\\\ b_{o}(r)&-b_{d}(r)\end{pmatrix}.$
(4.19)

In addition to the vector modes analogous to Ross:2009ar , we also have tensor
modes for the gauge field perturbation.

#### 4.2.1 Scalar modes

The field equations for the constant scalar perturbations read

$\displaystyle 2r^{2}j^{\prime\prime}(r)$
$\displaystyle=-8rj^{\prime}(r)-rf^{\prime}(r)+2z(z+3)j(r),$ (4.20)
$\displaystyle r^{2}f^{\prime\prime}(r)$
$\displaystyle=-2(z+1)rf^{\prime}(r)+2(z-1)rj^{\prime}(r)+2\left(3z^{2}+z-4\right)j(r),$
(4.21) $\displaystyle 2rk^{\prime}(r)$
$\displaystyle=-rf^{\prime}(r)+2(z-1)rj^{\prime}(r)-2z(z-1)j(r).$ (4.22)

For $z\geq 1$, the solutions are

$\displaystyle j(r)=$
$\displaystyle\mathfrak{c}_{1}r^{-(z+2)}+\mathfrak{c}_{2}r^{-\frac{1}{2}(z+2+\beta_{z})}+\mathfrak{c}_{3}r^{-\frac{1}{2}(z+2-\beta_{z})},$
(4.23) $\displaystyle f(r)=$
$\displaystyle-\frac{4\mathfrak{c}_{1}(z+1)}{z+2}r^{-(z+2)}+\frac{2\mathfrak{c}_{2}(z-1)(-2+3z+\beta_{z})}{2+z+\beta_{z}}r^{-\frac{1}{2}(z+2+\beta_{z})}$
$\displaystyle+\frac{2\mathfrak{c}_{3}(z-1)(-2+3z-\beta_{z})}{2+z-\beta_{z}}r^{-\frac{1}{2}(z+2-\beta_{z})}+\mathfrak{c}_{4},$
(4.24) $\displaystyle k(r)=$
$\displaystyle\frac{2\mathfrak{c}_{1}z(1+z)}{2+z}r^{-(z+2)}+\frac{4\mathfrak{c}_{2}(z-1)}{2+z+\beta_{z}}r^{-\frac{1}{2}(z+2+\beta_{z})}+\frac{4\mathfrak{c}_{3}(z-1)}{2+z-\beta_{z}}r^{-\frac{1}{2}(z+2-\beta_{z})}+\mathfrak{c}_{5},$
(4.25)

where $\beta_{z}\equiv\sqrt{9z^{2}+4z+12}$. In order to satisfy the
asymptotically Lifshitz boundary conditions we should set constants
$\mathfrak{c}_{4}=\mathfrak{c}_{5}=0$, which is merely a redefinition of
coordinates. Moreover, $\mathfrak{c}_{3}$ should set to be zero since it is a
growing mode as $r\rightarrow\infty$. Therefore, we are left with two free
parameters $\mathfrak{c}_{1},\mathfrak{c}_{2}$ for the constant scalar sector.
With the scalar mode solutions in hand, we can now calculate the on-shell
value of the action (4.12)

$\displaystyle S=-\frac{4\mathfrak{c}_{1}(z-1)z}{z+2}.$ (4.26)

Energy (4.13) is also composed of only scalar modes

$\displaystyle\mathcal{E}=-2r^{z+2}\left(2(z-1)j+rk^{\prime}\right)=4\mathfrak{c}_{1}(z^{2}+1).$
(4.27)

As a side note, even we let all the modes in the solution (4.23)-(4.25) the
on-shell action and the energy stays the same.

Before moving on to the vector and tensor modes, let us present the results on
the $\hat{c}_{\Lambda}(r)$ and $\hat{b}_{i3}(r)$ components of the gauge field
perturbation. The field equations for these components are

$\displaystyle(z+2)\hat{c}_{i}(r)+r\hat{c}^{\prime}_{i}(r)+\frac{r\epsilon_{i}{}^{j}}{\sqrt{z+1}}\left[(z+3)\hat{b}^{\prime}_{j3}(r)+r\hat{b}^{\prime\prime}_{j3}(r)\right]=$
$\displaystyle 0,$ (4.28)
$\displaystyle\hat{c}_{i}(r)+\frac{r\epsilon_{i}{}^{j}\,\hat{b}^{\prime}_{j3}(r)}{\sqrt{z+1}}=$
$\displaystyle 0.$ (4.29)

This system of equations is underdetermined, signalling the perturbations
$\hat{c}_{i}$ and $\hat{b}_{i3}$ are pure gauge. In stress-energy tensor
(4.13)-(4.16), these functions appear with time and spatial derivatives,
therefore, do not contribute to the constant perturbations, maintaining the
gauge independence of conserved quantities. Finally, $\hat{c}_{3}(r)$ is fixed
to zero by its field equation, ensuring a symmetric spatial stress tensor.

#### 4.2.2 Vector modes

We next consider the contribution from the vector modes. The perturbations
$\hat{v}_{1i}(r)$, $\hat{v}_{2i}(r)$ are coupled

$\displaystyle
r^{2}\hat{v}_{1i}^{\prime\prime}-r^{4-2z}\hat{v}_{2i}^{\prime\prime}+(3z+1)r\hat{v}_{1i}^{\prime}-(z+3)r^{3-2z}\hat{v}_{2i}^{\prime}$
$\displaystyle=0,$ (4.30) $\displaystyle
r^{2}\hat{v}_{2i}^{\prime\prime}+r^{2z-1}\hat{v}_{1i}^{\prime}-(z-4)r\hat{v}_{2i}^{\prime}$
$\displaystyle=0,$ (4.31)

and $\hat{a}_{3}(r)$ has its own decoupled equation

$\displaystyle r^{2}\hat{a}_{3}^{\prime\prime}(r)$
$\displaystyle+(z+3)r\hat{a}_{3}^{\prime}(r)-2\hat{a}_{3}(r)=0.$ (4.32)

The solutions of $\hat{v}_{1i},\hat{v}_{2i}$ has two branches, for $z\neq 4$

$\displaystyle\hat{v}_{1i}(r)$
$\displaystyle=-\frac{\mathfrak{c}_{1i}}{z+2}\,r^{-(z+2)}+\frac{(1-2z)\mathfrak{c}_{2i}}{3z}\,r^{-3z}+\mathfrak{c}_{3i},$
(4.33) $\displaystyle\hat{v}_{2i}(r)$
$\displaystyle=-\frac{\mathfrak{c}_{1i}}{z-4}r^{-(4-z)}-\frac{\mathfrak{c}_{2i}}{z+2}r^{-(z+2)}+\mathfrak{c}_{4i},$
(4.34)

and $z=4$

$\displaystyle\hat{v}_{1i}(r)$
$\displaystyle=-\frac{\mathfrak{c}_{1i}}{6r^{6}}-\frac{7\mathfrak{c}_{2i}}{12r^{12}}+\mathfrak{c}_{3i},$
(4.36) $\displaystyle\hat{v}_{2i}(r)$
$\displaystyle=\mathfrak{c}_{1i}\log{r}-\frac{\mathfrak{c}_{2i}}{6r^{6}}+\mathfrak{c}_{4i}.$
(4.37)

Finally the solutions for $\hat{a}_{3}(r)$ reads

$\displaystyle\hat{a}_{3}(r)=$
$\displaystyle\mathfrak{a}_{31}\,r^{\frac{-(2\sqrt{2}+\sqrt{2}z+\sqrt{24+8z+2z^{2}})}{2\sqrt{2}}}+\mathfrak{a}_{32}\,r^{\frac{-(2\sqrt{2}+\sqrt{2}z-\sqrt{24+8z+2z^{2}})}{2\sqrt{2}}}.$
(4.38)

From the solutions we compute the energy flux and the momentum density as
follows

$\displaystyle\mathcal{E}^{i}=$
$\displaystyle-2\mathfrak{c}_{2j}\delta^{ij}(z-1),$ (4.39)
$\displaystyle\mathcal{P}_{i}=$ $\displaystyle 2\mathfrak{c}_{1i}(z-1).$
(4.40)

The solutions are similar to the ones in Ross:2009ar  so their analysis is the
same. The coefficient $\mathfrak{c}_{3i}$ represents the shifting of the
coordinate $t\rightarrow t+\mathfrak{c}_{3i}x^{i}$, and the coefficient
$\mathfrak{c}_{4i}$ represents the shift $x^{i}\rightarrow
x^{i}+\mathfrak{c}_{4i}t$, therefore they are a pure gauge and can be set to
zero. The linearisation is adequate for $\mathfrak{c}_{1i}$ in $\hat{v}_{2i}$
only for $2>z>1$ and at $z\geq 4$, we need to set it to zero since it becomes
a growing mode, spoiling the boundary conditions.

Finally, the solution $\hat{a}_{3}(r)$ corresponds to the perturbations in the
electric charge. Since the background solution is purely magnetic, this mode
does not contribute to the stress-energy complex as it appears with a boundary
coordinate derivative in (4.14). Therefore, we will ignore this solution in
constant perturbations.

#### 4.2.3 Tensor modes

There are two coupled systems of equations for tensor modes. The first one
involves the diagonal components

$\displaystyle
2r^{2}b_{d}^{\prime\prime}(r)+r^{2}t_{d}^{\prime\prime}(r)+r(z+1)t_{d}^{\prime}(r)+2r(z+3)b_{d}^{\prime}(r)+4(z+1)b_{d}(r)=$
$\displaystyle 0,$ (4.41) $\displaystyle
r^{2}t_{d}^{\prime\prime}(r)+r(3z+1)t_{d}^{\prime}(r)+4r(z-1)b_{d}^{\prime}(r)+4(z-1)b_{d}(r)=$
$\displaystyle 0,$ (4.42)

with the solutions

$\displaystyle t_{d}(r)=$
$\displaystyle\mathfrak{t}_{d1}\,r^{-(z+2)}+\mathfrak{t}_{d2}\,r^{-\frac{1}{2}(z+2+\xi_{z})}+\mathfrak{t}_{d3}\,r^{-\frac{1}{2}(z+2-\xi_{z})}+\mathfrak{t}_{d4},$
(4.43) $\displaystyle b_{d}(r)=$
$\displaystyle-\frac{\mathfrak{t}_{d1}(z+2)}{2(z+1)}r^{-(z+2)}+\frac{\mathfrak{t}_{d2}(2-3z+\xi_{z})}{4(z-1)}r^{-\frac{1}{2}(z+2+\xi_{z})}+\frac{\mathfrak{t}_{d3}(2-3z-\xi_{z})}{4(z-1)}r^{-\frac{1}{2}(z+2-\xi_{z})},$
(4.44)

where $\xi_{z}\equiv\sqrt{z^{2}-12z+4}$. The parameter $\xi_{z}$ is real for
$z\geq 2(3+2\sqrt{2})$. In this region it is possible to include the mode
$\mathfrak{t}_{d2}$. However, its sister mode $\mathfrak{t}_{d1}$ falls slower
than $r^{-(z+2)/2}$ so again linear analysis is insufficient to understand
this mode clearly. The constant term $\mathfrak{t}_{d4}$ corresponds to the
relative scaling of the $x,y$ coordinates and can be ignored.

