# Excluding static and spherically symmetric black holes in Einsteinian cubic
gravity
with unsuppressed higher-order curvature terms

Antonio De Felice1 and Shinji Tsujikawa2 1Center for Gravitational Physics and
Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto
University, 606-8502, Kyoto, Japan
2Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo
169-8555, Japan

###### Abstract

Einsteinian cubic gravity is a higher-order gravitational theory in which the
linearized field equations of motion match Einstein’s equations on a maximally
symmetric background. This theory allows the existence of a static and
spherically symmetric black hole solution where the temporal and radial metric
components are equivalent to each other ($f=h$), with a modified Schwarzschild
geometry induced by cubic curvature terms. We study the linear stability of
the static and spherically symmetric vacuum solutions against odd-parity
perturbations without dealing with Einsteinian cubic gravity as an effective
field theory where the cubic curvature terms are always suppressed relative to
the Ricci scalar. Unlike General Relativity containing one dynamical
perturbation, Einsteinian cubic gravity has three propagating degrees of
freedom in the odd-parity sector. We show that at least one of those dynamical
perturbations always behaves as a ghost mode. We also find that one dynamical
degree of freedom has a negative sound speed squared $-1/2$ for the
propagation of high angular momentum modes. Thus, the static and spherically
symmetric hairy black hole solutions realized by unsuppressed cubic curvature
terms relative to the Ricci scalar are excluded by ghost and Laplacian
instabilities.

††preprint: YITP-23-63, WUCG-23-06

##  I Introduction

General Relativity (GR) has been successful in describing the gravitational
interaction between submillimeter and solar-system scales [1, 2, 3, 4, 5]. At
extremely small distances close to the Planck length, however, it is expected
that GR is replaced by a quantum theory of gravity with an ultraviolet
completion. The attempt for the construction of a power-counting
renormalizable gravitational theory was advocated by Stelle [6] by taking into
account quadratic-order curvatures to the Einstein-Hilbert action. In string
theory, the low-energy effective action also contains quadratic curvature
corrections known as a Gauss-Bonnet (GB) term [7, 8]. Moreover, higher-order
curvature terms have played a prominent role in conformal field theory with
holography [9, 10, 11, 12].

In gravitational theories where the field equations of motion contain
derivatives higher than second order in the metric tensor $g_{\mu\nu}$, the
system can be unstable due to the emergence of Ostrogradski instability [13,
14] associated with a Hamiltonian unbounded from below. To avoid such a
problem, Lanczos [15] and Lovelock [16] constructed gravitational theories
with second-order field equations of motion for general, curved backgrounds.
Exploiting polynomial functions of the Riemann curvature tensors in four-
dimensional spacetime endowed with four-dimensional diffeomorphism invariance,
Lovelock showed that the symmetric and divergence-free tensors $A_{\mu\nu}$,
which depend on $g_{\mu\nu}$ and its derivatives up to second order, are
expressed by a linear combination of the Einstein tensor $G_{\mu\nu}$ and the
metric tensor $g_{\mu\nu}$. In spacetime dimensions higher than four, there is
the quadratic-order GB curvature scalar affecting the spacetime dynamics. In
four dimensions, the GB term corresponds to an Euler density which does not
contribute to the field equations of motion. To extract its effect in four
dimensions, the GB term needs to be coupled to other degrees of freedom (DOFs)
such as scalar or vector fields [17, 18, 19, 20, 21, 22, 23, 24, 25].

If we consider cubic-order curvature combinations constructed from the Riemann
tensors, there is also an additional Euler density in the action corresponding
to the surface integral in four dimensions [16]. According to the Lovelock
theorem, there are no other nontrivial cubic-order terms keeping the field
equations of motion up to second order in general, curved backgrounds. If we
consider some specific spacetime, however, it is possible to construct
nontrivial cubic-order theory whose graviton spectrum shares a similar
property to that in GR. On a maximally symmetric background, which includes
the Minkowski and de Sitter spacetimes, there is a unique cubic combination
${\cal P}$ which keeps the structure of linearized perturbation equations of
motion in Einstein gravity [26]. In this theory, which is dubbed Einsteinian
cubic gravity (ECG), the cubic curvature term can give rise to derivatives
higher than second order in general, curved backgrounds, so the spacetime
dynamics is generally different from that in GR [27].

On a static and spherically symmetric (SSS) background in a vacuum
configuration, ECG admits the existence of a black hole (BH) solution whose
temporal and radial metric components (denoted as $f$ and $h$, respectively)
coincide with each other [28, 29, 30, 31] (see also Refs. [32, 33, 34, 35]).
Since $f$ and $h$ are affected by the cubic curvature term, the background
geometry is still different from the Schwarzschild BH solution. One can also
construct a cubic gravity theory by imposing the condition $f=h$ (up to a time
reparametrization freedom in $f$) on the SSS background [36, 37, 38]. This
construction allows the existence of the Lagrangian ${\cal P}$ mentioned above
as well as two additional cubic combinations ${\cal C}$ and ${\cal
C}^{\prime}$ defined in Ref. [36]. As we will see later in Sec. II, both
${\cal C}$ and ${\cal C}^{\prime}$ vanish for $f=h$ and hence only the
Lagrangian ${\cal P}$ contributes to the single differential equation for
$f~{}(=h)$.

If we apply ECG to the cosmological dynamics on the Friedmann-Lemaître-
Robertson-Walker (FLRW) spacetime, the field equations of motion arising from
${\cal P}$ contain derivatives higher than second order. On the FLRW
background, there is a unique combination ${\cal P}-8{\cal C}$ that leads to
the second-order Friedmann equation [39, 40]. This theory–dubbed cosmological
Einsteinian cubic gravity (CECG)– was applied to the dynamics of inflation and
late-time cosmological epochs [39, 40, 41, 42, 43, 44, 45]. On an exact de
Sitter background, tensor perturbations with two polarized modes propagate as
in GR without pathological behavior. If one considers a spatially homogeneous
Bianchi type I manifold close to the isotropic de Sitter spacetime, however,
there are three dynamical propagating DOFs associated with linear
perturbations in the odd-parity sector. In Ref. [46], it was shown that, in
the regime of small anisotropies, such theory possesses at least one ghost
mode as well as short-time-scale tachyonic instability. Hence CECG cannot be
used to describe a viable geometric inflationary scenario.

Given that there is an instability problem of CECG on the anisotropic
cosmological background, we are now interested in the linear stability of SSS
vacuum solutions in ECG. For this purpose, we do not deal with ECG as a
trivial effective field theory (EFT) where the cubic Lagrangian is always
strongly suppressed relative to the Einstein-Hilbert term. We consider odd-
parity perturbations according to the Regge-Wheeler formulation [47] without
necessarily imposing the condition $f=h$. We show that the Lagrangian ${\cal
P}$ in ECG gives rise to three propagating DOFs in the odd-parity sector. We
find that there is at least one ghost mode and that the squared propagation
speed of one of the dynamical perturbations is negative for large multipoles
in the regime where the effective mass of perturbations is below the Planck
mass, or the cutoff of the theory $M$. The presence of these ghost and
Laplacian instabilities excludes the SSS vacuum solutions in ECG with
unsuppressed higher-order curvature terms, including the BH solution with
$f=h$.

