On holographic entanglement entropy of Horndeski black holes
HJE
On holographic entanglement entropy of Horndeski
Elena Caceres 0 1 3 4
Ravi Mohan 0 1 3 4
Phuc H. Nguyen 0 1 2 3 4
0 College Park, MD 20742 , U.S.A
1 Austin , TX 78712 , U.S.A
2 Maryland Center for Fundamental Physics, University of Maryland , USA
3 Theory Group, Department of Physics, University of Texas , USA
4 @ , a particular case of Horndeski gravity
5 To obtain the delta function, we used the identity @
We study entanglement entropy in a particular tensorscalar theory: Horndeski gravity. Our goal is twofold: investigate the LewkowyczMaldacena proposal for entanglement entropy in the presence of a tensorscalar coupling and address a puzzle existing in the literature regarding the thermal entropy of asymptotically AdS Horndeski black holes. Using the squashed cone method, i.e. turning on a conical singularity in the bulk, we derive the functional for entanglement entropy in Horndeski gravity. We analyze the divergence structure of the bulk equation of motion. Demanding that the leading divergence of the transverse component of the equation of motion vanishes we identify the surface where to evaluate the entanglement functional. We show that the surface obtained is precisely the one that minimizes said functional. By evaluating the entanglement entropy functional on the horizon we obtain the thermal entropy for Horndeski black holes; this result clari es discrepancies in the literature. As an application of the functional derived we nd the minimal surfaces numerically and study the entanglement plateaux.
AdSCFT Correspondence; Black Holes; Gaugegravity correspondence

2
3
4
5
6
A
B
3.1
3.2
3.3
4.1
4.2
5.1
5.2
5.3
1 Introduction Review of the squashed cone method Entanglement functional for Horndeski gravity
Derivation from Wald entropy formula
3.1.1
Minimization
Rederivation by squashed cone method
Canceling divergences of the equation of motion
Comments on the thermal entropy
Black hole entropy from IyerWald formalism
Black hole entropy from conical singularity
Numerical applications
3dimensional planar black hole
3dimensional, spherical black hole
4dimensional, spherical black hole
Conclusions and future directions
Minimal surfaces in 3d
Minimal surfaces in 4d
proved to be crucial as a tool to prove the RyuTakayanagi formula itself [10], as well as
to extend the formula to cases where the bulk theory of gravity is a higher derivative or
higher curvature theory [11, 12].
In this paper, we study the extension of the RyuTakayanagi formula in a di erent
direction: the bulk gravity theory includes nonminimally coupled matter. Curiously, while
the literature on holographic entanglement in higher curvature theories (such as Lovelock
{ 1 {
gravity) is already extensive [11{17], far less attention has been paid to the case of
nonminimal coupling. And the e orts have been concentrated in couplings of the type
From the viewpoint of black hole entropy, nonminimal coupling is an interesting and
puzzling topic. The
niteness of the BekensteinHawking entropy seems to indicate that
the UVdivergence of the entanglement entropy can be absorbed into Newton's constant.
For minimal coupling, this is indeed true, but nonminimal coupling has the potential to
spoil this nice structure [18, 19].
This paper is an attempt to ll in this gap in the literature. We study a theory with
a tensorscalar coupling of the form R
Horndeski gravity, despite being discovered a few decades ago [20], fell into oblivion and
only received much attention quite recently (for a selection of recent work related to
Horndeski gravity, see for example [21{24]). For the purpose of holography, asymptotically AdS
black hole solutions of Horndeski gravity have been worked out explicitly [25{27], and that
the parameter space of the couplings is reasonably well understood in terms of causality
and stability [29, 30].
In this paper, we use the squashed cone method to derive the entanglement entropy
functional;1 we nd that it receives a contribution proportional to the gradientsquare of
the scalar eld. Determining the precise form of the entanglement functional for Horndeski
gravity is one of the results of this paper. We also show that demanding that the leading
divergence of the tranverse, zz, component of the equations of motion vanishes implies that
the entangling surface is a minimal surface. This is our second result.
The question of whether the entangling surface minimizes the functional found from
the squashed cone method is a topic of recent attention [32]. Until recently, there exist two
main arguments that this should be the case: the argument based on the divergences of the
equation of motion, and the cosmicbrane argument. Both these arguments are presented
in [11] and are based on the equation of motion in the bulk. One of the novelties introduced
in [32] is an argument based directly on the action, without going through the equation of
motion. More speci cally, [32] considers a double variation of the bulk action (one with
respect to the replica index, and a second one which preserves the strength of the conical
defect), and derives the stationarity of the entanglement surface from it.
In higher derivative theories requiring that the surface is minimal is not enough to
cancel all the divergences. This is because in such theories generically all the components of
the eld equations diverge, not only the transverse one. Furthermore, there are subleading
divergences not present in Einstein gravity. Horndeski gravity is not a higher derivative
theory but we do nd a similar pattern of divergences. It is possible that just like in higher
derivative theories these divergences cancel if we allow for a more general ansatz [34].
One of the original motivations of the present work was related to a puzzle existing in
the literature of Horndeski black holes. In [27] the authors perform a careful calculation
of the thermal entropy of some particular Horndeski black holes solutions. They use the
Wald formula, Euclidean regularization and the IyerWald formalism and the results do not
1See also [31] for a di erent approach to deriving holographic entanglement entropy via eld rede nition,
which could be applicable to Horndeski gravity.
{ 2 {
HJEP10(27)45
agree with each other. Evaluating our result for the entanglement entropy on the horizon
we are able to shed some light on this issue.
The paper is organized as follows: in section 2, we brie y review the squashed cone
method to derive entanglement entropy. In section 3, we apply this machinery to Horndeski
gravity, and analyze the divergences of the bulk equation of motion. In section 4, we
comment on the thermal entropy and the confusion found in the literature regarding this
quantity. In section 5, we nd the RT surfaces numerically and study the phase transition
from connected surface to disconnected surface in the case of spherical black holes. We
conclude in section 6. We relegate the plots of the RT surfaces to appendices A and B.
