#### Mass-induced instability of SAdS black hole in Einstein–Ricci cubic gravity

Eur. Phys. J. C
Mass-induced instability of SAdS black hole in Einstein-Ricci cubic gravity
Yun Soo Myung 0
0 Institute of Basic Sciences and Department of Computer Simulation, Inje University , Gimhae 50834 , South Korea
We perform the stability analysis of Schwarzschild-AdS (SAdS) black hole in the Einstein-Ricci cubic gravity. It shows that the Ricci tensor perturbations exhibit unstable modes for small black holes. We call this the mass-induced instability of SAdS black hole because the instability of small black holes arises from the massiveness in the linearized Einstein-Ricci cubic gravity, but not a feature of higher-order derivative theory giving ghost states. Also, we point out that the correlated stability conjecture holds for the SAdS black hole by computing the Wald entropy of SAdS black hole in Einstein-Ricci cubic gravity.
1 Introduction
The study of higher-derivative gravity theories has attracted
critical attention in quantum gravity. Stelle’s seminal work [1]
has shown that the fourth-order gravity is renormalizable
and has a finite Newtonian potential at origin. However, this
gravity belongs to a nonunitary theory because it has a
massive spin-2 pole with negative residue which could be
interpreted as a state of negative norm (ghost). It turns out that
the infinite derivative gravity (non-local gravity) is
ghostfree and renormalizable around the Minkowski spacetime
background when one chooses the exponential form of an
entire function [2,3]. The most general class of theories that
are ghost-free on any background is Lovelock gravity [4]
whose terms of order k in the curvature are topological in
d = 2k dimensions and vanish identically for d < 2k.
Quasitopological gravities provide additional example in higher
dimensions than four [5–7], but all quasi-topological
theories are trivial in four dimensions.
Recently, the Einsteinian cubic gravity of LEC = R −
2 0 − λP/6 with P Riemann polynomials was introduced
to indicate that it is neither topological nor trivial in four
dimensions [8]. It was shown that black hole solutions of this
gravity have a number of interesting properties [9–11], but
these belong to either numerical or approximate solutions.
That is, the Einstein equation cannot be solved analytically.
An obstacle to studying these black holes is the lack of an
analytic solution. At this stage, we remind the reader that at
the critical points, the Einsteinian cubic gravity admits AdS
black boles in four and five dimensions [12].
On the other hand, it is important to note that Ricci
polynomials are much more manageable, compared to Riemann
polynomials. The Ricci cubic gravity [13] can be composed
of three terms from six cubic invariants in four dimensions,
which has a similar property to the Ricci quadratic gravity
(fourth-order gravity). It is known that the linearized theory
of any higher-order gravity around a maximally
symmetric background can be mapped into the linearized theory of
fourth-order gravity [14]. This may imply that if one performs
the linear stability analysis for a black hole obtained from a
general quadratic gravity, these results could apply to
analyzing the stability of the same black hole obtained from any
higher-order gravity. A crucial benefit of Ricci cubic gravity
is that the Schwarzschild black hole to Einstein gravity is a
solution to this theory. In this case, the solution represented
by mass r0 = 2M describes the gravitational field outside
of a static matter distribution, because its linearized theory
reduces to that of Einstein gravity. It was proved that the only
theories susceptible of admitting solutions with gtt grr = −1
and representing the exterior field of a spherically symmetric
distribution of mass are those that only propagate a
massless spin-2 mode with 2 DOF (degrees of freedom) on the
vacuum [15]. We note that the Schwarzschild-AdS (SAdS)
black hole to Einstein gravity with a cosmological constant is
a solution to the theory, but the SAdS is not a solution to the
Einsteinian cubic gravity. Importantly, one can construct a
covariant linearized gravity on the SAdS black hole in Ricci
cubic gravity, but the covariant linearized theory of Riemann
polynomials is allowed only on a maximally symmetric
vacuum of AdS4 spacetimes. However, the SAdS solution does
not describe the exterior field of a spherically symmetric
distribution of mass in Einstein–Ricci cubic gravity because its
linearized equation (
29
) describing 5 DOF could not reduce
to that of Einstein gravity.
Also, the (in)stability of SAdS black hole is known for
fourth-order gravity [16] and thus, this result will be
compared to that of Ricci cubic gravity. These are a few of reasons
why we wish to introduce the Ricci cubic gravity as a
higherorder gravity in the study of a black hole.
If a SAdS black hole is obtained from the Ricci cubic
gravity, one has to ask what this all mean for “physical black
hole”? A physical black hole can be selected by the
stability analysis. If it is stable against the metric perturbation,
one accepts it as a physical black hole. If not, one has to
reject it. This is a long-standing issue since 1957. First of
all, the linearized equation around the Schwarzschild black
hole is given by δ Rμν (h) = 0 in Einstein gravity. Then,
the metric perturbation hμν is classified depending on the
transformation property under parity, namely odd and even.
Using the Regge-Wheeler [17] and Zerilli gauge [18], one
obtains two distinct perturbations: odd and even parities. It
turns out that the Schwarzschild black hole is stable against
the metric perturbation [19,20]. Investigating the stability
analysis of the SAdS black hole in Einstein gravity with a
cosmological constant, one might use the linearized Einstein
equation δGμν (h) = 0. It turns out to be stable by following
the Regge-Wheeler prescription [21,22].
