# Charged Black Holes in Einsteinian Quartic Gravity

S. N. Sajadi, Leila Shahkarami , Farid Charmchi , S. H. Hendi
Department of Physics, School of Science, Shiraz University, Shiraz 71454,
Iran
Biruni Observatory, School of Science, Shiraz University, Shiraz 71454, Iran
School of Physics, Damghan University, Damghan, 41167-36716, Iran
School of Particles and Accelerators, Institute for Research in Fundamental
Sciences (IPM),
P.O.Box 19395-5746, Tehran, Iran Electronic address:
<EMAIL_ADDRESS>address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>address<EMAIL_ADDRESS>
###### Abstract

In this paper, we studied Einsteinian quartic gravity minimally coupled to
electrodynamics in four dimensions. First, by variation action, we obtain the
field equations, and by integration, we obtain a nonlinear third-order
differential equation for the metric function and as well as the
electromagnetic potential. Then, in the context of the Maxwell field, we
discussed the conditions under which the charged black hole exists. Then, we
have demonstrated the thermodynamics and stability of the solution for the
case of positive coupling of quartic theory. Finally, we showed that the
charged black hole solutions of EQG (unlike GR and like ECG) have no inner
horizon and do not conform to the extremal bound of GR. Also, the uniqueness
of BH solutions in this theory does not work anymore.

##  1 Introduction

Einstein’s general relativity is one of the most successful theories of
physics and its predictions have been well established in recent years
[1]-[4]. Despite these successes, it has had problems on various scales. In
cosmology, GR is faced with the problems of dark matter and dark energy and
does not unify with quantum on a small scale. As a result, the theory needs to
be modified. One of the most natural modifications is to add higher curvature
terms to Einstein-Hilbert’s action [5]-[9]. But, most higher curvature
theories have ghost degrees of freedom in their propagator. The Lovelock class
of theories is ghost-free but in four dimensions are trivial. In recent years,
a new class of higher derivative theories has been discovered that is ghost-
free and in four dimensions neither topological nor trivial known as
Generalized Quasi-Topological Gravity [10], [11], [12]. An important feature
of these theories is that their field equations admit static, spherically
symmetric solutions with a single metric function. The first such theory is
cubic [10] and the second one is quartic in curvature [12]. The Einstein
quartic gravity has six quartic curvature combination Lagrangians which lead
to the introduction of six different coupling constants. However, the
condition of spherical symmetry leads to a degeneracy between coupling
constants. In EQG, the second-order nonlinear differential equations for the
single metric function are obtained by integration of the field equations
obtained from the variation of the action. This field equation analytically
can not be solved but approximately using continued fraction expansions in
[13] has been solved and the black hole and black brane solutions have been
studied. Black holes correspond to extreme deformations of spacetime geometry
and can exist even without matter. These objects are the simplest objects in
the world because using the uniqueness theorem they are uniquely determined by
their mass, charge, and angular momentum. In [12], the black brane solutions
in arbitrary dimensions and, the thermodynamics of four-dimensional
asymptotically flat black hole solutions were studied and they found the first
law is satisfied. In the paper [13], the basic tests of Einstein Quartic
Gravity using the continued fraction expansion for black hole solution,
including the solar system tests, the motion of particles around a black hole,
and the properties of a black hole shadow have been studied. They found the
theory is compatible with solar system tests in the large values of the
coupling constants. In this paper, following the paper [17], we analytically
obtain the thermodynamical quantities using a local analysis around the
horizon of black holes, thermodynamical stability, nonuniqueness of charged
black holes, and the existence of horizonless solutions.

The paper is organized as follows: In the next section, we first review
Einsteinian quartic gravity and we get the field equations of the theory.
Next, we obtained the conditions in which the black hole could exist and their
thermodynamical stability. Then, we obtained the singular solutions for pure
Einsteinian quartic gravity. We finish the paper with some concluding remarks.

##  2 Basic Equations

The action of Einstein quartic gravity (EQG) coupled to nonlinear
electrodynamics in 4D can be written as

$S=\int d^{4}x\sqrt{-g}\mathcal{L},$ (1)

with Lagrangian

$\displaystyle\mathcal{L}$
$\displaystyle=R-2\Lambda-\sum\hat{\alpha}_{i}\hat{L}^{i}+\dfrac{1}{4}\left(-F\right)^{n},$
(2)

where $F=F_{ab}F^{ab}$ and $F_{ab}=2A_{[a;b]}$ is the electromagnetic tensor,
$\hat{L}_{i}$ are quasi-topological Lagrangian densities, whose analytical
expressions are given in [12], $\Lambda$ is cosmological constant and
$\hat{\alpha}_{i}$ are coupling constants of the theory. Using the variational
principle, one can find the following equation of motion

$\displaystyle E_{ab}$
$\displaystyle=P_{acde}R_{b}{}^{cde}-\dfrac{1}{2}g_{ab}\mathcal{L}-2\nabla^{c}\nabla^{d}P_{acdb}-2T_{ab}=0,$
(3)
$\displaystyle\nabla_{a}\left(F^{n-1}F^{ab}\right)=0,\;\;\;\;P_{abcd}=\dfrac{\partial\mathcal{L}}{\partial
R^{abcd}}.$ (4)

Here, we consider the following spherically symmetric and static line element
for describing the geometry of spacetime

$\displaystyle ds^{2}$
$\displaystyle=-N^{2}(r)f(r)dt^{2}+\dfrac{dr^{2}}{f(r)}+r^{2}\left(d\theta^{2}+\sin^{2}(\theta)d\phi^{2}\right),$
(5) $\displaystyle A(r)$ $\displaystyle=A_{0}(r)dt.$ (6)