2
Review of the squashed cone method
In this section, we review the squashed cone method, an application of the replica trick to
derive holographic entanglement entropy. Besides reviewing the background material, this
section also serves to
x the notation for the rest of the paper and to lay out the basic
equations to be used later when we apply the formalism to Horndeski gravity. First, recall
the microscopic de nition of entanglement entropy:
Note that the Riemann surface Mn has a discrete Zn symmetry. If we assume that this
discrete symmetry also exists for the bulk geometry Bn, then we can consider the quotient
Bn = Bn=Zn. Since Bn has to be regular in the interior, B^n is regular except at the xed
^
{ 3 {
SEE =
Tr( log )
obtain the entanglement entropy, the replica trick tells us to rst nd the nth Renyi entropy,
de ned by:
The entanglement entropy is the analytical continuation of Sn as n ! 1: SEE = limn!1 Sn.
From the path integral representation of , we can express Sn in terms of the partition
function Zn on an appropriate nsheeted Riemann surface Mn as:
where Z1 is the original partition function. Next, we use the basic holographic relation
ZCF T = e Sbulk between the eld theory partition function ZCF T and the bulk action Sbulk.
If the boundary is taken to be Mn, then this relation reads:
for an appropriate bulk geometry Bn. Substituting into (2.3), we nd:
1
n
1
points of the Zn symmetry, which now forms a codimension2 surface around which there
is a conical de cit. Moreover, we have:
where we do not include any contribution from the conical de cit in S[B^n]. The Renyi
entropy can now be written as:
S[Bn] = nS[B^n]
n=1
=0
SEE =
Thus, we can trade the integral outside the tube for the one inside:
Again, the entropy only receives contribution from the neartip region.
The fact that the bulk action localizes to the neartip region means it is enough to work
in an approximate metric near the tip. Such a coordinate system, analogous to Riemann
normal coordinates, has been worked out in [10] and [11]. The metric takes the form:
By taking the limit n ! 1, we nd the following formula for the entanglement entropy:
where we introduced the parameter
n 1 which characterizes the strength of the
conical de cit. Unlike the original replica index n which only makes sense for integer
values, the conical de cit
can be varied continuously. To summarize: the formula above
instructs us to solve the gravity equation of motion with some conical de cit for boundary
condition, then evaluate the onshell action and extract the term
rst order in . We stress
that the onshell action does not include any contribution from the conical singularity. In
other words, we can integrate over the whole spacetime outside a thin \tube" enclosing the
conical defect, then at the end shrink the diameter of the tube to zero.
On the face of it, formula (2.8) involves an integral over the whole spacetime (the
onshell action). It turns out, however, that the entropy only receives contribution from the
region near the tip of the cone. This can be seen in several ways. One way is to argue that
the variation of the action with
involves the integral of the equation of motion over all
of spacetime, but this integral clearly vanishes. This leaves us with a boundary term near
the tip of the cone, which contributes to the entropy. Alternatively, one can also argue as
follows. Let us call the action evaluated outside the \tube" described above Sout and the
action inside the \tube" Sin. Since the total action Sin + Sout is a variation away from an
onshell con guration (due to turning on ), it must be that:
ds2 = e2A[dzdz + e2AT (zdz
zdz)2] + (hij + 2Kaijxa + Qabijxaxb)dyidyj
+ 2ie2A(Ui + Vaixa)(zdz
zz)dyi + : : :
{ 4 {
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
Here use polarlike coordinates ( ; ) or complex coordinates z = ei , z = e i for the
directions transversal to the surface, and yi for the directions along the surface. The factor
e2A is given by:
e2A = (zz)
and encodes the conical defect. Also, Kzij and Kzij is the extrinsic curvature of the surface.
Note that we expand to second order in z, z in the metric (2.11). This is su cient when
the gravity equation of motion is second order in the metric, such as Einstein gravity or
Horndeski gravity. For such a theory terms at most quadratic in z or z in the metric
contribute to the curvature (and the onshell action) at
= 0. For higher derivative
theories, it is in general necessary to keep higher order terms in z(z) in the metric (2.11).
The Riemann tensor of the metric (2.11) contains the following terms rst order in :
The rst one above diverges as a (2dimensional) delta function, and the other two diverge
as z 1.2 Also, the Ricci tensor and Ricci scalar contain the following terms rst order in :
(2.12)
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
Rzzzz
Rzizj
Rzizj Rzz = Rzz Rzz
R
4 e 2A 2(z; z)
e2A 2(x; y)
2
z
z
Kzij
Kzij
2(x; y)
z
z
Kz
Kz
Using the formula (2.10) together with the approximate metric (2.11), [11] derived a formula
for holographic entanglement entropy for an arbitrary theory of gravity:
SEE = 2
Z
d p
d y h
"
+ X
8Kzij Kzkl
q + 1
#
Here h denotes the induced metric on the surface (we will use g for the bulk metric), z
and z are complex coordinates transversal to the surface. The rst term above is identical
to the Wald entropy, except that it is not evaluated on a black hole horizon (or Killing
horizon) here. As for the second term, it is an anomalylike contribution that only arises
in theories of gravity quadratic or higher order in the curvature. For Einstein gravity, only
the rst term above contributes and gives (one quarter of) the area, in agreement with the
RyuTakayanagi formula.
Note that the squashed cone method as described above gives us a functional for
entanglement entropy, which we are supposed to evaluate on the surface that is the xed
(2)(x; y). Here x and y are
the real and imaginary parts of z, respectively.
{ 5 {
g
R
1
2 (r r
(r r
)(r r
) + (r r
)(r r
) +
)
2
R
and the one for the scalar eld is:
r (( g
G
)r
) = 0
1
2
16
1
2
)
1
2
+
1
2
G
2
point of the Zn symmetry in the bulk. In practice nding this surface from its de nition
is di cult, but  as mentioned in the introduction  there exist general arguments that
the surface is also the one minimizing the functional.3
One such argument was given
in [10] for Einstein gravity, as follows. We go back to the parent space Bn by making
the period of the angle
in the metric (2.11) 2 n instead of 2 , and extract the leading
divergence of the Ricci tensor near z = z = 0. The leading divergences of Rzz and Rzz
are as previously found in (2.17) and (2.18), but the delta function divergence of Rzz no
longer exists because the parent space is regular at the xed point of the Zn symmetry.
Thus, the zz component of Einstein equation diverges as z Kz. But, as argued by [10], we
should not expect any divergence on physical grounds. If we demand that the coe cient
of the divergence vanishes, then we nd Kz = Kz = 0. This is precisely the condition of a
minimal surface (i.e. one which minimizes the area functional).