The second system is for the off-diagonal tensor modes which have the same
structure

$\displaystyle
2r^{2}b_{o}^{\prime\prime}(r)+r^{2}t_{o}^{\prime\prime}(r)+r(z+1)t_{o}^{\prime}(r)+2r(z+3)b_{o}^{\prime}(r)+4(z+1)b_{o}(r)=$
$\displaystyle 0,$ (4.45) $\displaystyle
r^{2}t_{o}^{\prime\prime}(r)+r(3z+1)t_{o}^{\prime}(r)+4r(z-1)b_{o}^{\prime}(r)+4(z-1)b_{o}(r)=$
$\displaystyle 0,$ (4.46)

where the solutions read

$\displaystyle t_{o}(r)=$
$\displaystyle\mathfrak{t}_{o1}\,r^{-(z+2)}+\mathfrak{t}_{o2}\,r^{-\frac{1}{2}(z+2+\xi_{z})}+\mathfrak{t}_{o3}\,r^{-\frac{1}{2}(z+2-\xi_{z})}+\mathfrak{t}_{o4},$
(4.47) $\displaystyle b_{o}(r)=$
$\displaystyle-\frac{\mathfrak{t}_{o1}(z+2)}{2(z+1)}r^{-(z+2)}+\frac{\mathfrak{t}_{o2}(2-3z+\xi_{z})}{4(z-1)}r^{-\frac{1}{2}(z+2+\xi_{z})}+\frac{\mathfrak{t}_{o3}(2-3z-\xi_{z})}{4(z-1)}r^{-\frac{1}{2}(z+2-\xi_{z})}.$
(4.48)

The $z$ bound on $\mathfrak{t}_{o2}$ and the fall rate argument of
$\mathfrak{t}_{o3}$ is identical. This time, the pure gauge mode
$\mathfrak{t}_{o4}$ is related to the rotation of $x,y$ coordinates.

Now, armed with the tensor solutions, we can finally work out the spatial
stress tensor and check the scaling Ward identity. The components of
$\Pi^{i}{}_{j}$ in terms of decomposition functions are as follows

$\displaystyle\Pi^{1}{}_{1}=$
$\displaystyle-r^{z+2}\Big{(}4(z-1)b_{d}+r\frac{d}{dr}[2(z-1)b_{d}-f+2(z-1)j+(z-2)k+zt_{d}]\Big{)},$
(4.49) $\displaystyle\Pi^{2}{}_{2}=$ $\displaystyle
r^{z+2}\Big{(}4(z-1)b_{d}+r\frac{d}{dr}[2(z-1)b_{d}+f-2(z-1)j-(z-2)k+zt_{d}]\Big{)},$
(4.50) $\displaystyle\Pi^{1}{}_{2}=$
$\displaystyle-r^{z+2}\Big{(}2r(z-1)b_{o}^{\prime}+4(z-1)b_{o}+rzt_{o}^{\prime}\Big{)}.$
(4.51)

Plugging in the solutions we have

$\displaystyle\Pi^{1}{}_{1}=$ $\displaystyle
2\mathfrak{c}_{1}z\left(z^{2}+1\right)+\frac{2\mathfrak{t}_{d1}z(z+2)}{z+1},$
(4.52) $\displaystyle\Pi^{2}{}_{2}=$ $\displaystyle
2\mathfrak{c}_{1}z\left(z^{2}+1\right)-\frac{2\mathfrak{t}_{d1}z(z+2)}{z+1},$
(4.53) $\displaystyle\Pi^{1}{}_{2}=$
$\displaystyle\Pi^{2}{}_{1}=\frac{2\mathfrak{t}_{o1}z(z+2)}{z+1}.$ (4.54)

Taking these components and the energy (4.27) into account, we see the
relation $z\mathcal{E}=\Pi^{i}{}_{i}$ is satisfied. However, a non-trivial
check for the conservation equations (3.7) is not possible here since all the
components of the stress-energy complex are constant, leading to a trivial
result. Therefore, we postpone that check to the next section, where we study
generalised perturbations. Finally, as we have checked all the solutions, it
is now easy to see that a smarter choice for the perturbative part of the
gauge field is

$\displaystyle
a_{\mu}dx^{\mu}=r\sigma\hat{v}_{2i}T^{i}dt+\frac{1}{2}r\sigma\left(2\,\hat{b}_{ij}T^{j}+\hat{h}_{ij}\delta^{j}_{k}T^{k}\right)dx^{i},$
(4.55)

or in flat indices

$\displaystyle
A^{I}_{\Lambda}T^{\Lambda}=\sigma\left(\delta^{I}_{i}+\hat{b}_{ij}\delta^{Ij}\right)T^{i}.$
(4.56)

### 4.3 General perturbations and solutions

Having dealt with the constant perturbations, let us switch the gears and
continue with the perturbations that depend on time and boundary coordinates
in the form of plane waves along $x$ direction. Again, exploiting the
background symmetry, we will group the perturbations as scalar and vector
parts. We also introduce factors of $k$ and $\omega$ to ensure that the field
equations will involve even powers of $k$, $\omega$ so that we can perform an
expansion, as the system of equations will get quite complicated to solve in
full form.

With this in mind, the scalar perturbations for the metric side read

$\displaystyle\hat{h}_{tt}=$ $\displaystyle f(r)e^{i(\omega t+kx)},$
$\displaystyle\hat{v}_{11}$ $\displaystyle=k\,\omega s_{1}(r)e^{i(\omega
t+kx)},$ (4.57) $\displaystyle\quad\hat{h}_{xx}=$
$\displaystyle(k_{L}(r)+k^{2}k_{T}(r))e^{i(\omega t+kx)},$
$\displaystyle\hat{h}_{yy}$ $\displaystyle=(k_{L}(r)-k^{2}k_{T}(r))e^{i(\omega
t+kx)},$ (4.58)

and on the matter side we have

$\displaystyle\hat{b}_{11}=$ $\displaystyle(b_{L}(r)+k^{2}b_{T}(r))e^{i(\omega
t+kx)},$ $\displaystyle\hat{b}_{22}$
$\displaystyle=(b_{L}(r)-k^{2}b_{T}(r))e^{i(\omega t+kx)},\quad$ (4.59)
$\displaystyle\hat{v}_{21}=$ $\displaystyle k\,\omega s_{2}(r)e^{i(\omega
t+kx)},$ $\displaystyle\hat{b}_{23}$ $\displaystyle=iks_{3}(r)e^{i(\omega
t+kx)},$ (4.60) $\displaystyle\hat{c}_{1}=$ $\displaystyle
iks_{4}(r)e^{i(\omega t+kx)}.$ (4.61)

On the other hand, the vector perturbations for the metric are as follows

$\displaystyle\hat{v}_{12}$
$\displaystyle={}k\,\omega\mathsf{v}_{1}(r)e^{i(\omega t+kx)},$ (4.62)
$\displaystyle\hat{h}_{12}$ $\displaystyle={}\mathsf{v}_{2}(r)e^{i(\omega
t+kx)},$ (4.63)

and finally the matter perturbations are chosen as

$\displaystyle\hat{v}_{22}$
$\displaystyle=k\,\omega\mathsf{v}_{3}(r)e^{i(\omega t+kx)},$
$\displaystyle\hat{b}_{13}$ $\displaystyle=ik\mathsf{v}_{6}(r)e^{i(\omega
t+kx)},$ (4.64) $\displaystyle\hat{a}_{3}$
$\displaystyle=i\omega\mathsf{v}_{4}(r)e^{i(\omega t+kx)},$
$\displaystyle\hat{c}_{2}$ $\displaystyle=ik\mathsf{v}_{7}(r)e^{i(\omega
t+kx)},$ (4.65) $\displaystyle\hat{b}_{12}$
$\displaystyle=\mathsf{v}_{5}(r)e^{i(\omega t+kx)},$
$\displaystyle\hat{c}_{3}$ $\displaystyle=\mathsf{v}_{8}(r)e^{i(\omega
t+kx)}.$ (4.66)

In the coming sections, we will find solutions to these perturbations starting
from the scalar modes and prove that the conservation laws are satisfied.

#### 4.3.1 Scalar modes

Let us start with the scalar mode functions that contribute to the stress
energy tensor complex in the following way

$\displaystyle\mathcal{E}=$
$\displaystyle-2r^{z+2}\left[2(z-1)b_{L}+rk_{L}^{\prime}-\frac{k^{2}(z-1)s_{3}}{r\sqrt{z+1}}\right]e^{i(\omega
t+kx)},$ (4.67) $\displaystyle\mathcal{E}^{x}=$ $\displaystyle-k\omega
r^{z+2}\left[r^{2z-1}s_{1}^{\prime}-rs_{2}^{\prime}+\frac{2(z-1)s_{3}}{r\sqrt{z+1}}\right]e^{i(\omega
t+kx)},$ (4.68) $\displaystyle\mathcal{P}_{x}=$ $\displaystyle-k\omega
r^{z+2}\left[rs_{1}^{\prime}-(z-1)r^{-2z+3}s_{2}^{\prime}-2(z-1)r^{1-2z}s_{4}\right]e^{i(\omega
t+kx)}$ (4.69) $\displaystyle\Pi^{x}{}_{x}=$
$\displaystyle-r^{z+2}\Bigg{[}k^{2}\left\\{4(z-1)b_{T}+2(z-1)rb_{T}^{\prime}+zrk_{T}^{\prime}+\frac{2(z-1)s_{4}}{r}\right\\}$
$\displaystyle+r(2(z-1)b_{L}^{\prime}-f^{\prime}+(z-2)k_{L}^{\prime})\Bigg{]}e^{i(\omega
t+kx)},$ (4.70) $\displaystyle\Pi^{y}{}_{y}=$
$\displaystyle-r^{z+2}\Bigg{[}-k^{2}\left\\{4(z-1)b_{T}+2(z-1)rb_{T}^{\prime}+zrk_{T}^{\prime}\right\\}$
$\displaystyle+r(2(z-1)b_{L}^{\prime}-f^{\prime}+(z-2)k_{L}^{\prime})\Bigg{]}e^{i(\omega
t+kx)}.$ (4.71)

Plugging in the plane wave ansatz, the conservation equations (3.7) will yield

$\displaystyle\omega\mathcal{E}+k\mathcal{E}^{x}=0,\quad\omega\mathcal{P}_{x}+k\Pi^{x}{}_{x}=0,\quad
z\mathcal{E}=$ $\displaystyle\Pi^{i}{}_{i}.$ (4.72)

It can be shown that the first conservation equation is equal to $tr$
component of the linearized Einstein equations (4.10), i.e.
$\mathcal{E}_{tr}$, and the second one is a combination of the
$\mathcal{E}_{x_{1}r}$ and $\mathcal{E}_{r}^{1}$, therefore the conservation
laws are guaranteed to be satisfied on-shell.

