# Revisiting linear stability of black hole odd-parity perturbations
in Einstein-Aether gravity

Shinji Mukohyamaa,b<EMAIL_ADDRESS>Shinji Tsujikawac
<EMAIL_ADDRESS>Anzhong Wangd<EMAIL_ADDRESS>aCenter for
Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto
University, 606-8502, Kyoto, Japan
bKavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo Institutes for Advanced Study, The University of Tokyo,
Kashiwa, Chiba 277-8583, Japan
cDepartment of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo
169-8555, Japan
dGCAP-CASPER, Physics Department, Baylor University, Waco, TX 76798-7316, USA

###### Abstract

In Einstein-Aether gravity, we revisit the issue of linear stabilities of
black holes against odd-parity perturbations on a static and spherically
symmetric background. In this theory, superluminal propagation is allowed and
there is a preferred timelike direction along the unit Aether vector field. If
we choose the usual spherically symmetric background coordinates with respect
to the Killing time $t$ and the areal radius $r$, it may not be appropriate
for unambiguously determining the black hole stability because the constant
$t$ hypersurfaces are not necessarily always spacelike. Unlike past related
works of black hole perturbations, we choose an Aether-orthogonal frame in
which the timelike Aether field is orthogonal to spacelike hypersurfaces over
the whole background spacetime. In the short wavelength limit, we show that
no-ghost conditions as well as radial and angular propagation speeds coincide
with those of vector and tensor perturbations on the Minkowski background.
Thus, the odd-parity linear stability of black holes for large radial and
angular momentum modes is solely determined by constant coefficients of the
Aether derivative couplings.

###### pacs:

04.50.Kd,95.30.Sf,98.80.-k

††preprint: YITP-24-65, IPMU24-0024, WUCG-24-05

##  I Introduction

General relativity (GR) is a fundamental pillar of modern physics for
describing the gravitational interaction. GR enjoys the invariance under
transformations in the Lorentz group. From the perspective of quantum gravity
and high-energy theories, however, there are some indications that Lorentz
invariance may not be an exact symmetry at all energies [1, 2, 3, 4, 5].
Lorentz violation at high energies may allow for the possibility of
regularizing field theories, while recovering Lorentz symmetry at low energies
[6]. Although broken Lorentz invariance for the standard model matter fields
is highly constrained from numerous experiments, the bounds on Lorentz
violation in the gravitational sector are not so stringent yet [7, 8, 9].

To accommodate broken Lorentz invariance for the gravitational fields without
losing the covariant property of GR, there is a way of introducing a unit
timelike vector field $u^{\mu}$ satisfying the relation $u^{\mu}u_{\mu}=-1$.
This is known as Einstein-Aether theory [10], in which a preferred threading
with respect to the Aether field is present. To maintain general covariance of
Einstein gravity, we require that the preferred threading is dynamical. Since
the timelike Aether field is nonvanishing at any spacetime points, it always
breaks local Lorentz invariance. In this sense, Einstein-Aether theory is
distinguished from other Lorentz-violating theories restoring Lorentz
invariance at some particular energy scales.

The covariant action of Einstein-Aether theory, which was introduced by
Jacobson and Mattingly [10], contains four derivative couplings of the Aether
field with dimensionless coupling constants $c_{1,2,3,4}$ besides the Ricci
scalar $R$. The unit vector constraint on the timelike Aether field can be
incorporated into the action as a Lagrange multiplier of the form
$\lambda(u^{\mu}u_{\mu}+1)$. We should mention that there was also an
equivalent approach based on a tetrad formalism advocated by Gasperini [11].
The Einstein-Aether framework can encompass several classes of vector-tensor
theories such as the spontaneous breaking of Lorentz invariance in string
theory [8] and cuscuton theories with a quadratic scalar potential [12, 13].
There are also extended versions of Einstein-Aether theory in which a
symmetry-breaking potential for the vector is introduced [14] or the Aether
coupling functions are generalized [15, 16]. The generalized Einstein-Aether
theory of Ref. [15] is subject to severe constraints on the coupling
functions, if $c_{1}+c_{3}\not=0$ [17].

The perturbative analysis of Einstein-Aether theory on the Minkowski
background (with all nonvanishing coupling constants $c_{1,2,3,4}$) shows that
there are one scalar, two vector, and two tensor propagating degrees of
freedom [18]. As we will review in Sec. II, their squared propagation speeds
are given, respectively, by Eqs. (10), (11), and (12), all of which are
different from that of light. The gravitational-wave event GW170817 of a
black-hole (BH)-neutron star binary, along with the gamma-ray burst 170817A,
put a stringent limit $|c_{T}-1|\lesssim 10^{-15}$ on the tensor propagation
speed $c_{T}$ [19], thereby translating to the bound $|c_{1}+c_{3}|\lesssim
10^{-15}$ [20, 21]. The coupling constants $c_{1,2,3,4}$ have been constrained
from other experiments and observations such as gravitational Cerenkov
radiation [22], big-bang nucleosynthesis [23], solar-system tests of gravity
[24], binary pulsars [25, 26, 27, 28], and gravitational waveforms [29, 30,
31]. Despite those numerous observational data, there are still wide regions
of parameter space that are compatible with all these constraints.

If we apply Einstein-Aether theory to the physics on a static and spherically
symmetric (SSS) background, it is known that there are some nontrivial BH
solutions endowed with the Aether hair [32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Besides the usual metric horizon at
which the time translation Killing vector $\zeta^{\mu}$ becomes null
($\zeta^{\mu}\zeta_{\mu}=0$), broken Lorentz invariance can give rise to the
existence of a universal horizon at which the Aether field $u_{\mu}$ is
orthogonal to $\zeta^{\mu}$, i.e., $u_{\mu}\zeta^{\mu}=0$ [51]. This universal
horizon, which lies inside the metric horizon, can be interpreted as a causal
boundary of any speeds of propagation. In other words, once a wave signal is
trapped inside the universal horizon, it does not escape from BHs toward
spatial infinity. These distinguished features in Einstein-Aether theory may
manifest themselves for inspiral gravitational waveforms emitted from BH
binaries and BH quasinormal modes in the ringdown phase. Thus, the upcoming
high-precision observations of gravitational waves will offer the possibility
of probing the signature of BHs with the Aether hair.

The BH perturbation theory, which was originally developed by Regge-Wheeler
[52] and Zerilli [53], plays a crucial role in computing the quasinormal modes
of BHs. Moreover, the linear stability of BHs is known by studying conditions
for the absence of ghosts and Laplacian instabilities in the small-scale
limit. In scalar-tensor Horndeski theories [54], for example, the second-order
actions of odd- and even-parity perturbations on the SSS background were
derived in Refs. [55, 56, 57] for exploring the linear stability of hairy BHs.
In Refs. [58, 59, 60], it was found that the angular propagation speeds of
even-parity perturbations, besides other stability conditions, are important
to exclude a large class of hairy BHs due to Laplacian instabilities. As a
result, the presence of a Gauss-Bonnet term coupled to the scalar field plays
a prominent role in the realization of linearly stable BH solutions in
Horndeski theories [61].

In Einstein-Aether theory, the second-order action of odd-parity perturbations
was derived in Ref. [62] by using a standard SSS coordinate introduced later
in Eq. (43). The odd-parity sector contains two propagating degrees of
freedom: (1) one tensor mode arising from the gravitational perturbation
$\chi$, and (2) one vector mode arising from the Aether perturbation $\delta
u$. The no-ghost conditions and propagation speeds for $\chi$ and $\delta u$
were obtained by dealing with the $t$ coordinate as a time clock [62]. In
Einstein-Aether theory, however, there is a preferred timelike direction along
the unit Aether field. Since the timelike property of $t$ coordinate is not
always ensured in this setup, the choice of $t$ and $r$ coordinates should not
be necessarily appropriate for discussing the linear stability of BHs.

On the SSS spacetime where the background Aether field does not have
vorticity, it is possible to locally choose a timelike coordinate $\phi$ in
the form $u_{\mu}|_{\rm background}=-\eta\partial_{\mu}\phi$, where $\eta$ is
a nonvanishing function. This scalar field $\phi$, which was named “khronon”
in Ref. [63], defines the timelike direction in the foliation structure of
spacetime. On the SSS background, one can introduce an Aether-orthogonal frame
in which the Aether field is orthogonal to spacelike hypersurfaces. Indeed, it
is known that [64] Einstein-Aether theory in such a configuration is
equivalent to the infrared limit of the non-projectable version of
Ho$\check{\rm r}$ava gravity [65]111It should be noted that the equivalence
between Einstein-Aether gravity and khronometric theory (or the infrared limit
of non-projectable Ho$\check{\rm r}$ava gravity [66]) holds only when the
Aether field has zero vorticity. In particular, their Hamiltonian structures
are different. In fact, while in Einstein-Aether gravity there are five
propagating local physical degrees of freedom, in khronometric theory the
number of propagating local physical degrees of freedom is three [51, 67], so
is in Ho$\check{\rm r}$ava gravity [68]..

Since the Aether-orthogonal frame is a proper choice of the timelike
coordinate orthogonal to spacelike hypersurfaces, we will revisit the linear
stability analysis of BHs in the odd-parity sector for this coordinate system.
In Sec. II, we briefly review current constraints on the coupling constants
$c_{1,2,3,4}$ of derivative couplings of the Aether field. In Sec. III, we
will see how a naive choice of the usual SSS coordinate (43) can cause
apparent instabilities and introduce the Aether-orthogonal frame as well as
relations between two different frames. In Sec. IV, we transform the second-
order action of odd-parity perturbations derived in Ref. [62] to that in the
Aether-orthogonal frame and show that, for large radial and angular momentum
modes, the no-ghost conditions and speeds of propagation are identical to
those of vector and tensor perturbations on the Minkowski background. Thus,
unlike the results in Ref. [62], the linear stability of BHs against odd-
parity perturbations does not add new conditions to those known in the
literature. Sec. V is devoted to conclusions.