3
Entanglement functional for Horndeski gravity
In this section, we apply the squashed cone method to Horndeski gravity. The Lagrangian
of the Horndeski theory is:4
L =
(R
2 ) +
( g
G
Here
is the inverse Newton's constant (G 1). The equation of motion for gravity is:
0 = (G
+
g )
We will derive the functional for holographic entanglement entropy in two ways: the rst
way is by using (2.20), and the second way is by going through the squashed cone method.
We will of course obtain the same answer in the end. The reason for this twopronged
derivation is as follows: the formula (2.20) was technically derived in the absence of matter
elds in [11], but is expected to apply even in the presence of matter elds. By deriving the
entanglement functional for Horndeski gravity twice, we verify that (2.20) indeed applies
and yields the same answer as the more careful derivation with the matter eld included
at the outset of the squashed cone method.
3It is highly desirable that the surface minimizes the functional, since this would immediately imply that
the surface satis es quantuminformation properties of entanglement entropy such as the concavity [35].
4A remark about convention here is in order.
We have multiplied the Lagrangian as written in the
EinsteinHilbert part of the action into (2.20), we obtain onequarter of the area times G 1.
papers [26, 27] by an overall factor of
161 . This overall factor has been chosen so that when we plug the
(3.1)
(3.2)
(3.3)
where ; = g
turned o ):
simpli es to:
g g ) ;
=
8
32
;
;
;z ;z
=
8
32
hij
;i ;j
SEE =
4
Z
d p h
d y h 1
4
hij
;i ;j i
Upon substituting the metric components into (3.4), the partial derivative @L=@Rzzzz
Upon further expanding ;
; = ;z ;z + hij ;i ;j , this further simpli es to:
The functional for holographic entanglement entropy then reads:
Since the Horndeski gravity action does not involve terms quadratic or higher order in the
curvature, only the Waldlike term in (2.20) contributes to the entropy, and the
anomalylike term does not. Let us di erentiate the Horndeski Lagrangian with respect to the
128
[g
=
(g g
g g )
; ;
g
; ; + g
; ;
g
; . We need to take the zzzz component of the expression above (in the
HJEP10(27)45
coordinate system of the metric (2.11)) and evaluate on the surface (where z = z = 0).
It is enough to truncate the metric (2.11) to zeroth order in z(z) (with the conical defect
Thus, the entropy receives a Waldlike correction proportional to the normsquared of the
gradient of the scalar eld on the surface. Equation (3.8) is one of the results of this paper.
By analogy with the RT formula, the entanglement functional is expected to be evaluated
in the surface that minimizes its value. In subsection 3.3 we will show that this is indeed
the right thing to do. More precisely, we will show that demanding that the divergences of
the equations of motion cancel yields a condition that is exactly the equation of the surface
that minimizes (3.8)!. This will be another important result of our work.
3.1.1
Minimization
Let us now explicitly minimize the functional (3.8) to derive the equation characterizing
the surface (the \surface equation"). To do this, we vary the embedding functions x (yi) of
the surface (where x
denotes the bulk coordinates and yi denotes the coordinates on the
surface), and compute the variation S of the functional due to x . When the embedding
is varied, the value of the functional is varied due to two e ects: (
1
) the change of the
induced metric, and (2) the change of the value of the scalar eld on the surface. Thus,
we have:
S =
Z
ddy
hij +
L
L
hij
{ 7 {
; ;
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
with hij and
related to x as:
hij = g
=
Next, consider the second term in (3.9). Under the z ( z), we have:
L
=
8
p
h hij DiDj
;a a
We now adopt the coordinate system x = (yi; z; z) of the metric (2.11). This coordinate
system has the nice feature that it splits into coordinates on the surface (yi) and coordinates
transversal to the surface (z, z). It is enough to consider the variations z and z (i.e. in
the normal direction), since these actually change the embedding, whereas the variations
yi merely correspond to coordinate transformations on the surface and should not change
the value of the functional. Under the variation z ( z), the rst term in (3.9) can be cast
in terms of the extrinsic curvature of the surface as:
L
hij
1
hij = 2Kaij hij
a
L
where a = z; z (see for example [17] for more details). For our speci c functional (3.8), this
becomes after some algebra:
L
hij
hij =
p
h
4G
4
together, and equating the result to zero, we nally nd the surface equation:
1
4
gravity. Note also that the horizon of a static, spherically symmetric black hole solution
also satis es the surface equation above, assuming that it is also a Killing horizon (as is the
case with more familiar black holes). Indeed, for a Killing horizon the extrinsic curvature
Kij vanishes, and spherical symmetry also implies that the scalar eld is constant on the
horizon, so that Di vanishes.
Thus, the thermal entropy of the black hole should be found by evaluating the
functional (3.8) on the horizon. An immediate interesting consequence of this is that
the thermal entropy is equal to the usual BekensteinHawking entropy (assuming again
spherical symmetry):
where A is the area of the horizon. Indeed, spherical symmetry implies that the Waldlike
correction in (3.8) simply vanishes.
Sthermal =
A
4G
{ 8 {
As pointed out in [27], however, the thermal entropy of Horndeski black holes is
surprisingly subtle, with di erent computation methods yielding di erent answers. We will
revisit this issue in section 4, but we assume for the rest of this paper that (3.16) is the
correct one. We also stress that the Waldlike correction only vanishes for the thermal
entropy, and not for entanglement entropy of a boundary subregion as in general the scalar
eld is not constant on the RyuTakayanagi surface.
Rederivation by squashed cone method
We proceed to rederive the functional (3.8) using the squashed cone method, with the
scalar eld put in at the outset. We will need to know the expansion of the scalar eld near
the surface, in the coordinate system of the metric (2.11). This expansion is dependent,
since the scalar eld has to readjust itself to the conical defect, and this
dependence has
to be fed into the onshell action to see if it gives rise to any additional term compared to
the functional (3.8). In the end, we will nd that there are no additional terms, thus the
expectation that the functional (3.8) applies even with matter elds is borne out.