Varying the action (2) with respect to $N(r)$, $f(r)$ and $A_{0}(r)$, we are
left with three field equations. One of these field equations which arises
from variation with respect to $f(r)$, is satisfied by $N(r)=c$. Then, we have
two field equations as follows

$\displaystyle
5r^{6}(2n-1)A_{0}^{\prime}{}^{2n}+20r^{5}f^{\prime}+20r^{4}(\Lambda
r^{2}+f-1)-\mathcal{K}[24r^{2}f^{\prime}ff^{\prime\prime\prime}(2-2f+rf^{\prime})+48r^{2}ff^{\prime\prime
2}(1-$ $\displaystyle f+f^{\prime}r)+16rf^{\prime 3}(7f-1)+144ff^{\prime
2}(1-f)-48rff^{\prime}f^{\prime\prime}(3rf^{\prime}-4f+4)-2r^{2}f^{\prime
4}]=0,$ (7) $\displaystyle
r\left(2n-{1}\right)A_{0}^{\prime\prime}A_{0}^{\prime}{}^{2n-2}+2A_{0}^{\prime}{}^{2n-1}=0,$
(8)

where prime is the derivative to $r$ and $\mathcal{K}$ is a function of
coupling constans $\hat{\alpha}_{i}$. The solution of electrodynamic field
equation (8) is given as

$A_{0}(r)=c_{2}+\dfrac{(2n-1)r\left(\dfrac{c_{1}}{r}\right)^{\frac{2}{2n-1}}}{-3+2n}.$
(9)

Inserting (9) into the modified Einstein field equations (7) and by
integrating it, one can get a nonlinear differential equation for $f(r)$ as

$\displaystyle 72(2n-3)\left(-\dfrac{r^{2}}{4}f^{\prime
4}-\dfrac{r}{3}(2+f)f^{\prime 3}+2f(f-1)f^{\prime
2}+rff^{\prime}f^{\prime\prime}(2-2f+rf^{\prime})\right)\mathcal{K}-$
$\displaystyle
60\left(n-\dfrac{1}{2}\right)^{2}c_{1}^{\frac{4n}{2n-1}}r^{\frac{8n-6}{2n-1}}-60(2n-3)\left(\dfrac{\Lambda
r^{3}}{3}+fr-r+2M\right)r^{3}=0,$ (10)

the parameter $M$ is the integration constant which is related to the mass of
the spacetime. The equation (2), is not solved analytically. Therefore, we
solve it approximately. In the approximate calculation method, one first gets
the solution close to the horizon of the black hole. So, expanding the
function $f(r)$ around the event horizon $r_{+}$

$\displaystyle f(r)$
$\displaystyle=f_{1}(r-r_{+})+f_{2}(r-r_{+})^{2}+f_{3}(r-r_{+})^{3}+...$ (11)

and then inserting these expressions into equations (2), one can consider the
two lowest order equations as

$\displaystyle
15(2n-1)^{2}c_{1}^{\dfrac{4n}{2n-1}}r_{+}^{\dfrac{8n-6}{2n-1}}+18(2n-3)r_{+}\left(\mathcal{K}r_{+}f_{1}^{4}+\dfrac{8}{3}\mathcal{K}f_{1}^{3}+\dfrac{10}{9}\Lambda
r_{+}^{5}+\dfrac{20}{3}Mr_{+}^{2}-\dfrac{10}{3}r_{+}^{3}\right)=0,$ (12)
$\displaystyle-5(2n-1)c_{1}^{\dfrac{4n}{2n-1}}r_{+}^{\dfrac{8n-6}{2n-1}}+20r_{+}^{4}-16\mathcal{K}f_{1}^{3}r_{+}-20f_{1}r_{+}^{5}-2\mathcal{K}f_{1}^{4}r_{+}^{2}-20\Lambda
r_{+}^{6}=0,$ (13)

by substituting $c_{1}=\sqrt{2Q}$ in the above equations, and solving them
simultaneously the following solutions for $Q$ and $r_{+}$ can be reached:

$\displaystyle Q^{2}$
$\displaystyle=\dfrac{(2n-3)^{2}}{240n^{4}(2n-1)^{2}\Lambda^{2}}\left(\dfrac{20\mathcal{A}^{6}}{\mathcal{B}}\right)^{\dfrac{1}{n}}\times$
$\displaystyle\dfrac{65536\mathcal{A}\mathcal{J}n^{6}+65536\mathcal{G}n^{5}+65536\mathcal{H}n^{4}-172800f_{1}\mathcal{N}n^{3}-1166400f_{1}^{3}\mathcal{P}n^{2}+340200\mathcal{S}f_{1}^{5}n+18225f_{1}^{7}\mathcal{X}}{128\mathcal{A}\mathcal{T}n^{4}+128\mathcal{U}n^{3}+128\mathcal{V}n^{n}+1080f_{1}\mathcal{W}n+135f_{1}^{3}n},$
(14) $\displaystyle r_{+}$ $\displaystyle={\it
RootOf}(20n{\it\\_Z}^{5}\Lambda+(30nf_{1}-15f_{1}){\it\\_Z}^{4}-30{\it\\_Z}^{3}+(-60nM+90M){\it\\_Z}^{2}+(12\mathcal{K}f_{1}^{4}-$
$\displaystyle 6n\mathcal{K}f_{1}^{4}){\it\\_Z}+24\mathcal{K}f_{1}^{3}),$ (15)

where $\mathcal{A}$, $\mathcal{B}$, $\mathcal{G}$… are provided in appendix A,
and ${\it RootOf}$ is a command used as a placeholder for roots of equations
in maple [16]. The other constants in (11), provided in appendix B. In the
follwoing first we consider the case $n=1$ and $\mathcal{K}>0$. Therefore, by
setting $M=1,\Lambda=0$ from (12)-(15) we have