This paper is organized as follows. In Sec. II, we revisit the SSS BH solution
present in ECG. In Sec. III, we derive the second-order action of odd-parity
perturbations and show how the ghost and Laplacian instabilities arise for the
large frequency and momentum modes. Sec. IV is devoted to conclusions.

##  II Cubic gravity

General cubic gravity theories consist of a combination of cubic products of
the Riemann tensor $R_{\alpha\beta\gamma\delta}$, Ricci tensor $R_{\mu\nu}$,
and Ricci scalar $R$. We consider the following eight cubic Lagrangians:

$\displaystyle\mathcal{L}_{1}={{R_{\alpha}}^{\beta}}{}_{\gamma}{}^{\delta}{{R_{\beta}}^{\mu}}{}_{\delta}{}^{\nu}{{R_{\mu}}^{\alpha}}{}_{\nu}{}^{\gamma}\,,\qquad\mathcal{L}_{2}=R_{\alpha\beta}{}^{\gamma\delta}R_{\gamma\delta}{}^{\mu\nu}R_{\mu\nu}{}^{\alpha\beta}\,,\qquad\mathcal{L}_{3}=R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma}{}_{\mu}R^{\delta\mu}\,,\qquad\mathcal{L}_{4}=RR_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}\,,$
$\displaystyle\mathcal{L}_{5}=R_{\alpha\beta\gamma\delta}R^{\alpha\gamma}R^{\beta\delta}\,,\qquad\mathcal{L}_{6}=R_{\alpha}{}^{\beta}R_{\beta}{}^{\gamma}R_{\gamma}{}^{\alpha}\,,\qquad\mathcal{L}_{7}=RR_{\alpha\beta}R^{\alpha\beta}\,,\qquad\mathcal{L}_{8}=R^{3}\,.$
(1)

Taking into account the Einstein-Hilbert term $M_{\rm Pl}^{2}R/2$, where
$M_{\rm Pl}$ is the reduced Planck mass, we express the total action as

${\cal S}=\frac{M_{\rm pl}^{2}}{2}\int{\rm
d}^{4}x\sqrt{-g}\left(R+\sum_{i=1}^{8}c_{i}\mathcal{L}_{i}\right)\,,$ (2)

where $g$ is a determinant of the metric tensor $g_{\mu\nu}$, and $c_{i}$’s
are constants.

ECG is constructed to possess a transverse and massless graviton spectrum as
in GR on a maximally symmetric background [26]. Requiring also that the
relative coefficients of different curvature terms are the same in all
dimensions, there is a unique combination ${\cal P}$ which is neither trivial
nor topological in four dimensions. This nontrivial cubic interaction
corresponds to the choice of coefficients $c_{1}=12$, $c_{2}=1$, $c_{5}=-12$,
and $c_{6}=8$, i.e.,

${\cal
P}=12\mathcal{L}_{1}+\mathcal{L}_{2}-12\mathcal{L}_{5}+8\mathcal{L}_{6}\,.$
(3)

In general curved spacetime including the SSS background as well as the FLRW
background, the equations of motion following from the Lagrangian ${\cal P}$
are higher than the second order. In such cases, higher-order derivatives can
induce extra DOFs in comparison to those in GR.

In Ref. [36], the authors took a different approach to the construction of
cubic gravity theories (dubbed generalized quasi-topological gravity) by
demanding that the vacuum SSS solution is fully characterized by a single
field equation. In general, the line element on the SSS background is given by

${\rm d}s^{2}=-f(r){\rm d}t^{2}+h^{-1}(r){\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta\,{\rm d}\varphi^{2}\right)\,,$ (4)

where $f$ and $h$ are functions of the radial coordinate $r$. If $f(r)$ is
proportional to $h(r)$ such that $f(r)=Nh(r)$, where $N$ is a constant, the
time and radial components of the field equations coincide with each other.
This gives the following constraints among the coefficients in Eq. (2):

$c_{4}=\frac{3c_{1}-36c_{2}-14c_{3}}{56},\quad
c_{5}=-\frac{3c_{1}+48c_{2}+14c_{3}}{7},\quad
c_{7}=\frac{6c_{1}+96c_{2}+14c_{3}-21c_{6}}{28},\quad
c_{8}=-\frac{3c_{1}+20c_{2}-7c_{6}}{56}.$ (5)

The six-dimensional Euler density ${\cal X}_{6}$ corresponds to the
coefficients $c_{1}=8$, $c_{2}=-4$, $c_{3}=24$, and $c_{6}=-16$, but this is a
topological term that does not affect the field equations of motion. One of
the remaining three cubic interactions is the Lagrangian ${\cal P}$ given by
Eq. (3). There are also two additional terms

$\displaystyle{\cal C}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{3}-\frac{1}{4}\mathcal{L}_{4}-2\mathcal{L}_{5}+\frac{1}{2}\mathcal{L}_{7}\,,$
(6) $\displaystyle{\cal C}^{\prime}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{6}-\frac{3}{4}\mathcal{L}_{7}+\frac{1}{8}\mathcal{L}_{8}\,,$
(7)

which correspond to the coefficients $c_{1}=0$, $c_{2}=0$, $c_{3}=1$,
$c_{6}=0$ and $c_{1}=0$, $c_{2}=0$, $c_{3}=0$, $c_{6}=1$, respectively. As we
will see below, for $f=h$, the contributions of the Lagrangians ${\cal C}$ and
${\cal C}^{\prime}$ to the field equations of motion vanish. On the FLRW
spacetime, the Lagrangian ${\cal P}$ gives rise to the Friedmann equation
higher than second-order. The specific combination ${\cal P}-8{\cal C}$ leads
to the second-order field equations on the FLRW background [39, 40]. If we
apply this latter cubic theory to inflation, the presence of small
anisotropies close to the de Sitter background generates the instability of
cosmological solutions [46].

In both ECG and CECG, the dynamical DOFs are more than those in GR (two tensor
polarizations) around general, curved backgrounds. The property that the
propagating DOFs around the maximally symmetric background degenerate to those
in GR implies that the disappearing DOFs are, in general, strongly coupled
[44, 48]. In such theories, the maximally symmetric background corresponds to
a singular surface at which the coefficients of higher-order kinetic terms
appearing in the perturbation equations of motion are degenerate. This
degeneracy leads to the divergence of couplings of the canonically normalized
fields, which is a signal of the strong coupling. On the background different
from the maximally symmetric spacetime, some pathological behavior like the
instability of perturbations usually arises as it happens for the cosmological
background [46, 44]. In this paper, we would like to study whether this is
also the case for BHs on the SSS background.