Throughout the paper, we will use the natural unit in which the speed of light
$c$ and the reduced Planck constant $\hbar$ are unity. We also adopt the
metric signature $(-,+,+,+)$.

##  II Einstein-Aether theory and current constraints

Einstein-Aether theory is given by the action [10]

${\cal S}=\frac{1}{16\pi G_{\ae}}\int{\rm d}^{4}x\sqrt{-g}\left[R+{\cal
L}_{\ae}+\lambda(g_{\mu\nu}u^{\mu}u^{\nu}+1)\right],$ (1)

where $G_{\ae}$ is a constant corresponding to the gravitational coupling, $R$
is the Ricci scalar, $g$ is the determinant of metric tensor $g_{\mu\nu}$,
$\lambda$ is a Lagrange multiplier, $u^{\mu}$ is the Aether vector field, and

${\cal
L}_{\ae}=-{M^{\alpha\beta}}_{\mu\nu}\nabla_{\alpha}u^{\mu}\nabla_{\beta}u^{\nu}\,,$
(2)

with

${M^{\alpha\beta}}_{\mu\nu}:=c_{1}g^{\alpha\beta}g_{\mu\nu}+c_{2}\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}+c_{3}\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu}-c_{4}u^{\alpha}u^{\beta}g_{\mu\nu}\,.$
(3)

The Greek indices run from 0 to 3, $\nabla_{\alpha}$ is a covariant derivative
operator with respect to $g_{\mu\nu}$, and $c_{1,2,3,4}$ are four
dimensionless coupling constants.

Varying the action (1) with respect to $\lambda$, it follows that

$g_{\mu\nu}u^{\mu}u^{\nu}=-1\,.$ (4)

This constraint ensures the existence of a timelike unit vector field at any
spacetime points, so that there is a preferred threading responsible for the
breaking of Lorentz invariance. Varying Eq. (1) with respect to $u^{\mu}$, we
obtain

$\nabla_{\mu}{J^{\mu}}_{\alpha}+\lambda
u_{\alpha}+c_{4}u^{\beta}\nabla_{\beta}u^{\mu}\nabla_{\alpha}u_{\mu}=0\,,$ (5)

where

$\displaystyle{J^{\mu}}_{\alpha}$ $\displaystyle:=$
$\displaystyle{M^{\mu\nu}}_{\alpha\beta}\nabla_{\nu}u^{\beta}\,.$ (6)

Multiplying Eq. (5) by $u^{\alpha}$ and using Eq. (4), the Lagrange multiplier
can be expressed as

$\lambda=u^{\alpha}\nabla_{\mu}{J^{\mu}}_{\alpha}+c_{4}(u^{\beta}\nabla_{\beta}u^{\mu})(u^{\rho}\nabla_{\rho}u_{\mu})\,.$
(7)

The gravitational field equations derived by the variation of (1) with respect
to $g_{\mu\nu}$ are

$\displaystyle G_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\left[u_{(\alpha}{J^{\mu}}_{\beta)}+u^{\mu}J_{(\alpha\beta)}-u_{(\alpha}{J_{\beta)}}^{\mu}\right]$
(8)
$\displaystyle+c_{1}\left(\nabla_{\alpha}u^{\nu}\nabla_{\beta}u_{\nu}-\nabla^{\nu}u_{\alpha}\nabla_{\nu}u_{\beta}\right)$
$\displaystyle+c_{4}(u^{\rho}\nabla_{\rho}u_{\alpha})(u^{\nu}\nabla_{\nu}u_{\beta})$
$\displaystyle+\frac{1}{2}g_{\alpha\beta}{\cal L}_{\ae}+\lambda
u_{\alpha}u_{\beta},$

where $G_{\alpha\beta}$ is the Einstein tensor.

In general, the theory contains three different species of propagating degrees
of freedoms, i.e., spin-0 (scalar), spin-1 (vector), and spin-2 (tensor)
modes. On the Minkowski background with the line element

${\rm d}s^{2}=\eta_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-{\rm
d}t^{2}+\delta_{ij}{\rm d}x^{i}{\rm d}x^{j}\,,$ (9)

the Aether field is aligned along the $t$ direction, as
$u^{\mu}=\delta^{\mu}_{0}$. According to the perturbative analysis on the
background (9), the squared propagation speeds of spin-0, spin-1, and spin-2
modes are given, respectively, by [18, 21]

$\displaystyle c_{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{c_{123}(2-c_{14})}{c_{14}(1-c_{13})(2+c_{13}+3c_{2})}\,,$
(10) $\displaystyle c_{V}^{2}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}\,,$ (11)
$\displaystyle c_{T}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{1-c_{13}}\,,$ (12)

where $c_{ij}:=c_{i}+c_{j}$ and $c_{ijk}:=c_{i}+c_{j}+c_{k}$. The coefficients
of the kinetic terms for each mode are

$\displaystyle q_{S}$ $\displaystyle=$
$\displaystyle\frac{(1-c_{13})(2+c_{13}+3c_{2})}{c_{123}}\,,$ (13)
$\displaystyle q_{V}$ $\displaystyle=$ $\displaystyle c_{14}\,,$ (14)
$\displaystyle q_{T}$ $\displaystyle=$ $\displaystyle 1-c_{13}\,.$ (15)

So long as the denominators in Eqs. (10)-(13) do not vanish, there are one
scalar, two vector, and two tensor propagating degrees of freedom in general.

If we require that the theory: (i) be self-consistent, such as free of ghosts
and Laplacian instabilities; and (ii) be compatible with all the experimental
and observational constraints obtained so far, it was found that the coupling
constants must satisfy the following conditions [21]

$\displaystyle\left|c_{13}\right|\lesssim 10^{-15}\,,$ (19) $\displaystyle
0<c_{14}\leq 2.5\times 10^{-5}\,,$ $\displaystyle c_{14}\leq c_{2}\leq
0.095\,,$ $\displaystyle c_{4}\leq 0\,.$

It should be noted that the recent studies of the neutron star binary systems
showed that one of the parameterized post-Newtonian parameters,
$\alpha_{1}=-4c_{14}$, is further restricted to $|\alpha_{1}|<10^{-5}$ [28].
This translates to the limit

$0\lesssim c_{14}\lesssim 2.5\times 10^{-6},$ (20)

which is stronger than the bound derived from lunar laser ranging experiments
by one order of magnitude [21].

##  III Disformal transformation and Aether-orthogonal frame

###  III.1 Disformal transformation

Under a redefinition of the metric accompanied with the Aether field in the
form $\tilde{g}_{\mu\nu}=g_{\mu\nu}+Bu_{\mu}u_{\nu}$, where $B$ is a constant,
the structure of the action (1) is preserved with a change of the coupling
constants $\tilde{c}_{1,2,3,4}$ in the transformed frame [69]. This
redefinition stretches the metric tensor in the Aether direction by a factor
$1-B$. On choosing $B=1-c_{I}^{2}$ for the Minkowski metric
$g_{\mu\nu}=\eta_{\mu\nu}$, where the subscript $I$ is either $S,V,T$ with the
squared propagation speeds $c_{I}^{2}$ given by Eqs. (10)-(12), it is possible
to transform to a metric frame $\tilde{g}_{\mu\nu}$ in which one of the speeds
is equivalent to 1 [32, 70].

One can perform a more general disformal transformation [71] of the form

$\bar{g}_{\mu\nu}=\Omega^{2}\left(g_{\mu\nu}+Bu_{\mu}u_{\nu}\right)\,,$ (21)

where the conformal factor $\Omega$ and the disformal factor $B$ are
constants, and the Aether field $u_{\mu}$ satisfies the unit-vector constraint
(4). The corresponding inverse metric and determinant are

$\displaystyle\bar{g}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{\Omega^{2}}\left(g^{\mu\nu}-\frac{B}{1-B}u^{\mu}u^{\nu}\right),$
$\displaystyle\sqrt{-\bar{g}}$ $\displaystyle=$
$\displaystyle\Omega^{4}\sqrt{-(1-B)g}\,,$ (22)

where we are assuming that $1-B>0$. The Aether field has a different (but
constant) norm with respect to $\bar{g}^{\mu\nu}$ as

$\bar{g}^{\mu\nu}u_{\mu}u_{\nu}=-\frac{1}{\Omega^{2}(1-B)}\,.$ (23)

Hence, it makes sense to define

$\bar{u}_{\mu}=\Omega\sqrt{1-B}\,u_{\mu}\,,\qquad\bar{u}^{\mu}=\bar{g}^{\mu\nu}\bar{u}_{\nu}\,.$
(24)

The first covariant derivative of the Aether field with respect to
$\bar{g}_{\mu\nu}$ is given by [72]

$\bar{\nabla}_{\mu}u_{\nu}=\nabla_{\mu}u_{\nu}-B\delta\Gamma^{\rho}_{\mu\nu}u_{\rho}\,,$
(25)

where

$\displaystyle\delta\Gamma^{\rho}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
u_{(\mu}F_{\nu)}^{\ \
\rho}+\frac{u^{\rho}}{1-B}\left[\nabla_{(\mu}u_{\nu)}+Bu_{\lambda}u_{(\mu}\nabla^{\lambda}u_{\nu)}\right]\,,$
$\displaystyle\quad F_{\mu\nu}$ $\displaystyle=$
$\displaystyle\nabla_{\mu}u_{\nu}-\nabla_{\nu}u_{\mu}\,.$ (26)