$\displaystyle-\dfrac{3\mathcal{K}f_{1}^{4}}{10r_{+}}-\dfrac{4\mathcal{K}f_{1}^{3}}{5r_{+}^{3}}+\dfrac{Q^{2}}{r_{+}}-2+r_{+}=0,$
(16)
$\displaystyle-\dfrac{\mathcal{K}f_{1}^{4}}{10r_{+}^{2}}-\dfrac{4\mathcal{K}f_{1}^{3}}{5r_{+}^{3}}-\dfrac{Q^{2}}{r_{+}^{2}}+1-r_{+}f_{1}=0,$
(17)

and solving them for $Q$ and $r_{+}$ one can get

$\displaystyle Q^{2}$
$\displaystyle=\dfrac{3\mathcal{K}\mathcal{E}f_{1}^{4}+8\mathcal{K}f_{1}^{3}-10\mathcal{E}^{3}+20\mathcal{E}^{2}}{10\mathcal{E}},$
(18) $\displaystyle r_{+}$ $\displaystyle={\it
RootOf}(2{f_{1}}^{4}\mathcal{K}{\\_Z}+8{f_{1}}^{3}\mathcal{K}+10{{\\_Z}}^{2}-10{{\\_Z}}^{3}+5{{\\_Z}}^{4}f_{1}),$
(19)

here $\mathcal{E}$ is

$\mathcal{E}={\it
RootOf}\left(2\mathcal{K}{f_{1}}^{4}{\\_Z}+8\mathcal{K}{f_{1}}^{3}-10{{\\_Z}}^{3}+10{{\\_Z}}^{2}+5{{\\_Z}}^{4}f_{1}\right).$
(20)

In figures (1) and (2) the behavior of horizon radius $r_{+}$ in terms of $Q$
for $f_{1}>0$ and for different positive large and small values of
$\mathcal{K}$ are illustrated, respectively. The innermost red curve
corresponds to $\mathcal{K}=0$ (RN-solution), and the black dot at the end of
the curve indicates the extremal solution, for which $r_{+}=Q=M=1$. The
figures show that there are the real solutions in which $Q>1$, suggesting that
charged BHs in EQG need not comply with the extremality condition of Einstein-
Maxwell theory. One can find a finite interval $1<Q<Q_{max}$ where there are
two real solutions with $f_{1}>0$. $Q_{max}$ is a function of $\mathcal{K}$,
which can be easily determined by finding the root of $dQ/dr_{+}$. So, for
$\mathcal{K}>\mathcal{K}_{b}=0.1538716564$ there are the real solutions over
extremal regime with positive temperature ($T=4\pi f_{1}$).

Figure 1: The plots of horizon radius $r_{+}$ in terms of $Q$ for large
$\mathcal{K}={\color[rgb]{1,0,0}0},0.1,0.5,1,10,20,{\color[rgb]{1,.5,0}100}$
(left). Zoom in around the middle part of the left figure (middle). Zoom in
around the right part of the middle figure (right). (The quantities are in
units of the mass $M$)

Figure 2: The plots of horizon radius $r_{+}$ in terms of $Q$ for small
$\mathcal{K}={\color[rgb]{1,0,0}0},0.00001,0.00005,0.0001,0.0005,0.001,0.005,0.01,0.05,0.1,0.5,{\color[rgb]{0,1,1}1}$
(left). Zoom in around the middle part of the left figure (middle). Zoom in
around the right part of the middle figure (right). (The quantities are in
units of the mass $M$)

Solving the equations (16) and (17) for $T$ and $M$, one can get

$\displaystyle T=\dfrac{1}{4\pi}\left[{\it
RootOf}\left(5Q{r_{+}}+512{\pi}^{4}{{\\_Z}}^{4}\mathcal{K}{r_{+}}+1024{\pi}^{3}{{\\_Z}}^{3}\mathcal{K}-20{{r_{+}}}^{3}+80{{r_{+}}}^{4}\pi{\\_Z}\right)\right],$
(21) $\displaystyle M=-\dfrac{1}{5r_{+}^{2}}[2048\pi^{3}({\it
RootOf}(5Qr_{+}+512\pi^{4}{\\_Z}^{4}\mathcal{K}{r_{+}}+1024\pi^{3}{\\_Z}^{3}\mathcal{K}-20r_{+}^{3}+80r_{+}^{4}\pi{\\_Z}))^{3}\mathcal{K}+$
$\displaystyle 5Qr_{+}-40r_{+}^{3}+120\pi r_{+}^{4}{\it
RootOf}(5Q{r_{+}}+512{\pi}^{4}{\\_Z}^{4}\mathcal{K}{r_{+}}+1024\pi^{3}{\\_Z}^{3}\mathcal{K}-20r_{+}^{3}+80r_{+}^{4}\pi{\\_Z})],$
(22)

and by using Wald’s formula, it is possible to evaluate the entropy of these
black hole solutions [14], [15]