In this system of equations, there are nine functions to solve, and there are
twelve equations in total, so three of the equations can be written as a
linear combination of others. In Appendix B, we give all the field equations
and the relations in detail. The main difference from the constant
perturbations is that we lose the Euler type equation structure. Thus, we are
not able to solve the system in its full form, and our line of attack will be
expanding the perturbation functions in even powers of $k,\omega$, i.e.
$F=\sum_{m,n}k^{2m}\omega^{2n}F^{(m,n)}$ where $m,n=0,1,2\cdots$. Then, the
components of the stress energy tensor can be written as

$\displaystyle\mathcal{E}=$
$\displaystyle\sum_{m,n}k^{2m}\omega^{2n}(\mathcal{E}_{1}^{(m,n)}+k^{2}\mathcal{E}_{2}^{(m,n)})e^{i(\omega
t+kx)},$ (4.73) $\displaystyle\mathcal{E}^{x}=$
$\displaystyle\sum_{m,n}k^{2m+1}\omega^{2n+1}\mathcal{E}^{x(m,n)}e^{i(\omega
t+kx)},$ (4.74) $\displaystyle\mathcal{P}_{x}=$
$\displaystyle\sum_{m,n}k^{2m+1}\omega^{2n+1}\mathcal{P}^{(m,n)}_{x}e^{i(\omega
t+kx)}$ (4.75) $\displaystyle\Pi^{x}{}_{x}=$
$\displaystyle\sum_{m,n}k^{2m}\omega^{2n}(\Pi^{(m,n)}_{L}+k^{2}\Pi^{(m,n)}_{S}+k^{2}\Pi^{(m,n)}_{T})e^{i(\omega
t+kx)},$ (4.76) $\displaystyle\Pi^{y}{}_{y}=$
$\displaystyle\sum_{m,n}k^{2m}\omega^{2n}(\Pi^{(m,n)}_{L}-k^{2}\Pi^{(m,n)}_{T})e^{i(\omega
t+kx)},$ (4.77)

where the following are defined to evaluate (4.72) in a more organized way

$\displaystyle\mathcal{E}_{1}^{(m,n)}=$
$\displaystyle-2r^{z+2}\left[2(z-1)b_{L}^{(m,n)}+rk_{L}^{(m,n)\prime}\right],$
(4.78) $\displaystyle\mathcal{E}_{2}^{(m,n)}=$
$\displaystyle-2r^{z+2}\left[-\frac{(z-1)s_{3}^{(m,n)}}{r\sqrt{z+1}}\right],$
(4.79) $\displaystyle\mathcal{E}^{x(m,n)}=$
$\displaystyle-r^{z+2}\left[r^{2z-1}s_{1}^{(m,n)\prime}-rs_{2}^{(m,n)\prime}+\frac{2(z-1)s_{3}^{(m,n)}}{r\sqrt{z+1}}\right],$
(4.80) $\displaystyle\mathcal{P}_{x}^{(m,n)}=$
$\displaystyle-r^{z+2}\left[rs_{1}^{(m,n)\prime}-(z-1)r^{-2z+3}s_{2}^{(m,n)\prime}-2(z-1)r^{1-2z}s_{4}^{(m,n)}\right],$
(4.81) $\displaystyle\Pi^{(m,n)}_{L}=$
$\displaystyle-r^{z+2}\left[2(z-1)rb_{L}^{(m,n)\prime}-rf^{(m,n)\prime}+(z-2)rk_{L}^{(m,n)\prime}\right],$
(4.82) $\displaystyle\Pi^{(m,n)}_{T}=$
$\displaystyle-r^{z+2}\left[4(z-1)b_{T}^{(m,n)}+2(z-1)rb_{T}^{(m,n)\prime}+zrk_{T}^{(m,n)\prime}\right],$
(4.83) $\displaystyle\Pi^{(m,n)}_{S}=$
$\displaystyle-r^{z+2}\left[\frac{2(z-1)s_{4}^{(m,n)}}{r}\right].$ (4.84)

Before moving on to the solutions, let us impose conservation equations (4.72)
to (4.74)-(4.77) and infer at which order we would get non-zero contribution
for the components (4.78)-(4.83).

$\displaystyle\mathcal{E}_{1}^{(m+1,n)}+\mathcal{E}_{2}^{(m,n)}+\mathcal{E}^{x(m,n)}$
$\displaystyle=0,$ (4.85)
$\displaystyle\mathcal{P}_{x}^{(m+1,n)}+\Pi^{(m+1,n+1)}_{L}+\Pi^{(m,n+1)}_{T}+\Pi^{(m,n+1)}_{S}$
$\displaystyle=0,$ (4.86) $\displaystyle
z(\mathcal{E}_{1}^{(m+1,n)}+\mathcal{E}_{2}^{(m,n)})-2\Pi^{(m+1,n)}_{L}-\Pi^{(m,n)}_{L}$
$\displaystyle=0.$ (4.87)

From the first equation it follows that $\mathcal{E}^{(0,n)}_{1}=0$. Then,
using second and third relations, it is easy to see
$\Pi^{(0,n)}_{L}=\mathcal{P}_{x}^{(0,n)}=0$.

Close scrutiny of the expanded field equations will reveal the following
structure: at every order $(m,n)$ there will be a part of field equations that
will correspond to the homogeneous solutions which are the same with constant
perturbations. Then through the expansion, these homogeneous solutions are
sourced by lower-order expansion ones, leading to the particular integrals.
However, the particular integrals are always suppressed by powers of $r$,
therefore not contributing to the stress energy tensor. As a simple
represantative of this structure, consider the following $(0,0)$ and $(1,0)$
order expansions of the equation (B.1)

$\displaystyle
r^{2}\left(k_{L}^{(0,0)\prime\prime}+f^{(0,0)\prime\prime}\right)+r\left[(z+3)k_{L}^{(0,0)\prime}+2(z+1)f^{(0,0)\prime}\right]-4(z^{2}-1)b_{L}^{(0,0)}=$
$\displaystyle 0$ (4.88) $\displaystyle
r^{2}\left(k_{L}^{(1,0)\prime\prime}+f^{(1,0)\prime\prime}\right)+r\left[(z+3)k_{L}^{(1,0)\prime}+2(z+1)f^{(1,0)\prime}\right]-4(z^{2}-1)b_{L}^{(1,0)}=$
$\displaystyle\frac{f^{(0,0)}}{2r^{2}}-\frac{2(z^{2}-1)}{\sqrt{z+1}}\frac{s_{3}^{(0,0)}}{r}.$
(4.89)

Therefore, once the zeroth-order equations are solved, the first order
homogenous solutions are identical with extra particular solutions sourced by
the zeroth-order ones. Moreover, unlike constant perturbations, the solutions
now have extra constraints which will relate different modes to ensure the
conservation equations are satisfied.

Taking all these into account, we have the following solution structure at the
order $(m,n)$ for the equations 444In this and the following section, we are
going to use the boldfaced characters for labelling modes, e.g.
$\mathbf{k}^{\text{\tiny 103}}_{\text{\tiny T}}$ where first two upper indices
tags the corresponding expansion parameter for wavenumber $k$ and frequency
$\omega$ respectively, i.e. $k=1$ and $\omega=0$ in this case. The last index
is the mode number.

For $b_{L}^{(m,n)},k_{L}^{(m,n)},f^{(m,n)}$

$\displaystyle b_{L}^{(m,n)}(r)$ $\displaystyle=\mathbf{b}^{\text{\tiny
mn1}}_{\text{\tiny L}}\,r^{-(z+2)}+\mathbf{b}^{\text{\tiny mn2}}_{\text{\tiny
L}}\,r^{-\frac{1}{2}(z+2+\beta_{z})}+\mathcal{O}(r^{p})$ (4.90) $\displaystyle
f^{(m,n)}(r)$ $\displaystyle=\mathbf{f}^{\text{\tiny
mn1}}\,r^{-(z+2)}+\mathbf{f}^{\text{\tiny
mn2}}\,r^{-\frac{1}{2}(z+2+\beta_{z})}+\mathcal{O}(r^{p})$ (4.91)
$\displaystyle k_{L}^{(m,n)}(r)$ $\displaystyle=\mathbf{k}^{\text{\tiny
mn1}}_{\text{\tiny L}}\,r^{-(z+2)}+\mathbf{k}^{\text{\tiny mn2}}_{\text{\tiny
L}}\,r^{-\frac{1}{2}(z+2+\beta_{z})}+\mathcal{O}(r^{p})$ (4.92)

for $k_{T}^{(m,n)},b_{T}^{(m,n)}$

$\displaystyle k_{T}^{(m,n)}(r)$ $\displaystyle=\mathbf{k}^{\text{\tiny
mn1}}_{\text{\tiny T}}\,r^{-(z+2)}+\mathbf{k}^{\text{\tiny mn2}}_{\text{\tiny
T}}\,r^{-\frac{1}{2}(z+2-\xi_{z})}+\mathbf{k}^{\text{\tiny mn3}}_{\text{\tiny
T}}\,r^{-\frac{1}{2}(z+2+\xi_{z})}+\mathcal{O}(r^{p}),$ (4.93) $\displaystyle
b_{T}^{(m,n)}(r)$ $\displaystyle=\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny
T}}\,r^{-(z+2)}+\mathbf{b}^{\text{\tiny mn2}}_{\text{\tiny
T}}\,r^{-\frac{1}{2}(z+2-\xi_{z})}+\mathbf{b}^{\text{\tiny mn3}}_{\text{\tiny
T}}\,r^{-\frac{1}{2}(z+2+\xi_{z})}+\mathcal{O}(r^{p}).$ (4.94)

for $s_{2}^{(m,n)}(r),s_{1}^{(m,n)}(r)$

$\displaystyle s_{1}^{(m,n)}(r)$ $\displaystyle=\mathbf{s}^{\text{\tiny
mn1}}_{\text{\tiny 1}}\,r^{-(z+2)}+\mathbf{s}^{\text{\tiny mn2}}_{\text{\tiny
1}}\,r^{-3z}+\mathcal{O}(r^{p})$ (4.95) $\displaystyle s_{2}^{(m,n)}(r)$
$\displaystyle=\mathbf{s}^{\text{\tiny mn1}}_{\text{\tiny
2}}\,r^{-(z+2)}+\mathbf{s}^{\text{\tiny mn2}}_{\text{\tiny
2}}\,r^{-(4-z)}+\mathcal{O}(r^{p})$ (4.96)

for $s_{3}^{(m,n)}(r),s_{4}^{(m,n)}(r)$

$\displaystyle s_{3}^{(m,n)}(r)$ $\displaystyle=\mathbf{s}^{\text{\tiny
mn1}}_{\text{\tiny 3}}\,r^{-(z+1)}+\mathcal{O}(r^{p})$ (4.97) $\displaystyle
s_{4}^{(m,n)}(r)$ $\displaystyle=\mathbf{s}^{\text{\tiny mn1}}_{\text{\tiny
4}}\,r^{-(z+1)}+\mathcal{O}(r^{p})$ (4.98)

where $\mathcal{O}(r^{p})$ corresponds to particular integrals that do not
contribute to the stress energy tensor. Actually we will only track down and
solve the modes which contribute to the stress energy complex when evaluated
at infinity. To that end, ”solving” here will mean finding out the relation
between the contributing modes. After meticulousy working out the expanded
field equations, we have the following solutions