We would like to mention that the Regge-Wheeler
prescription is limited to the second-order gravity. Thus, one
could not implement the Regge-Wheeler prescription to
perform the stability analysis of black hole found in higher-order
gravity. For a higher-order gravity, Whitt [23] has argued that
provided both massive spin-0 and spin-2 gravitons are
nontachyonic, the Schwarzschild black hole is classically stable
in fourth-order gravity when using the linearized Ricci
tensor equation. In this case, the linearized Ricci tensor could
represent a massive spin-2 field. Considering an auxiliary
field formulation for decreasing a fourth-order gravity to a
second-order theory of gravity [24], one found that the
linearized equation for Ricci tensor δ Rμν is transformed exactly
into that for auxiliary field ψμν . Hence, one does not worry
about the ghost (an unhealthy massive spin-2 field) problem
arising from the fourth-order gravity because the linearized
Ricci tensor δ Rμν as a healthy massive spin-2 field
satisfies a second-order equation [25]. Visiting this stability issue
again, it has shown that the small black hole in Einstein-Weyl
gravity is unstable against s(l = 0)-mode Ricci tensor
perturbation, while the large black hole is stable against s-mode
perturbation [26]. Actually, this was performed by comparing
the linearized Ricci tensor equation with the linearized
metric equation around the five-dimensional black string where
the Gregory-Laflamme instability appeared [27].
In addition, one observes that there was a close
connection between thermodynamic instability and classical
[Gregory-Laflamme] instability for the black strings/branes.
This Gubser- Mitra proposal [28] was referred to as the
correlated stability conjecture (CSC) [29]. The CSC states that
the classical instability of a black string/brane with
translational symmetry and infinite extent sets in precisely, when
the corresponding thermodynamic system becomes
thermodynamically unstable (that is, either Hessian matrix1 < 0
or heat capacity < 0). Here the additional assumption of
translational symmetry and infinite extent has been added
to ensure that finite size effects do not spoil the
thermodynamic nature of the argument and to exclude a well-known
case of the Schwarzschild black hole. A famous example of
holding in the CSC is the five-dimensional black string. It is
known that the Schwarzschild black hole is classically
stable, but thermodynamically unstable because of its negative
heat capacity. Also, the SAdS black hole is stable against the
metric perturbation, whereas the small (large) black hole with
r+ < r∗ = /√3(r+ > r∗) is thermodynamically unstable
(stable) because of negative (positive) heat capacity.
Therefore, the last two examples show violation of the CSC and
inapplicability of s-mode perturbation because a massless
spin-2 mode perturbation starts from l = 2.
However, the situation is changed when one analyzes a
black hole found in a fourth-order gravity (a massive
gravity) where the infinite extent with translational symmetry is
absent clearly. An important thing being different from the
Einstein gravity is the appearance of a massive spin-2 mode.
A massive spin-2 mode allows us to define a mass squared M 2
and to analyze the black hole with s(l = 0)-mode
perturbation. Considering a setting e r0 t ei rk0 z for a black string
perturbation [27], there exists a critical wave number kc where for
k < kc(k > kc), the black sting is unstable (stable) against
metric perturbations. There is an unstable (stable) mode for
any wavelength large (smaller) than the critical wavelength
λGL = 2πkcr0 . Here, the mass M of a massive spin-2 mode
plays a role of k/r0 because the mass operator is given by
∂z . This implies that the massiveness (M 2 = 0) takes over a
black string located in z direction effectively. The
GregoryLaflamme instability is an s-wave unstable mode from the
four-dimensional perspective [25]. In this respect, the dRGT
massive gravity [30] having a Schwarzschild solution is
subject directly to an s-wave instability [31,32]. Here, we pay
our attention to an equivalence between 4D black hole in
linearized massive gravity and 5D black string in linearized
Einstein gravity from the four-dimensional perspective.
Furthermore, the Gregory-Laflamme instability condition
(massiveness) picks up the small AdS black hole with r+ <
r∗ which is thermodynamically unstable in fourth-order
gravity. It is known that the CSC holds for the SAdS black hole in
Einstein-Weyl gravity by establishing a connection between
the thermodynamic instability and the Gregory-Laflamme
instability [33]. Also, the CSC holds for the BTZ black hole
regardless of the horizon radius r+ in three-dimensional new
massive gravity. Hence it is quite interesting to check whether
the CSC holds for a black hole found in higher-order gravity
theory.
In this work, we will investigate classical instability and
thermodynamics of SAdS black holes in Einstein–Ricci
cubic gravity. Hereafter, we rename Ricci cubic gravity as
Einstein–Ricci cubic gravity for a precise definition. We
perform the stability analysis of SAdS black hole in Einstein–
Ricci cubic gravity by introducing the Gregory-Laflamme
scheme. Computing the Wald entropy, we derive other
thermodynamic quantities by making use of the first-law of
thermodynamics in Einstein–Ricci cubic gravity. Finally,
we wish to establish a connection between the
GregoryLaflamme instability and the thermodynamic instability of
SAdS black holes in α = −3β Einstein–Ricci cubic gravity.
This will provide another example for which the CSC holds.
2 Einstein–Ricci cubic gravity
We start with the Einstein–Ricci cubic gravity (ER) in four
dimensional spacetimes [13]
1
SER ≡ 16π
1
= 16π
4 √
d x
−gLER
d4x √−g κ(R − 2 0)
+(e1 R2 + e2 Rμν Rμν )R + e3 Rνμ Rρν Rμρ
with κ = 1/G the inverse of Newtonian constant, 0 the bare
cosmological constant, and (e1, e2, e3) three cubic
parameters. Here we observe from the second term of (
1
) that
the fourth-order gravity is embedded into the Einstein–Ricci
cubic gravity. From the action (
1
), the Einstein equation is
derived to be
1
Pμαβγ Rν αβγ − 2 gμν LER − 2∇α∇β Pμαβν = 0,
where the P-tensor is defined by
(
1
)
(
2
)
(
3
)
Pμνρσ = ∂∂RLμEνRρσ .