$S=-2\pi\int d^{2}x\sqrt{h}\dfrac{\delta\mathcal{L}}{\delta
R_{abcd}}\epsilon_{ab}\epsilon_{cd}=\pi
r_{+}^{2}\left[1+\dfrac{32\pi^{2}\mathcal{K}T^{2}}{3r_{+}^{4}}\left(3+4\pi
r_{+}T\right)\right],$ (23)

where $h$ is the determinant of the induced metric on the horizon and
$\epsilon_{ab}\epsilon^{ab}=-2$. The above relation reduces to $\pi r_{+}^{2}$
if the higher-order terms disappear ($\mathcal{K}=0$). Now, we turn to
calculate the electric potential. The electric potential $\Phi$, measured at
infinity with respect to horizon is defined by

$\Phi=\left.A_{\mu}\xi^{\mu}{}\right|_{r\to\infty}-\left.A_{\mu}\xi^{\mu}{}\right|_{r\to
r_{+}}.$ (24)

Using $\xi=\partial_{t}$ and (9), one can find

$\Phi=\dfrac{(2n-1)r\left(\dfrac{c_{1}}{r}\right)^{\frac{2}{2n-1}}}{3-2n}.$
(25)

By using the thermodynamical quantities (21)-(25), it is easy to check that
the first law of thermodynamics ($dM=TdS+\Phi dQ$) is satisfied similar to
what happens in quadratic and cubic gravity [18]-[19].

Now, we would like to check the thermodynamical stability of the four-
dimensional asymptotically flat black hole solutions of EQG. Useful
information regarding the local stability can be extracted from the behavior
of the specific heat at a constant charge, $C_{Q}=T(\partial S/\partial
T)_{Q}$ (canonical ensemble). It can be seen from the figure (3a) for $M<1$,
that there are two BH branches with opposite signs specific heats. The larger
black holes display positive specific heat, while smaller black holes have
negative specific heat. This shows that the larger BH is locally stable. In
figure (3b), the corresponding temperatures as a function of $M$ (at a fixed
charge) are shown. This figure shows that the temperature is positive for two
branches of solutions and as the mass parameter of quartic terms grows, the
temperature and $T_{max}$ of black holes decreases. In the figure (4), in
order to look at the global stability, the behavior of free energy ($F=M-TS$)
has been shown. As can be seen, the free energy of larger BHs is lower which
shows that globally is stable.

Figure 3: The plots of $C$ in terms of $M$ for
$\mathcal{K}={\color[rgb]{1,0,0}0},1,2,4,7,10,15,40,60,80,{\color[rgb]{0,1,0}100}$
(left). The plots of $T$ in terms of $M$ for
$\mathcal{K}={\color[rgb]{1,0,0}0},1,2,4,7,10,15,{\color[rgb]{1,.75,.75}20}$
(right). (The quantities are in units of the charge $Q$) Figure 4: The plots
of $F$ in terms of $M$ for
$\mathcal{K}={\color[rgb]{1,0,0}0},1,2,4,7,10,15,{\color[rgb]{1,.75,.75}20}$.

As we have seen for $\mathcal{K}>\mathcal{K}_{b}$, there are the black hole
solutions with the same $M$ and $Q$ but different event horizon radii. Here,
we show that both solutions for $r_{+}$ are associated with asymptotically
flat spacetimes. So, we consider the large $r$ limit of metric as

$\displaystyle f(r)=1-\dfrac{2M}{r}+\epsilon F(r)$ (26)

by inserting the expansions into the field equations (2), one can get

$\displaystyle
F^{\prime\prime}-\dfrac{(5r-8M)}{r(r-2M)}F^{\prime}-\dfrac{(688\mathcal{K}M^{3}-144M^{2}\mathcal{K}r+5r^{9})}{72M^{2}\mathcal{K}r^{2}(r-2M)}F=\dfrac{(3104M^{4}\mathcal{K}-1728M^{3}\mathcal{K}r+5Q^{2}r^{8})}{288\mathcal{K}M^{2}r^{3}(2M-r)},$
(27)

the solution for homogenous equation at large $r$ reads

$F_{h}(r)=Ar^{3}I_{\frac{3}{4}}\left(\dfrac{\sqrt{5}r^{4}}{48M\sqrt{\mathcal{K}}}\right)+Br^{3}K_{\frac{3}{4}}\left(\dfrac{\sqrt{5}r^{4}}{48M\sqrt{\mathcal{K}}}\right)$
(28)

$I_{\nu}(x)$ and $K_{\nu}(x)$ are the first and second kinds of modified
Bessel functions. To obtain the special solution we assume

$F_{s}(r)=\sum_{n=2}\dfrac{F_{n}}{r^{n}}=\dfrac{F_{2}}{r^{2}}+\dfrac{F_{3}}{r^{3}}+...$
(29)

solving order by order for $F_{n}$, finally one can get

$\displaystyle
f(r)=1-\dfrac{2M}{r}+\dfrac{Q^{2}}{r^{2}}-\dfrac{432\mathcal{K}M^{3}}{5r^{9}}+\dfrac{4\mathcal{K}M^{2}(194M^{2}+324Q^{2})}{5r^{10}}-\dfrac{104\mathcal{K}MQ^{2}(26M^{2}+12Q^{2})}{5r^{11}}+$
$\displaystyle\dfrac{96\mathcal{K}Q^{4}(4Q^{2}+35M^{2})}{5r^{12}}-\dfrac{432\mathcal{K}MQ^{6}}{r^{13}}+Ar^{3}I_{\frac{3}{4}}\left(\dfrac{\sqrt{5}r^{4}}{48M\sqrt{\mathcal{K}}}\right)+Br^{3}K_{\frac{3}{4}}\left(\dfrac{\sqrt{5}r^{4}}{48M\sqrt{\mathcal{K}}}\right)+\mathcal{O}\left(\dfrac{1}{r^{14}}\right).$
(30)