On the SSS background given by the line element (4), lets us consider cubic
gravity theories given by the action

${\cal S}=\frac{M_{\rm pl}^{2}}{2}\int{\rm d}^{4}x\sqrt{-g}\left(R+\alpha{\cal
P}+\kappa{\cal C}+\mu{\cal C}^{\prime}\right)\,,$ (8)

where $\alpha$, $\kappa$, $\mu$ are constants. Then, the quantities (3), (6),
and (7) reduce, respectively, to

$\displaystyle{\cal P}$ $\displaystyle=$
$\displaystyle\frac{1}{f^{4}r^{4}}[3f^{\prime
4}h^{3}r^{2}-3f^{3}h^{\prime}(2f^{\prime\prime}hr+f^{\prime}h^{\prime}r-2f^{\prime}h)(h^{\prime}r-2h+2)-6f^{\prime
2}fh^{2}r\\{f^{\prime\prime}hr+f^{\prime}(h^{\prime}r+h-1)\\}$ (9)
$\displaystyle+6rf^{\prime}f^{2}h\\{f^{\prime}h^{\prime}(h^{\prime}r+h)+f^{\prime\prime}h(h^{\prime}r+2h-2)\\}]\,,$
$\displaystyle{\cal C}$ $\displaystyle=$
$\displaystyle-\frac{3}{8f^{4}r^{4}}\left(f^{\prime}h-h^{\prime}f\right)^{2}\left[(2f^{\prime\prime}fh-
hf^{\prime 2}+f^{\prime}h^{\prime}f)r^{2}-4f^{2}(h-1)\right]\,,$ (10)
$\displaystyle{\cal C}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal C}\,,$ (11)

where a prime represents the derivative with respect to $r$. If $f$ is equal
to $h$, both ${\cal C}$ and ${\cal C}^{\prime}$ vanish. Varying the action (8)
with respect to $f$ and $h$, we obtain the third-order differential equations
for $h$ and $f$, respectively111If we relax the assumption of staticity, it
can be shown that for this theory the Birkhoff theorem does not hold in
general. Therefore, there could be other solutions that in principle have some
relevance but will be in general time-dependent. We will not investigate their
existence, however, as we will already set strong constraints on the theory
just by looking at their static limit.. Setting $f(r)=h(r)$, the two
differential equations coincide with each other, giving

$rf^{\prime}+f-1+\frac{6\alpha}{r^{3}}[rf^{\prime
2}(4f-1)+rf\\{r^{2}f^{\prime\prime
2}+4f^{\prime\prime}(f-1)-2rf^{\prime\prime\prime}(f-1)\\}+f^{\prime}f(r^{3}f^{\prime\prime\prime}-4r^{2}f^{\prime\prime}-4f+4)]=0\,.$
(12)

For $\alpha=0$, the solution to Eq. (12) is $f=h=1-r_{h}/r$, where $r_{h}$ is
an integration constant corresponding to the horizon radius. For $\alpha\neq
0$, there are corrections to the Schwarzschild metric. At large distances away
from the horizon ($r\gg r_{h}$), we derive the solution to Eq. (12) under the
following expansion

$f(r)=h(r)=1-\frac{r_{h}}{r}+\sum_{i=2}c_{i}\left(\frac{r_{h}}{r}\right)^{i}\,,$
(13)

where $c_{i}$’s are constants. Substituting Eq. (13) into Eq. (12) and
computing the coefficients at each order, we obtain
$c_{2}=c_{3}=c_{4}=c_{5}=0$, $c_{6}=54\alpha/r_{h}^{4}$, and
$c_{7}=-46\alpha/r_{h}^{4}$. Then, the leading-order correction to $f$ arises
at sixth order in the expansion (12), so that222Here we do not study whether
the series converges, as in any case, the solution loses its validity for
large enough values of $r$, but we assume that it can be considered as a good
approximation in the physical range of $r$ of interest to the real numerical
solution having suitable asymptotically flat boundary conditions.

$f(r)=h(r)=1-\frac{r_{h}}{r}+\frac{54\alpha r_{h}^{2}}{r^{6}}\left[1+{\cal
O}\left(\frac{r_{h}}{r}\right)\right]\,.$ (14)

The solution (14) has been derived by requiring that $f(r)=h(r)$ without
explicitly imposing any requirement on the value of $\alpha$. The same
solution also follows without assuming the condition $f(r)=h(r)$ in ECG with

${\cal P}\neq 0\,,\qquad{\cal C}=0\,,\qquad{\cal C}^{\prime}=0\,.$ (15)

In this case, we write the large-distance solutions as

$f(r)=1-\frac{r_{h}}{r}+\sum_{i=2}c_{i}\left(\frac{r_{h}}{r}\right)^{i}\,,\qquad
h(r)=1-\frac{r_{h}}{r}+\sum_{i=2}d_{i}\left(\frac{r_{h}}{r}\right)^{i}\,.$
(16)

We substitute Eq. (16) into the third-order differential equations of $f$ and
$h$ and obtain the coefficients $c_{i}$ and $d_{i}$. This gives rise to the
same coefficients $c_{i}$ derived above, with $c_{i}=d_{i}$, so that the
large-distance solution is again given by Eq. (14). Hence the BH solution with
$f(r)=h(r)$ generically arises in ECG.

##  III Odd-parity perturbations on the SSS background

We study the stability of SSS vacuum solutions in ECG given by the action

${\cal S}=\frac{M_{\rm pl}^{2}}{2}\int{\rm d}^{4}x\sqrt{-g}\left(R+\alpha{\cal
P}\right)\,.$ (17)

In our analysis, we do not restrict ourselves to the EFT regime where
$\alpha{\cal P}$ is always suppressed relative to $R$. The modification to the
Schwarzschild BH in GR can be significant by allowing for the possibility that
$\alpha{\cal P}$ can be as large as $R$. This is analogous to Starobinsky’s
model given by the Lagrangian $L=R+\beta R^{2}$ [49], where the cosmic
acceleration occurs in the regime $\beta R^{2}\gtrsim R$. If we stick to the
EFT regime with $\beta R^{2}\ll R$, one cannot accommodate the physics of
inflation driven by the quadratic curvature term.

On the background (4), we decompose the metric tensor into
$g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu}$, where $g_{\mu\nu}^{(0)}$ and
$h_{\mu\nu}$ correspond to the background and perturbed parts, respectively.
Although we are primarily interested in the stability of the BH solution (14),
we do not impose the condition $f=h$ for the background metric from the
beginning. Under the rotation in the $(\theta,\varphi)$ plane, the metric
perturbations $h_{\mu\nu}$ can be separated into odd- and even-parity modes
[47, 50]. Expanding $h_{\mu\nu}$ in terms of the spherical harmonics
$Y_{lm}(\theta,\varphi)$, the odd- and even-modes have parities $(-1)^{l+1}$
and $(-1)^{l}$, respectively. In the odd-parity sector, the components of
$h_{\mu\nu}$ are given by [51, 52, 53, 54, 55]