Explicitly, it takes the form
κ
Pμνρσ = 2 (gμρ gνσ − gμσ gνρ )
3e1 R2(gμρ gνσ − gμσ gνρ )
+ 2
e2
+ 2 Rαβ Rαβ (gμρ gνσ − gμσ gνρ )
e2
+ 2 R(gμρ Rνσ − gμσ Rνρ − gνρ Rμσ + gνσ Rμρ )
At this stage, we propose that a SAdS black hole solution to
the Einstein gravity,
dsS2AdS = g¯μν d x μd x ν = − f (r )dt 2 + fd(rr2) + r 2d 22
is also a solution to Eq. (
2
). Here the metric function is given
by
f (r ) = 1 − rr0 − 3 r 2,
3
= − 2
with the curvature radius of AdS4 spacetimes. The effective
cosmological constant is related to the bare cosmological
constant as
κ
− (16e1 + 4e2 + e3) 3 = κ 0.
We note that a black hole mass parameter is determined as
r +2
r0 = r+ 1 + 2
3e3 (gμρ Rνγ Rσγ − gμσ Rνγ Rργ
+ 4
−gνρ Rμγ Rσγ + gνσ Rμγ Rργ ).
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
which is not surely the horizon radius r+. Hereafter we
denote all background quantities with the “overbar”. The
background Ricci tensor and Ricci scalar are given by
R¯μν =
g¯μν , R¯ = 4 .
In this case, one notes that the background P-tensor takes a
maximally symmetric form
P¯μνρσ = 21 κ +48e1 2 +12e2 2 +3 2e3 (g¯μρ g¯νσ −g¯μσ g¯νρ ).
It is easy to show that the SAdS black hole (
5
) to the
Einstein equation of Gμν − g¯μν = 0 is also the solution to
the Einstein–Ricci cubic gravity when one substitutes (
10
)
together with (
9
) into (
2
). However, it is important to note
that the background Riemann tensor for the SAdS black hole
is not given by the AdS4-curvature tensor
R¯μνρσ = R¯μAνdρSσ4 = 3 (g¯μρ g¯νσ − g¯μσ g¯νρ ),
which means that the SAdS spacetimes (
5
) is not a maximally
symmetric vacuum.
3 Linearized Einstein–Ricci cubic gravity
To perform the stability analysis, we introduce the metric
perturbation around the SAdS black hole as
gμν = g¯μν + hμν .
Then, we may define the linearized Ricci tensor and scalar
as
δ R˜μν = δ Rμν −
hμν , δ R = δ(gμν Rμν ) = g¯μν δ R˜μν , (
13
)
where
δ Rμν = 21 ∇¯ ρ ∇¯ μhνρ + ∇¯ ρ ∇¯ ν hμρ − ∇¯ 2hμν − ∇¯ μ∇¯ ν h ,
δ R = g¯μν δ Rμν − hμν R¯μν = ∇¯ μ∇¯ ν hμν − ∇¯ 2h −
Taking the trace of Eq. (
17
) leads to the linearized Ricci
scalar equation
2 (α + 3β) ¯ δ R + [−κ +
2(α + 4β)]δ R = 0.
(
19
)
One notes that Eq. (
17
) is a coupled second-order equation
for δ R˜μν and δ R, which seems to be difficult to be solved.
One way to avoid this difficulty is to split Eq. (
17
) into the
traceless and trace parts by choosing α and β appropriately.
For this purpose, we introduce a traceless Ricci tensor as
1
δ Rˆμν = δ R˜μν − 4 g¯μν δ R, δ Rˆ = 0.
Then, Eqs. (
17
) and (
19
) lead to
α( ¯ L − 2
+ μ22)δ Rˆμν
1
= − (α + 2β) ∇¯ μ∇¯ ν − 4 g¯μν ¯ δ R,
2 (α + 3β)( ¯ − μ02)δ R = 0,
(
12
)
(
14
)
(
20
)
(
21
)
(
22
)
.
(
23
)
(
24
)
(
25
)
(
26
)
where the mass squared μ22 for spin-2 mode, and the mass
squared μ20 for spin-0 mode are given by
2
μ2 = 2
−
First of all, decoupling of all massive modes requires either
= 0 or α = β = 0. The former case corresponds to the
Ricci-flat spacetimes on which cubic curvature tensor gives
no contribution to the linearized Ricci tensor equation. On
the other hand, the latter case yields the quasi-topological
Ricci cubic gravity whose linearized equation is exactly the
same form as in the Einstein gravity with a cosmological
constant [13]. Hence, these cases lead to a linearized
secondorder gravity.
A nontrivial decoupling between traceless and trace parts
may occur when choosing a condition of α = −2β. This
case corresponds to the linearized fourth-order (Einstein–
Ricci quadratic) gravity [34] because one parameter e3 can
be represented by e1 and e2. In this case, Eqs. (
21
) and (
22
)
lead to the massive spin-2 and massive spin-0 equations,
separately,
Here the mass squared μ22 and μ20 are given by
μ22 = κ3α2 − 32 , μ02 = κ3α2 + 32 .