$\displaystyle\mathbf{f}^{\text{\tiny 0n1}}$ $\displaystyle=0,\quad n\geq 0,$
$\displaystyle\mathbf{f}^{\text{\tiny mn1}}$
$\displaystyle=-\frac{4(z+1)\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny
L}}}{z+2},\quad m>0,n\geq 0,$ $\displaystyle\mathbf{k}^{\text{\tiny
0n1}}_{\text{\tiny L}}$ $\displaystyle=0,\quad n\geq 0,$
$\displaystyle\mathbf{k}^{\text{\tiny mn1}}_{\text{\tiny L}}$
$\displaystyle=\frac{2z(z+1)\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny
L}}}{z+2},\quad m>0,n\geq 0,$ $\displaystyle\mathbf{b}^{\text{\tiny
0n1}}_{\text{\tiny L}}$ $\displaystyle=0,\quad n\geq 0,$
$\displaystyle\mathbf{b}^{\text{\tiny m01}}_{\text{\tiny T}}$
$\displaystyle=\frac{1}{2}(z^{2}+1)\mathbf{b}^{\text{\tiny(m+1)01}}_{\text{\tiny
L}},\quad m\geq 0,$ $\displaystyle\mathbf{k}^{\text{\tiny m01}}_{\text{\tiny
T}}$
$\displaystyle=-\frac{(z+1)(z^{2}+1)\mathbf{b}^{\text{\tiny(m+1)01}}_{\text{\tiny
L}}}{z+2},\quad m\geq 0,$ $\displaystyle\mathbf{k}^{\text{\tiny
mn1}}_{\text{\tiny T}}$ $\displaystyle=-\frac{2(z+1)\mathbf{b}^{\text{\tiny
mn1}}_{\text{\tiny T}}}{z+2},\quad m\geq 0,n>0,$
$\displaystyle\mathbf{s}^{\text{\tiny 0n1}}_{\text{\tiny 1}}$
$\displaystyle=0,\quad n\geq 0,$ $\displaystyle\mathbf{s}^{\text{\tiny
mn1}}_{\text{\tiny 1}}$
$\displaystyle=\frac{z\left((z^{2}+1)\mathbf{b}^{\text{\tiny
m(n+1)1}}_{\text{\tiny L}}-2\mathbf{b}^{\text{\tiny(m-1)(n+1)1}}_{\text{\tiny
T}}\right)}{(z+2)(z-1)},\quad m>0,n\geq 0,$
$\displaystyle\mathbf{s}^{\text{\tiny mn2}}_{\text{\tiny 1}}$
$\displaystyle=-\frac{2(2z-1)(z^{2}+1)\mathbf{b}^{\text{\tiny(m+1)n1}}_{\text{\tiny
L}}}{3z(z-1)},\quad m\geq 0,n\geq 0,$ $\displaystyle\mathbf{s}^{\text{\tiny
mn1}}_{\text{\tiny 2}}$
$\displaystyle=-\frac{2(z^{2}+1)\mathbf{b}^{\text{\tiny(m+1)n1}}_{\text{\tiny
L}}}{(z+2)(z-1)},\quad m\geq 0,n\geq 0,$ $\displaystyle\mathbf{s}^{\text{\tiny
0n2}}_{\text{\tiny 2}}$ $\displaystyle=0,\quad n\geq 0,$
$\displaystyle\mathbf{s}^{\text{\tiny mn2}}_{\text{\tiny 2}}$
$\displaystyle=-\frac{z\left((z^{2}+1)\mathbf{b}^{\text{\tiny
m(n+1)1}}_{\text{\tiny L}}-\mathbf{b}^{\text{\tiny(m-1)(n+1)1}}_{\text{\tiny
T}}\right)}{(z+1)(z-4)},\quad m>0,n\geq 0.$

Thus, we find that the scalar modes are parametrised by two coefficients
$\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny L}}$ for $m>0,n\geq 0$ and
$\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny T}}$ for $m\geq 0,n>0$. The
solution $\mathbf{s}^{\text{\tiny mn2}}_{\text{\tiny 2}}$ has a singularity at
$z=4$, but that do not ”leak” into the conserved quantities computed below, so
we avoid going over the same analysis for $z=4$. Moreover, at $z\geq 4$ that
mode is already constant or growing so it should be discarded. The modes
$\mathbf{s}^{\text{\tiny mn1}}_{\text{\tiny 3}}$ and $\mathbf{s}^{\text{\tiny
mn1}}_{\text{\tiny 4}}$ do not contribute to the stress energy tensor. As a
matter of fact, these correspond to the pure gauge components that are
discussed in constant pertubations so they also seem to stay as pure gauge in
this case.

Plugging the solutions (4.90)-(4.98) in (4.78)-(4.84) yield the following
results for the components of the stress energy tensor

$\displaystyle\mathcal{E}^{(0,n)}_{1}$ $\displaystyle=0,\quad n\geq 0,$
$\displaystyle\mathcal{E}^{(m,n)}_{1}$
$\displaystyle=4(z^{2}+1)\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny L}},\quad
m>0,\,n\geq 0,$ $\displaystyle\mathcal{E}^{(m,n)}_{2}$ $\displaystyle=0,\quad
m\geq 0,n\geq 0,$ $\displaystyle\mathcal{E}^{x(m,n)}$
$\displaystyle=-4(z^{2}+1)\mathbf{b}^{\text{\tiny(m+1)n1}}_{\text{\tiny
L}},\quad m\geq 0,\,n\geq 0,$ $\displaystyle\mathcal{P}_{x}^{(0,n)}$
$\displaystyle=0,\quad n\geq 0,$ $\displaystyle\mathcal{P}_{x}^{(m,n)}$
$\displaystyle=-2z\left[(z^{2}+1)\mathbf{b}^{\text{\tiny
m(n+1)1}}_{\text{\tiny L}}-2\mathbf{b}^{\text{\tiny(m-1)(n+1)1}}_{\text{\tiny
T}}\right],\quad m>0,\,n\geq 0,$ $\displaystyle\Pi^{(0,n)}_{L}$
$\displaystyle=0,\quad n\geq 0,$ $\displaystyle\Pi^{(m,n)}_{L}$
$\displaystyle=2z(z^{2}+1)\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny L}},\quad
m>0,\,n\geq 0,$ $\displaystyle\Pi^{(m,0)}_{T}$
$\displaystyle=-2z(z^{2}+1)\mathbf{b}^{\text{\tiny(m+1)01}}_{\text{\tiny
L}},\quad m\geq 0,$ $\displaystyle\Pi^{(m,n)}_{T}$
$\displaystyle=-4z\mathbf{b}^{\text{\tiny mn1}}_{\text{\tiny T}},\quad m\geq
0,\,n>0,$ $\displaystyle\Pi^{(m,n)}_{S}$ $\displaystyle=0,\quad m\geq,\,n\geq
0,$

which are all evaluated at infinity. It is now straightforward to verify that
the conservation (4.85), (4.86) and scaling (4.87) equations are satisfied.

#### 4.3.2 Vector modes

The next and final step in general perturbations is the vector mode solutions.
The relevant components of the energy momentum complex read

$\displaystyle\mathcal{E}^{y}$ $\displaystyle=-k\omega
r^{z+2}\left[r^{2z-1}\mathsf{v}_{1}^{\prime}-r\mathsf{v}_{3}^{\prime}-\frac{2(z-1)}{\sqrt{z+1}}\left(\frac{\mathsf{v}_{6}}{r}+r^{z-2}\mathsf{v}_{4}\right)\right]e^{i(\omega
t+kx)},$ (4.99) $\displaystyle\mathcal{P}_{y}$ $\displaystyle=-k\omega
r^{z+2}\left[r\mathsf{v}_{1}^{\prime}-(2z-1)r^{3-2z}\mathsf{v}_{3}^{\prime}-2(z-1)r^{1-2z}\mathsf{v}_{8}\right]e^{i(\omega
t+kx)},$ (4.100) $\displaystyle\Pi^{x}{}_{y}$
$\displaystyle=-r^{z+2}\left[zr\mathsf{v}_{2}^{\prime}+2(z-1)\left(r\mathsf{v}_{5}^{\prime}+2\mathsf{v}_{5}+\sqrt{z+1}\mathsf{v}_{7}+\frac{k^{2}\mathsf{v}_{8}}{r}\right)\right]e^{i(\omega
t+kx)}.$ (4.101)

This time, the only conservation law needed to be fullfilled is the
conservation of momenta along $y$ direction

$\displaystyle\omega\mathcal{P}_{y}+k\Pi^{x}{}_{y}=0,$ (4.102)

which can be written as a combination of $\mathcal{E}^{2}_{r}$ and
$\mathcal{E}_{x_{2}r}$. On the other hand, $\mathcal{E}^{y}$ is not
constrained. We take the expansions of (4.99)-(4.101) in the form

$\displaystyle\mathcal{E}^{y}={}$
$\displaystyle\sum_{m,n}k^{2m}\omega^{2n}\mathcal{E}^{y(m,n)},$ (4.103)
$\displaystyle\mathcal{P}_{y}={}$
$\displaystyle\sum_{m,n}k^{2m+1}\omega^{2n+1}\mathcal{P}_{y}^{(m,n)},$ (4.104)
$\displaystyle\Pi^{x}{}_{y}={}$
$\displaystyle\sum_{m,n}k^{2m}\omega^{2n}\Pi^{x(m,n)}_{y},$ (4.105)

which amount to the conservation equation

$\displaystyle\mathcal{P}_{y}^{(m,n)}+\Pi^{x(m,n+1)}_{y}=0.$ (4.106)

The structure of expanded field equations remains similar so, following the
same strategy as before we write down the contributing solutions