$\displaystyle h_{tt}=h_{tr}=h_{rr}=0\,,\qquad h_{ab}=0\,,$ $\displaystyle
h_{ta}=\sum_{l,m}Q(t,r)E_{ab}\nabla^{b}Y_{lm}(\theta,\varphi)\,,\qquad
h_{ra}=\sum_{l,m}W(t,r)E_{ab}\nabla^{b}Y_{lm}(\theta,\varphi)\,,$
$\displaystyle
h_{ab}=\frac{1}{2}\sum_{l,m}U(t,r)\left[E_{a}{}^{c}\nabla_{c}\nabla_{b}Y_{lm}(\theta,\varphi)+E_{b}{}^{c}\nabla_{c}\nabla_{a}Y_{lm}(\theta,\varphi)\right]\,,$
(18)

where $Q$, $W$, and $U$ depend on $t$ and $r$, and the subscripts $a$ and $b$
denote either $\theta$ or $\varphi$. $E_{ab}$ is an antisymmetric tensor with
nonvanishing components $E_{\theta\varphi}=-E_{\varphi\theta}=\sin\theta$. In
a strict sense, we should write subscripts $l$ and $m$ for the variables $Q$,
$W$, and $U$, but we omit them for brevity. Under the gauge transformation
$x_{\mu}\to x_{\mu}+\xi_{\mu}$, where $\xi_{t}=0$, $\xi_{r}=0$, and
$\xi_{a}=\sum_{l,m}\Lambda(t,r)E_{ab}\nabla^{b}Y_{lm}(\theta,\varphi)$, metric
perturbations transform as $Q\to Q+\dot{\Lambda}$, $W\to
W+\Lambda^{\prime}-2\Lambda/r$, and $U\to U+2\Lambda$. There are several
gauge-invariant combinations like

$\hat{W}\equiv W-\frac{1}{2}U^{\prime}+\frac{1}{r}U\,,\qquad\hat{Q}\equiv
Q-\frac{1}{2}\dot{U}\,.$ (19)

We fix the residual gauge DOF by choosing

$U=0\,,$ (20)

under which $\hat{W}=W$ and $\hat{Q}=Q$.

For the purpose of expanding the action (17) up to second order in odd-parity
perturbations, we will focus on the axisymmetric modes ($m=0$) without loss of
generality because nonaxisymmetric modes ($m\neq 0$) can be restored under a
suitable rotation. For the integrals with respect to $\theta$ and $\varphi$,
we exploit the following properties

$\displaystyle\int_{0}^{2\pi}{\rm d}\varphi\int_{0}^{\pi}{\rm
d}\theta\,(Y_{l0,\theta})^{2}\sin\theta=l(l+1)\,,\qquad\int_{0}^{2\pi}{\rm
d}\varphi\int_{0}^{\pi}{\rm
d}\theta\,\left[\frac{(Y_{l0,\theta})^{2}}{\sin\theta}+(Y_{l0,\theta\theta})^{2}\sin\theta\right]=l^{2}(l+1)^{2}\,,$
$\displaystyle\int_{0}^{2\pi}{\rm d}\varphi\int_{0}^{\pi}{\rm
d}\theta\,\left[\left(\frac{2}{\sin\theta}-\frac{3}{\sin^{3}\theta}\right)(Y_{l0,\theta})^{2}+\frac{3}{\sin\theta}(Y_{l0,\theta\theta})^{2}+\sin\theta\,(Y_{l0,\theta\theta\theta})^{2}\right]=\frac{1}{4}l^{2}(l+1)^{2}(4l^{2}-2l-3)\,.$
(21)

After the integration with respect to $\theta$ and $\varphi$, the second-order
action of perturbations, which is expressed in the form ${\cal
S}^{(2)}=\int{\rm d}t{\rm d}r\,L$, contains time derivatives such as
$\ddot{W}^{2}$, $\dot{Q}^{\prime 2}$, and $\dot{Q}^{2}$. They can be factored
out as

$L_{K}\equiv\frac{3\alpha M_{\rm
pl}^{2}\sqrt{h}[2f^{\prime}hr+f(2-2h-rh^{\prime})]l(l+1)}{2f^{5/2}r^{2}}\left(\ddot{W}-{\dot{Q}}^{\prime}+\frac{2\dot{Q}}{r}\right)^{2}\,,$
(22)

where $L_{K}\in L$. Then, the rest of the Lagrangian $L-L_{K}$ does not
contain products like $\ddot{W}\dot{Q}^{\prime}$ and $\ddot{W}\dot{Q}$. We
introduce a Lagrange multiplier $\chi$ that helps us to understand the
presence and behavior of propagating DOFs. Then, the Lagrangian equivalent to
$L$ can be written as $L-L_{K}+\tilde{L}_{K}$ where

$\tilde{L}_{K}=\frac{3\alpha M_{\rm
pl}^{2}\sqrt{h}[2f^{\prime}hr+f(2-2h-rh^{\prime})]l(l+1)}{2f^{5/2}r^{2}}\left[2\chi\left(\ddot{W}-{\dot{Q}}^{\prime}+\frac{2\dot{Q}}{r}\right)-\chi^{2}\right]\,.$
(23)

Indeed, the variation of (23) with respect to $\chi$ leads to
$\chi=\ddot{W}-{\dot{Q}}^{\prime}+2\dot{Q}/r$, so that (23) reduces to (22).
We note that the gauge-invariant field
$\chi=\ddot{W}-{\dot{Q}}^{\prime}+2\dot{Q}/r$ corresponds to a time derivative
of the dynamical perturbation $\tilde{\chi}=\dot{W}-Q^{\prime}+2Q/r$ in GR
[47, 53, 55] (up to a free function of $r$). Integrating the Lagrangian
$\chi\ddot{W}$ by parts in Eq. (23), we obtain the product
$-\dot{W}\dot{\chi}$ between the two first derivatives. As we will see below,
there are three propagating DOFs $\vec{{\cal X}}=(W,\chi,Q)$ in this system.
In other words, we generally need to give six independent initial conditions
to determine the time evolution of $\vec{{\cal X}}$.

After the integration by parts, the total second-order can be expressed in the
form ${\cal S}^{(2)}=\int{\rm d}t{\rm d}r\,L$, where

$\displaystyle L$ $\displaystyle=$ $\displaystyle
a_{1}\dot{W}^{2}+a_{2}\dot{Q}^{2}+2a_{3}\dot{W}\dot{\chi}+a_{4}\left(\dot{W}^{\prime}-Q^{\prime\prime}+\frac{2Q^{\prime}}{r}\right)^{2}+a_{5}W^{\prime
2}+a_{6}Q^{\prime 2}+a_{7}W^{2}+a_{8}\chi^{2}+a_{9}Q^{2}$ (24)
$\displaystyle+a_{10}W^{\prime}\dot{Q}+a_{11}\dot{W}Q^{\prime}+a_{12}\dot{\chi}Q^{\prime}+a_{13}\dot{W}Q+a_{14}\chi\dot{Q}\,,$

with $a_{1},\cdots,a_{14}$ being functions of $r$ alone. Varying the
Lagrangian (24) with respect to $W$, $\chi$, and $Q$, we obtain the field
equations of motion for these perturbations. They contain the derivatives up
to the fourth order, e.g., $\ddot{W}^{\prime\prime}$,
$\dot{Q}^{\prime\prime\prime}$.