Asymptotic flatness demands that we should set $A=0$. Using the continued
fraction expansion, the near horizon expansion (11) can be connected to the
asymptotic expansion (2) to obtain a general metric that works perfectly for
the outside of the black hole. Here, we solve the differential equation (2),
using the NDSolve technique in Mathematica which includes the Explicit Runge-
Kutta method to obtain the solutions which asymptotically approach (2). In
figure (5), for the non-extremal horizon, $T\neq 0$ we have obtained the
behavior of metric function for the same conserved charges. The figure reveals
the absence of an inner horizon in both black hole spacetimes. So the causal
structure of these BHs is different from that of the RN solutions and instead
is similar to that of the Schwarzschild solution. This shows that the
uniqueness of black holes in EQG-Maxwell theory no longer works.

Figure 5: The plots of the $f(r)$ for two distinct BH solutions with tha same
$M$ and $Q=1.3$ and $\mathcal{K}=10$ (left). The plots of the $f(r)$ for naked
singularities for the same $M$ and different $c_{0}=0.8,0.6,0.4,0.2,0.02$
(right). The red line is the corresponding RN over-extremal solution for
$Q=1.3,\mathcal{K}=0$. (The quantities are in units of the mass $M$)

Here, we are going to focusing on the center of black hole solutions ($r=0$)
and obtaining the horizonless solutions of theory. To do so, we assume an
expansion $f(r)=\sum_{i}c_{i}r^{i}$ where the coefficients $c_{i}$ can be
determined by solving (2) order by order in powers of $r$. So, near the origin
for the case of $\Lambda=Q=0$ we have

$\displaystyle f(r)$
$\displaystyle=c_{0}+c_{2}r^{2}-\dfrac{5Mr^{3}}{36\mathcal{K}c_{0}c_{2}(c_{0}-1)}+\dfrac{(384\mathcal{K}^{2}c_{0}c_{2}^{5}(c_{0}-1)^{2}-60\mathcal{K}c_{0}c_{2}^{2}(c_{0}-1)^{2}-25M^{2})r^{4}}{2304\mathcal{K}^{2}c_{0}^{2}c_{2}^{3}(c_{0}-1)^{2}}$
$\displaystyle-\dfrac{5M\left(\mathcal{K}c_{0}^{3}c_{2}^{2}-2\mathcal{K}c_{0}^{2}c_{2}^{2}+\mathcal{K}c_{0}c_{2}^{2}+\dfrac{5}{12}M^{2}\right)r^{5}}{1152\mathcal{K}^{3}c_{0}^{3}c_{2}^{5}(c_{0}-1)^{3}}+\mathcal{O}\left(r^{6}\right),$
(31)

there is one additional free parameter compared to the expansion around a
horizon and asymptotic ($c_{2}$). This parameter should be determined to get
asymptotically flat solutions. The Kretschmann scalar near the origin behaves
as

$K=R_{abcd}R^{abcd}\approx\dfrac{4(c_{0}-1)^{2}}{r^{4}}+\dfrac{8c_{2}(c_{0}-1)}{r^{2}}-\dfrac{10M}{9c_{2}c_{0}\mathcal{K}r}+\mathcal{O}(r^{0}).$
(32)

The curvature can be made small by choosing $c_{0}$ near 1, but as can be
seen, it can not be made regular without sending $M$ to zero and recovering
flat spacetime.
Starting from the near origin metric (2) and asymptotic solution (2), we
obtained the singular solutions with positive mass for the field equation (2)
and shown them in figure (5)b. As can be seen from the figure, unlike cubic
gravity the plots are ascending and do not have a minimum due to the
divergence of the coefficients of an arbitrary expansion of a metric function.

Finally, we want to look at the equations (16) and (17) in the case of
negative $\mathcal{K}$ and positive temperature $T>0$. For this case, as can
be seen, from figure (6), for every value of $Q$ there can be up to three
solutions for $r_{+}$. But, these horizons cannot incorporate an
asymptotically flat spacetime, because from the asymptotic metric function
(2), in cases where $\mathcal{K}<0$, the homogeneous solution contains
oscillating terms that spoil the asymptotic flatness.

Figure 6: The plots of horizon radius $r_{+}$ in terms of $Q$ for large
$-\mathcal{K}={\color[rgb]{1,0,0}0},0.1,0.5,1,10,20,{\color[rgb]{1,.5,0}100}$.

##  3 Conclusion

In this work, we studied the charged black hole solutions of Einsteinian
quartic gravity –a modification to Einstein gravity that includes fourth-order
curvature corrections– in the presence of Maxwell electrodynamics. We find
that similar to Einsteinian cubic gravity [17] the theory admits the charged
black hole solutions with a charge greater than mass. We also find that, in
the case of a positive coupling constant, there can be two asymptotically flat
charged black holes with the same charge and mass. We show that only the
larger black hole is thermodynamically stable because the smaller one has
negative specific heat and free energy. The surprising feature of these
charged black holes is the absence of an inner horizon, in contrast with the
usual Reissner-Nordstrom solution. For the same values of mass and charge,
there are two black hole solutions with different horizons. Besides the black
holes, we showed that there exists a naked singularity solution in pure
(uncharged) Einsteinian quartic gravity.