The Einstein-Hilbert action transforms as [72]

$\displaystyle\int{\rm d}^{4}x\sqrt{-\bar{g}}\,\bar{R}=$
$\displaystyle\int{\rm
d}^{4}x\sqrt{-g}\left[\Omega^{2}\sqrt{1-B}R-\frac{\Omega^{2}B}{\sqrt{1-B}}\left\\{(\nabla_{\mu}u^{\mu})^{2}-\nabla_{\rho}u^{\sigma}\nabla_{\sigma}u^{\rho}\right\\}\right.$
$\displaystyle\left.+\frac{\Omega^{2}B^{2}}{2\sqrt{1-B}}\left(u^{\mu}u^{\nu}F_{\mu\rho}F_{\nu}^{\
\rho}+\frac{1}{2}F_{\mu\nu}F^{\mu\nu}\right)\right]\,.$ (27)

On using these relations, in the absence of matter fields, Einstein-Aether
theory for the combination ($g_{\mu\nu}$, $u_{\mu}$) with the constant
parameters ($G_{\ae}$, $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$) is equivalent to
Einstein-Aether theory for the combination ($\bar{g}_{\mu\nu}$,
$\bar{u}_{\mu}$) with a different set of parameters ($\bar{G}_{\ae}$,
$\bar{c}_{1}$, $\bar{c}_{2}$, $\bar{c}_{3}$, $\bar{c}_{4}$), where [69]

$\displaystyle\bar{G}_{\ae}=\Omega^{2}\sqrt{1-B}\,{G}_{\ae}\,,\quad\bar{c}_{14}=c_{14},$
$\displaystyle\bar{c}_{123}=(1-B)c_{123}\,,\quad\bar{c}_{13}-1=(1-B)(c_{13}-1)\,,$
$\displaystyle\bar{c}_{1}-\bar{c}_{3}-1=\frac{1}{1-B}\left(c_{1}-c_{3}-1\right)\,.$
(28)

More explicitly, the coefficients ${\bar{c}}_{i}$ are related to $c_{i}$, as

$\displaystyle\bar{c}_{1}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-2(c_{1}+c_{3})B+(c_{1}+c_{3}-1)B^{2}}{2(1-B)}\,,$
(29) $\displaystyle\bar{c}_{2}$ $\displaystyle=$ $\displaystyle
c_{2}(1-B)-B\,,$ (30) $\displaystyle\bar{c}_{3}$ $\displaystyle=$
$\displaystyle\frac{2c_{3}-(c_{1}+c_{3}-1)B(2-B)}{2(1-B)}\,,$ (31)
$\displaystyle\bar{c}_{4}$ $\displaystyle=$
$\displaystyle\frac{2c_{4}+2(c_{3}-c_{4})B-(c_{1}+c_{3}-1)B^{2}}{2(1-B)}\,.$
(32)

We consider the Minkowski background characterized by the metric tensor
$g_{\mu\nu}=\eta_{\mu\nu}$ and perform the disformal transformation (21). Upon
choosing

$B=1-c_{I}^{2}\,,\quad{\rm where}\quad I=S,V,T\,,$ (33)

and using Eqs. (10)-(15) and Eqs. (29)-(32), the squared propagation speeds
$\bar{c}_{I}^{2}$ in the frame ($\bar{g}_{\mu\nu}$, $\bar{u}_{\mu}$) yield

$\bar{c}_{I}^{2}=1\,,$ (34)

for each subscript $I=S,V,T$. Furthermore, the coefficients of the time
kinetic terms yield

$\bar{Q}_{I}=c_{I}^{2}Q_{I}\,,$ (35)

where222For the vector perturbation, the no-ghost condition changes from
$q_{V}>0$ to $Q_{V}>0$ if one swaps the roles of the dynamical variable and
its canonical momentum by a canonical transformation. See e.g., Appendix B of
[73] or/and Section IV of [74] for a technique to perform canonical
transformations at the level of the Lagrangian.

$Q_{S}\equiv q_{S}\,,\qquad Q_{V}\equiv\frac{q_{V}}{c_{V}^{2}}\,,\qquad
Q_{T}\equiv q_{T}\,.$ (36)

Here, we have assumed

$c_{I}^{2}>0\,,$ (37)

in order to avoid the gradient instability of perturbations. Under this
assumption, the inequality $1-B>0$ holds and thus the disformal transformation
does not change the Lorentzian signature of the metric.

###  III.2 Aether-orthogonal frame

If the background Aether field has zero vorticity, which is the case for any
spherically symmetric configurations, one can locally choose the time
coordinate $\phi$ such that

$\left.u_{\mu}\right|_{\rm
background}=-\eta\partial_{\mu}\phi=-\eta\delta^{\phi}_{\mu}\,,$ (38)

where $\eta$ is a nonvanishing function. This choice of the time coordinate
$\phi$ for the metric frame $g_{\mu\nu}$ is called the Aether-orthogonal
frame. The unit-vector constraint (4) gives $\left.g^{\phi\phi}\right|_{\rm
background}=-\eta^{-2}$ and $\left.u^{\phi}\right|_{\rm
background}=\eta^{-1}$.

At each point of physical interest, one can choose a local Lorentz frame and
then perform a local Lorentz transformation so that the metric and Aether
field are of the form

$g_{\mu\nu}|_{\rm local}=-\eta^{2}{\rm d}\phi^{2}+\delta_{ij}{\rm d}x^{i}{\rm
d}x^{j}\,,\quad\quad u_{\mu}|_{\rm local}=-\eta\delta_{\mu}^{\phi}\,.$ (39)

At leading order in the geometrical optics approximation, i.e., for modes
whose wavelengths are much shorter than the time and length scales of the
background, the background in the vicinity of the point of interest can be
approximated by Eq. (39). In particular, in the vicinity of the point of
interest, one can decompose the perturbations into spin-$0$ (scalar, $I=S$),
spin-$1$ (vector, $I=V$) and spin-$2$ (tensor, $I=T$) modes. Let us consider
the perturbations $\delta\chi_{I}$ corresponding to the $I$-excitation
($I=S,V,T$). Then, by definition of $Q_{I}$ and $c_{I}^{2}$, the kinetic and
gradient terms of $\delta\chi_{I}$ should locally have the following structure

$L_{{\rm
kin},I}=\frac{1}{2}C_{I}Q_{I}\left[(\eta^{-1}\partial_{\phi}\delta\chi_{I})^{2}-c_{I}^{2}\delta^{ij}\partial_{i}\delta\chi_{I}\partial_{j}\delta\chi_{I}\right]+\cdots\,,$
(40)

where $C_{I}$ are positive definite coefficients, which should not be confused
with $C_{1},C_{2},\cdots$ introduced later in Sec. IV, and $\cdots$ represents
higher-order terms in the geometrical optics approximation. By undoing the
local Lorentz transformation and going back to the original coordinate system
before choosing the local Lorentz frame, the leading kinetic and gradient
terms (40) can be written in a general coordinate system as

$L_{{\rm
kin},I}=-\frac{1}{2}C_{I}\bar{Q}_{I}\bar{g}_{I}^{\mu\nu}\partial_{\mu}\delta\chi_{I}\partial_{\nu}\delta\chi_{I}+\cdots\,,$
(41)

at leading order in the geometrical optics approximation, where $\cdots$ again
represents higher-order terms in the geometrical optics approximation. We have
assumed that the time and spatial scales involved in the coordinate
transformation between the original coordinate system and the local Lorentz
frame in the vicinity of the point of interest are sufficiently longer than
the wavelengths of the modes of interest.

The local structure (41) clearly states that the perturbations
$\delta\chi_{I}$ in the Aether-orthogonal frame do not behave as ghosts so
long as $Q_{I}>0$. Indeed, if we choose the time coordinate $\phi$ in the
Aether-orthogonal frame, then the leading time kinetic term is

$\displaystyle L_{{\rm kin},I}$ $\displaystyle\ni$
$\displaystyle-\frac{1}{2}C_{I}\bar{Q}_{I}\bar{g}_{I}^{\phi\phi}(\partial_{\phi}\delta\chi_{I})^{2}+\cdots$
(42) $\displaystyle=$
$\displaystyle\frac{C_{I}c_{I}^{2}Q_{I}}{2\Omega_{I}^{2}\eta^{2}}(\partial_{\phi}\delta\chi_{I})^{2}+\cdots\,,$

which is positive.

Rigorously speaking, we have only given a heuristic argument for the local
structure (41) without a proof, which requires analysis similar to that in
Ref. [75]. In the rest of the present paper, we shall show that the kinetic
and gradient terms for odd-parity perturbations (including $I=V$ and $I=T$
modes) around spherically symmetric BHs indeed have the local structure (41).
The same analysis for even-parity perturbations (including $I=S,V,T$ modes)
will be left for a future work.

###  III.3 Spherically symmetric background and apparent instabilities

Let us consider the SSS background given by the line element

${\rm d}s^{2}=g_{\mu\nu}{\rm d}x^{\mu}{\rm d}x^{\nu}=-f(r){\rm
d}t^{2}+\frac{{\rm d}r^{2}}{h(r)}+r^{2}\Omega_{pq}{\rm d}\vartheta^{p}{\rm
d}\vartheta^{q}\,,$ (43)

together with the Aether-field configuration

$u^{\mu}\partial_{\mu}=a(r)\partial_{t}+b(r)\partial_{r}\,,$ (44)

where $f,h,a,b$ are functions of $r$. In Eq. (43) the angular part contains
two angles $\theta$ and $\varphi$, such that $\Omega_{pq}{\rm
d}\vartheta^{p}{\rm d}\vartheta^{q}={\rm d}\theta^{2}+\sin^{2}\theta\,{\rm
d}\varphi^{2}$.