Let us rst recall how the coordinate system of (2.11) is constructed. In the parent
space Bn (i.e. before the quotienting by Zn), we set up polarlike coordinate ( ~; ~) in the
plane transversal to the surface. Since the bulk has to be regular at the surface and also
has the replica Zn symmetry, any ~ dependence has to be through ~ne in~. In the quotient
B^n, we rede ne coordinates as follows:
where the coe cients
0;0, 0;1 etc. are functions of yi. Let us explain how the di erent
powers in the expansion above arise, especially the subleading terms in each of the square
brackets above. Consider for example the term with
0;1. This term comes from the power
~2(n 1) in the parent space, which is consistent with regularity and replica symmetry in
the bulk. Similarly, the power ( ~2)2n 2 gives rise to (zz)2 etc. Note that at
= 0, each
square bracket above collapses to a constant and we nd:
=
0 + zz + zz + z2 z2 + z2 z2 + zzzz : : :
The metric in terms of
and
then takes the form (2.11). For the scalar eld, we can
proceed similarly. Regularity and the replica symmetry dictate that the scalar eld near
the surface is a function of ~2 and ~ne in in the parent space Bn. In the quotient space,
we nd  to second order in z or z  a rather complicated expansion:
= [ 0;0 + 0;1(zz) + 0;2(zz)2 +: : : ]+[ z;0 + z;1(zz) + z;2(zz)2 +: : : ]z
+[ z;0 + z;1(zz) + z;2(zz)2 +: : : ]z +[ z2;0 + z2;1(zz) + z2;2(zz)2 +: : : ]z2
+[ z2;0 + z2;1(zz) + z2;2(zz)2 +: : : ]z2 +[ zz;0 + zz;1(zz) + zz;2(zz)2 +: : : ](zz)1
k=0 z;k and so on. In other words, each term in the series
at
= 0 is a whole in nite series at
= 0. This is sometimes dubbed the \splitting
problem" [32, 33].
Before proceeding with the squashed cone method, we would like to make two remarks
about the expansion (3.19) for the sake of clarity. First, the coe cients
0;0 etc. in this
expansion are taken to be independent of . This is not strictly true; for example, the
term ~nein~ in the parent space becomes nnz
(1 +
transformation (3.17) and (3.18). For general n, the
+ O( 2))z after the coordinate
corrections of the coe cients are
not ignorable. However, throughout this paper we work with the
0 (or n
1) regime,
and these
corrections are small. Moreover, even if we keep the
corrections, they will
not contribute to the entanglement entropy anyway. Note also that this discarding of the
corrections is not speci c to the scalar eld expansion: the same treatment was applied to
the metric expansion (2.11), i.e. coe cients in this expansion in general receive correction
but they have been discarded for the reasons above.
Our second remark is that the expansion (3.19) implies that the entanglement entropy
functional depends on the scalar eld through the coe cients 0;0, 0;1, z;0, z;1. This may
appear peculiar for the following reason. Our experience with the usual RyuTakayanagi
formula for Einstein gravity, as well as the JacobsonMyers functional for GaussBonnet
gravity, would suggest that the entanglement functional should be a function of the value
of the scalar eld or its derivatives at
= 0. On the other hand, each of the coe cient
0;0, 0;1 etc. does not really have an intrinsic meaning at
P
k z;k etc. do: the rst is the value of the scalar eld on the minimal surface, and the
second is the derivative of the scalar eld in the z direction on the minimal surface).
Fortunately, we nd that the entanglement functional indeed depends on the coe
cients in (3.19) only through the sums P
k 0;k =
0, P
k z;k = z etc., in other words the
functional only depends on quantities which \make sense" at
= 0. Roughly speaking, this
is because we need to look for terms rst order in
in the action. While one can extract
many terms rst order in
from the expansion (3.19), the only terms rst order in
that
matter come from the curvature at the tip of the cone, i.e. the coupling of the scalar eld
to the conical singularity. But the scalar eld that appear in such terms can be evaluated
at
= 0 since the curvature is already rst order in , therefore only the expansion (3.20)
= 0 (only the sums P
k 0;k,
really matters.
Using the scalar eld expansion together with equations (2.17), (2.16), (2.18) and (2.19)
for the singular terms of the Ricci tensor and Ricci scalar, we then nd the term rst order
in
of the nonminimal coupling part of the action:
where the subscript N M C stands for nonminimal coupling, and the superscript (
1
) means
rst order in . We can easily argue that the terms in S(
1
)
NMC containing Kz and Kz vanish.
This is because the integral is regular at the origin and we integrate over an in nitesimally
small region around the tip of the cone. Thus, we are left with only the term with the
delta function. The above then simpli es to:
0;i by ;i since the functional above is to be evaluated on the
surface z = z = 0, where those two quantities agree. By the formula (2.10), this contributes
to the entropy an amount:
SE(NEMC) =
16
Z p
hdd 2yhij ;i ;j
We now combine the contribution above to the area functional from the EinsteinHilbert
part of the action, to obtain the total functional:
SEE =
in agreement with the functional previously derived from (2.20) which in this case is
equivalent to the Wald entropy formula.
3.3
Canceling divergences of the equation of motion
In this subsection, we go back to the parent space Bn and extract the leading order
divergence in the equation of motion. We will continue to work with the metric (2.11) but
with
+ 2 n so that there is no conical singularity. Similarly, we will use the
expansion (3.19) of the scalar eld, but this expansion is now understood as an expansion in the
parent space rather than the quotient space.
The divergences of the bulk equation of motion come from two contributions: (
1
) rst,
there are divergences from the components Rzizj and Rzizj of the Riemann tensor, like in
Einstein gravity; (2) Secondly, the components rzrz
and rzrz
of the second covariant
derivative of the scalar eld also contribute divergences. This is because the Christo el
symbols zzz and zzz also contain 1=z divergences:
Therefore, the following components of r r
diverge:
z
zz
z
zz
rzrz
rzrz
z
z ;z
z ;z
(3.22)
(3.23)
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
Consider rst the zz component of the gravity equation. Term by term, we
nd the
following contributions to the =z divergence:
Gzz
)(rzr
Rz z
2 (rzrz )
1
Kzhij 0;i 0;j
Kzij 0;i 0;j
z zz
2 z z(4 zz + hij DiDj 0)
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
and all other terms in the equation of motion are nonsingular.
Let us sketch out how to obtain some of the expressions above in details. In most of
the equations above (namely, equations (3.30), (3.32) and (3.33)), the ( =z) factor comes
from the metric (through the curvature tensor or the Christo el symbols), and for the
scalar part it su ces to use the scalar eld expansion at
= 0 (equation (3.20)).5
In the case of equation (3.31), however, the divergence coming from the metric is
actually slightly sublinear, but the scalar expansion (3.19) for
a slight enhancement of the divergence from sublinear to linear. To see this, note that
when we expand the sum over
in (3.31), the only singular term is 2gzz(rzrz )(rzrz ).