## Acknowledgements

We would like to thank Ali Hajilou for useful discussions on Mathematica
programming. SNS and SHH also thank the support of Iran National Science
Foundation 99022223.

##  Appendix A Explicit Constants

Here, we present the explicit constants that appear in (14):

$\displaystyle\mathcal{A}$ $\displaystyle={\it
RootOf}(20n{\it\\_Z}^{5}\Lambda+(30nf_{1}-15f_{1}){\it\\_Z}^{4}-30{\it\\_Z}^{3}+(-60nM+90M){\it\\_Z}^{2}+$
$\displaystyle(12\mathcal{K}f_{1}^{4}-6n\mathcal{K}f_{1}^{4}){\it\\_Z}+24\mathcal{K}f_{1}^{3}),$
$\displaystyle\mathcal{B}$
$\displaystyle=80(((-\frac{3}{4}f_{{1}}^{2}+\Lambda)n+\frac{3}{8}f_{1}^{2})\mathcal{A}^{4}+(\frac{3}{4}f_{1}-3\Lambda
nM)\mathcal{A}^{3}-\frac{2}{5}f_{1}((\Lambda\mathcal{K}f_{1}^{3}-{\frac{15}{4}}M)n+{\frac{45}{8}}M)\mathcal{A}^{2}$
$\displaystyle-\frac{4}{5}((-\frac{3}{16}f_{1}^{2}+\Lambda)n+\frac{3}{8}f_{1}^{2})f_{1}^{3}\mathcal{K}\mathcal{A}-\frac{3}{5}\mathcal{K}f_{1}^{4})(-\frac{3}{2}+n){n}^{-2}{\Lambda}^{-1}(2n-1)^{-1},$
$\displaystyle\mathcal{J}$
$\displaystyle=(-{\frac{3645}{1024}}\mathcal{K}-{\frac{135}{64}}\mathcal{K}^{2}{\Lambda}^{3}){f_{{1}}}^{11}+{\frac{81}{64}}\mathcal{A}\Lambda(\mathcal{K}{\Lambda}^{3}+{\frac{15}{8}})\mathcal{K}{f_{{1}}}^{10}-\frac{3}{4}\Lambda((\mathcal{K}{\Lambda}^{4}+{\frac{135}{64}}\Lambda)\mathcal{A}^{2}-{\frac{27}{4}}\mathcal{K}{\Lambda}^{3}-{\frac{405}{32}})\mathcal{K}{f_{{1}}}^{9}$
$\displaystyle+(({\frac{1485}{128}}\mathcal{K}{\Lambda}^{3}+{\frac{18225}{1024}}+{\Lambda}^{6}\mathcal{K}^{2})\mathcal{A}^{3}+(-3\mathcal{K}^{2}{\Lambda}^{5}-{\frac{405}{64}}\mathcal{K}{\Lambda}^{2})\mathcal{A}+\frac{9}{2}{\Lambda}^{2}M({\frac{405}{64}}+\mathcal{K}{\Lambda}^{3})\mathcal{K}){f_{{1}}}^{8}+(({\frac{135}{32}}\mathcal{K}{\Lambda}^{3}$
$\displaystyle+4{\Lambda}^{6}\mathcal{K}^{2})\mathcal{A}^{2}-{\frac{1215}{32}}M(\mathcal{K}{\Lambda}^{3}+{\frac{15}{16}})\mathcal{A}-3\mathcal{K}^{2}{\Lambda}^{5}-{\frac{405}{64}}\mathcal{K}{\Lambda}^{2}){f_{{1}}}^{7}-{\frac{225}{8}}\Lambda(({\frac{27}{16}}+\mathcal{K}{\Lambda}^{3})\mathcal{A}^{3}+(-\frac{4}{5}\mathcal{K}{\Lambda}^{3}M$
$\displaystyle-{\frac{27}{32}}M)\mathcal{A}^{2}+(-{\frac{32}{225}}\mathcal{K}^{2}{\Lambda}^{5}-{\frac{3}{20}}\mathcal{K}{\Lambda}^{2})\mathcal{A}+3/2\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{6}-{\frac{285}{8}}\Lambda((\mathcal{K}{\Lambda}^{3}+{\frac{675}{152}})\Lambda
M\mathcal{A}^{3}+{\frac{3}{38}}\mathcal{A}^{2}{\Lambda}^{3}\mathcal{K}$
$\displaystyle+(-{\frac{405}{152}}M-{\frac{39}{19}}\mathcal{K}{\Lambda}^{3}M)\mathcal{A}+{\frac{45}{38}}{\Lambda}^{3}\mathcal{K}{M}^{2}){f_{{1}}}^{5}+{\frac{135}{8}}((\mathcal{K}{\Lambda}^{3}+{\frac{15}{8}})\mathcal{A}^{3}+(-{\frac{15}{4}}M-{\frac{22}{9}}\mathcal{K}{\Lambda}^{3}M)\mathcal{A}^{2}+$
$\displaystyle\frac{11}{3}{M}^{2}({\frac{405}{88}}+\mathcal{K}{\Lambda}^{3})\mathcal{A}+\frac{1}{3}\mathcal{K}{\Lambda}^{2}M){\Lambda}^{2}{f_{{1}}}^{4}+30\mathcal{A}{\Lambda}^{2}M((\mathcal{K}{\Lambda}^{4}+{\frac{135}{16}}\Lambda)\mathcal{A}^{2}-{\frac{45}{8}}\mathcal{A}M\Lambda-{\frac{135}{64}}-\mathcal{K}{\Lambda}^{3}){f_{{1}}}^{3}$
$\displaystyle+{\frac{2475}{8}}\mathcal{A}{\Lambda}^{3}M(\mathcal{A}^{2}\Lambda
M-{\frac{15}{11}}M+{\frac{3}{22}}\mathcal{A}){f_{{1}}}^{2}-{\frac{225}{4}}\mathcal{A}{\Lambda}^{4}M(\frac{15}{2}{M}^{2}+\mathcal{A}^{2}-4\mathcal{A}M)f_{{1}}-{\frac{225}{2}}\mathcal{A}{\Lambda}^{4}{M}^{2}(\mathcal{A}^{2}\Lambda$