However, the number of degrees of freedom (DOF) for δ Rˆμν
cannot be reduced to five because the contracted Bianchi
with h = hρρ . In this case, the linearized Einstein tensor can
be written by
1
δGμν = δ R˜μν − 2 g¯μν δ R.
Introducing two new parameters α = 4e2 + 3e3 and β =
2(6e1 + e2), the linearized Einstein equation can be rewritten
compactly as
κδGμν +
1
2(3α + 4β)δ R˜μν − 2
2αg¯μν δ R
α ¯ LδGμν
−
− (α + 2β)(∇¯ μ∇¯ ν − g¯μν ¯ )δ R = 0,
where the background Lichnerowicz operators are defined
by acting on scalar and tensor, respectively,
¯ Lδ R = − ¯ δ R,
¯ Lδ R˜μν = − ¯ δ R˜μν − 2 R¯μρνσ δ R˜ ρσ
+ R¯ μρδ R˜ρν + R¯νρ δ R˜ρμ.
The linearized equation (
17
) is a second-order equation for
δGμν and δ R, but it becomes a fourth-order equation for
hμν . This is surely our expectation that the linearized theory
of any higher-order gravity is a fourth-order theory of
gravity. Here, we have a fourth-order equation (
17
) because of
the inclusion of the cosmological constant 0 in the action
(
1
). Putting = 0( 0 = 0) yields Ricci-flat spacetimes on
which cubic curvature terms give no contributions to the
linearized equations, leading to δ Rμν = 0. This is one reason
why we included the cosmological constant in the beginning
action (
1
), compared to the fourth-order gravity. In addition,
it is worth noting that Eq. (
17
) leads to (2.15) in Ref. [14]
when replacing R¯μνρσ by R¯μAνdρSσ4 in (
11
).
identity of ∇¯ μδGμν = 0 does not imply the transverse
condition,
1
∇¯ μδ Rˆμν = 4 g¯μν ∇¯ ν δ R
∇¯ μδ Rˆμν = 0
due to δ R = 0 in α = −2β Einstein–Ricci cubic gravity.
A promising choice may be done by requiring the
nonpropagation of the Ricci scalar. From Eq. (
19
), imposing
the condition of α = −3β, we have a constraint of
nonpropagating Ricci scalar
δ R = 0.
Also, one finds from (
23
) that the mass squared μ20 of
massive spin-0 blows up, which means that the massive spin-0 is
decoupled from the theory. This case corresponds to the
linearized Einstein-Weyl gravity [16,33] because the linearized
Ricci scalar is decoupled from the theory. Considering δGμν
in (
16
) together with δ R = 0, Eq. (
17
) leads to the massive
spin-2 equation for δ Rˆμν as
M22 = 2
Here, one requires M22 > 0 to avoid the tachyonic instability
of δ Rˆμν propagating on the SAdS black hole [35]. Taking into
account Eq. (
28
), the contracted Bianchi identity provides a
desired transverse condition
∇¯ μδ Rˆμν = 0.
Hence, the DOF of δ Rˆμν becomes five from the counting of
10 − 4 − 1 = 5 in α = −3β Einstein–Ricci cubic gravity.
First of all, in 0 = 0 Einstein–Ricci cubic gravity (Einstein
gravity), the linearized equation around the Schwarzschild
black hole is given by δ Rμν (h) = 0 with δ Rμν (h) in (
14
).
Then, the metric perturbation hμν is classified depending on
the transformation properties under parity, namely odd and
even. Using the Regge-Wheeler [17] and Zerilli gauge [18],
one obtains two distinct perturbations: odd with 2 DOF and
even with 4 DOF. This implies that even though one starts
with 6 DOF under the Regge-Wheleer gauge, the propagating
DOF is two for a massless spin-2 metric tensor hμν . It turned
out that the Schwarzschild black hole is stable against the
metric perturbation [19,20].
Performing the stability analysis of the SAdS black hole in
α = β = 0 Einstein–Ricci cubic gravity (quasi-topological
Ricci cubic gravity) [13], one might use the linearized
equation δGμν (h) = 0 with δGμν (h) in (
16
). It turns out to be
stable by following the Regge-Wheeler prescription [21,22,36].
In these cases, the s(l = 0)-mode analysis is not necessary
to show the stability of the Schwarzschild and SAdS black
holes because the massless spin-2 mode starts from l = 2.
4.2 SAdS black hole in Einstein-Weyl gravity
The Regge-Wheeler prescription is no longer suitable for
performing the stability analysis of the SAdS black hole in
fourth-order gravity, since it focused on a linearized
secondorder gravity. One may explore the black hole stability by
means of the second-order equation for the linearized Ricci
tensor δ Rμν , instead of the fourth-order equation for the
metric perturbation hμν [23,25]. The s-mode analysis is an
essential tool to detect the instability of small SAdS black
holes [26] obtained from Einstein-Weyl gravity given by
3b
LEW = κ(R − 2 ) − 2 Cμνρσ C μνρσ
with Cμνρσ the Weyl tensor. Considering thermodynamics
of the SAdS black hole in (
5
), we usually denote the small
r(l∗a)rgwei)thSAr∗dS=bla/c√k3hoanleds b2y =the−c3o/ndi.tiHoneroefrr∗+is<a rp∗o(sri+tio>n
where the heat capacity blows up [see Fig. 4]. Its linearized
equation is given by
with the mass squared of massive spin-2 mode
M 2
κ 2
= 3b + 3
κ 2
= 3b − 2 .