The unit-vector constraint (4) gives the following relation

$b=\epsilon\sqrt{(a^{2}f-1)h}\,,$ (45)

where $\epsilon=\pm 1$. The existence of the Aether-field profile (45) with
$b\neq 0$ requires that $(a^{2}f-1)h>0$.

A metric (Killing) horizon is defined at the radius $r=r_{\rm g}$ at which the
time translation Killing vector $\zeta^{\mu}=\delta^{\mu}_{t}$ is null, i.e.,
$\zeta^{\mu}\zeta_{\mu}=0$. This translates to the condition

$f(r_{\rm g})=0\,.$ (46)

So long as $h$ also vanishes on the metric horizon, the product $a^{2}fh$ is
at least a positive constant at $r=r_{\rm g}$ to satisfy the condition
$(a^{2}f-1)h>0$. This means that, as $r\to r_{\rm g}^{+}$, the temporal
Aether-field component diverges as $a^{2}\propto(fh)^{-1}$ for the coordinate
system (43).

The leading-order kinetic and gradient terms of perturbations $\delta\chi_{I}$
(where $I=S,V,T$) for the effective metric
$\bar{g}^{I}_{\mu\nu}=\Omega_{I}^{2}[g_{\mu\nu}+(1-c_{I}^{2})u_{\mu}u_{\nu}]$
(under the geometric optics approximation) are

$\displaystyle L_{{\rm kin},I}$
$\displaystyle=-\frac{1}{2}C_{I}\bar{Q}_{I}\bar{g}_{I}^{\mu\nu}\partial_{\mu}\delta\chi_{I}\partial_{\nu}\delta\chi_{I}=\frac{C_{I}\bar{Q}_{I}}{2\Omega_{I}^{2}}\left[\frac{1}{f}(\partial_{t}\delta\chi_{I})^{2}-h(\partial_{r}\delta\chi_{I})^{2}-\frac{1}{r^{2}}\Omega^{pq}\partial_{p}\delta\chi_{I}\partial_{q}\delta\chi_{I}+\frac{1-c_{I}^{2}}{c_{I}^{2}}\left(a\partial_{t}\delta\chi_{I}+b\partial_{r}\delta\chi_{I}\right)^{2}\right]$
$\displaystyle\ni\frac{1}{2}fC_{I}Q_{I}^{\rm
apparent}\left[\frac{1}{f}(\partial_{t}\delta\chi_{I})^{2}-(c_{I,\Omega}^{\rm
apparent})^{2}\frac{1}{r^{2}}\Omega^{pq}\partial_{p}\delta\chi_{I}\partial_{q}\delta\chi_{I}\right]\,,$
(47)

where

$\displaystyle Q_{I}^{\rm apparent}$ $\displaystyle=$
$\displaystyle\frac{\bar{Q}_{I}}{\Omega_{I}^{2}}\left(\frac{1}{f}+\frac{1-c_{I}^{2}}{c_{I}^{2}}a^{2}\right)\,,$
(48) $\displaystyle\left(c_{I,\Omega}^{\rm apparent}\right)^{2}$
$\displaystyle=$
$\displaystyle\left(1+\frac{1-c_{I}^{2}}{c_{I}^{2}}a^{2}f\right)^{-1}\,.$ (49)

Therefore, the coefficient $Q_{I}^{\rm apparent}$ of the kinetic term with
respect to the Killing time $t$ may become negative, despite the fact that the
kinetic term with respect to the time coordinate in the Aether-orthogonal
frame is always positive as in Eq. (42). Also, the apparent angular sound
speed squared $(c_{I,\Omega}^{\rm apparent})^{2}$ would have nontrivial
position dependence, although the propagation speed squared $c_{I}^{2}$
relative to the Aether-orthogonal frame is a constant given by the theory.

These behaviors are due to the deviation of the Killing time slicing from the
Aether-orthogonal frame. The sound cones are not only narrowed or widened but
also tilted relative to the Killing time slicing333See Ref. [76] for similar
behaviors of sound cones for open string modes in the context of inhomogeneous
tachyon condensation in string theory.. On the other hand, the sound cones are
not tilted relative to the Aether-orthogonal frame.

Near the metric horizon $r=r_{\rm g}$, we have the following expansions

$f=f_{1}(r-r_{\rm g})+\cdots\,,\quad a=-\frac{1}{2f_{1}\alpha_{0}}(r-r_{\rm
g})^{-1}+\cdots\,,$ (50)

where $f_{1}$ and $\alpha_{0}$ are constants [62]. Note that $\alpha_{0}$ is
the value of $\alpha$ at $r=r_{\rm g}$, where $\alpha$ is defined later in Eq.
(56). In this regime, the quantities (48) and (49) can be estimated as444For
$I=V$ and $c_{13}=0$, the result is to be compared with (5.32) and (5.33) of
Ref. [62] up to an arbitrary positive overall factor for $q_{V}^{\rm
apparent}$.

$\displaystyle Q_{I}^{\rm apparent}$
$\displaystyle=\frac{\bar{Q}_{I}}{4f_{1}^{2}\alpha_{0}^{2}\Omega_{I}^{2}}\frac{1-c_{I}^{2}}{c_{I}^{2}}(r-r_{\rm
g})^{-2}+\cdots\,,$ (51) $\displaystyle(c_{I,\Omega}^{\rm apparent})^{2}$
$\displaystyle=4f_{1}\alpha_{0}^{2}\frac{c_{I}^{2}}{1-c_{I}^{2}}(r-r_{\rm
g})+\cdots\,.$ (52)

Therefore, if $c_{I}^{2}>1$, the apparent no-ghost condition is violated near
the horizon. Moreover, the apparent angular sound speed squared vanishes at
the horizon. Let us stress again that the kinetic term with respect to the
time coordinate in the Aether-orthogonal frame is always positive for
$Q_{I}>0$ and that the propagation speed squared $c_{I}^{2}$ relative to the
Aether-orthogonal frame is constant given by the theory. In Sec. IV, we will
address the linear stability of BHs in Einstein-Aether theory by choosing the
Aether-orthogonal frame as a timelike coordinate.

###  III.4 Aether-orthogonal frame on the spherically symmetric background

The SSS background is described by the line element (43) with the Aether-field
profile (44). Alternatively, we can choose the following Eddington-Finkelstein
coordinate [48]

${\rm d}s^{2}=-f(r){\rm d}v^{2}+2B(r){\rm d}v{\rm d}r+r^{2}\Omega_{pq}{\rm
d}\vartheta^{p}{\rm d}\vartheta^{q}\,,$ (53)

which is related to the coordinate (43) as

${\rm d}v={\rm d}t+\frac{{\rm d}r}{\sqrt{fh}}\,,\qquad
B(r)=\sqrt{\frac{f}{h}}\,.$ (54)

The transformation to the coordinate (53) is valid in the regime $f/h>0$,
i.e., for the same signs of $f$ and $h$. On this background, we take the
Aether-field configuration

$u^{\mu}\partial_{\mu}=-\alpha(r)\partial_{v}-\beta(r)\partial_{r}\,.$ (55)

The $r$-dependent functions $\alpha$ and $\beta$ are related to $a$ and $b$ in
Eq. (44), as

$a+\frac{b}{\sqrt{fh}}=-\alpha\,,\qquad b=-\beta\,.$ (56)

From Eq. (45), we obtain the following relation

$a=-\frac{1+f\alpha^{2}}{2f\alpha}\,,\qquad
b=\sqrt{fh}\frac{1-f\alpha^{2}}{2f\alpha}\,.$ (57)

The sign of $b$ depends on that of $(1-f\alpha^{2})/(f\alpha)$. Then, the
Aether field $u^{\mu}$ has nonvanishing components

$u^{v}=-\alpha\,,\qquad u^{r}=\sqrt{fh}\frac{1-f\alpha^{2}}{2f\alpha}\,.$ (58)

Even though $a$ is divergent on the metric horizon ($r=r_{\rm g}$), the
quantity $\alpha$ can take a finite value $\alpha_{0}$ due to the relation
$fa=-1/(2\alpha_{0})$ at $r=r_{\rm g}$, see Eq. (50). On using the metric
components $g_{vv}=-f$, $g_{vr}=g_{rv}=B$, and $g_{rr}=0$, the nonvanishing
components of $u_{\mu}$ are $u_{v}=f\alpha-B\beta$ and $u_{r}=-B\alpha$, so
that

$u_{v}=\frac{1+f\alpha^{2}}{2\alpha}\,,\qquad
u_{r}=-\alpha\sqrt{\frac{f}{h}}\,.$ (59)

A vector field $s_{\mu}$ that is orthogonal to $u^{\mu}$ obeys the relation
$s_{\mu}u^{\mu}=0$. This has the following nonvanishing components

$s_{v}=\frac{1-f\alpha^{2}}{2\alpha}\,,\qquad
s_{r}=\alpha\sqrt{\frac{f}{h}}\,.$ (60)

Moreover, it satisfies the relation $s_{\mu}s^{\mu}=1$.