Expanding the second covariant derivatives into partial derivatives and Christo el symbols,
we further nd that the leading divergent part of this expression is 4 2
z ;z ;zz, which is
slightly subleading compared to the =z divergence. Using (3.19), however, we nd that
6
= 0 actually results in
;zz secretly contains a factor of
Thus, there is an enhancement from a slightly subleading divergence (of the order of 2
to a leading divergence ( z ), and we obtain (3.31). Note also the delicate cancellation of
the terms with
zz in (3.31) and (3.32).
Now we add up (3.29){(3.33) and demanding that the =z divergence cancels. We then
nd the condition:
1 +
4
where we dropped the subscript 0 in 0 (and replaced it simply by ), and also replaced z
by ;z it is now implicitly understood that this is an equation at
= 0 and at z = z = 0.
5More properly, suppose we insist on using the expansion (3.19) at nite . Then equations (3.30), (3.32)
and (3.33) each will take the form of the product of ( =z) and a Taylor series in (zz) . In other words,
we have a hierarchy of divergences. This is a manifestation of the \splitting problem" on the level of the
divergences of the equation of motion.
However, we are trying to demonstrate that the coe cient of the ( =z) yields the minimal surface as
! 0. Thus we have simply evaluated the coe cient of ( =z) at
= 0, which collapses the hierarchy and
amounts to plugging in the scalar eld expansion at zero epsilon at the outset.
(3.34)
z
where : : : stands for subleading divergences (including linear). But the quadratic
divergences in these two terms exactly cancel out each other!
Finally, we consider the ij component. This one has a genuine quadratic divergence:
4 hij (r r
)(r r
2 hij e 4A
2
zz z z
Similar, the equation of motion for the scalar eld also has a quadratic divergence:
G
r r
e 4A(Kz z + Kz z)
Comparing with the surface equation (3.15), we see that this is precisely the same equation.
Thus, the LewkowyczMaldacena prescription of xing the surface by the divergence of the
equation of motion works at least with respect to the 1=z divergence of the zz component
of the equation of motion.
We end this section by mentioning the divergences appearing in the other components
of the gravity equation of motion as well as the scalar equation. The divergence of the zz
component is the same as that of the zz component, except for the substitution z ! z.
The zi component has no rst order divergence, but it does have slightly subleading ones
(of order z1 ).
As for the zz component, two of the terms actually have a quadratic divergence:
The presence of the subleading divergences as well as quadratic divergences is somewhat
troublesome, but it is a feature that Horndeski theory shares with higher derivative/higher
curvature gravity (see for example [15]). In fact, the divergence structure is strikingly
similar: the paper [15] shows that the ij component of the equation of motion also su ers
from a quadratic divergence in GaussBonnet gravity. We will leave the question of how
to get rid of these other divergences to future work, but the work [34] is a promising
step in this direction: the authors of [34] show that an ansatz more general than (2.11)
is needed to cancel the subleading divergences. We will come back to this point in the
Conclusion section.
4
Comments on the thermal entropy
In this section, we revisit the issue of the thermal entropy for the black hole solution in
section 5. As pointed out in the literature [27], the standard methods of deriving the
entropy (Wald's entropy formula, the IyerWald formalism, and the Euclidean method)
seem to yield con icting answers. We will make the case that the correct entropy should
be the one given in (3.16), i.e. the Wald entropy (which happens to coincide with the usual
BekensteinHawking entropy).
As shown in [27], Horndeski gravity admits black hole solutions given by:
ds2 =
h(r)dt2 +
where d is the dimension, and
=
1; 0; 1 correspond to a hyperbolic, planar and spherical
horizon, respectively. Also, the constants g and
are related to the couplings
and
in
the action by:
=
=
1
2
(d
1
2
1)(d
2)g2
(d
1)(d
2)g2 1 +
2
4.1
Black hole entropy from IyerWald formalism
Following [27], let us review the computation of black hole entropy used the IyerWald
formalism. This formalism gives us 2 statements related to the entropy: (
1
) the rst law of
black hole mechanics, and (2) the statement relating the integral of the Noether charge (of
di eomorphism invariance) over the bifurcation surface of the black hole to the entropy.
To obtain the rst statement (the rst law), we compute a closed di erential form
Q
, where Q is an onshell perturbation of the Noether charge (of di eomorphism
invariance),
is the bifurcate timelike Killing vector eld of the black hole, and
is the
boundary term of the gravity action. The rst law of black hole thermodynamics then
comes out as the equation:
Z
1
Q
=
Z
H
Q
where, on the left hand side, we integrate this di erential form on a sphere at in nity, and
on the righthand side we integrate on the bifurcation surface of the black hole. In the case
of Einstein gravity, the lefthand side above coincides with the mass perturbation
M , and
the righthand side coincides with T S.
The second statement of the IyerWald formalism tells us that:
In other words, the integral of Q over the bifurcation surface equals the product of the
temperature and the entropy.
Z
H
Q = T S
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
Let us now examine what happens to these 2 statements in the case of Horndeski
gravity. The Noether charge for a stationary black hole metric of the form (4.1) is given
in [27]. If we write it as Q = QEinstein + Q
where QEinstein is the contribution from
Einstein gravity and the minimal coupling, and Q is the contribution from the
nonminimal coupling, we have:
QEinstein =
From the above, it is easily seen that Q vanishes on the horizon. To see this, we use (4.4)
to substitute for ( 0)2, and note the fact that h=f is regular on the horizon. Thus the
integral of Q on the bifurcation surface reduces to that of QEinstein, and equals the product
T S with S given by the Wald entropy!
The fact that the identity (4.8) is consistent with the Wald entropy, i.e. consistent with
the squashed cone method, should not be surprising. Indeed, there exist quite rigorous
arguments (see for example [36, 37]) that the black hole entropy as derived by the conical
singularity method always coincides with the entropy obtained from the integral of the
Noether charge.