Here, one requires M 2 > 0 to avoid the tachyonic
instability of δGμν propagating on the SAdS black hole
background [35]. We depict M as a function of b with κ = 1
in Fig. 1. It is worth noting that M 2 is zero at b = b∗ =
2/6 = 16.6, which corresponds to the critical gravity. The
tachyon-free condition implies the allowed range for b as
0 < b < b∗.
However, M 2 > 0 is not a necessary and sufficient condition
to guarantee a stable SAdS black hole. If M 2 < 0(b > b∗),
one does not need to consider a further analysis because it
implies the tachyonic instability.
(
32
)
(
33
)
(
34
)
(
35
)
Here, one has δ R = 0 which means that there is no
massive spin-0 (Ricci scalar) propagation in Einstein-Weyl
gravity. Therefore, the linearized Einstein tensor δGμν
satisfies the transverse-traceless condition of ∇¯ μδGμν = 0
and δG = − δ R = 0. Hence, its DOF is five from
counting of 10 − 1 − 4 = 5. The even-parity metric
perturbation in Einstein gravity is used for a single s-mode
analysis in the Einstein-Weyl gravity. Its form is given by
δGtt , δGtr , δGrr , and δGθθ as displayed in the matrix form
⎛ δGtt (r ) δGtr (r )
δGeμν = e t ⎜⎜⎝ δGt0r (r ) δGr0r (r )
0 0
0
0
δGθθ (r )
0
0 ⎞
0
0
sin2 θ δGθθ (r )
Even though one starts with 4 DOF, they are related to each
other when one uses the Bianchi identity of ∇¯ μδGμν = 0
together with δG = −δ R = 0. Hence, we derive one
decoupled second-order equation for δGtr ,
A(r ; r0, , 2, M 2) ddr22 δGtr + B ddr δGtr + C δGtr = 0,
where A, B and C were given by (
20
) in [33,37]. See Fig. 2
that is obtained by solving Eq. (
37
) numerically. We note
that the small black holes of horizon radii r+ = 1, 2, 4
correspond to r0 = 1.01, 2.08, 4.64 with = 10, respectively.
M ends up at M c ≡ O(
1
)/r0|r+=1,2,4 = 0.84, 0.40, 0.18
whose horizontal lines appear in Fig. 1. From the
observation of Fig. 2 with O(
1
) 0.85, we find unstable modes for
given r+ = 1, 2, 4 as
⎟⎟ .
⎠
(
36
)
(
37
)
(
38
)
0 < M < M c =
O(
1
)
.
r0
As the horizon size r+ increases, the instability region
becomes narrow and narrow. We call this instability as
the Gregory-Laflamme instability [27,37] because the
fourdimensional linearized equation for hμν around the
fivedimensional black string background leads to (
33
) when
replacing hμν by δGμν . We check that for r+ = 6 > r∗ =
5.7, the maximum value of is less than 10−4, which implies
that there is no unstable modes for large black hole with
r+ > r∗. The case of M = 0 yields the critical gravity
avoiding massive spin-2 and spin-0 modes when choosing
the transverse-traceless gauge for hμν [35]. At the critical
point, the massless spin-2 modes have zero energy whereas
the massive spin-2 modes are replaced by the log modes.
The presence of log modes implies another instability of the
SAdS black hole in critical gravity.
For given r+ = 1, 2, 3, from Figs. 1 and 2, the stable
condition of the SAdS black hole in Einstein-Weyl gravity is
given by
M > M c.
(
39
)
It show clearly that the Gregory-Laflamme instability of
small black holes in the Einstein-Weyl gravity is due to the
massiveness of M ∈ (0, M c), but not a feature of fourth-order
gravity giving ghost states. Taking into account the number
of degrees of freedom (DOF), it is helpful to show that the
SAdS black hole is physically stable in the Einstein
gravity [21,22], whereas the small SAdS black hole is unstable in
the Einstein-Weyl gravity. The number of DOF of the metric
perturbation is two in the Einstein gravity, while the number
of DOF of massive spin-2 δGμν is five in the Einstein-Weyl
gravity. The s(l = 0)-mode analysis of the massive spin-2
with five DOF shows the Gregory-Laflamme instability. The
s-mode analysis is useful for handling the massive spin-2
mode in the Einstein-Weyl gravity, but is not suitable for the
massless spin-2 mode in the Einstein gravity.
4.3 SAdS black hole in Einstein–Ricci cubic gravity
First of all, we consider the α = −2β Einstein–Ricci cubic
gravity. We wish to solve Eq. (
24
) for the traceless Ricci
tensor δ Rˆμν . Actually, it is observed that Eq. (
24
) becomes
Eq. (
33
) when substituting δ Rˆμν and μ22 by δGμν and M 2.
However, we note that δGμν is a transverse-traceless tensor
in Einstein-Weyl gravity, while δ Rˆμν is a traceless tensor in
the α = − 2β Einstein–Ricci cubic gravity. Hence, the DOF
of δGμν are five, whereas the DOF of δ Rˆμν is nine. The
nontransversality for δ Rˆμν does not make a further progress on
the s-mode analysis of stability.