Now, we introduce the two coordinates $\phi$ and $\psi$, as

$\displaystyle{\rm d}\phi$ $\displaystyle=$ $\displaystyle{\rm
d}v+\frac{u_{r}}{u_{v}}{\rm d}r={\rm
d}v-\frac{2\alpha^{2}}{1+f\alpha^{2}}\sqrt{\frac{f}{h}}\,{\rm d}r\,,$ (61)
$\displaystyle{\rm d}\psi$ $\displaystyle=$ $\displaystyle-{\rm
d}v-\frac{s_{r}}{s_{v}}{\rm d}r=-{\rm
d}v-\frac{2\alpha^{2}}{1-f\alpha^{2}}\sqrt{\frac{f}{h}}\,{\rm d}r,$ (62)

which mean that $\phi$ is constant on a hypersurface orthogonal to $u_{\mu}$
and that $\psi$ is constant on a hypersurface orthogonal to $s_{\mu}$. Since
$\partial_{\mu}\phi=\delta^{v}_{\mu}-2\alpha^{2}/(1+f\alpha^{2})\sqrt{f/h}\,\delta^{r}_{u}$
from Eq. (61), the background Aether field can be expressed in the form

$u_{\mu}=\frac{1+f\alpha^{2}}{2\alpha}\partial_{\mu}\phi\,,$ (63)

and hence $\eta=-(1+f\alpha^{2})/(2\alpha)=fa$ in Eq. (38). Thus, the
coordinate system $(\phi,\psi)$ corresponds to the Aether-orthogonal frame in
which $\phi$ is the time measured by observers comoving with the Aether field.

Substituting the first of Eq. (54) into Eqs. (61) and (62), the relation
between the Aether-orthogonal frame and the coordinate (43) is given by [38]

$\displaystyle{\rm d}\phi$ $\displaystyle=$ $\displaystyle{\rm
d}t+\frac{1}{\sqrt{fh}}\frac{1-f\alpha^{2}}{1+f\alpha^{2}}{\rm d}r\,,$ (64)
$\displaystyle{\rm d}\psi$ $\displaystyle=$ $\displaystyle-{\rm
d}t-\frac{1}{\sqrt{fh}}\frac{1+f\alpha^{2}}{1-f\alpha^{2}}{\rm d}r\,.$ (65)

Solving these equations for ${\rm d}t$ and ${\rm d}r$ and substituting them
into Eq. (43), the line element is expressed as

${\rm d}s^{2}=-\frac{\left(1+f\alpha^{2}\right)^{2}}{4\alpha^{2}}{\rm
d}\phi^{2}+\frac{\left(1-f\alpha^{2}\right)^{2}}{4\alpha^{2}}{\rm
d}\psi^{2}+r^{2}\Omega_{pq}{\rm d}\vartheta^{p}{\rm d}\vartheta^{q}.$ (66)

Since $g_{\phi\phi}$ is always non-positive, $\phi$ is a timelike coordinate.
Similarly, $\psi$ is a spacelike coordinate due to the positivity of
$g_{\psi\psi}$. Note that a universal horizon is defined as the radius $r_{\rm
UH}$ at which the time translation Killing vector $\zeta^{\mu}$ is orthogonal
to $u_{\mu}$, i.e., $\zeta^{\mu}u_{\mu}=0$. This corresponds to the radius
satisfying

$\left(1+f\alpha^{2}\right)|_{r=r_{\text{UH}}}=0\,,$ (67)

at which the metric component $g_{\phi\phi}$ in Eq. (66) is vanishing. It is
clear that Eq. (67) has solutions only when $f<0$, that is, the universal
horizon always exists inside the metric horizon. For more details, see Ref.
[38].

From Eqs. (64) and (65), we have

$\displaystyle\frac{\partial\phi}{\partial
t}=1\,,\qquad\frac{\partial\phi}{\partial
r}=\frac{1}{\sqrt{fh}}\frac{1-f\alpha^{2}}{1+f\alpha^{2}}\,,$ (68)
$\displaystyle\frac{\partial\psi}{\partial
t}=-1\,,\qquad\frac{\partial\psi}{\partial
r}=-\frac{1}{\sqrt{fh}}\frac{1+f\alpha^{2}}{1-f\alpha^{2}}\,.$ (69)

Then, for a given function ${\cal F}$ of $t$ and $r$, the $t$ and $r$
derivatives of ${\cal F}$ are given, respectively, by

$\displaystyle\frac{\partial{\cal F}}{\partial t}={\cal F}_{,\phi}-{\cal
F}_{,\psi}\,,$ (70) $\displaystyle\frac{\partial{\cal F}}{\partial
r}=\frac{1}{\sqrt{fh}}\left(\frac{1-f\alpha^{2}}{1+f\alpha^{2}}{\cal
F}_{,\phi}-\frac{1+f\alpha^{2}}{1-f\alpha^{2}}{\cal F}_{,\psi}\right)\,,$ (71)

where ${\cal F}_{,\phi}:=\partial{\cal F}/\partial\phi$ and ${\cal
F}_{,\psi}:=\partial{\cal F}/\partial\psi$. The relations (70) and (71) will
be used in Sec. IV.

##  IV Odd-parity stability in the Aether-orthogonal frame

In this section, we study the linear stability of SSS BHs against odd-parity
perturbations in the Aether-orthogonal frame. The second-order action in the
odd-parity sector was derived in Ref. [62] for the line element (43). Since
the constant $t$ hypersurfaces are not always spacelike, the coordinate choice
(43) is not suitable for studying the linear stability of BHs. We will express
the second-order action of odd-parity perturbations by using the derivatives
with respect to $\phi$ and $\psi$. In the following, we will discuss the two
cases: (A) $l\geq 2$, and (B) $l=1$, in turn, where $l$’s are spherical
multipoles.

###  IV.1 $l\geq 2$

On the SSS background (43) with $\Omega_{pq}{\rm d}\vartheta^{p}{\rm
d}\vartheta^{q}={\rm d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}$, metric
perturbations $h_{\mu\nu}$ can be separated into odd-parity (axial) and even-
parity (polar) sectors [52, 53]. We express $h_{\mu\nu}$ in terms of the
spherical harmonics $Y_{lm}(\theta,\varphi)$. We will focus on axial
perturbations with the parity $(-1)^{l+1}$. We choose the gauge in which the
components $h_{ab}$ vanish, where the subscripts $a$ and $b$ denote either
$\theta$ or $\varphi$. Then, the nonvanishing components of odd-parity metric
perturbations are given by

$\displaystyle h_{ta}$ $\displaystyle=$
$\displaystyle\sum_{l,m}Q_{lm}(t,r)E_{ab}\nabla^{b}Y_{lm}(\theta,\varphi)\,,$
(72) $\displaystyle h_{ra}$ $\displaystyle=$
$\displaystyle\sum_{l,m}W_{lm}(t,r)E_{ab}\nabla^{b}Y_{lm}(\theta,\varphi)\,,$
(73)

where $Q_{lm}$ and $W_{lm}$ are functions of $t$ and $r$. The tensor $E_{ab}$
is antisymmetric with nonvanishing components
$E_{\theta\varphi}=-E_{\varphi\theta}=\sin\theta$.

In the odd-parity sector, the Aether field has the following components

$\displaystyle u_{t}=-a(r)f(r)\,,\qquad u_{r}=\frac{b(r)}{h(r)}\,,$
$\displaystyle u_{a}=\sum_{l,m}\delta
u_{lm}(t,r)E_{ab}\nabla^{b}Y_{lm}(\theta,\varphi)\,,$ (74)

where $\delta u_{lm}$ is a function of $t$ and $r$.

In the following, we will set $m=0$ without loss of generality. We also omit
the subscripts $l$ and $m$ from the perturbations $Q_{lm}$, $W_{lm}$, and
$\delta u_{lm}$. Expanding the action (1) up to quadratic order and
integrating it with respect to $\theta$ and $\varphi$, we obtain the second-
order action containing the fields $Q$, $W$, $\delta u$ and their $t,r$
derivatives. The dynamical field associated with the gravitational (tensor)
perturbation is given by [62]

$\chi:=\dot{W}-Q^{\prime}+\frac{2}{r}Q+\frac{C_{2}\dot{\delta u}+C_{3}\delta
u^{\prime}+C_{4}\delta u}{C_{1}}\,,$ (75)

where a dot and prime represents the derivative with respect to $t$ and $r$,
respectively, and

$\displaystyle C_{1}=\frac{(1-c_{13})h}{2r^{2}f}\,,\quad
C_{2}=-\frac{c_{13}b}{2r^{2}f}\,,\quad C_{3}=-\frac{c_{13}ah}{2r^{2}},$
$\displaystyle
C_{4}=\frac{[(2c_{14}-c_{13})(fa^{\prime}+af^{\prime})r+2c_{13}af]h}{2r^{3}f}\,.$
(76)

Taking into account the field $\chi$ as a form of the Lagrange multiplier and
varying the corresponding second-order action with respect to $W$ and $Q$, we
can eliminate $W$, $Q$, and their derivatives from the action. Then, the
resulting quadratic-order action can be expressed in the form [62]

${\cal S}_{\rm odd}=\sum_{l}L\int{\rm d}t{\rm d}r\,{\cal L}_{\rm odd}\,,$ (77)

where

$L=l(l+1)\,,$ (78)

and

$\displaystyle{\cal L}_{\rm odd}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{16\pi
G_{\ae}}\sqrt{\frac{f}{h}}\;\;(\dot{\vec{\mathcal{X}}}^{t}{\bm{K}}\dot{\vec{\mathcal{X}}}+\dot{\vec{\mathcal{X}}}^{t}{\bm{R}}\vec{\mathcal{X}}^{\prime}$
(79) $\displaystyle\qquad\qquad\qquad+\vec{\mathcal{X}}^{\prime
t}{\bm{G}}\vec{\mathcal{X}}^{\prime}+\vec{\mathcal{X}}^{t}{\bm{M}}\vec{\mathcal{X}})\,.$