In the following, we study the propagation of fast oscillating modes by
assuming the solutions in the form

$\vec{\cal X}=\vec{\cal X}_{0}e^{i(\omega t-kr)}\,,\qquad{\rm
with}\qquad\vec{\cal X}_{0}=(W_{0},\chi_{0},Q_{0})\,.$ (25)

Because of staticity, here, the coefficients $\vec{{\cal X}}_{0}$ are supposed
to be functions slowly varying in the $r$-direction, so that in the WKB domain
they satisfy for instance the condition $|{\vec{{\cal
X}}}_{0}^{\prime}|\ll|k\vec{{\cal X}}_{0}|$. For the same reason, we will also
suppose that $|\omega^{\prime}|\ll|k\omega|\simeq|\omega^{2}|$. As a
consequence, we will also consider the wavenumber $k$ and the multipole $l$ in
the ranges $kr_{h}\gg 1$ and $l\gg 1$.

Then, each coefficient in Eq. (24) has the following $l$ dependence:

$\displaystyle a_{1}=b_{1}l^{4}\,,\quad a_{2}=b_{2}l^{4}\,,\quad
a_{3}=b_{3}l^{2}\,,\quad a_{4}=b_{4}l^{2}\,,\quad a_{5}=b_{5}l^{4}\,,\quad
a_{6}=b_{6}l^{4}\,,\quad a_{7}=b_{7}l^{6}\,,\quad a_{8}=b_{8}l^{2}\,,$
$\displaystyle a_{9}=b_{9}l^{6}\,,\quad a_{10}=b_{10}l^{4}\,,\quad
a_{11}=b_{11}l^{4}\,,\quad a_{12}=b_{12}l^{2}\,,\quad
a_{13}=b_{13}l^{4}\,,\quad a_{14}=b_{14}l^{2}\,,$ (26)

where

$\displaystyle b_{1}=\frac{3\alpha M_{\rm
Pl}^{2}h^{1/2}[fr(2f^{\prime\prime}hr+f^{\prime}h^{\prime}r-6f^{\prime}h)+2f^{2}(h^{\prime}r+2h-2)-f^{\prime
2}hr^{2}]}{4f^{5/2}r^{4}}\,,$ $\displaystyle b_{2}=\frac{3\alpha M_{\rm
Pl}^{2}[(2f^{\prime\prime}f-f^{\prime
2})hr+h^{\prime}f(f^{\prime}r-2f)]}{2f^{7/2}h^{1/2}r^{3}}\,,$ $\displaystyle
b_{3}=\frac{r^{2}}{f}b_{1}-\frac{hr^{2}}{2}b_{2}+\frac{b_{0}}{2fh}\,,\qquad
b_{4}=fhb_{3}+\frac{3}{2}b_{0}\,,\qquad
b_{5}=f^{2}h^{2}b_{2}-\frac{f}{r^{2}}b_{0}\,,\qquad
b_{6}=b_{1}+\frac{2}{hr^{2}}b_{0}\,,$ $\displaystyle
b_{7}=-\frac{b_{5}}{2hr^{2}}\,,\qquad b_{8}=b_{3}\,,\qquad
b_{9}=\frac{f}{2r^{2}}b_{2}\,,\qquad b_{10}=-\frac{3\alpha M_{\rm
Pl}^{2}\sqrt{h}(3f^{\prime}h+h^{\prime}f)}{f^{3/2}r^{3}}\,,$ $\displaystyle
b_{11}=-2b_{1}-2fhb_{2}-b_{10}-\frac{b_{0}}{hr^{2}}\,,\qquad
b_{12}=-2b_{3}\,,$ (27)

with

$b_{0}=\frac{3\alpha M_{\rm
Pl}^{2}h^{3/2}(f^{\prime}h-h^{\prime}f)}{f^{3/2}r}\,.$ (28)

We do not show the explicit forms of $b_{13}$ and $b_{14}$, as they are not
needed in the following discussion.

Exploiting the approximation of large values of $\omega$, $k$, and $l$ in the
field equations of motion, we ignore terms proportional to $i\omega$, $ik$ and
$l$ relative to those proportional to $\omega^{2}$, $k^{2}$ and $l^{2}$. Then,
the perturbation equations of motion following from the Lagrangian (24) can be
expressed in the form

${\bm{A}}\vec{{\cal X}}_{0}^{\rm T}=0\,,$ (29)

where ${\bm{A}}$ is the $3\times 3$ matrix whose components are given by

$\displaystyle
A_{11}=2l^{2}\left[(b_{4}k^{2}+b_{1}l^{2})\omega^{2}+b_{5}k^{2}l^{2}+b_{7}l^{4}\right]\,,\qquad
A_{22}=2l^{2}b_{8}\,,\qquad
A_{33}=2l^{2}\left(b_{2}l^{2}\omega^{2}+b_{4}k^{4}+b_{6}k^{2}l^{2}+b_{9}l^{4}\right)\,,$
$\displaystyle A_{12}=A_{21}=2l^{2}b_{3}\omega^{2}\,,\qquad
A_{13}=A_{31}=l^{2}(2b_{4}k^{3}-b_{10}kl^{2}-b_{11}kl^{2})\omega\,,\qquad
A_{23}=A_{32}=-l^{2}b_{12}k\,\omega\,.$ (30)

###  III.1 Ghost instability

To derive the no-ghost conditions, we pick up the matrix components in
${\bm{A}}$ containing terms proportional to $\omega^{2}$, i.e.,

${\bm{K}}=2\omega^{2}\left(\begin{array}[]{ccc}K_{11}&K_{12}&0\\\
K_{12}&0&0\\\ 0&0&K_{33}\end{array}\right)\,,$ (31)

where

$K_{11}=l^{2}(b_{4}k^{2}+b_{1}l^{2})\,,\qquad K_{12}=l^{2}b_{3}\,,\qquad
K_{33}=l^{4}b_{2}\,.$ (32)

The absence of ghosts requires the following three conditions

$K_{11}>0\,,\qquad-K_{12}^{2}>0\,,\qquad-K_{12}^{2}K_{33}>0\,.$ (33)

The explicit form of $K_{12}$ is given by

$K_{12}=-\frac{3\alpha M_{\rm
Pl}^{2}h^{1/2}[r(2f^{\prime}h-h^{\prime}f)+2f(1-h)]}{2f^{5/2}r^{2}}l^{2}\,.$
(34)

So long as $\alpha\neq 0$, the second condition of Eq. (33) is always violated
and hence we have at least one ghost mode. The existence of the dynamical
perturbation $\chi$, which is kinetically coupled to $W$, gives rise to the
ghost DOF. This result generally holds without restricting the background BH
solution to the form (14), as we have not assumed the condition $f=h$.