Now, let us consider the α = − 3β Einstein–Ricci cubic
gravity. Its linearized equation is given by Eq. (
29
) with the
mass squared (
30
). Eq. (
29
) becomes Eq. (
33
) exactly when
substituting δ Rˆμν and M22 by δGμν and M 2. Importantly, the
traceless Ricci tensor δ Rˆμν satisfies the transverse condition
(
31
). Hence, its DOF is determined to be five as for δGμν in
Einstein-Weyl gravity. At this stage, the even-parity metric
perturbation could be chosen for a single s-mode analysis in
the α = − 3β Einstein–Ricci cubic gravity and whose form
is given by
⎛ δ Rˆtt (r ) δ Rˆtr (r )
δ Rˆ μeν = e t ⎜⎜ δ Rˆtr (r ) δ Rˆrr (r )
⎜ 0 0
⎝ 0 0
0
0
δ Rˆθθ (r )
0
0 ⎞
0 ⎟⎟ .
0 ⎟
sin2 θ δ Rˆθθ (r ) ⎠
Once one starts with 4 DOF, they are related to each other
when using the Bianchi identity of ∇¯ μδ Rˆμν = 0 together
with δ Rˆ = 0. Thus, we derive one decoupled second-order
equation for δ Rˆtr ,
A(r ; r0, , 2, M22) ddr22 δ Rˆtr + B
d
dr δ Rˆtr +C δ Rˆtr = 0, (
41
)
where A, B and C were given by (
20
) in [33]. A physical
mode of δ Rˆtr grows exponentially in time as e t with > 0,
spatially vanishes at the AdS infinity, and regular at the future
horizon [37]. Here M2 is the mass of massive spin-2 mode
given by
M2 =
2 1
3α − 2
with κ = 1. We depict M2 as a function of α in Fig. 3. The
massive spin-2 mass M2 is zero at α = α∗ = 4/3 = 3333,
where the critical gravity appears. We require M22 > 0 to
avoid the tachyonic instability of δ Rˆμν propagating on the
(
40
)
(
42
)
As the horizon size r+ increases, the instability region
starting from the origin becomes narrow and narrow. For given
r+ = 1, 2, 4, three horizontal lines M2c which are ending
points split unstable (M2 < M2c) and stable (M2 > M c)
2
black holes.
It indicates that the Gregory-Laflamme instability of small
black holes in the α = −3β Einstein–Ricci cubic
gravity is due to the massiveness (0 < M2 < M c) of massive
2
spin-2 mode, but not a feature of higher-order gravity
giving a ghost. This ghost may appear only when expressing
the linearized equation (
29
) in terms of the metric
perturbation hμν , giving seven DOF. Here the massive ghost (an
unhealthy massive spin-2 mode) does not appear because we
used the linearized Ricci tensor δ Rˆμν to represent a healthy
SAdS black hole background [35]. The tachyon-free
condition implies the allowed range for α as
If M22 < 0(α > α∗), we do not need to perform a further
analysis for the stability because it indicates the tachyonic
instability.
It is emphasized that M22 > 0 is not a necessary and
sufficient condition to obtain a stable SAdS black hole. We need
to follow Gregory-Laflamme scheme to distinguish between
stable and unstable black holes by solving Eq. (
41
)
numerically. Observing Fig. 2 when replacing M by M2, we note that
M2 ends up at M c
2 ≡ O(
1
)/r0|r+=1,2,4 = 0.84, 0.40, 0.18
with O(
1
) = 0.85 whose horizontal lines appear in Fig. 3.
Here, we find unstable modes for small black holes with
r+ = 1, 2, 4 as
0 < M2 < M2c =
O(
1
) .
r0
massive spin-2 mode. However, this does not mean that our
perturbation analysis misses the ghost instability. The ghost
is present in the spectrum of the theory because the theory
provides a fourth-order linearized equation when
expressing in terms of hμν . The matter is how to represent a massive
spin-2 mode propagating on the SAdS black hole spacetimes.
Expressing the linearized equation in term of the linearized
Ricci tensor instead of the metric perturbation, it becomes
a second-order linearized equation. This is one tip to
handle a fourth-order linearized gravity. Considering an
auxiliary field formulation for decreasing a fourth-order
gravity to a second-order theory of gravity [24], one found that
the linearized equation for Ricci tensor δ Rμν is transformed
exactly into that for auxiliary field ψμν . Using an auxiliary
field enables one to distinguish the perturbation related to a
massive ghost. Furthermore, we wish to point out that the
linearized Ricci tensor could represent a massive spin-2 mode
with five DOF [38,39]. We stress here that the
GregoryLaflamme instability based on δ Rμν has nothing to do with
the ghost instability and it reflects a feature of massive gravity
described by the linearized equation (
29
).
5 Wald entropy and black hole thermodynamics
It is known that the correlated stability conjecture proposed
by Gubser-Mitra [28] does not hold for the SAdS black hole
found in Einstein gravity, but it holds for the SAdS black hole
found in Einstein-Weyl gravity [33]. In order to confirm the
classical instability found in the previous section, one has to
explore the thermodynamic property of the SAdS black hole
obtained from the Einstein–Ricci cubic gravity.