The vector field $\vec{\mathcal{X}}$ is given by

$\vec{\mathcal{X}}=\left(\begin{array}[]{c}\chi\\\ \delta
u\end{array}\right),\quad\vec{\mathcal{X}}^{t}=\left(\chi,\delta u\right)\,,$
(80)

where $\chi$ and $\delta u$ are the dynamical perturbations arising from the
gravitational and Aether sectors, respectively. ${\bm{K}}$, ${\bm{R}}$,
${\bm{G}}$, and ${\bm{M}}$ are the $2\times 2$ symmetric and real matrices,
among which only ${\bm{M}}$ has off-diagonal components. Nonvanishing
components of these matrices are

$\displaystyle K_{11}$ $\displaystyle=$
$\displaystyle\frac{4C_{1}^{2}C_{10}}{(L-2)(a^{2}C_{9}^{2}-4C_{8}C_{10})},$
$\displaystyle K_{22}$ $\displaystyle=$
$\displaystyle\frac{C_{1}C_{5}-C_{2}^{2}}{C_{1}}\,,$ $\displaystyle R_{11}$
$\displaystyle=$ $\displaystyle-\frac{aC_{9}}{C_{10}}K_{11}\,,\qquad
R_{22}=\frac{C_{1}C_{6}-2C_{2}C_{3}}{C_{1}}\,,$ $\displaystyle G_{11}$
$\displaystyle=$ $\displaystyle\frac{C_{8}}{C_{10}}K_{11}\,,\qquad
G_{22}=\frac{C_{1}C_{7}-C_{3}^{2}}{C_{1}}\,,$ $\displaystyle M_{11}$
$\displaystyle=$ $\displaystyle-C_{1}\,,$ $\displaystyle M_{22}$
$\displaystyle=$ $\displaystyle
LC_{12}+\frac{L[C_{8}C_{11}^{2}+C_{9}^{2}(C_{10}+aC_{11})]}{a^{2}C_{9}^{2}-4C_{8}C_{10}}\,,$
(81)

where the explicit form of $M_{12}~{}(=M_{21})$ is not shown here, and

$\displaystyle C_{5}$ $\displaystyle=$
$\displaystyle\frac{c_{1}+c_{4}a^{2}f}{r^{2}f},\qquad
C_{6}=\frac{2c_{4}ab}{r^{2}}\,,$ $\displaystyle C_{7}$ $\displaystyle=$
$\displaystyle\frac{[c_{4}(a^{2}f-1)-c_{1}]h}{r^{2}}\,,$ $\displaystyle C_{8}$
$\displaystyle=$ $\displaystyle-\frac{[c_{13}(a^{2}f-1)+1]h}{2r^{4}}\,,$
$\displaystyle C_{9}$ $\displaystyle=$
$\displaystyle\frac{c_{13}b}{r^{4}}\,,\qquad
C_{10}=\frac{1-c_{13}a^{2}f}{2r^{4}f},$ $\displaystyle C_{11}$
$\displaystyle=$ $\displaystyle\frac{c_{13}a}{r^{4}}\,,\qquad
C_{12}=-\frac{c_{1}}{r^{4}}\,.$ (82)

Since $M_{12}$ does not depend on $L$, it does not affect the angular
propagation speeds in the large $l$ limit (which will be discussed below).

In Ref. [62], the linear stability conditions of BHs against odd-parity
perturbations were derived by using the coordinates $t$ and $r$. As we showed
in Sec. III.3, unless a proper coordinate is chosen, we may encounter
artificial ghosts or Lagrangian instabilities in theories with superluminal
propagation. To overcome this problem, we use the Aether-orthogonal frame
introduced in Sec. III.4, where $\phi$ defines the causality and chronology:
all particles must move along the increasing direction of $\phi$ [51, 77]. As
a result, the future light cone defined by each particle with any given speed
lies to the future of spacelike hypersurfaces ($\phi={\rm constant}$), as
explained explicitly in Ref. [75].

We transform the action (77) to that in the coordinate system (66). We convert
the $t$ and $r$ derivatives of $\chi$ and $\delta u$ to their $\phi$ and
$\psi$ derivatives by exploiting the relations (70) and (71). We also use Eq.
(57) to express $a$ and $b$ with respect to $\alpha$. Then, the second-order
action of odd-parity perturbations yields

${\cal S}_{\rm odd}=\sum_{l}L\int{\rm d}\phi{\rm d}\psi\,\hat{{\cal L}}_{\rm
odd}\,,$ (83)

where

$\hat{{\cal L}}_{\rm odd}=\frac{1}{16\pi
G_{\ae}}\sqrt{\frac{h}{f}}(\vec{\mathcal{X}}^{t}_{,\phi}\hat{{\bm{K}}}\vec{\mathcal{X}}_{,\phi}+\vec{\mathcal{X}}^{t}_{,\psi}\hat{{\bm{G}}}\vec{\mathcal{X}}_{,\psi}+\vec{\mathcal{X}}^{t}\hat{{\bm{M}}}\vec{\mathcal{X}})\,.$
(84)

Nonvanishing components of the $2\times 2$ matrices $\hat{{\bm{K}}}$,
$\hat{{\bm{G}}}$, and $\hat{{\bm{M}}}$ are given by

$\displaystyle\hat{K}_{11}$ $\displaystyle=$
$\displaystyle\frac{2(1-c_{13})^{2}\alpha^{2}r^{2}}{(L-2)(1+f\alpha^{2})^{2}}\,,$
(85) $\displaystyle\hat{K}_{22}$ $\displaystyle=$
$\displaystyle\frac{4c_{14}\alpha^{2}}{(1+f\alpha^{2})^{2}}\frac{f}{h}\,,$
(86) $\displaystyle\hat{G}_{11}$ $\displaystyle=$
$\displaystyle-\frac{(1+f\alpha^{2})^{2}}{(1-f\alpha^{2})^{2}}c_{T}^{2}\hat{K}_{11}\,,$
(87) $\displaystyle\hat{G}_{22}$ $\displaystyle=$
$\displaystyle-\frac{(1+f\alpha^{2})^{2}}{(1-f\alpha^{2})^{2}}c_{V}^{2}\hat{K}_{22}\,,$
(88) $\displaystyle\hat{M}_{11}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(1-c_{13}\right)\,,$ (89)
$\displaystyle\hat{M}_{22}$ $\displaystyle=$
$\displaystyle-L\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2(1-c_{13})r^{2}}\frac{f}{h}\,,$
(90)

besides the off-diagonal components $\hat{M}_{12}=\hat{M}_{21}$ (which are of
order $L^{0}$). The quantities $c_{T}^{2}$ and $c_{V}^{2}$ are defined,
respectively, by Eqs. (12) and (11). We recall that the Lagrangian (79)
possesses products of the $t$ and $r$ derivatives, but the Lagrangian (84)
does not contain products of the $\phi$ and $\psi$ derivatives.

The absence of ghosts for dynamical perturbations $\chi$ and $\delta u$
requires the two conditions $\hat{K}_{11}>0$ and $\hat{K}_{22}>0$. The former
is satisfied except for $c_{13}=1$ (in which case $\hat{K}_{11}=0$). The
second holds under the inequality

$c_{14}>0\,.$ (91)

This is equivalent to the no-ghost condition of vector perturbations on the
Minkowski background [18, 21]. On the other hand, it does not coincide with
the no-ghost condition derived on the SSS background (43) with the coordinates
$t$ and $r$ [62]. The latter coordinate choice is not suitable for discussing
the linear stability of BHs, since the constant $t$ hypersurfaces are not
always spacelike.

To study the propagation of small-scale perturbations with large angular
frequencies $\omega$ and momenta $k$, we assume the solutions to the
perturbation equations for $\chi$ and $\delta u$ in the form

$\vec{\mathcal{X}}=\vec{\mathcal{X}}_{0}e^{-i(\omega\phi-k\psi)}\,,$ (92)

with $\vec{\mathcal{X}}_{0}=(\chi_{0},\delta u_{0})$, where $\chi_{0}$ and
$\delta u_{0}$ are constants.