Things do go wrong for the rst law, however: it can be seen from (4.4) that the
derivative of the scalar eld in the radial direction diverges at the horizon, and the scalar
eld itself has a branch cut singularity there. As the analysis in [27] shows, what happens is
that the variational identity (4.7) continues to hold (since it follows from Stokes theorem),
but the two sides of this equation can no longer be identi ed as
M and T S (with S taken
to be the Wald entropy). Explicitly, we obtain [27] for the planar case ( = 0):
Z
1
Z
H
=
=
(d
(d
above essentially arise from the singularity of the scalar eld
mentioned above. In view of this di culty, the authors of [27] proceed by simply de ning
the lefthand side of (4.7) as
M and the righthand side as T S. From (4.11) and (4.12),
the authors of [27] then obtain the following de nitions for the mass and the entropy:
M~ =
S~ =
(n
1
4G
While the de nitions above for M and S have the virtue that the rst law is automatically
satis ed, it is important to keep in mind that they are merely de nitions. In the case of
Einstein gravity, there exist indepdendent, nontrivial checks for the mass and the entropy:
(4.11)
(4.12)
(4.13)
(4.14)
the mass in that case coincides with the Komar integral, and the entropy is of course
the BekensteinHawking entropy, which obeys the second law for example. In the case of
Horndeski gravity, there are no independent checks of (4.13) and (4.14).
On the other hand, we have seen that from the viewpoint of holographic entanglement,
it is much more natural to take the thermal entropy to be the usual Wald entropy since this
is what we obtain from entanglement entropy as the size of the boundary region approaches
the whole boundary. To summarize, for planar black holes we have the following mass and
thermal entropy:
M =
(n
2)
The corresponding expressions for
= 1 are somewhat more complicated, and can be found
in [27].
Black hole entropy from conical singularity
Even though the analysis above should assure us that the Wald entropy is correct, there
is potentially a loophole because the singular behavior of the scalar eld on the horizon
contradicts the assumption of regularity of the scalar eld used in the derivation of the
entanglement entropy functional. Indeed, near the horizon, the scalar eld expands as:
(4.15)
Thus, the assumption of regularity used to derive the expansion (3.19) technically does not
apply in the case of the thermal entropy. In this subsection, we take a closer look at this
and argue that  despite this singularity of the scalar eld on the horizon  the conical
singularity method should still yield the Wald entropy.
First, we need to rede ne the radial coordinate from the Schwarzschildlike r to the
coordinate
used in (3.19) to facilitate comparison between the two nearhorizon
expansions. Note that the coordinate
in (3.19) satis es two properties: the horizon is at
= 0,
and the nearhorizon metric looks like at space in polar coordinates: ds2 = d 2 + 2d 2
when we turn o the conical singularity ( = 0). Note also that the usual steps taken to
derive the Hawking temperature of a black hole involves precisely a coordinate rede
nition with the two properties above, and this is the procedure we will follow to obtain the
transformation from r to . Near the horizon, the functions f (r) and h(r) in the metric
expand as:
tion:
f (r) = f1(r
h(r) = h1(r
r0) + O(r
r0) + O(r
r0)2
r0)2
for some coe cients f1 and h1. From the above we obtain the desired coordinate rede
niThe near horizon expansion (4.16) in terms of
now reads:
= p
2 p
f1
r
r0
=
0 + 1 + : : :
(4.17)
(4.18)
(4.19)
(4.20)
for some coe cient 1. The expansion above is supposed to replace the expansion (3.19)
at
= 0, so let us compare the two. The expansion above is a function of
alone,
whereas (3.19) at
= 0 allows for angular dependence. This is expected due to the U(
1
)
symmetry of the black hole (i.e. time translation symmetry) which is not present in (3.19).
Secondly, the leading power of is rst order in the expansion above, whereas it is quadratic
in (3.19) at
= 0. This is, of course, due to the fact that one expansion is regular near
= 0 and the other is not. We also remark that, in terms of , the singularity of the
scalar eld is only a \kink" near
= 0 as opposed to a divergence (4.16). That the nature
of this singularity depends on the coordinate used should not be surprising; moreover it
was noted in [27] already that the singularity is milder than it seems, in the sense that
coordinateinvariant quantities do not su er from divergences across the horizon.
The challenge now is to
gure out how expansion (4.20) is a ected when we turn on
, because we need the expansion at nite
to derive entanglement entropy. Equivalently,
we need to nd out the leading power of ~ in the parent space. We remark that this is best
done by rst computing the parentspace analog of the Horndeski black hole, then go to
the quotient space and expanding near the horizon. This is clearly a very nontrivial task in
general, and few exact solutions of this type are known (however an explicit parentspace
analog of the hyperbolic black hole is known [28]). Fortunately, we can deduce the powers
of ~ in the parent space based on (4.20): the smallest power of ~ consistent with a rst
power in
at
= 0 is a rst power in ~. Thus, in the parent space:
In the quotient space, this translates to:
=
0 + ~1 ~ + : : :
=
= 0, reduces to (4.20). Of course, there are in nitely many higher powers of ~
which are also consistent with rst order in
at
= 0. For example, ~n gives
(with no
dependence) etc.
Can the anomalous power of
in (4.22) result in a new contribution to the black hole
entropy? It is true that one can extract a term
rst order in
from the expansion above:
=
1 log . However, the fact that log
vanishes as
! 0 means this term cannot
give rise to any new contribution when we plug it into the action and integrate over a small
region near the tip of the cone. For example, the minimal coupling term gives:
Z
0
Z
0
d d
(g
; ; ) /
d
(
); = 0
(4.23)
similarly for the nonminimally coupled term.
To summarize, we think that the correct entropy should be the Wald entropy for 3
reasons: (
1
) the Wald entropy is consistent with the IyerWald formalism, in the sense that
it is consistent with the integral of Q over the horizon; (2) a rederivation of the black hole
entropy from the conical method does not seem to yield any new contribution on top of the
Wald entropy, and (3) the Wald entropy coincides with the limit of entanglement entropy
as the boundary region approaches the whole boundary.
Note that the metric is exactly the BTZ black hole. The horizon is located at r+ = p =g.
We will nd it convenient to go to the Fe ermanGraham coordinate:
and rescale the boundary coordinates as
= t=2 and y = x=(2g), the metric then becomes:
ds2 =
1
g2 2
(g2
2)2d 2 + d 2 + (g2 +
2)2dy2
and the scalar eld in terms of z satis es:
=
gr + pg2r2
g
d
d
=
p
g
(5.1)
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
Horndeski gravity given in [27].