5.1 Einstein-Weyl gravity
We start with the Einstein-Weyl gravity because the
thermodynamic quantities of the SAdS black hole was
completely computed by employing the Abbot-Deser-Tekin
method [40]. It was well-known that the Wald entropy of
Einstein-Weyl gravity (
32
) is given by
SWEW
A
= 4 1 + 2b
6b
= 1 − 2 SBH
with SBH = πr +2 the Bekenstein-Hawking entropy of the
SAdS black hole in the Einstein gravity. The other
thermodynamic quantities of mass, heat capacity, and free energy
are given by [33,40]
M EW = 1 − 6b2
F EW = 1 − 6b2
(46)
(47)
(48)
(49)
Fig. 4 Plot of heat capacity CSAdS with√l = 10 in Einstein gravity. The
heat capacity blows up at r+ = r∗ = / 3 = 5.7. The thermodynamic
stability is based on the sign of heat capacity. The small (unstable) black
hole with r+ < r∗ is defined by the negative heat capacity, whereas the
large (stable) black hole with r
capacity. This picture persists to+C>EWr∗wiitshdbefi<ned2/b6yatnhdeCpoα=si−ti3vβe wheitaht
α < 4/3
where the thermodynamic quantities including the Hawking
temperature for a SAdS black hole in Einstein gravity take
the forms
MSAdS = 21 r+ + r +32 , CSAdS = 2πr +2 33rr +22 +
+ −
2
2 ,
1 r 3 1 1
FSAdS = 4 r+ − +2 , TH = 4π r
+
where “d” denotes the differentiation with respect to the
horizon radius r+ only.
We briefly sketch the thermodynamic stability of the SAdS
black hole found in Einstein-Weyl gravity [33]. First we
consider the case of b < 2/6(M 2 > 0) which is dominantly
described by the Einstein-Hilbert term. Since the heat
capacity CSAdS blows up at r+ = r∗ = /√3 = 5.7 [see Fig. 4], we
divide the black hole into the small black hole with r+ < r∗
and the large black hole with r+ > r∗. We know that the
small black hole is thermodynamically unstable because the
heat capacity C EW < 0, while the large black hole is
thermodynamically stable because C EW > 0. For the other case
of b > 2/6(M 2 < 0) which is dominantly described by
the Weyl term, the situation reverses. The small black hole is
thermodynamically stable because C EW > 0, while the large
black hole is thermodynamically unstable because C EW < 0.
In this case, the mass squared of massive spin-2 is negative,
which implies the tachyonic instability. So, this case should
be excluded from the consideration. We would like to
mention that there is no connection between classical stability
and thermodynamic instability for small SAdS black hole in
Einstein gravity. However, let us see how things are improved
in Einstein-Weyl gravity. For a small black hole with r+ < r∗
and b/ 2/6, the heat capacity is negative which means that
it is thermodynamically unstable. On the other hand, we
observe from (
38
) that a small black hole with r+ < r∗ is
unstable against the s-mode massive spin-2 perturbation. The
Gregory-Laflamme instability condition picks up the small
SAdS black hole which is thermodynamically unstable in
Einstein-Weyl gravity. This implies that the CSC [28] holds
for the SAdS black hole found in Einstein-Weyl gravity.
5.2 Einstein–Ricci cubic gravity
First of all, we wish to compute the Wald entropy of the
SAdS black hole in Einstein–Ricci cubic gravity. The Wald
entropy is defined by the following integral performed on
2-dimensional spacelike bifurcation surface [41–43]:
1
SW = − 8
1
= − 8
δLER
δ Rμνρσ
(0)
μν ρσ d V22
P¯ μνρσ μν ρσ d V22,
where d V22 = r 2 sin θ dθ dφ is the volume element on and
μν is the binormal vector to normalized as μν μν = − 2.
We note that the superscript (0) denotes that the functional
derivative with respect to Rμνρσ is evaluated on-shell. The
background (on-shell) P-tensor P¯ μνρσ is given by (
10
). Now,
the Wald entropy takes the form
1
SW = 4
1 + (α + 4β) 2 r +2 sin θ dθ dφ
A
= 4 1 + (α + 4β) 2
with A = 4πr +2 and κ = 1/G = 1. In the case of α = −3β
Einstein–Ricci cubic gravity, the Wald entropy takes the form
However, we do not know the other thermodynamic
quantities of the SAdS black hole in Einstein–Ricci cubic gravity.
We propose that the first-law should be satisfied in α = −3β
Einstein–Ricci cubic gravity as
d M α=−3β = THd SWα=−3β .
0
r+
Using (54) together with (52) and (53), we derive the mass
of the SAdS black hole
M α=−3β (r+) =
dr+TH(r+)d SWα=−3β (r+)
Now we are in a position to mention the thermodynamic
stability of the SAdS black hole in α = − 3β Einstein–Ricci
cubic gravity. First, we consider the case of α < 4/3 <
1(M22 > 0) which is dominantly described by the
EinsteinHilbert term. Since the heat capacity C α=−3β blows up at
r+ = r∗ = /√3 = 0.57 [see Fig. 4], we divide still
the black hole into the small black hole with r+ < r∗ and
the large black hole with r+ > r∗. Then, it suggests that
the small black hole is thermodynamically unstable because
C α=−3β < 0, while the large black hole is
thermodynamically stable because C α=−3β > 0. For the other case of
α > 4/3(M22 < 0) which is dominantly described by
the Ricci cubic terms, the thermodynamic stability reverses.
The small black hole is thermodynamically stable because
C α=−3β > 0, while the large black hole is
thermodynamically unstable because C α=−3β < 0. However, we note that
the mass squared M22 of massive spin-2 mode is negative.
This corresponds to the tachyonic instability and thus, this
case is unacceptable.