$\displaystyle\mathsf{v}_{1}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{1}}\,r^{-(z+2)}+\mathbf{v}^{\text{\tiny
mn2}}_{\text{1}}\,r^{-3z}+\mathcal{O}(r^{p}),$ (4.107)
$\displaystyle\mathsf{v}_{2}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{2}}\,r^{-(z+2)}+\mathcal{O}(r^{p}),$ (4.108)
$\displaystyle\mathsf{v}_{3}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{3}}\,r^{-(z+2)}+\mathbf{v}^{\text{\tiny
mn2}}_{\text{3}}\,r^{-(4-z)}+\mathcal{O}(r^{p}),$ (4.109)
$\displaystyle\mathsf{v}_{4}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{4}}\,r^{-2z}+\mathcal{O}(r^{p}),$ (4.110)
$\displaystyle\mathsf{v}_{5}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{5}}\,r^{-(z+2)}+\mathcal{O}(r^{p}),$ (4.111)
$\displaystyle\mathsf{v}_{6}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{6}}\,r^{-(z+1)}+\mathcal{O}(r^{p}),$ (4.112)
$\displaystyle\mathsf{v}_{7}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{7}}\,r^{-(z+2)}+\mathcal{O}(r^{p})$ (4.113)
$\displaystyle\mathsf{v}_{8}^{(m,n)}(r)$
$\displaystyle=\mathbf{v}^{\text{\tiny
mn1}}_{\text{8}}\,r^{-(z+1)}+\mathbf{v}^{\text{\tiny
mn2}}_{\text{8}}\,r^{-(3-z)}+\mathcal{O}(r^{p}).$ (4.114)

In addition to the $\mathbf{v}^{\text{\tiny mn2}}_{\text{3}}$ which is growing
for $z\geq 4$, we need to keep an eye on $\mathbf{v}^{\text{\tiny
mn2}}_{\text{8}}$ mode for $z\geq 3$. Feeding these into field equations and
grinding through several orders we arrive at

$\displaystyle\mathbf{v}^{\text{\tiny 0n1}}_{\text{1}}$
$\displaystyle=\mathbf{v}^{\text{\tiny m02}}_{\text{1}}=0,\quad m\geq
0,\,n\geq 0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn2}}_{\text{1}}$
$\displaystyle=-\frac{z(z^{2}+z+1)(2z^{2}+9z+13)}{(z+2)(2z^{4}+2z^{3}-z^{2}+3z-2)}\mathbf{v}^{\text{\tiny(m+1)(n-1)1}}_{\text{1}},\quad
m\geq 0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny m01}}_{\text{2}}$
$\displaystyle=\mathbf{v}^{\text{\tiny 0n1}}_{\text{2}}=0,\quad m\geq
0,\,n\geq 0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{2}}$
$\displaystyle=-\frac{z(z-1)(4z+11)(z+1)^{2}}{(z+2)(2z^{4}+2z^{3}-z^{2}+3z-2)}\mathbf{v}^{\text{\tiny
m(n-1)1}}_{\text{1}},\quad m>0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny
m01}}_{\text{3}}$ $\displaystyle=\mathbf{v}^{\text{\tiny
0n2}}_{\text{3}}=0,\quad m\geq 0,\,n\geq 0,$
$\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{3}}$
$\displaystyle=-\frac{(z^{2}+z+1)(4z^{2}+7z-3)}{(2z^{4}+2z^{3}-z^{2}+3z-2)}\mathbf{v}^{\text{\tiny(m+1)(n-1)1}}_{\text{1}},\quad
m\geq 0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn2}}_{\text{3}}$
$\displaystyle=-\frac{z+2}{z-4}\mathbf{v}^{\text{\tiny mn1}}_{\text{1}},\quad
m>0,\,n\geq 0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{4}}$
$\displaystyle=0,\quad m\geq 0,\quad n\geq 0,$
$\displaystyle\mathbf{v}^{\text{\tiny m01}}_{\text{5}}$
$\displaystyle=\mathbf{v}^{\text{\tiny 0n1}}_{\text{5}}=0,\quad m\geq
0,\,n\geq 0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{5}}$
$\displaystyle=-\frac{3(z^{4}+8z^{3}+17z^{2}+12z-2)}{2(z+2)(2z^{4}+2z^{3}-z^{2}+3z-2)}\mathbf{v}^{\text{\tiny
m(n-1)1}}_{\text{1}},\quad m>0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny
m01}}_{\text{6}}$ $\displaystyle=0,\quad m\geq 0,$
$\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{6}}$
$\displaystyle=\frac{2\sqrt{z+1}(z^{2}+z+1)(2z^{3}+9z^{2}+10z-3)}{(z+2)(2z^{4}+2z^{3}-z^{2}+3z+2)}\mathbf{v}^{\text{\tiny(m+1)(n-1)1}}_{\text{1}},\quad
m\geq 0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny m01}}_{\text{7}}$
$\displaystyle=\mathbf{v}^{\text{\tiny 0n1}}_{\text{7}}=0,\quad m\geq
0,\,n\geq 0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{7}}$
$\displaystyle=\frac{(z+1)^{3/2}(z^{2}+z+1)}{(2z^{4}+2z^{3}-z^{2}+3z-2)}\mathbf{v}^{\text{\tiny
m(n-1)1}}_{\text{1}},\quad m>0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny
m01}}_{\text{8}}$ $\displaystyle=0,\quad m\geq 0,$
$\displaystyle\mathbf{v}^{\text{\tiny mn1}}_{\text{8}}$
$\displaystyle=-\frac{2(z+1)(z^{2}+z+1)(2z^{3}+9z^{2}+10z-3)}{(2z^{5}+6z^{4}+3z^{3}+z^{2}+4z-4)}\mathbf{v}^{\text{\tiny(m+1)(n-1)1}}_{\text{1}},\quad
m\geq 0,\,n>0,$ $\displaystyle\mathbf{v}^{\text{\tiny mn2}}_{\text{8}}$
$\displaystyle=0,\quad m\geq 0,\,n\geq 0.$

In this case the solutions are parametrized by only $\mathbf{v}^{\text{\tiny
mn1}}_{\text{1}}$ for $m>0,\,n\geq 0$. Similar to the constant perturbations,
the mode $\mathbf{v}^{\text{\tiny mn1}}_{\text{4}}$ corresponding to the
electric charge does not contribute here. Moreover, the mode
$\mathbf{v}^{\text{\tiny mn2}}_{\text{8}}$ which we mentioned before has no
nontrivial solution. Gathering all the solutions, we compute the vector
components of stress tensor

$\displaystyle\mathcal{P}_{y}^{(0,n)}$ $\displaystyle=0,\quad n\geq 0,$
(4.115) $\displaystyle\mathcal{P}_{y}^{(m,n)}$
$\displaystyle=-2(z+2)(z-1)\mathbf{v}^{\text{\tiny mn1}}_{\text{1}},\quad
m>0,\,n\geq 0,$ (4.116) $\displaystyle\Pi^{x}{}_{y}^{(m,0)}$
$\displaystyle=\Pi^{x}{}_{y}^{(0,n)}=0,\quad m\geq 0,\,\,n\geq 0,$ (4.117)
$\displaystyle\Pi^{x}{}_{y}^{(m,n)}$
$\displaystyle=2(z+2)(z-1)\mathbf{v}^{\text{\tiny m(n-1)1}}_{\text{1}},\quad
m>0,\,n>0,$ (4.118) $\displaystyle\mathcal{E}^{y(m,0)}$ $\displaystyle=0,\quad
m\geq 0,$ (4.119) $\displaystyle\mathcal{E}^{y(m,n)}$
$\displaystyle=\frac{6z(z-1)(z+3)(z^{2}+z+1)}{2z^{4}+2z^{3}-z^{2}+3z-2}\mathbf{v}^{\text{\tiny(m+1)(n-1)1}}_{\text{1}},\quad
m\geq 0,\,n>0.$ (4.120)

It is now evident that (4.106) is satisfied.

In summary, the preceding two sections showed that the conservation laws are
satisfied on-shell. We perform this by first verifying that the conservation
equations are a combination of full field equations. Then, we solve the
expanded field equations for the modes that contribute to the stress-energy
tensor and compute the values of components explicitly. It is worth
emphasizing that we argued only from the structure of the expanded field
equations that the particular solutions would be suppressed. We did not solve
the equations explicitly since the number of terms and complications of the
equations proliferate as we go on higher orders in $(n,m)$. It is possible
(especially for the vector modes) that some of the particular solutions may
grow large and spoil the behavior of stress-energy tensor at infinity. In any
case, as demonstrated in Ross:2009ar , it is possible to kill these
divergences by adding $S_{deriv}$ to the action and adjust the free parameters
accordingly.

## 5 Numerical black holes and thermodynamics

The final task we undertake will be to study the conserved quantities and
thermodynamics of the numerical black hole solutions found in
Devecioglu:2014iia . The solutions admit Lifshitz spacetime as a background
and for a fixed horizon radius, depend on one parameter, the gauge field
strength at the horizon. The behavior of solutions differs among horizon
topologies; that being said, in this work we will restrict our attention to
the planar ones, as the stress-energy complex is constructed for the planar
boundary. In Fan:2015yza , it was discussed (through linear analysis) that
unlike their AdS counterparts in other models Fan:2014ixa  which can have non-
zero YM charges, the Lifshitz black holes are analogous to the Minkowski
family Bizon:1990sr , i.e., they do not possess a global YM charge. In this
section, our first aim is to compute the numerical value of the energy of the
solutions and show they are finite, obeying scale relations. Then, employing
the energies for different horizon radius, we will demonstrate that the
alternative Smarr relation and the first law of thermodynamics hold without
any YM hair modification, extending the result of Fan:2015yza  into the non-
linear regime.

Let us start with the redefinitions of metric and gauge field functions which
will help us to cast them in more convenient forms for numerical purposes. The
metric is chosen in the following way

$\displaystyle
ds^{2}=-r^{2z}f(r)^{2}dt^{2}+r^{2}(dx^{2}+dy^{2})+\dfrac{g(r)^{2}dr^{2}}{r^{2}},$
(5.1)

and the gauge field (2.7) redefinitions are as follows

$\displaystyle R(r)=\sigma rh(r),\quad R(r)^{\prime}=\sigma j(r).$ (5.2)

The function $j(r)$ is defined to reduce the boundary value problem to an
initial value problem and employ the shooting method for numerical solution of
field equations. Thus, as $r\rightarrow\infty$, all functions will asymptote
to unity to have Lifshitz background. The first order field equations for
these choices read555The reader is referred to Devecioglu:2014iia  for details
of the numerical solutions.