3dimensional planar black hole
In this section, we nd numerically the entanglement entropy of black hole solutions of
Let us rst specialize to the 3dimensional, planar black hole (d = 3,
= 0). The metric
and the scalar eld pro le simplify to:
=
p
g
log (gr + p(gr)2
) + 0
dr2
g2r2
ds2 =
(g2r2
)dt2 +
+ r2dx2
Let us now parametrize the RyuTakayanagi surface as X
= (
= const; y; (y)). The
functional to be minimized then takes the form:
SEE =
dx
Z
p( 0)2 + (g2 +
2)2
g
1 + ~
( 0)2
( 0)2 + (g2 +
2)2
where we de ned ~ =
G4 . We minimized the functional above numerically and plot in
gure 1a the entanglement entropy versus the halfwidth ymax of the boundary interval
(which ranges from
ymax to ymax). For completeness, we present 3 cases: the Einstein
gravity case
= 0, a case with
> 0 and a case with
< 0, even though from the
viewpoint of bulk causality
is required to be nonpositive [29]. In gure 1b, we present
the plot of a few RT surfaces for a few di erent values of .
Let us comment that all three curves in gure 1a are concave. Concavity is one of the
hallmark features of entanglement entropy (in fact, of entropy of any kind), and implies
the property of strong subadditivity. From the holographic viewpoint, concavity can be
expected from the fact that the holographic entanglement entropy is the minimization of
a functional. Indeed, the proof of strong subadditivity in the usual Einstein gravity case
can be generalized in a straightforward way to any extensive functional [35].
2.5
2.0
0.6
0.4
0.2
of the boundary interval. b) Some representative minimal surfaces. In both plots we have used
g =
=
= G = 1, and the values of
are:
= 0 (black),
=
0:05 (light green),
=
0:125
(dark green),
= 0:05 (light purple) and
= 0:125 (dark purple).
3dimensional, spherical black hole
Next, we consider the 3dimensional spherical black hole (d = 3; = 1). The metric is still
HJEP10(27)45
ds2 =
0 + g2r2 dt2 +
dr2
(
0 + g2r2)
+ r2d 2
0 =
and the scalar eld pro le is identical to that of the 3dimensional planar black hole.
As usual, for a given boundary interval of halfwidth 0 there are two minimal surfaces
satisfying the homology constraint: a connected one and a disconnected one which includes
the horizon as a connected component. Intuition from Einstein gravity suggests that there
exists a phase transition from the connected one to the disconnected as we increase the
size 0 of the boundary region.
In gure 2a, we vary the size 0 of the boundary interval from 0 to
and plot the
value of the functional on evaluated on the connected surface and disconnected surface. As
expected, the curve for the connected surface increases monotonically while the curve for
the disconnected one decreases with increasing 0
. The two curves intersect at an angle
c (around 2.8 rad), at which point the phase transition happens (since the disconnected
one yields a smaller value of the functional beyond this angle, and the homology constraint
instructs us to use whichever surface has the smaller value).
Like in Einstein gravity, this phase transition has an interesting consequence for the
ArakiLieb inequality, which tells us that the di erence between the entropy of a subregion
A and the entropy of its complement is at most the entropy of the mixed state (the thermal
entropy in this case):
Sthermal
For 0
c, the phase transition implies that the di erence above is exactly equal to the
thermal entropy, hence the ArakiLieb inequality is saturated. Put di erently, if we were
(5.7)
(5.8)
(5.9)
to plot jSA
for 0 > c
.
2.80
2.75
HJEP10(27)45
surfaces as a function of the boundary region 0. We have used 0 = g =
=
b) Critical angle c versus , also with g = 0 =
= G = 1.
SAC j=Sthermal as a function of 0, we would see an entanglement plateau
Next, we keep the values of the constants g,
and
xed (to unity) and vary the value
of the coupling . For each such
we computed the angle c of the phase transition, and we
plot in gure 2b the critical angle as a function of . In this plot we focus on the negative
regime, since this is the regime consistent with bulk causality (see for instance [29]). Note
that keeping g,
and
xed means both the metric and the scalar eld pro le are kept
xed as we vary . However, because of the relations (4.5) and (4.6), this means the value
of
and
are actually varied, so that the di erent curves belong to di erent Horndeski
theories. Also, for
= 0, gure 2b is consistent with the analytical value of the critical
angle which was computed analytically in [38] for the usual BTZ black hole:
1
arccoth(2coth( r+))
1
(5.10)
In terms of the entanglement plateau, this means that the plateau becomes larger and
larger as we make
more and more negative.
5.3
4dimensional, spherical black hole
In this subsection, we present the phase transition for a higherdimensional case: the
4dimensional black hole.
We relegate the plots of the minimal surfaces themselves to
appendix B. From this appendix, a noteworthy feature of the RT surfaces is that the
connected surface stops existing for su ciently large 0
. Equivalently, the disconnected
surface does not exist for su ciently small 0. This feature is potentially worrisome, since
it implies that the competition between the two kinds of surfaces only exist within a range
of 0 smaller than (0; ). The phase transition, thus, must occur within this band.
In
gure 3b, we plot the angle c of the phase transition as a function of the
nonminimal coupling
at xed g (which corresponds to the AdS lengthscale at in nity) and
xed temperature T . The plot is quite similar to the 3dimensional one 2b. Like in the
3dimensional case, we focus on the negative
regime.
2.00
1.99
0.8
0.6
0.2
0.4
(b)
surfaces as a function of the boundary region 0
. We have used g =
=
= 2:711055. b) Critical angle c versus , with g =
=
= 1.
6
Conclusions and future directions
In this paper we obtained the holographic entanglement entropy functional for a particular
class of gravity with tensorscalar coupling, Horndeski gravity. We nd that the
entanglement entropy receives a Waldlike contribution proportional to the gradientsquare of the
scalar eld. We show that, as in LewkowyczMaldacena, demanding that the divergence of
the zz component of the bulk equation of motion vanishes allows us to identify the surface
where to evaluate the entanglement functional. This surface turns out to be the one that
minimizes said functional.