One finds from (
44
) that for M22 > 0, a small (large)
black hole with r+ < r∗(r+ > r∗) is unstable (stable)
against the s-mode massive spin-2 perturbation δ Rˆμν . The
Gregory-Laflamme instability condition picks up the small
SAdS black hole which is thermodynamically unstable in
α = − 3β Einstein–Ricci cubic gravity. This indicates that
the CSC proposed by Gubser-Mitra [28] holds for the SAdS
black hole found in α = − 3β Einstein–Ricci cubic gravity
too. However, the other case of M22 < 0 corresponds to the
tachyonic instability and its thermodynamic stability
contradicts to the conventional one. Hence, the CSC does not hold
for M22 < 0.
6 Discussions
We would like to mention the Einstein–Ricci cubic gravity
according to the relation between α = 4e2 + 3e3 and β =
2(6e1 + 2e2).
1. α = β = 0 case. This case describes a quasi-topological
Ricci cubic gravity because its linearized equation around
the SAdS black hole reduces to that of Einstein gravity
with cosmological constant. There is no counterpart in
fourth-order gravity. The absence of massive spin-0 and
spin-2 modes implies that the fourth-order terms give
no contribution to this linearized theory as if they were
purely topological, implying that this linearized theory
is ghost-free. The SAdS black hole is stable regardless
of the horizon radius r+ in α = β = 0 Einstein–Ricci
cubic gravity because its linearized equation takes the
form of δGμν (h) = 0 without any mass terms. However,
the small black hole with r+ < r∗ is
thermodynamically unstable while the large black hole with r+ > r∗ is
thermodynamically unstable. The CSC does not hold for
the α = β = 0 Einstein–Ricci cubic gravity which is a
ghost-free theory.
2. α = − 2β case. This case allows a decoupling of
massive spin-0 mode from massive spin-0 mode. This
corresponds to the fourth-order gravity. Here δ Rˆμν is only
a traceless tensor whose DOF is nine. We note that the
absence of transversality condition ∇¯ μδ Rˆμν = 0 makes
the stability analysis of SAdS black hole difficult. Its
Wald entropy is given by SWα=− 2β = SBH[1 − α 2].
All thermodynamic quantities are similar to those for
α = − 3β Einstein–Ricci cubic gravity except
replacing [1 − α 2/3] by [1 − α 2]. For α < 4/9, the small
black hole with r+ < r∗ is thermodynamically unstable
while the large black hole with r+ > r∗ is
thermodynamically unstable. In this case, we may not discuss the CSC
because we could not explore the stability issue of the
SAdS black hole explicitly in α = −2β Einstein–Ricci
cubic gravity.
3. α = − 3β case. This is related to the Einstein-Weyl
gravity. We show by adopting the Gregory-Laflamme scheme
that the small SAdS black hole with r+ < r∗ is
classically unstable, while the large SAdS black hole with
r+ > r∗ is classically stable. It indicates clearly that
the Gregory-Laflamme instability arises from the
massiveness (0 < M2 < M c) of massive spin-2 mode, but
2
not from a feature of Einstein–Ricci cubic (fourth-order)
gravity giving ghost states. On the other hand, its
thermodynamic quantities are computed by making use of the
Wald entropy and the first-law of thermodynamics. For
the case of M22 > 0 which is dominantly described by the
Einstein-Hilbert term, one finds that the small black hole
with r+ < r∗ is thermodynamically unstable, whereas
the large black hole with r+ > r∗ is thermodynamically
stable. This shows that the CSC proposed by
GubserMitra [28] works for the SAdS black hole obtained from
the α = −3β Einstein–Ricci cubic gravity. The other
case of M22 < 0 corresponds to the tachyonic instability
and its thermodynamic stability is unacceptable. Hence,
the CSC does not hold for M22 < 0.
4. α = − 3β case with M 2
2 = 0(α = α∗). In this case,
considering the transverse and traceless gauge of ∇¯ μhμν = 0
and h = 0, the linearized Einstein tensor takes the from of
δGμν = −( ¯ L − 2 )hμν /2. Then, its linearized
equation is given by ( ¯ L − 2 )2hμν = 0 that corresponds
to the critical gravity. It turned out be an unstable theory,
even though massive spin-2 and massive spin-0 modes are
decoupled from the theory [35]. On the other hand, all
thermodynamic quantities disappear except the Hawking
temperature. Therefore, we could not discuss the CSC for
this case.
Consequently, we have obtained the SAdS black hole
solution (
5
) from the Einstein–Ricci cubic gravity (
1
). A physical
black hole could be found by performing the analysis of
stability which means a perturbation analysis based on physical
field around the black hole because the black hole is a
solution in the curved spacetimes [20, 44]. In this work, we have
shown that the α = − 3β Einstein–Ricci cubic gravity passes
the stability analysis of SAdS black hole. The α = β = 0
case provides a quasi-topological Ricci cubic gravity whose
linearized theory is just the linearized Einstein gravity,
implying that the black hole solution (
5
) becomes a physical black
hole in α = β = 0 Einstein–Ricci cubic gravity. However,
we could not establish the (in) stability for other relations
between α and β. In addition, it is worth noting that the
α = −3β Einstein–Ricci cubic gravity has a similar
property to the Einstein-Weyl gravity because any higher-order
gravity could be mapped into the quadratic gravity at the
linearized level.
Finally, if one considers the Riemann cubic gravity
(Einsteinian cubic gravity), covariant linearized gravity is
possible only in maximally symmetric vacua. Also, black hole
solutions are numerical or approximate solutions in the
Einsteinian cubic gravity and an obstacle to studying these black
hole is the lack of an analytic solution [45].
Acknowledgements This work was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korea government
(MOE) (No. NRF-2017R1A2B4002057).
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
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