It is important to understand how many ghost DOFs are generated in ECG, at
least for the vacuum case. A simple way to see this is to make the field
redefinitions: $W=W_{2}-K_{12}\,{\chi_{2}}/K_{11}$, $\chi=\chi_{2}$, and
$Q=Q_{2}$. The Jacobian of the transformation always has a determinant equal
to unity, so that the transformation is well defined. On using these new
fields, the kinetic matrix becomes diagonal with elements $K_{11}$,
$-K_{12}^{2}/K_{11}$, and $K_{33}$. For the BH solution (14), which is
approximately valid for $r\gg r_{h}$, we find that

$K_{33}\simeq-\frac{9\alpha M_{\rm
Pl}^{2}r_{h}l^{4}}{r^{5}}\,,\qquad\frac{K_{33}}{K_{11}}\simeq\frac{2l^{2}}{k^{2}r^{2}}>0\,.$
(35)

This shows that, at least for the approximate BH solution, $K_{11}$ and
$K_{33}$ share the same sign. Therefore, in the range of validity of this
solution, we would have one ghost only if $\alpha<0$, whereas we would have
two ghost DOFs if instead $\alpha>0$. This calculation also shows that the
kinetic term of, e.g., the mode $Q$, tends to vanish as $r_{h}/r\to 0$. This
limit then needs to be taken with care.

###  III.2 Radial propagation

The propagation speeds $c_{r}={\rm d}r_{*}/{\rm d}\tau$ along the radial
direction, where $r_{*}=\int{\rm d}r/\sqrt{h}$ is the rescaled radial
coordinate and $\tau=\int\sqrt{f}\,{\rm d}t$ is the proper time, are expressed
as $c_{r}=(fh)^{-1/2}(\partial\omega/\partial k)$. We compute the eigenvalue
of the matrix ${\bm{A}}$ and expand it with respect to the large momentum $k$.
This amounts to considering the modes in the range $kr_{h}\gg l\gg 1$. The
leading-order $k^{8}$ dependent term vanishes on account of the relation
$b_{12}=-2b_{3}$, so the dominant terms in ${\rm det}\,{\bm{A}}$ are those
proportional to $k^{6}$. Then, we obtain the following three solutions to the
squared radial propagation speeds:

$\displaystyle c_{r1}^{2}$ $\displaystyle=$
$\displaystyle\frac{4b_{4}b_{8}f^{4}r^{4}}{9\alpha^{2}M_{\rm
Pl}^{4}h^{2}[(h^{\prime}r+2h-2)f-2f^{\prime}hr]^{2}}=1+\frac{3r(f^{\prime}h-h^{\prime}f)}{(h^{\prime}r+2h-2)f-2f^{\prime}hr}\,,$
(36) $\displaystyle c_{r2}^{2}$ $\displaystyle=$
$\displaystyle\frac{-(b_{1}+b_{6}+b_{10}+b_{11})+\sqrt{{\cal
D}_{1}}}{2b_{2}fh}=1\,,$ (37) $\displaystyle c_{r3}^{2}$ $\displaystyle=$
$\displaystyle\frac{-(b_{1}+b_{6}+b_{10}+b_{11})-\sqrt{{\cal
D}_{1}}}{2b_{2}fh}=1-\frac{2f(f^{\prime}h-h^{\prime}f)}{2f^{\prime\prime}fhr-f^{\prime}r(hf^{\prime}-fh^{\prime})-2h^{\prime}f^{2}}\,,$
(38)

where

$\displaystyle{\cal
D}_{1}=b_{1}^{2}+2b_{1}b_{6}-4b_{2}b_{5}+b_{6}^{2}+b_{10}^{2}+2(b_{1}+b_{6}+b_{11})b_{10}+2(b_{1}+b_{6})b_{11}+b_{11}^{2}=\frac{9\alpha^{2}M_{\rm
Pl}^{4}h(f^{\prime}h-h^{\prime}f)^{2}}{f^{3}r^{6}}\,.$ (39)

For the BH solution (14) we have $f=h$, in which case all the three squared
propagation speeds (36)-(38) reduce to 1.

###  III.3 Angular instability

The propagation speeds $c_{\Omega}=r{\rm d}\theta/{\rm d}\tau$ along the
angular direction are expressed as $c_{\Omega}=r\omega/(\sqrt{f}l)$. Expanding
${\rm det}\,{\bm{A}}$ with respect to large $l$ in the range $l\gg kr_{h}\gg
1$, the leading-order terms, which are proportional to $l^{14}$, give the
following squared angular propagation speeds:

$\displaystyle c_{\Omega 1}^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}(b_{1}b_{8}+\sqrt{{\cal D}_{2}})}{2b_{3}^{2}f}=1\,,$
(40) $\displaystyle c_{\Omega 2}^{2}$ $\displaystyle=$
$\displaystyle\frac{r^{2}(b_{1}b_{8}-\sqrt{{\cal
D}_{2}})}{2b_{3}^{2}f}=\frac{[2f^{\prime\prime}fhr-f^{\prime}(f^{\prime}hr-h^{\prime}fr+2fh)]r}{2f[r(h^{\prime}f-2f^{\prime}h)+2f(h-1)]}\,,$
(41) $\displaystyle c_{\Omega 3}^{2}$ $\displaystyle=$
$\displaystyle-\frac{b_{9}r^{2}}{b_{2}f}=-\frac{1}{2}\,,$ (42)

where

$\displaystyle{\cal D}_{2}$ $\displaystyle=$ $\displaystyle
b_{8}(b_{1}^{2}b_{8}+4b_{3}^{2}b_{7})$ (43) $\displaystyle=$
$\displaystyle\frac{81\alpha^{4}M_{\rm
Pl}^{8}h^{2}[r(2f^{\prime}h-h^{\prime}f)-2f(h-1)]^{2}[2f^{\prime\prime}fhr^{2}-f^{\prime}r(f^{\prime}hr-h^{\prime}fr-2fh)-2f^{2}(h^{\prime}r+2h-2)]^{2}}{64f^{10}r^{12}}\,.$

Using the BH solution (14) in the regime $r\gg r_{h}$, the second propagation
speed squared (41) has the following behavior

$c_{\Omega 2}^{2}=1-\frac{720\alpha r_{h}}{r^{5}}+{\cal O}(r^{-10})\,,$ (44)

which quickly approaches 1 at large distances.

The third propagation speed squared (42) is negative, so there is Laplacian
instability along the angular direction. This instability arises from the
unhealthy propagation of the perturbation $Q$. In the region slightly away
from the horizon, the typical time for the instability to occur is of order
$T\simeq\mathcal{O}(1)/|\omega|\simeq r_{h}/(\sqrt{f}l)\ll r_{h}$. We also
note that the condition $f=h$ was not used for the derivation of $c_{\Omega
3}^{2}$. Due to this angular instability besides the appearance of a ghost,
the SSS BH solutions with unsuppressed cubic-order terms are not stable.

###  III.4 Instability versus EFT mass breaking limit

In what follows, we study the regime of validity of the BH solution and
conditions under which our previous analysis can be trusted, especially
regarding the presence of instabilities. Indeed, we know that one or two
ghosts are present and that the field $Q$ acquires a negative propagation
speed squared along the angular direction. The last instability is purely (and
already) classical, and as such, probably the most serious one. We would like
to see here whether this instability for the field $Q$ occurs anywhere in the
spatial slicing with $l$ in the range $l\gg kr_{h}$. In order to study this,
we consider the partial differential equations, focusing on the unstable mode,
which corresponds to the mass term of the perturbation $Q$.