$\displaystyle rf(r)^{\prime}$
$\displaystyle=-f(r)\Big{(}(z-1)-\dfrac{j(r)^{2}}{2}(z-1)+\dfrac{g(r)^{2}h(r)^{4}}{4}(z^{2}-1)-\dfrac{g(r)^{2}}{4}(3+2z+z^{2})+\dfrac{3}{2}\Big{)},$
(5.3) $\displaystyle rj(r)^{\prime}$
$\displaystyle=j(r)+g(r)^{2}h(r)^{3}(z+1)-\dfrac{g(r)^{2}j(r)}{2}(z^{2}+2z+3)+\dfrac{g(r)^{2}h(r)^{4}j(r)}{2}(z^{2}-1),$
(5.4) $\displaystyle rg(r)^{\prime}$
$\displaystyle=\dfrac{g(r)j(r)^{2}}{2}(z-1)+\dfrac{g(r)^{3}h(r)^{4}}{4}(z^{2}-1)-g(r)^{3}(3+2z+z^{2})+\dfrac{3g(r)}{2},$
(5.5) $\displaystyle rh(r)^{\prime}$ $\displaystyle=j(r)-h(r).$ (5.6)

Here, the equation (5.3) is linear in function $f(r)$ so the overall
normalization of $f$ is not fixed. Thus, the numerical value of
$f(r)\rightarrow f_{\infty}$ will be normalized to unity at infinity by
dividing $f_{\infty}$, which corresponds to rescaling of the time coordinate.
In numerical integration, we will set the initial value at the horizon, i.e.,
$f(R_{0})=1$ then deal with the normalization at the end.

Before moving on to numerical solutions, let us shortly mention about the
horizon expansions which will shed light on the initial values of functions
and shooting parameter. Provided $g_{tt}$ and $g_{rr}$ components of (5.1)
have a simple zero and a simple pole, the black hole will be non-extremal.
This argument leads to the following expansions at the horizon

$\displaystyle f(r)$
$\displaystyle=\sqrt{r-R_{0}}\,\sum_{n=0}^{\infty}f_{n}(r-R_{0})^{n},$ (5.7)
$\displaystyle g(r)$
$\displaystyle=\dfrac{1}{\sqrt{r-R_{0}}}\,\sum_{n=0}^{\infty}g_{n}(r-R_{0})^{n}.$
(5.8)

Incorporating these into field equations, one can study the series solutions
Devecioglu:2014iia  and extract info on initial values.

The next step is to numerically integrate the system of equations (5.3)-(5.6)
by fixing one of the free parameters, i.e. the event horizon radius $R_{0}$
and make the functions $g,h,j$ converge to unity asymptotically, by fine-
tuning the initial value of $h(R_{0})=h_{0}$. Setting $R_{0}=10$ we find
solutions for $z=2,3,4$ and represented them in Figures 1,2. The value of the
shooting parameter $h_{0}$ varies with different $z$ but stays the same for
large and small planar black holes. On the other hand, for other topologies,
the shooting parameter and the behavior of functions vary for small and large
black holes with respect to the length scale $L$ (which should appear in (5.1)
but fixed to $L=1$ in our discussion).

Figure 1: The figure plots the gauge field functions $h(r)$ and $j(r)$ as a
function of radial distance $r$. This is an example of a large black hole with
$R_{0}=10$ for different values of $z$. Figure 2: The figure shows the metric
functions $f(r)$ and $g(r)$ as a function of radius $r$ with $R_{0}=10$.

Having solutions in hand, we now appeal to the stress tensor definitions
(3.12), (3.13) and compute the non-zero components

$\displaystyle E$
$\displaystyle=r^{z+2}f\left(z+3-(z-1)h^{4}-\frac{4}{g}\right),$ (5.9)
$\displaystyle\Pi^{x}{}_{x}$
$\displaystyle=\Pi^{y}{}_{y}=\frac{r^{z+2}}{g}\left[2rf^{\prime}+f\left(g\left((z-1)h^{4}-z-3\right)+2\left(z+1-r(z-1)hh^{\prime}-(z-1)h^{2}\right)\right)\right].$
(5.10)

As in the case of field equation (5.3) the energy and diagonal components of
the spatial stress tensor are linear in $f$ and will be normalised with
$f_{\infty}$ of the corresponding $z$ value. After evaluating the numerical
solutions on (5.9) and (5.10), we plot the results in Figures 3, 4.

Figure 3: The energy of a black holes with $R_{0}=10$. Figure shows the energy
for different cases with $z=2,3,4$. Figure 4: The figure illustrates the
$\Pi^{x}{}_{x}=\Pi^{y}{}_{y}$ components of the black holes with $R_{0}=10$.
The smaller figure is a close up to demonstrate the behaviour for $z=2$.

First relation we check is whether the presence of a black hole breaks the
scale invariance or not. It is evident from the values in the Table 1 that the
scaling Ward identity $z\mathcal{E}=\Pi^{i}{}_{i}$ still holds within the
bounds of numerical errors.

|  $z=2$ |  $z=3$ |  $z=4$
---|---|---|---
$z\mathcal{E}$ |  $3.940\times 10^{4}$ |  $5.923\times 10^{5}$ |  $7.921\times 10^{6}$
$\Pi^{i}{}_{i}$ |  $3.952\times 10^{4}$ |  $5.927\times 10^{5}$ |  $7.923\times 10^{6}$
Table 1: The numerical values of $z\mathcal{E}$ and $\Pi^{i}{}_{i}$ at
$r\rightarrow\infty$.

As a second and final check, we verify the thermodynamic relations of these
solutions. The energy $\mathcal{E}$ defined here is shown to agree with the
thermodynamic energy density obtained by the Euclidean version of the black
holes Ross:2009ar . The proof is quite general so, it is possible to extend
and apply it to our case. For the planar Lifshitz black holes, i.e. $k=0$, it
was also shown that Brenna:2015pqa  the thermodynamic quantities are
consistent with an alternative Smarr relation

$\displaystyle(D+z-2)E=(D-2)TS,$ (5.11)

and an accompanying first law

$\displaystyle dE=TdS.$ (5.12)

These two equalities have the extended versions when the solutions admit
global charges, but as we discussed before, in Fan:2015yza  by studying the
linearized solutions, it was argued that the exclusion of the
$\mathfrak{c}_{3}$ mode (as it diverges for $r\rightarrow\infty$) implies the
lack of global YM charge. In accordance with this reasoning (5.11), (5.12)
should hold in their simplest form. In order to check this claim, we first
compute the Hawking temperature through Wick rotation with an assumption of
regularity at the horizon

$\displaystyle
T=\frac{r^{z+1}}{4\pi}\sqrt{f(r)^{\prime}g(r)^{\prime}}\Bigg{|}_{r=R_{0}},$
(5.13)

and the entropy is $S=\pi R_{0}^{2}$ where we suppose suitable identifications
and volume renormalizations. From near horizon expansions (5.7), (5.8) it is
easy to see $T\sim R_{0}^{z}$. Then the first law (5.12) implies the energy
should scale as $E\sim R_{0}^{z+2}$, which is in agreement with the
$L^{D+z-2}$ scaling that can be obtained from the Euler’s relation in (5.11).
This is different from $L^{D-3}$ scaling in AdS as alluded in Brenna:2015pqa .
On the numerical side, we verify this feature by plotting the energy of the
black holes with respect to their horizon radius having different dynamical
exponents in Fig. 5.

Figure 5: The circle, square, and triangle points are numerical values
generated from (5.9) for different horizon values, while solid, dashed, dotted
curves are quartic, quintic and sextic polynomials, respectively. As
demonstrated, the numerical values fit nicely to their corresponding
polynomial curves.

It remains to verify the Smarr relation; to that end, we opted to compute the
left and right-hand side of (5.11) numerically and represent them in the Table
2. For the sake of clarity, we also calculate the ratio $2TS/(z+2)E$ (which is
not unity, as we did not include the physical constants in our numerical
computations) to show that the Smarr relation holds (5.11).

|  $z=2$ |  $z=3$ |  $z=4$
---|---|---|---
$(z+2)\mathcal{E}$ |  $7.425\times 10^{8}$ |  $9.188\times 10^{9}$ |  $1.105\times 10^{11}$
$2TS$ |  $9.296\times 10^{13}$ |  $1.149\times 10^{15}$ |  $1.381\times 10^{16}$
$2TS/(z+2)E$ |  $1.251\times 10^{5}$ |  $1.250\times 10^{5}$ |  $1.250\times 10^{5}$
Table 2: The numerical values of $(z+2)\mathcal{E}$ and $2TS$ for $z=2,3,4$.
The final row of the table contains the ratio $2TS/(z+2)E$. Here we did not
divide the first two rows with $f_{\infty}$ as their ratio will be the same.

With this final result, we demonstrated that the thermodynamic relations hold
without an additional YM charge. As advertised earlier in Fan:2015yza , this
makes the numerical black holes of Devecioglu:2014iia  hairy solutions. The
stability of these black holes is another important question that can save the
no-hair theorem.

In short, the energy definition is shown to work well in the non-linear
regime, producing finite results with the scaling properties left intact.
Moreover, the thermodynamical energy is in agreement with the Smarr formula
and the first law, confirming the $SU(2)$ hair of the black holes.

## 6 Conclusions

In this work, we have computed the holographic stress-energy tensor of the EYM
model with Lifshitz backgrounds as a solution. After setting the stage for a
properly defined tensor, the main results can be outlined as follows.

First, we study the linearised field equations to show that the action and the
stress-energy tensor stay finite for a more general class of asymptotically
Lifshitz spacetimes. The choice for the form of perturbation plays an
essential role in decomposing field equations into scalar, vector, and tensor
modes. For constant perturbations, we were able to solve the linearised
equations in their full forms and choose the modes that asymptote to the
Lifshitz backgrounds. We can summarise the modes as:

  * •

The modes $\mathfrak{c}_{1},\mathfrak{t}_{d1},\mathfrak{t}_{o1}$ have fall
rate $(z+2)$ and contribute to the $\mathcal{E},\Pi^{i}{}_{j}$ components of
the stress tensor.

  * •

The mode $\mathfrak{c}_{1i}$ have the fall rates $(z+2)$, $(4-z)$ and
contribute to $\mathcal{P}_{i}$.

  * •

The mode $\mathfrak{c}_{2i}$ have the fall rates $(z+2)$, $3z$ and contribute
to $\mathcal{E}^{i}$.

  * •

The mode $\mathfrak{c}_{2}$ has a fall rate $(z+2+\beta_{z})$ for $z>1$, and
the modes $\mathfrak{t}_{d2},\mathfrak{t}_{o2}$ decay as $(z+2+\xi_{z})$ for
$z\geq 2(3+2\sqrt{2})$. These do not contribute to the stress tensor and
related to the expectation value of the operator associated with the $SU(2)$
gauge field.

On the other hand, in general perturbations, the complicated structure does
not allow for an exact solution. Nevertheless, it was possible to solve the
system through an expansion in $k$ and $\omega$. In both instances (constant
and generalised), the conservation and scaling laws are shown to be satisfied
on-shell.