The radial propagation speeds can be known by considering the modes with
$\omega r_{\rm g}\approx kr_{\rm g}\gg l\gg 1$. In this regime, we substitute
Eq. (92) into the perturbation equations following from Eq. (84). This leads
to the two dispersion relations

$\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle-\frac{\hat{G}_{11}}{\hat{K}_{11}}k^{2}=\frac{(1+f\alpha^{2})^{2}}{(1-f\alpha^{2})^{2}}c_{T}^{2}k^{2}\,,$
(93) $\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle-\frac{\hat{G}_{22}}{\hat{K}_{22}}k^{2}=\frac{(1+f\alpha^{2})^{2}}{(1-f\alpha^{2})^{2}}c_{V}^{2}k^{2}\,,$
(94)

which correspond to those of $\chi$ and $\delta u$, respectively. Considering
a given point $P$, in the neighborhood of which we can always express the line
element (66) in the form

${\rm d}s^{2}=-{\rm d}\tilde{\phi}^{2}+{\rm
d}\tilde{\psi}^{2}+r^{2}\Omega_{pq}{\rm d}\vartheta^{p}{\rm
d}\vartheta^{q}\,,$ (95)

where

${\rm d}\tilde{\phi}^{2}=\frac{(1+f\alpha^{2})^{2}}{4\alpha^{2}}{\rm
d}\phi^{2}\,,\qquad{\rm
d}\tilde{\psi}^{2}=\frac{(1-f\alpha^{2})^{2}}{4\alpha^{2}}{\rm d}\psi^{2}\,.$
(96)

Note that $\tilde{\phi}$ corresponds to a proper time for this coordinate.
Then, the radial propagation speed squared yields

$c_{r}^{2}=\left(\frac{{\rm d}\tilde{\psi}}{{\rm
d}\tilde{\phi}}\right)^{2}=\frac{(1-f\alpha^{2})^{2}}{(1+f\alpha^{2})^{2}}\left(\frac{{\rm
d}\psi}{{\rm
d}\phi}\right)^{2}=\frac{(1-f\alpha^{2})^{2}}{(1+f\alpha^{2})^{2}}\frac{\omega^{2}}{k^{2}},$
(97)

where we used $({\rm d}\psi/{\rm d}\phi)^{2}=\omega^{2}/k^{2}$. Then, from
Eqs. (93) and (94), the radial squared propagation speeds of $\chi$ and
$\delta u$ are given, respectively, by

$\displaystyle c_{r1}^{2}=c_{T}^{2}=\frac{1}{1-c_{13}}\,,$ (98) $\displaystyle
c_{r2}^{2}=c_{V}^{2}=\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}\,.$
(99)

Thus, they are identical to the squared tensor and vector propagation speeds
on the Minkowski background, respectively. These values are different from
those derived on the SSS background (43) with the coordinates $t$ and $r$
[62]. To avoid the Laplacian instabilities along the radial direction, we
require the two conditions $c_{T}^{2}>0$ and $c_{V}^{2}>0$. Under the no-ghost
condition (91), they amount to the inequalities

$\displaystyle c_{13}<1\,,$ (100) $\displaystyle
2c_{1}-c_{13}(2c_{1}-c_{13})>0\,.$ (101)

Under the inequality (100), the other no-ghost condition $\hat{K}_{11}>0$ is
also satisfied.

To derive the angular propagation speeds, we consider the eikonal limit
$l\approx\omega r_{\rm g}\gg kr_{\rm g}\gg 1$. We substitute the solution (92)
into the perturbation equations of motion by noting that the off-diagonal
matrix components $\hat{M}_{12}=\hat{M}_{21}$ are of order $L^{0}$. Then, in
the eikonal limit, we obtain

$\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle-\frac{\hat{M}_{11}}{\hat{K}_{11}}=\frac{(L-2)(1+f\alpha^{2})^{2}}{4(1-c_{13})\alpha^{2}r^{2}}\,,$
(102) $\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle-\frac{\hat{M}_{22}}{\hat{K}_{22}}=\frac{L(1+f\alpha^{2})^{2}[2c_{1}-c_{13}(2c_{1}-c_{13})]}{8c_{14}(1-c_{13})\alpha^{2}r^{2}}\,,$
(103)

which correspond to the dispersion relations of $\chi$ and $\delta u$,
respectively. In terms of the proper time $\tilde{\phi}$ in the coordinate
(95), the propagation speed squared in the $\theta$ direction is given by

$\displaystyle c_{\Omega}^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{r{\rm d}\theta}{{\rm
d}\tilde{\phi}}\right)^{2}=\frac{4\alpha^{2}}{(1+f\alpha^{2})^{2}}\left(\frac{r{\rm
d}\theta}{{\rm d}\phi}\right)^{2}$ (104) $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}}{(1+f\alpha^{2})^{2}}\frac{r^{2}\omega^{2}}{l^{2}},$

where we used ${\rm d}\theta/{\rm d}\phi=\omega/l$. Substituting Eqs. (102)
and (103) into Eq. (104) and taking the limit $l\gg 1$, the squared
propagation speeds of $\chi$ and $\delta u$ are given, respectively, by

$\displaystyle c_{\Omega 1}^{2}=\frac{1}{1-c_{13}}=c_{T}^{2}\,,$ (105)
$\displaystyle c_{\Omega
2}^{2}=\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}=c_{V}^{2}\,.$
(106)

These values are equivalent to $c_{r1}^{2}$ and $c_{r2}^{2}$ derived in Eqs.
(98) and (99), respectively. Thus, the perturbations $\chi$ and $\delta u$
propagate with the same sound speeds as those in the Minkowski spacetime both
along the radial and angular directions. The linear stability of BHs is
ensured under the three conditions (91), (100), and (101).

###  IV.2 $l=1$

For the dipole mode ($l=1$ and $L=2$), the metric components $h_{ab}$ vanish
identically and hence there is a residual gauge degree of freedom to be fixed.
We choose the gauge $W=0$ and introduce the Lagrangian multiplier $\chi$ given
by Eq. (75). For the coordinate system (43), the second-order action is
expressed in the form ${\cal S}_{\rm odd}=\int{\rm d}t{\rm d}r\,{\cal L}_{\rm
odd}$, where

$\displaystyle{\cal L}_{\rm odd}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{8\pi
G_{\ae}}\sqrt{\frac{f}{h}}\biggl{[}C_{1}\biggl{\\{}2\chi\biggl{(}-Q^{\prime}+\frac{2Q}{r}+\frac{C_{2}\dot{\delta
u}+C_{3}\delta u^{\prime}+C_{4}\delta
u}{C_{1}}\biggr{)}-\chi^{2}\biggr{\\}}-\frac{(C_{2}\dot{\delta u}+C_{3}\delta
u^{\prime}+C_{4}\delta u)^{2}}{C_{1}}$ (107)
$\displaystyle\qquad\qquad\quad+C_{5}\dot{\delta u}^{2}+C_{6}\dot{\delta
u}\delta u^{\prime}+C_{7}\delta u^{\prime 2}+(2C_{12}+C_{13})\delta
u^{2}\biggr{]}\,,$

with

$\displaystyle C_{13}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{r^{2}}-\frac{c_{13}[(rh^{\prime}+2h-2)f+rhf^{\prime}]}{2r^{4}f}$
(108) $\displaystyle-\frac{2c_{4}(fa^{\prime}+f^{\prime}a)ah}{r^{3}}\,,$

and $\lambda$ is given in Appendix. Varying the Lagrangian (107) with respect
to $Q$, we obtain

$\left(\sqrt{\frac{f}{h}}r^{4}C_{1}\chi\right)^{\prime}=0\,.$ (109)

We can choose an appropriate boundary condition at spatial infinity, such that
Eq. (109) gives $\chi=0$. Then, the Lagrangian (107) reduces to

$\displaystyle{\cal L}_{\rm odd}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{8\pi
G_{\ae}}\sqrt{\frac{f}{h}}\biggl{[}\left(C_{5}-\frac{C_{2}^{2}}{C_{1}}\right){\delta\dot{u}}^{2}+\left(C_{7}-\frac{C_{3}^{2}}{C_{1}}\right){\delta
u^{\prime}}^{2}$ (110)
$\displaystyle+\left(C_{6}-\frac{2C_{2}C_{3}}{C_{1}}\right){\delta\dot{u}}{\delta
u}^{\prime}+{\cal M}\delta u^{2}\biggr{]}\,,$

where

${\cal
M}=2C_{12}+C_{13}-\frac{C_{4}^{2}}{C_{1}}+\left(\frac{C_{3}C_{4}}{C_{1}}\right)^{\prime}\,.$
(111)

Now, we convert the Lagrangian (110) to that in the Aether-orthogonal frame.
For this purpose, we replace the derivatives ${\delta\dot{u}}$ and $\delta
u^{\prime}$ with $\delta u_{,\phi}$ and $\delta u_{,\psi}$ by using the
relations (70) and (71). Then, the resulting second-order action reduces to
${\cal S}_{\rm odd}=\int{\rm d}\phi{\rm d}\psi\,\hat{\cal L}_{\rm odd}$, where

$\hat{\cal L}_{\rm odd}=\frac{1}{8\pi G_{\ae}}\sqrt{\frac{f}{h}}\left({\cal
K}\delta u_{,\phi}^{2}+{\cal G}\delta u_{,\psi}^{2}+r^{2}{\cal M}\delta
u^{2}\right)\,,$ (112)

with

$\displaystyle{\cal K}$ $\displaystyle=$
$\displaystyle\frac{4\alpha^{2}c_{14}}{(1+f\alpha^{2})^{2}}\,,$ (113)
$\displaystyle{\cal G}$ $\displaystyle=$
$\displaystyle-\frac{2\alpha^{2}[2c_{1}-c_{13}(2c_{1}-c_{13})]}{(1-f\alpha^{2})^{2}(1-c_{13})}\,.$
(114)

Thus, the Aether perturbation $\delta u$ is the only propagating degree of
freedom for $l=1$. The ghost is absent under the condition ${\cal K}>0$, which
translates to

$c_{14}>0\,,$ (115)

and is the same as Eq. (91) derived for $l\geq 2$. The radial squared
propagation speed measured in terms of the proper time $\tilde{\phi}$ reads

$c_{r}^{2}=-\frac{(1-f\alpha^{2})^{2}}{(1+f\alpha^{2})^{2}}\frac{{\cal
G}}{{\cal
K}}=\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}=c_{V}^{2}\,,$ (116)

which is equivalent to the squared propagation speed of vector perturbations
in the Minkowski spacetime. Thus, for $l=1$, there are no additional stability
conditions to those derived for $l\geq 2$.