Focusing on the term which makes $Q$ unstable and also looking for the case
$r\gg r_{h}$, we find that

$\ddot{Q}\simeq\left(\frac{l^{2}}{2r^{2}}-\frac{r_{h}^{2}}{36\alpha}\,\frac{r^{3}}{r_{h}^{3}}\right)Q+\dots$
(45)

where the dots stand for other terms which are irrelevant to the following
discussion. For example, in Eq. (45), we also have a term of the form

$\frac{r^{5}}{36l^{2}\alpha r_{h}}\,Q^{\prime\prime}\,,$ (46)

but in the following, we will show that, for the Laplacian instability to
occur, we need to be in a region of the spatial geometry (and parameter space)
for which $l^{2}\gg r^{5}/(18|\alpha|r_{h})$. This sets the coefficient of
$Q^{\prime\prime}$ very small. Furthermore, since we are considering the case
in which the angular instability occurs, we also need to assume that the
angular derivatives overcome the $r$-derivatives, that is,
$l^{2}|Q|/r^{2}\gg|Q^{\prime\prime}|$. For the same reason, the coefficients
in the terms $-r^{5}{\dot{W}^{\prime}}/(36l^{2}\alpha r_{h})$ and
$-r^{4}{\dot{W}}/(18l^{2}\alpha r_{h})$ are suppressed and can be thought to
be subdominant, also because the field $W$ does not exhibit instabilities
along the angular direction (as it propagates with a speed approximately equal
to unity).

In any case, from Eq. (45), we can define an effective squared mass, the self-
coupling term independent of $l$ (or $k$), for the variable $Q$ as follows

$\mu^{2}_{Q}=\frac{r_{h}^{2}}{36|\alpha|}\,\frac{r^{3}}{r_{h}^{3}}\,,$ (47)

where we suppose for the moment that $\alpha>0$. We see that a negative value
of $\alpha$ would make the instability even worse, but our goal here is not
trying to solve the partial differential equations, even approximately, for
$r\to\infty$. This is because, in fact, it is clear that the term proportional
to $l^{2}$ in Eq. (45) decreases as $r$ increases, whereas the opposite
happens for $\mu^{2}_{Q}$. Evidently, this increase in the mass of the
perturbation $Q$ cannot approach infinity. In fact, we would expect that the
theory reaches an EFT cutoff so that it needs to be ultraviolet (UV)
completed. In other words, this behavior implies that the asymptotically flat
limit for the theory needs a UV completion. Let us call this cutoff mass scale
$M\lesssim M_{\rm Pl}$. Then, in this case, the instability really occurs only
if

$\frac{r_{h}^{2}}{36|\alpha|}\,\frac{r^{3}}{r_{h}^{3}}\ll M^{2}\,.$ (48)

This implies that, if the instability occurs at about $r=\beta r_{h}$, then we
require that

$M_{\rm Pl}^{2}\gtrsim
M^{2}\gg\beta^{3}\frac{r_{h}^{2}}{36|\alpha|}\,\qquad{\rm
or}\qquad|\alpha|\gg\frac{\beta^{3}}{36(4\pi)^{2}}\,\frac{M_{S}^{2}}{M_{\rm
Pl}^{4}M^{2}}\,,$ (49)

where $M_{S}$ is the BH mass related to $r_{h}$ as $r_{h}=M_{S}/(4\pi M_{\rm
Pl}^{2})$. If the inequality (49) does not hold, we should expect that the
solution is outside the EFT domain. In this case, it should either be replaced
by another solution (maybe time-dependent) or instead, the spherically
symmetric description of gravity in this theory already requires an
appropriate UV completion.

At the same time, we require that the BH solution (14), if it is inside the
EFT domain, should not be so different from the standard Schwarzschild
solution. This statement holds true if

$\frac{54|\alpha|r_{h}^{2}}{r^{6}}\ll\frac{r_{h}}{r}\,,$ (50)

in particular for $r=\beta r_{h}$. This condition leads to

$|\alpha|\ll\frac{\beta^{5}}{54(4\pi)^{4}}\,\frac{M_{S}^{4}}{M_{\rm
Pl}^{8}}\,.$ (51)

Putting these two requirements on $\alpha$ together, we find

$\alpha_{\rm min}^{r=\beta
r_{h}}\equiv\frac{\beta^{3}}{36(4\pi)^{2}}\,\frac{M_{S}^{2}}{M_{\rm
Pl}^{4}M^{2}}\ll|\alpha|\ll\alpha_{\rm max}^{r=\beta
r_{h}}\equiv\frac{\beta^{5}}{54(4\pi)^{4}}\,\frac{M_{S}^{4}}{M_{\rm
Pl}^{8}}\,.$ (52)

This condition makes sense only if $\alpha_{\rm min}^{r=\beta
r_{h}}\ll\alpha_{\rm max}^{r=\beta r_{h}}$, that is $M\gg(M_{\rm
Pl}/M_{S})M_{\rm Pl}$, i.e. $M\gg 5.3\times 10^{-21}$ GeV for a solar mass BH.
A larger BH would make this bound weaker. Hence, in the following, we will
consider the range $(M_{\rm Pl}/M_{S})M_{\rm Pl}\ll M\lesssim M_{\rm Pl}$. It
should be noted that this relation depends on the BH mass. Expressing the
dimensionful coupling $\alpha$ in terms of a length scale $\ell_{\alpha}$ as
$\alpha=\ell_{\alpha}^{4}$ and saturating the cutoff mass scale to $M\simeq
M_{\rm Pl}$ in order to be able to discuss a more concrete example, the
inequality (52) translates to

$\frac{\beta^{3/4}}{2\sqrt{6\pi}}\left(\frac{M_{S}}{M_{\rm
Pl}}\right)^{1/2}\ell_{\rm
Pl}\ll|\ell_{\alpha}|\ll\frac{\beta^{5/4}}{4\pi(54)^{1/4}}\frac{M_{S}}{M_{\rm
Pl}}\ell_{\rm Pl}\,,$ (53)

where $\ell_{\rm Pl}=1/M_{\rm Pl}=8.1\times 10^{-35}$ m is the reduced Planck
length. If we consider a BH with $M_{S}=M_{\odot}=2.0\times 10^{30}$ kg and
$\beta=2$, for instance, the inequality (53) corresponds to $3.4\times
10^{-16}~{}{\rm m}\ll|\ell_{\alpha}|\ll 2.6\times 10^{3}$ m. For
$|\ell_{\alpha}|$ close to the upper limit of (53), the cubic curvature terms
can give rise to appreciable deviation from BHs in GR in the vicinity of the
horizon ($r_{h}\simeq 3\times 10^{3}$ m). For increasing $M_{S}$ and $\beta$,
both the upper and lower limits of $|\ell_{\alpha}|$ tend to be larger.