In the second part of the work, we turned our attention to the numerical black
hole solutions of Devecioglu:2014iia . Focusing on the planar ones, we find
the energy and the spatial stress tensor for different $z$ values and show
that they are finite while satisfying the scaling identity
$z\mathcal{E}=\Pi^{i}{}_{i}$. The results prove that the stress tensor
definition is reliable in the non-linear regime. A more non-trivial check was
the thermodynamic equalities satisfied by these black holes. The Smarr
relation and the first law hold without the YM parameter so that the solutions
are hairy, like their asymptotically flat and AdS counterparts.

There are several directions for further exploration in this model. First, on
the holography side, using the results of the mode analysis as a starting
point, it is now possible to look for the divergences in the one-point
functions and perform holographic renormalisation. It will also be interesting
to see the structure of TNC geometry that will be conjured on the boundary.
Along these lines, it can be helpful to investigate the eleven-dimensional
model Pope:1985bu , which might help to extract information about the sources
and VeVs.

Finally, hairy black holes of AdS are used in modelling the second-order
transitions Gubser:2008px , where an abelian gauge symmetry is spontaneously
broken near a black hole horizon in ADS using a condensate of $SU(2)$ gauge
fields. A similar setup where the AdS solution is replaced with a hairy
Lifshitz black hole can now be considered by turning on the electric part
$Q(r)$ in the gauge field ansatz. We plan to return these issues in the
future.

###### Acknowledgements.

I am grateful to Simon F. Ross for the detailed explanation of his work,
comments and suggestions. I also thank Gökhan Alkaç, Özgür Sarıoğlu for
discussions. Finally I thank Gökhan Alkaç for his critical reading of the
manuscript. This work was supported by the National Natural Science Foundation
of China under Grant No. 11875136 and the Major Program of the National
Natural Science Foundation of China under Grant No. 11690021.

##  Appendix A Linearized quantities

Here we present some of the linearized objects that are used to compute the
perturbative action and the field equations.

$\displaystyle K=\bar{K}+K^{L}=$
$\displaystyle(z+2)+\frac{r}{2}\partial_{r}[\hat{h}_{tt}+\hat{h}^{i}{}_{i}],$
(A.1) $\displaystyle R=\bar{R}+R^{L}=$
$\displaystyle-2(3+z(2+z))-r\left((3+2z)\partial_{r}\hat{h}_{tt}+(4+z)\partial_{r}\hat{h}^{i}{}_{i}+r\partial_{r}^{2}[\hat{h}_{tt}+\hat{h}^{i}{}_{i}]\right),$
(A.2) $\displaystyle F^{\Lambda}_{\mu\nu}F^{\mu\nu}_{\Lambda}=$
$\displaystyle\bar{F}^{\Lambda}_{\mu\nu}\bar{F}^{\mu\nu}_{\Lambda}+(F^{\Lambda}_{\mu\nu}F^{\mu\nu}_{\Lambda})^{L}=2(z+1)\left(3+z+2(2+z)\hat{b}^{i}{}_{i}+r\partial_{r}[\hat{h}^{i}{}_{i}+2\hat{b}^{i}{}_{i}]\right),$
(A.3) $\displaystyle F^{\Lambda}_{ab}F^{ab}_{\Lambda}=$ $\displaystyle
4(z+1)^{2}(1+2\hat{b}^{i}{}_{i}).$ (A.4)

##  Appendix B Field equations for generalised perturbations

### 2.1 Field equations for the scalar modes

In this section we provide the scalar mode field equations for the generalised
perturbations. First, let us call the the linearized equations for the gravity
sector as $\mathcal{E}_{\mu\nu}$ and for the matter sector
$\mathcal{E}_{\mu}^{\Lambda}$. The field equations for the scalar modes read

$\displaystyle
r^{2}\left(f^{\prime\prime}+k_{L}^{\prime\prime}\right)+2(z+1)rf^{\prime}+(z+3)rk_{L}^{\prime}-4\left(z^{2}-1\right)b_{L}={}\frac{k^{2}}{2r^{2}}f-\frac{2k^{2}(z^{2}-1)s_{3}}{r\sqrt{z+1}}$
$\displaystyle-\frac{\omega^{2}}{r^{2z}}k_{L}-\frac{k^{2}\omega^{2}}{r^{2}}s_{1}$
(B.1) $\displaystyle
r^{2}(2b_{L}^{\prime\prime}+k_{L}^{\prime\prime})+r(z+3)(2b_{L}^{\prime}+k_{L}^{\prime})+rf^{\prime}-4(z+1)b_{L}={}-\frac{k^{2}}{r}\left(rs_{4}^{\prime}+zs_{4}+3\sqrt{z+1}s_{3}\right)$
$\displaystyle+\frac{k^{2}}{2r^{2}}\left(2b_{L}+k_{L}\right)-\frac{\omega^{2}}{r^{2z}}\left(2b_{L}+k_{L}\right)$
$\displaystyle+\frac{k^{2}\omega^{2}}{r^{2z}}s_{2}-\frac{k^{4}}{r^{2}}(b_{T}+k_{T})$
(B.2) $\displaystyle
2rk_{L}^{\prime}+rf^{\prime}-2(z-1)b_{L}^{\prime}+2(z-1)b_{L}={}\frac{k^{2}}{2r^{2}}\left(f+k_{L}\right)-\frac{\omega^{2}}{r^{2z}}k_{L}$
$\displaystyle+\frac{k^{2}(z-1)}{r}(\sqrt{z+1}s_{3}+s_{4})$
$\displaystyle-\frac{k^{4}}{2r^{2}}k_{T}+k^{2}\omega^{2}\left(\frac{s_{2}}{r^{2z}}-\frac{s_{1}}{r^{2}}\right)$
(B.3) $\displaystyle
r^{2}k_{T}^{\prime\prime}+4r(z-1)b_{T}^{\prime}+(3z+1)rk_{T}^{\prime}+4(z-1)b_{T}-\frac{f}{2r^{2}}+\frac{2(z-1)s_{4}}{r}={}-\omega^{2}\left(\frac{k_{T}}{r^{2z}}+\frac{s_{1}}{r^{2}}-\frac{s_{2}}{r^{2z}}\right)$
(B.4) $\displaystyle
r^{2}(2b_{T}^{\prime\prime}+k_{T}^{\prime\prime})+(z+1)rk_{T}^{\prime}+2(z+3)rb_{T}^{\prime}+4(z+1)b_{T}+s_{4}^{\prime}+\frac{zs_{4}}{r}-\frac{\sqrt{z+1}s_{3}}{r}+\frac{b_{L}}{r^{2}}+\frac{k_{L}}{2r^{2}}$
$\displaystyle={}-\frac{\omega^{2}}{r^{2z}}(2b_{T}+k_{T}-s_{2})$
$\displaystyle+\frac{k^{2}}{2r^{2}}(2b_{T}+k_{T})$ (B.5) $\displaystyle
r^{3-z}s_{2}^{\prime\prime}-r^{1-z}s_{4}^{\prime}-(z-4)r^{2-z}s_{2}^{\prime}-r^{z}s_{1}^{\prime}+\frac{(z-2)s_{4}}{r^{z}}-\frac{\sqrt{z+1}s_{3}}{r^{z}}-\frac{1}{2r^{z+1}}(2b_{L}+k_{L})={}$
$\displaystyle
k^{2}\Big{(}\frac{b_{T}}{r^{z+1}}+\frac{k_{T}}{2r^{z+1}}+\frac{s_{2}}{r^{2}}\Big{)}$
(B.6) $\displaystyle
r^{2z}s_{1}^{\prime\prime}-r^{2}s_{2}^{\prime\prime}-(z+3)rs_{2}^{\prime}+r^{2z-1}(3z+1)s_{1}^{\prime}-\frac{2(z-1)s_{4}}{r}+\frac{2(z^{2}-1)s_{3}}{r\sqrt{z+1}}+\frac{k_{L}}{r^{2}}$
$\displaystyle={}\frac{k^{2}k_{T}}{r^{2}}$ (B.7)
$\displaystyle\frac{2r^{2}(z^{2}-1)s_{3}^{\prime}}{\sqrt{z+1}}+r(f^{\prime}+k_{L}^{\prime}+2(z^{2}-1)s_{4})+(z-1)(2b_{L}+f+k_{L})={}$
$\displaystyle
k^{2}(2(z-1)b_{T}+(z-1)k_{T}+rk_{T}^{\prime})+\omega^{2}(2(z-1)s_{1}+rs_{1}^{\prime}-r^{3-2z}s_{2}^{\prime})$
(B.8)
$\displaystyle\frac{2r}{\sqrt{z+1}}(rs_{3}^{\prime\prime}+(z+3)s_{3}^{\prime})+2(rs_{4}^{\prime}+(z+3)s_{3}^{\prime})={}\frac{k^{2}}{r^{2}}(2rb_{T}+rk_{T}+\frac{2s_{3}}{\sqrt{z+1}})$
$\displaystyle+\omega^{2}\left(\frac{2s_{1}}{r}-\frac{2s_{2}}{r^{z}}-\frac{2s_{3}}{r^{2z}\sqrt{z+1}}\right)$
(B.9)

$\displaystyle
rk_{L}^{\prime}+2(z-1)b_{L}={}-k^{2}\left(\frac{r^{2z-1}s_{1}^{\prime}}{2}-\frac{1}{2}r^{2}s_{2}^{\prime}\right)$
(B.10) $\displaystyle
2r^{2}\sqrt{z+1}s_{3}^{\prime}+r(2b_{L}^{\prime}+k_{L}^{\prime})+2(z+1)rs_{4}+2b_{L}+f+k_{L}={}$
$\displaystyle-\frac{k^{2}}{r}(r(2b_{T}-k_{T})+2s_{4}+r^{2}(2b_{T}^{\prime}+k_{T}^{\prime}))$
$\displaystyle+2\omega^{2}(s_{1}+r^{1-2z}(s_{4}+r^{2}s_{2}^{\prime}))$ (B.11)
$\displaystyle
rk_{L}^{\prime\prime}+r(z+3)k_{L}^{\prime}+2(z-1)rb_{L}^{\prime}+2(z^{2}+z-2)b_{L}={}k^{2}\left(\frac{k_{L}}{2r^{2}}+\frac{(z^{2}-1)s_{3}}{r\sqrt{z+1}}-\frac{(z-1)s_{4}}{r}\right)-\frac{k^{4}}{2r^{2}}k_{T}$
(B.12)