###  IV.3 Cases of specific coefficients

For the multiples $l\geq 2$, we consider several specific cases in which some
of the coefficients $c_{1,2,3,4}$ are vanishing. From Eq. (86), the matrix
component $\hat{K}_{22}$ vanishes for $c_{14}=0$. The matrix components
$\hat{G}_{22}$ and $\hat{M}_{22}$ can be expressed as

$\displaystyle\hat{G}_{22}$ $\displaystyle=$
$\displaystyle-\frac{2[2c_{1}-c_{13}(2c_{1}-c_{13})]\alpha^{2}}{(1-c_{13})(1-f\alpha^{2})^{2}}\frac{f}{h}\,,$
(117) $\displaystyle\hat{M}_{22}$ $\displaystyle=$
$\displaystyle-\frac{L[2c_{1}-c_{13}(2c_{1}-c_{13})]}{(1-c_{13})r^{2}}\frac{f}{h}\,.$
(118)

Then, the Aether field either behaves as a vector-type instantaneous mode or
exhibits a strong coupling problem for

$c_{14}=0\,,\quad{\rm and}\quad 2c_{1}-c_{13}(2c_{1}-c_{13})\neq 0\,,$ (119)

depending on the behavior of the system at nonlinear level, the analysis of
which is beyond the scope of the present paper. If we demand that $c_{13}=0$
for the consistency with the observational bound (19), then either a vector-
type instantaneous mode or a strong coupling problem may arise for $c_{14}=0$
and $c_{1}\neq 0$, depending on the nonlinear behavior of the system. This is
the case for the stealth Schwarzshild BH solution discussed in Refs. [48, 62,
78].

If we consider Einstein-Aether theory with $c_{14}=0$, $c_{1}=0$, and
$c_{13}=0$, i.e.,

$c_{1}=0\,,\qquad c_{2}\neq 0\,,\qquad c_{3}=0\,,\qquad c_{4}=0\,,$ (120)

we have the following matrix components

$\displaystyle\hat{K}_{11}=\frac{2\alpha^{2}r^{2}}{(L-2)(1+f\alpha^{2})^{2}}\,,$
$\displaystyle\hat{G}_{11}=-\frac{2\alpha^{2}r^{2}}{(L-2)(1-f\alpha^{2})^{2}}\,,$
$\displaystyle\hat{M}_{11}=-\frac{1}{2}\,,$
$\displaystyle\hat{K}_{22}=\hat{G}_{22}=\hat{M}_{22}=0\,.$ (121)

The number of propagating degrees of freedom in the odd-parity sector is 1 at
linear level. This indicates either the absence of vector modes, the presence
of a vector-type instantaneous mode or the strong coupling problem for $\delta
u$, depending on the nonlinear behavior of this system. Fortunately, in this
case, we know the nonlinear behavior since Einstein-Aether theory with the
coefficients (120) and zero vorticity is equivalent to a class of cuscuton
theories with a quadratic potential [13]. This means that the absence of the
time kinetic term and the gradient term shown above for the specific
coefficients (120) simply corresponds to the absence of vector modes as far as
the equivalence to the cuscuton theory holds. There is a single dynamical
degree of freedom $\chi$ with the propagation speeds given by

$c_{r1}^{2}=c_{\Omega 1}^{2}=1\,,$ (122)

which are both luminal.

##  V Conclusions

In this paper, we addressed the linear stability of BHs against odd-parity
perturbations in Einstein-Aether theory given by the action (1). In this
theory, there is a preferred threading aligned with a unit timelike vector
field. If the background Aether field $u_{\mu}$ has vanishing vorticity, one
can introduce a scalar (Khronon) field $\phi$ whose gradient
$\partial_{\mu}\phi$ is timelike and proportional to $u_{\mu}$. This property
holds for the SSS background given by the line element (43).

In Einstein-Aether theory, the constant $t$ hypersurfaces in the coordinate
(43) are not always spacelike outside the universal horizon, which now is the
boundary of a BH and is always inside the metric horizon, when superlunimal
speeds are allowed [51]. In this sense, the derivation of linear stability
conditions using $t$ as a time clock can lead to inconsistent results. The
proper coordinate choice for obtaining no-ghost conditions and propagation
speeds of dynamical perturbations should be the Aether-orthogonal frame in
which the Khronon field $\phi$ is treated as a time clock, in which case the
constant time hypersurfaces are always spacelike over the whole region outside
the universal horizon, as shown explicitly by the metric (66). In Sec. III.3,
we argued how the coordinate choice different from the Aether-orthogonal frame
can give rise to apparent ghost and Laplacian instabilities.

In Sec. IV, we derived the second-order action of odd-parity perturbations by
transforming the action derived for the SSS coordinate (43) in Ref. [62] to
the one in the Aether-orthogonal frame with the line element (66). For this
purpose, we exploited transformation properties (70)-(71) of the derivatives
of perturbations between the two sets of coordinates. For the multipoles
$l\geq 2$, there are two dynamical perturbations $\chi$ and $\delta u$ arising
from the gravitational and vector-field sectors, respectively. The resulting
second-order Lagrangian is of the form (84), which does not contain products
of the $\phi$ and $\psi$ derivatives (unlike the Lagrangian (79) containing
products of the $t$ and $r$ derivatives). The stability analysis of BHs in the
Aether-orthogonal frame shows that the ghost is absent under the inequality
$c_{14}>0$, which is the same no-ghost condition of vector perturbations on
the Minkowski background. In large momentum limits, the radial squared
propagation speeds of $\chi$ and $\delta u$ are equivalent to those of the
tensor and vector perturbations on the Minkowski background. This is also the
case for the angular squared propagation speeds of $\chi$ and $\delta u$ in
the eikonal limit $l\gg 1$. For $l=1$, the vector-field perturbation alone
propagates with the same stability conditions of $\delta u$ as those derived
for $l\geq 2$.

We thus showed that the proper odd-parity stability analysis of BHs based on
the Aether-orthogonal frame gives rise to the same no-ghost conditions and
propagation speeds of dynamical perturbations as those on the Minkowski
background. In Sec. IV.3, we discussed several specific cases of coupling
constants in which the strong coupling problem may arise or the number of
degrees of freedom reduces. It will be of interest to classify surviving BH
solutions free from the linear instability and strong coupling problems. For
this purpose, we plan to extend the stability analysis in the Aether-
orthogonal frame to perturbations in the even-parity sector.

## Acknowledgements

The work of SM was supported in part by Japan Society for the Promotion of
Science (JSPS) Grants-in-Aid for Scientific Research No. 24K07017 and the
World Premier International Research Center Initiative (WPI), MEXT, Japan. ST
was supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No.
22K03642 and Waseda University Special Research Project No. 2023C-473. AW is
partially supported by a US NSF grant with the grant number: PHY2308845.

## Appendix A: The Lagrange multiplier in Spherical Spacetimes

The quantity $\lambda$ in Eq. (108) is given by

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{1}{4f^{2}r^{2}\left(a^{2}f-1\right)}\bigg{\\{}fr^{2}\left[a^{2}\left(6c_{1}+3c_{2}+2c_{13}-8c_{14}\right)h\left(f^{\prime}\right)^{2}+c_{2}\left(2hf^{\prime\prime}+f^{\prime}h^{\prime}\right)\right]$
(A.1)
$\displaystyle+2a^{2}f^{4}\left[-2a^{2}\left(2\left(c_{2}+c_{13}\right)h-c_{2}rh^{\prime}\right)+4\left(c_{14}-c_{1}\right)hr^{2}\left(a^{\prime}\right)^{2}+a\left(-c_{1}+c_{2}+c_{13}\right)r\left(2hra^{\prime\prime}+a^{\prime}\left(rh^{\prime}+4h\right)\right)\right]$
$\displaystyle-f^{2}\Big{[}rh^{\prime}\left(a^{2}\left(-2c_{1}+3c_{2}+2c_{13}\right)rf^{\prime}-4c_{2}\right)$
$\displaystyle+2h\big{(}a^{4}\left(3c_{1}+c_{2}+c_{13}-4c_{14}\right)r^{2}\left(f^{\prime}\right)^{2}+a^{2}r\left(\left(-2c_{1}+3c_{2}+2c_{13}\right)rf^{\prime\prime}+2\left(-2c_{1}+c_{2}+2c_{13}\right)f^{\prime}\right)$
$\displaystyle-a\left(11c_{1}-5c_{2}-5c_{13}-8c_{14}\right)r^{2}a^{\prime}f^{\prime}+4\left(c_{2}+c_{13}\right)\big{)}\Big{]}-\left(2c_{2}+c_{13}\right)hr^{2}\left(f^{\prime}\right)^{2}$
$\displaystyle+2f^{3}\Big{[}a^{4}r\left(\left(-c_{1}+c_{2}+c_{13}\right)rf^{\prime}h^{\prime}+2h\left(\left(-c_{1}+c_{2}+c_{13}\right)rf^{\prime\prime}+\left(-2c_{1}+c_{2}+2c_{13}\right)f^{\prime}\right)\right)$
$\displaystyle+a^{2}\left(8\left(c_{2}+c_{13}\right)h-4c_{2}rh^{\prime}\right)-2\left(-2c_{1}+c_{2}+c_{13}+2c_{14}\right)hr^{2}\left(a^{\prime}\right)^{2}$
$\displaystyle+a^{3}\left(-11c_{1}+3c_{2}+3c_{13}+8c_{14}\right)hr^{2}a^{\prime}f^{\prime}+a\left(c_{1}-c_{2}-c_{13}\right)r\left(2hra^{\prime\prime}+a^{\prime}\left(rh^{\prime}+4h\right)\right)\Big{]}\bigg{\\}}.~{}~{}~{}$

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