Now, if the relation (52) or (53) holds, then we can trust the solution as a
valid EFT, but it would be unstable, because of the negative angular squared
speed of propagation. Indeed, this instability would take place if

$\frac{l^{2}}{2r^{2}}\gg\mu_{Q}^{2}\,,\qquad{\rm or}\qquad
l^{2}\gg\frac{r^{5}}{18|\alpha|r_{h}}\gg 1\,.$ (54)

Assuming also that $l^{2}/(2r^{2})\ll M^{2}\lesssim M_{\rm Pl}^{2}$, we find

$3\,\frac{\alpha_{\rm max}^{r=\beta r_{h}}}{|\alpha|}\ll
l^{2}\ll\frac{\beta^{2}}{8\pi^{2}}\frac{M_{S}^{2}\,M^{2}}{M_{\rm Pl}^{4}}\,.$
(55)

These last inequalities are compatible with the conditions (52).

We do not find it interesting to discuss the behavior of the BH solution in
the limit $r\to\infty$, as in this case, we soon hit the EFT breaking scale,
and hence the analysis would lose its validity.

After the initial submission of this paper, we noticed that Bueno et al. put a
paper on the arXiv claiming that the BH instability can be avoided within the
EFT regime of ECG [56]. First of all, we would like to stress that, unlike the
discussion given above, the EFT approach taken in Ref. [56] means that the
cubic Lagrangian $\ell_{\alpha}^{4}{\cal P}$ is always suppressed relative to
the Einstein-Hilbert term. In this case, the cutoff mass scale $M$ for the
validity of the theory is related to the cubic coupling constant
$\ell_{\alpha}^{4}$ as $M=1/\ell_{\alpha}$. As also the authors admit, the
difference in our approach consists of relaxing this bound and considering
instead general values for the cutoff $M$. In this sense, our approach is more
general, but it is true, on the other hand, that our bound does not apply to
all the parameter spaces in ECG theories. For the choice $M=1/\ell_{\alpha}$,
the condition (48) for the occurrence of Laplacian instability translates to
$r^{3}\ll 36\ell_{\alpha}^{2}r_{h}$. The EFT approach requires that the
corrections induced by the cubic Lagrangian are small outside the BH horizon.
Since $\ell_{\alpha}\ll r_{h}$ in this case, the inequality (48) holds in the
regime $r\ll r_{h}$, i.e., deep inside the horizon.

As we already mentioned at the beginning of this section, unlike Ref. [56], we
have not dealt with the ECG as an EFT where the cubic Lagrangian
$\ell_{\alpha}^{4}{\cal P}$ is always suppressed relative to $R$. In this
case, the cutoff mass scale $M$ in Eq. (48) is not necessarily restricted to
taking values of the order of $1/\ell_{\alpha}$. Then, the Laplacian
instability is present outside the horizon ($\beta\geq 1$) for the coupling
constant $\ell_{\alpha}$ in the wide range of (53). In other words, whenever
the cubic Lagrangian starts to be comparable to the Ricci scalar, the problem
of Laplacian instability emerges besides the ghost and strong coupling
problems. Since we always need to be in the EFT domain with
$|\ell_{\alpha}^{4}{\cal P}|\ll|R|$ to avoid this pathological behavior, it is
not possible to deal with any nonperturbative phenomenon in the vicinity of
the horizon [48]. Whenever nonperturbative effects come into play in BH
physics, we hit the aforementioned problems due to the breakdown of EFT.

##  IV Conclusions

ECG is a cubic-order gravitational theory constructed to share the massless
graviton spectrum similar to that in GR on a maximally symmetric background.
Since the Lagrangian ${\cal P}$ is beyond the domain of Lovelock theories, the
field equations of motion contain derivatives higher than second order in
general, curved geometries. At the same time, this suggests that there should
be additional propagating DOFs to those in GR for the spacetime different from
the maximally symmetric background. On the SSS background given by the line
element (4), we studied the propagation of odd-parity perturbations and the
resulting linear stability conditions of propagating DOFs. In this procedure,
we did not deal with the ECG as a rather trivial EFT where the cubic
Lagrangian is always strongly suppressed relative to the Einstein-Hilbert
term.

At the background level, the metric components of SSS vacuum solutions in ECG
obey the third-order single differential Eq. (12) with $f=h$. At large
distances away from the horizon, the SSS BH solution is approximately given by
Eq. (14). On using the large-distance expansion (16) of metrics without
imposing the condition $f=h$, we also obtain the same result as Eq. (14). This
shows the universality of SSS BH solutions with $f=h$.

In Sec. III, we expanded the action (17) in ECG up to quadratic order in odd-
parity perturbations to see the propagation of dynamical DOFs. There are some
higher-order derivatives appearing as the form (22) in the action. By
introducing a gauge-invariant Lagrange multiplier
$\chi=\ddot{W}-{\dot{Q}}^{\prime}+2\dot{Q}/r$, one can express Eq. (22) as the
equivalent Lagrangian (23). Then, the total Lagrangian is expressed by the
form (24), which consists of three dynamical perturbations $\vec{{\cal
X}}=(W,\chi,Q)$. In comparison to GR, which contains only one dynamical
perturbation $\tilde{\chi}=\dot{W}-Q^{\prime}+2Q/r$ in the odd-parity sector,
there are two more propagating DOFs in ECG.

For high frequencies and large radial and angular momenta, we derived the
perturbation equations in the form (29) by assuming the solution as the WKB
form (25). The ghosts are absent under the three conditions (33), but the
second condition is always violated for $\alpha\neq 0$. This is the outcome of
a kinetic mixing of $\dot{\chi}$ and $\dot{W}$ without the $\dot{\chi}^{2}$
term. We showed that all the radial propagation speeds reduce to 1 for the BH
solution (14). Along the angular direction, the squared propagation speed of
the perturbation $Q$ is negative ($c_{\Omega 3}^{2}=-1/2$) for the coupling
constant $\alpha$ in the range (52). Thus, the SSS BH in ECG with unsuppressed
higher-order curvature terms is excluded by both ghost and Laplacian
instabilities.

The results found in this paper are analogous to what happens in the
anisotropic cosmological background in CECG [46]. In this case, there are also
three propagating DOFs, with the appearance of ghosts and tachyonic
instabilities on the quasi-de Sitter background with small anisotropies. We
also note that a strong coupling problem arises for both ECG and CECG due to
the degeneracy of propagating DOFs around the maximally symmetric background
[44, 48]. These pathologies may be avoided by restricting ourselves to the EFT
domain in which the cubic Lagrangians are strongly suppressed relative to the
Einstein-Hilbert term, but the problems of ghosts, instabilities, and strong
couplings manifest themselves in the regime where the cubic curvature terms
become comparable to $R$. These results suggest that going beyond the Lovelock
domain with unsuppressed cubic curvature terms can cause problems in the
spacetime geometry different from the maximally symmetric background. It will
be of interest to explore further whether the similar property persists or not
for higher-order gravitational theories containing quadratic and quartic
curvature terms.

## Acknowledgements

We thank Jose Beltrán Jiménez and Alejandro Jiménez-Cano for useful
correspondence. The work of ADF was supported by the Japan Society for the
Promotion of Science Grants-in-Aid for Scientific Research No. 20K03969 and by
grant PID2020-118159GB-C41 funded by MCIN/AEI/10.13039/501100011033. ST is
supported by the Grant-in-Aid for Scientific Research Fund of the JSPS No.
22K03642 and Waseda University Special Research Project No. 2023C-473.

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