In terms of components (B.1)-(B.12) corresponds to
$(\mathcal{E}_{x_{1}x_{1}}+\mathcal{E}_{x_{2}x_{2}})$,
$(\mathcal{E}^{1}_{x_{1}}+\mathcal{E}_{x_{2}}^{2})$, $\mathcal{E}_{rr}$,
$(\mathcal{E}_{x_{1}x_{1}}-\mathcal{E}_{x_{2}x_{2}})$,
$(\mathcal{E}^{1}_{x_{1}}-\mathcal{E}^{2}_{x_{2}})$, $\mathcal{E}^{1}_{t}$,
$\mathcal{E}_{tx_{1}}$, $\mathcal{E}_{x_{1}r}$, $\mathcal{E}^{3}_{x_{2}}$,
$\mathcal{E}_{tr}$, $\mathcal{E}^{1}_{r}$, $\mathcal{E}_{tt}$ in their order
of appearance. Note that, there are twelve equations and nine functions.
Therefore three relations constraint this system as follows

$\displaystyle\partial_{r}[\eqref{eq:7}]-\eqref{eq:9}+\frac{z+1}{r}\eqref{eq:7}+\frac{k^{2}}{2}\eqref{eq:11}$
$\displaystyle=0,$ (B.13) $\displaystyle
r\partial_{r}[\eqref{eq:10}]-(z^{2}-1)r\eqref{eq:12}+\eqref{eq:1}+(z+2)\eqref{eq:10}+(z-1)\eqref{eq:8}+k^{2}\eqref{eq:4}+\omega^{2}\frac{\eqref{eq:11}}{r^{2z-2}}$
$\displaystyle=0,$ (B.14) $\displaystyle
r\partial_{r}[\eqref{eq:3}]+(z-1)\eqref{eq:2}-\eqref{eq:1}-z\eqref{eq:9}+(z+2)\eqref{eq:3}-\frac{k^{2}}{2r^{2}}\eqref{eq:10}-\frac{\omega^{2}}{r^{2z+1}}\eqref{eq:7}$
$\displaystyle=0.$ (B.15)

We approach the system of equations by first solving (B.8) for $s_{4}$
algebraically. Then, feeding this solution into (B.1)-(B.9) we expand the
equations in $k,\omega$ and solve the homogeneous parts order by order.

### 2.2 Field equations for the vector modes

The equations for the vector modes are as follows

$\displaystyle
r^{2}\mathsf{v}_{2}^{\prime\prime}+4(z-1)r\mathsf{v}_{5}^{\prime}+(3z+1)r\mathsf{v}_{2}^{\prime}+4(z-1)\mathsf{v}_{5}={}-\frac{2k^{2}}{r}(z-1)\mathsf{v}_{7}-\frac{\omega^{2}}{r^{2z}}\mathsf{v}_{2}-k^{2}\omega^{2}\left(\frac{\mathsf{v}_{1}}{r^{2}}-\frac{\mathsf{v}_{3}}{r^{2z}}\right)$
(B.16) $\displaystyle
2r^{2}\mathsf{v}_{5}^{\prime\prime}+r^{2}\mathsf{v}_{2}^{\prime\prime}-2\sqrt{z+1}r\mathsf{v}_{8}^{\prime}+2(z+3)r\mathsf{v}_{5}^{\prime}+(z+1)r\mathsf{v}_{2}^{\prime}-2\sqrt{z+1}(z+2)\mathsf{v}_{8}+4(z+1)\mathsf{v}_{5}={}$
$\displaystyle
k^{2}\left(\frac{\mathsf{v}_{2}}{r^{2}}+\frac{2\mathsf{v}_{5}}{r^{2}}-\frac{2\sqrt{z+1}\mathsf{v}_{6}}{r}\right)-\omega^{2}\left(\frac{\mathsf{v}_{2}}{r^{2z}}+\frac{2\sqrt{z+1}\mathsf{v}_{4}}{r^{z}}+\frac{2\mathsf{v}_{5}}{r^{2z}}\right)$
(B.17) $\displaystyle
r^{3-z}\mathsf{v}_{3}^{\prime\prime}-r^{1-z}\mathsf{v}_{7}^{\prime}-(z-4)r^{2-z}\mathsf{v}_{3}^{\prime}-r^{z}\mathsf{v}_{1}^{\prime}+(z-2)r^{-z}\mathsf{v}_{7}+r^{-z}\sqrt{z+1}\mathsf{v}_{6}-r^{-(z+1)}\mathsf{v}_{5}+\frac{2\sqrt{z+1}}{r}\mathsf{v}_{4}$
$\displaystyle+(z-1)\mathsf{v}_{3}-\frac{\mathsf{v}_{2}}{2r^{z+1}}={}\frac{k^{2}\mathsf{v}_{3}}{r^{z+1}}$
(B.18) $\displaystyle
2\sqrt{z+1}(r\mathsf{v}_{8}^{\prime}+(z+2)\mathsf{v}_{8})={}k^{2}\left(\frac{\sqrt{z+1}\mathsf{v}_{6}}{r}-\frac{\mathsf{v}_{2}}{2r^{2}}-\frac{\mathsf{v}_{5}}{r^{2}}-\frac{z\mathsf{v}_{7}}{r}-\mathsf{v}_{7}^{\prime}\right)+\omega^{2}\frac{2\sqrt{z+1}\mathsf{v}_{4}}{r^{z}}+k^{2}\omega^{2}\frac{\mathsf{v}_{3}}{r^{2z}}$
(B.19)
$\displaystyle\sqrt{z+1}r\mathsf{v}_{6}^{\prime}-\mathsf{v}_{5}^{\prime}-\frac{\mathsf{v}_{2}^{\prime}}{2}-\frac{2\sqrt{z+1}\mathsf{v}_{8}}{r}-(z+1)\mathsf{v}_{7}-\frac{\mathsf{v}_{5}}{r}+\frac{\mathsf{v}_{2}}{2r}={}k^{2}\frac{\mathsf{v}_{7}}{r^{2}}+\omega^{2}\Big{(}\frac{\mathsf{v}_{3}^{\prime}}{r^{2z-2}}-\frac{\mathsf{v}_{1}}{r}-\frac{\mathsf{v}_{7}}{r^{2z}}\Big{)}$
(B.20) $\displaystyle
r^{2}\mathsf{v}_{4}^{\prime\prime}+r^{1-z}\mathsf{v}_{8}^{\prime}+r(z+3)\mathsf{v}_{4}^{\prime}-(z-2)r^{-z}\mathsf{v}_{8}-2\mathsf{v}_{4}={}k^{2}\Big{(}\frac{\sqrt{z+1}\mathsf{v}_{1}}{r^{2-z}}+\frac{\mathsf{v}_{4}}{r^{2}}+\frac{\mathsf{v}_{6}}{r^{z+1}}+\frac{\sqrt{z+1}\mathsf{v}_{3}}{r^{z}}\Big{)}$
(B.21) $\displaystyle
r^{2}\mathsf{v}_{1}^{\prime\prime}-r^{4-2z}\mathsf{v}_{3}^{\prime\prime}-(z+3)r^{3-2z}\mathsf{v}_{3}^{\prime}-(z-1)\Big{(}2r^{1-2z}\mathsf{v}_{7}+2r^{1-2z}\sqrt{z+1}\mathsf{v}_{6}+2r^{-z}\sqrt{z+1}\mathsf{v}_{4}\Big{)}$
$\displaystyle+r(3z+1)\mathsf{v}_{1}^{\prime}-r^{-2z}\mathsf{v}_{2}=k^{2}\left(\frac{\mathsf{v}_{1}}{r^{2}}-\frac{\mathsf{v}_{3}}{r^{2z}}\right)$
(B.22)
$\displaystyle\sqrt{z+1}(z-1)\Big{(}r\mathsf{v}_{6}^{\prime}-\frac{\mathsf{v}_{8}}{r}-\sqrt{z+1}\mathsf{v}_{7}+\frac{\mathsf{v}_{5}}{r\sqrt{z+1}}+\frac{\mathsf{v}_{2}}{2r\sqrt{z+1}}\Big{)}+\frac{\mathsf{v}_{2}^{\prime}}{2}=$
$\displaystyle{}\omega^{2}\left(\frac{(1-z)\mathsf{v}_{1}}{r}-\frac{\mathsf{v}_{1}^{\prime}}{2}-\frac{\mathsf{v}_{3}^{\prime}}{2r^{2z-2}}\right)$
(B.23) $\displaystyle
r^{2}\mathsf{v}_{6}^{\prime\prime}-\mathsf{v}_{8}^{\prime}-\sqrt{z+1}r\mathsf{v}_{7}^{\prime}+(z+3)r\mathsf{v}_{6}^{\prime}-(z+2)\sqrt{z+1}\mathsf{v}_{7}-\frac{z\mathsf{v}_{8}}{r}+\frac{\sqrt{z+1}\mathsf{v}_{5}}{r}+\frac{\sqrt{z+1}\mathsf{v}_{2}}{2r}={}$
$\displaystyle\omega^{2}\left(-\frac{\sqrt{z+1}\mathsf{v}_{1}}{r}-\frac{\mathsf{v}_{4}}{r^{z+1}}+\frac{\mathsf{v}_{6}}{r^{2z}}\right)$
(B.24) $\displaystyle
2(z+1)r\mathsf{v}_{8}=k^{2}\left(\mathsf{v}_{6}+r\mathsf{v}_{6}^{\prime}-2\sqrt{z+1}\mathsf{v}_{7}-\frac{\mathsf{v}_{8}}{r}\right)+\omega^{2}\left(r^{1-2z}\mathsf{v}_{8}+r^{2-z}\mathsf{v}_{4}^{\prime}+r^{1-z}z\mathsf{v}_{4}\right)$
(B.25)

where the corresponding components are $\mathcal{E}_{x_{1}x_{2}}$,
$\mathcal{E}_{x_{2}}$, $\mathcal{E}^{2}_{t}$,
$(\mathcal{E}^{1}_{y}-\mathcal{E}^{2}_{x})$, $\mathcal{E}^{2}_{r}$,
$\mathcal{E}^{3}_{t}$, $\mathcal{E}_{tx_{2}}$, $\mathcal{E}_{x_{2}r}$,
$\mathcal{E}^{3}_{x_{1}}$, $\mathcal{E}^{3}_{r}$. This time there are two
constraint equations

$\displaystyle\partial_{r}[\eqref{veq:8}]-\sqrt{z+1}\eqref{veq:24}+\frac{z+1}{r}\eqref{veq:8}+\frac{k^{2}}{r}\eqref{veq:7}-\frac{\omega^{2}}{r^{z}}\eqref{veq:6}$
$\displaystyle=0,$ (B.26) $\displaystyle
r\partial_{r}[\eqref{veq:G23}]-(z-1)\sqrt{z+1}\eqref{veq:7}-(z-1)\eqref{veq:5}+(z+3)\eqref{veq:G23}-\frac{\eqref{veq:1}}{2r}+\frac{\omega^{2}}{2r}\eqref{veq:9}$
$\displaystyle=0,$ (B.27)

as the system is now composed of ten equations and eight functions. After we
determine $\mathsf{v}_{4}$ algebraically from (B.23), we expand and solve the
system (B.16)-(B.23).

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