$\displaystyle-\frac{3}{2}\mathcal{A}M\Lambda-\dfrac{1}{2}),$
$\displaystyle\mathcal{G}$
$\displaystyle={\frac{891}{128}}({\frac{405}{176}}+\mathcal{K}{\Lambda}^{3})\mathcal{A}\mathcal{K}{f_{{1}}}^{11}-{\frac{27}{8}}((\mathcal{K}{\Lambda}^{4}+{\frac{45}{16}}\Lambda)\mathcal{A}^{2}-\dfrac{5}{2}\mathcal{K}{\Lambda}^{3}-{\frac{135}{32}})\mathcal{K}{f_{{1}}}^{10}+\dfrac{3}{2}\mathcal{A}\Lambda((\mathcal{K}{\Lambda}^{4}+{\frac{945}{256}}\Lambda)\mathcal{A}^{2}$
$\displaystyle-{\frac{27}{2}}\mathcal{K}{\Lambda}^{3}-{\frac{1215}{32}})\mathcal{K}{f_{{1}}}^{9}+((-{\frac{2835}{128}}\mathcal{K}{\Lambda}^{3}-{\frac{54675}{1024}})\mathcal{A}^{4}+({\frac{8505}{256}}\mathcal{K}{\Lambda}^{2}+9\mathcal{K}^{2}{\Lambda}^{5})\mathcal{A}^{2}-9{\Lambda}^{2}(\mathcal{K}{\Lambda}^{3}+{\frac{3105}{256}})M\mathcal{KA}$
$\displaystyle-{\frac{81}{4}}\,{K}^{2}{\Lambda}^{4}-{\frac{1215}{32}}\mathcal{K}\Lambda){f_{{1}}}^{8}+((-{\frac{18225}{1024}}-{\frac{945}{32}}\mathcal{K}{\Lambda}^{3})\mathcal{A}^{3}+{\frac{3645}{32}}M(\mathcal{K}{\Lambda}^{3}+\frac{5}{4})\mathcal{A}^{2}+(18\mathcal{K}^{2}{\Lambda}^{5}+{\frac{8505}{128}}\mathcal{K}{\Lambda}^{2})\mathcal{A}$
$\displaystyle-18{\Lambda}^{2}M({\frac{405}{64}}+\mathcal{K}{\Lambda}^{3})\mathcal{K}){f_{{1}}}^{7}+{\frac{1755}{32}}\Lambda((\mathcal{K}{\Lambda}^{3}+{\frac{675}{208}})\mathcal{A}^{4}+(-{\frac{38}{39}}\mathcal{K}{\Lambda}^{3}M-{\frac{315}{208}}M)\mathcal{A}^{3}-{\frac{9}{13}}\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}$
$\displaystyle+{\frac{57}{13}}\mathcal{AK}{\Lambda}^{2}M+{\frac{128}{585}}\mathcal{K}^{2}{\Lambda}^{4}+{\frac{6}{13}}\mathcal{K}\Lambda){f_{{1}}}^{6}+{\frac{135}{4}}\Lambda(\Lambda
M(K{\Lambda}^{3}+{\frac{45}{4}})\mathcal{A}^{4}+({\frac{45}{32}}+{\frac{11}{8}}\mathcal{K}{\Lambda}^{3})\mathcal{A}^{3}+(-{\frac{405}{32}}M$
$\displaystyle-7\mathcal{K}{\Lambda}^{3}M)\mathcal{A}^{2}+{\frac{29}{8}}{\Lambda}^{2}\mathcal{K}(-{\frac{6}{29}}+{M}^{2}\Lambda)\mathcal{A}+5\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{5}-{\frac{135}{4}}{\Lambda}^{2}((\mathcal{K}{\Lambda}^{3}+{\frac{45}{8}})\mathcal{A}^{4}+(-{\frac{45}{4}}M$
$\displaystyle-\dfrac{10}{3}\mathcal{K}{\Lambda}^{3}M)\mathcal{A}^{3}+(-{\frac{5}{12}}\mathcal{K}{\Lambda}^{2}+3{\Lambda}^{3}\mathcal{K}{M}^{2}+{\frac{225}{8}}{M}^{2})\mathcal{A}^{2}+{\frac{15}{4}}\mathcal{AK}{\Lambda}^{2}M-5{\Lambda}^{2}\mathcal{K}{M}^{2}){f_{{1}}}^{4}-{\frac{6075}{8}}(\mathcal{A}^{4}M\Lambda$
$\displaystyle+({\frac{4}{135}}\mathcal{K}{\Lambda}^{3}-{\frac{11}{18}}{M}^{2}\Lambda+\dfrac{1}{24})\mathcal{A}^{3}+(-{\frac{16}{135}}\mathcal{K}{\Lambda}^{3}M-\dfrac{1}{2}M)\mathcal{A}^{2}+{\frac{4}{45}}{\Lambda}^{3}\mathcal{A}{M}^{2}\mathcal{K}+{\frac{4}{135}}\mathcal{K}{\Lambda}^{2}M){\Lambda}^{2}{f_{{1}}}^{3}$
$\displaystyle-{\frac{2025}{4}}\mathcal{A}^{2}{\Lambda}^{3}((-\dfrac{1}{8}+{M}^{2}\Lambda)\mathcal{A}^{2}+{\frac{19}{24}}\mathcal{A}M-3{M}^{2}){f_{{1}}}^{2}+{\frac{675}{2}}\mathcal{A}^{2}{\Lambda}^{3}M(-\frac{1}{4}-{\frac{19}{8}}\mathcal{A}M\Lambda+3{M}^{2}\Lambda+\mathcal{A}^{2}\Lambda)f_{{1}}$
$\displaystyle+{\frac{2025}{8}}\mathcal{A}^{2}{\Lambda}^{4}(\mathcal{A}^{2}\Lambda
M+(\frac{2}{9}-{M}^{2}\Lambda)\mathcal{A}-\frac{4}{3}M),$ (33)