We also pointed out the existence of other divergences that
deserve more study: quadratic divergences in other components of the equation of motion
and subleading divergences in the zz component. As an application of the entanglement
functional found, we present explicit minimal surfaces for black hole solutions and show
that they exhibit similar features to the ones observed in SchwarzchildAdS: the connected
surface ceases to exist for su ciently large boundary region and there exist subdominant
saddles. We also study the phase transition due to the exchange of dominance between
connected and disconnected surfaces. We show that the size of the entanglement plateau
(at xed temperature) depends on the nonminimal coupling.
The thermal entropy of the Horndeski black holes we study here was not well
established. It has been previously investigated in the literature but di erent methods of
calculating it yielded di erent results. We used the entanglement entropy functional
derived here to shed light on this issue. We identi ed an oversight in the literature and
determined the correct thermal entropy.
Let us conclude with some future directions.
Other divergences. As previously mentioned in section 3.3, the cancellation of the Tzz
divergence implies that the entanglement functional should be evaluated on the
surface that minimizes it. However, similar to what occurs in higher derivative theories,
there are divergences in other components of the equations of motion. In [34] the
authors showed that in the case of GaussBonnet gravity these other divergences cancel
if a more general ansatz is taken. This ansatz includes two types of new terms
compared to (2.11): terms that break replicasymmetry, and terms that can be gauged
away at
= 0 (but not at nonzero ). It is the latter kind of new terms that are
responsible for the cancellation of subleading divergences in the case of GaussBonnet
gravity. It would be very interesting to investigate if a similar cancellation occurs in
the case of scalartensor gravities.
Splitting problem. As mentioned previously, the splitting problem has to do with
the fact that the scalar eld on the minimal surface splits into a sum of di erent
contributions when we turn on . Investigating this splitting pattern further is of
interest. Such an investigation was carried out in the simpler case of dilaton gravity
in [32] by solving the equation of motion at
0 near the tip of the cone. We also
expect that a resolution of the splitting problem will shed some light on the problem
of cancelling divergences of the equation of motion mentioned in the previous point.
Field theory dual. Identifying the precise dual of Horndeski gravity is an open question
that deserves study. In particular, carrying out the holographic renormalization
programme for Horndeski gravity seems an important and attainable goal.
Causal wedge. The causal structure of Horndeski gravity has been extensively
studied [29, 30]. In the context of holography, it is understood that the RT surface should
lie on the causal shadow in order for the boundary theory to be causally well de ned.
In [39] it was proven that this is indeed the case if we consider Einstein gravity. It
would be interesting to verify if this is also the case in Horndeski or more general
scalartensor theories. This causality constraint could be used to rule out certain
scalartensor theories from having QFT duals.
Conformally coupled theories Conformally coupled black holes have been knnown for
quite some time [40{43]. It would be interesting to derive, following the same
approach as in the present work, the entanglement functional relevant for those theories.
Quantum corrections 1=N quantum corrections to the entanglement entropy involve
Sbulk i.e. the entanglement entropy of the entanglement wedge with the rest of the
spacetime, in a manner reminiscent of the generalized entropy of black holes. The
scalartensor coupling will surely contribute to Sbulk.
Acknowledgments
We would like to thank Joan Camps for reading the manuscript and giving us useful
comments. This material is based upon work supported by the National Science Foundation
under Grant Number PHY1620610 and was performed in part at Aspen Center for Physics,
which is supported by National Science Foundation grant PHY1607611. E.C. would like to
thank Centro de Ciencias de Benasque Pedro Pascual for its hospitality while this work was
completed. E. C. was also supported by Mexico's National Council of Science and
Technology (CONACYT) grant CB201401238734 and by a grant from the Simons Foundation.
d = 4, g =
Minimal surfaces in 3d
We plot in gure 1b a few RyuTakayanagi surfaces for the 3dimensional planar black hole
for various values of
at xed g,
and . In this appendix, we elaborate further on this
plot. For the Einstein case
= 0, the surface (black curve in gure 1b) is given by the
analytical expression (we set g =
= 1):
z(y) =
s
1
K cosh (2y)
1 + K cosh (2y)
(A.1)
Here K is an integration constant.
Although this may not be apparent from
gure 1b, the nearboundary region of these
surfaces is qualitatively di erent depending on whether
is negative or positive. For
positive , the surface is always perpendicular to the boundary, while for negative
this
is not the case: the surface does not approach the boundary at right angle for negative .
This fact can appear surprising, but it comes from the contribution of the scalar eld to
the entanglement entropy: the
term is basically the normsquared of the gradient of the
scalar eld on the surface, and this term occurs with a negative sign in the functional (3.8).
Note that the normsquared of any vector is positive in a Riemannian metric. Hence,
the sign of the
term in the functional (3.8) is the opposite of the sign of the coupling
itself. For negative , the functional is minimized if the magnitude of hij ;i ;j is minimized
on the surface. But since
is only a function of z, the quantity hij ;i ;j has maximal
magnitude if the surface approaches the boundary perpendicularly. Therefore, in the case
of negative , the surface should approach the boundary at some angle less than
=2 to
keep the magnitude of hij ;i ;j small. On the other hand, for positive , the functional
becomes smaller if the magnitude of hij ;i ;j becomes larger. In this case, the surface
should approach the boundary perpendicularly because this is when hij ;i ;j is maximal.
As for the global black hole, we present in the left panel of gure 4 the connected
minimal surface for various boundary interval (from very small to the whole boundary
circle) for a negative value of . Like the usual minimal surface of the Einsteingravity
BTZ black hole, the connected surface can wrap around the horizon all the way.
G = 1,
Minimal surfaces in 4d
In this appendix, we present the plots of the RT surfaces for the 3+1 dimensional (spherical)
Horndeski black hole, with the boundary region taken to be a disk
0. These surfaces
behave qualitatively di erent from the ones in 2+1 dimensions in 2 ways:
The connected surface does not exist for all values of 0. There exists a critical value
m such that no connected RT surface exists for 0 >
m. In general, the threshold
m depends on the numerical values of the coupling constants, in particular . In the
righthand side of gure 4, we plot a few connected RT surfaces for various sizes of
the boundary region, up to the critical value m.
There exists subdominant saddles which come closer to the horizon than the threshold
surface 0 =
m. For the same boundary region, we may have more than one RT
surfaces: the dominant one and the subdominant one. We illustrate this in gure 5.
We note that both features above (the existence of m and of subdominant saddles)
are also present in Einstein gravity, as explained in [38].
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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