$\displaystyle\mathcal{H}$
$\displaystyle=-{\frac{405}{64}}\mathcal{AK}({\frac{135}{32}}+\mathcal{K}{\Lambda}^{3}){f_{{1}}}^{11}+{\frac{27}{16}}\mathcal{K}((\mathcal{K}{\Lambda}^{4}+{\frac{495}{64}}\Lambda)\mathcal{A}^{2}-13/2\mathcal{K}{\Lambda}^{3}-{\frac{675}{32}}){f_{{1}}}^{10}-{\frac{6075}{1024}}\mathcal{A}\Lambda\mathcal{K}(-{\frac{512}{225}}\mathcal{K}{\Lambda}^{3}$
$\displaystyle+\mathcal{A}^{2}\Lambda-{\frac{88}{5}}){f_{{1}}}^{9}+(({\frac{273375}{4096}}+{\frac{6885}{512}}\mathcal{K}{\Lambda}^{3})\mathcal{A}^{4}-{\frac{6075}{128}}\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}+{\frac{69255}{512}}\mathcal{AK}{\Lambda}^{2}M+{\frac{81}{4}}\mathcal{K}^{2}{\Lambda}^{4}+{\frac{3645}{32}}\mathcal{K}\Lambda){f_{{1}}}^{8}$
$\displaystyle+(({\frac{4185}{128}}\mathcal{K}{\Lambda}^{3}+{\frac{91125}{2048}})\mathcal{A}^{3}-{\frac{12555}{128}}({\frac{1125}{496}}+\mathcal{K}{\Lambda}^{3})M\mathcal{A}^{2}-{\frac{18225}{128}}\mathcal{AK}{\Lambda}^{2}+{\frac{13365}{64}}\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{7}-{\frac{45}{2}}((\mathcal{K}{\Lambda}^{3}$
$\displaystyle+{\frac{675}{64}})\mathcal{A}^{4}-{\frac{9}{8}}M({\frac{135}{32}}+\mathcal{K}{\Lambda}^{3})\mathcal{A}^{3}-{\frac{81}{32}}\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}+{\frac{213}{16}}\mathcal{AK}{\Lambda}^{2}M+{\frac{81}{16}}\mathcal{K}\Lambda)\Lambda\,{f_{{1}}}^{6}-{\frac{42525}{128}}\Lambda(\mathcal{A}^{4}M\Lambda$
$\displaystyle+({\frac{8}{63}}\mathcal{K}{\Lambda}^{3}+3/7)\mathcal{A}^{3}+(-{\frac{16}{35}}\mathcal{K}{\Lambda}^{3}M-{\frac{15}{7}}M)\mathcal{A}^{2}+{\frac{8}{35}}\mathcal{K}{\Lambda}^{2}(-1+{M}^{2}\Lambda)\mathcal{A}+{\frac{16}{15}}\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{5}$
$\displaystyle+{\frac{18225}{64}}{\Lambda}^{2}(\mathcal{A}^{4}-2\mathcal{A}^{3}M+(-{\frac{32}{405}}\mathcal{K}{\Lambda}^{2}+{\frac{23}{6}}{M}^{2})\mathcal{A}^{2}+{\frac{8}{15}}\mathcal{AK}{\Lambda}^{2}M-{\frac{8}{15}}\Lambda({M}^{2}\Lambda-\frac{2}{9})\mathcal{K}){f_{{1}}}^{4}+$
$\displaystyle{\frac{10125}{16}}(\mathcal{A}^{4}M\Lambda+(-\frac{3}{5}{M}^{2}\Lambda+{\frac{9}{40}})\mathcal{A}^{3}-{\frac{9}{8}}M\mathcal{A}^{2}-{\frac{4}{225}}\mathcal{AK}{\Lambda}^{2}+{\frac{4}{25}}\mathcal{K}{\Lambda}^{2}M){\Lambda}^{2}{f_{{1}}}^{3}+{\frac{6075}{32}}(({M}^{2}\Lambda-2/3){A}^{2}$
$\displaystyle+11/3\mathcal{A}M-9{M}^{2})\mathcal{A}^{2}{\Lambda}^{3}{f_{{1}}}^{2}-{\frac{2025}{8}}\mathcal{A}^{2}{\Lambda}^{3}(\mathcal{A}^{2}\Lambda
M+(-9/4{M}^{2}\Lambda+1/6)\mathcal{A}-5/6M+9/4{M}^{3}\Lambda)f_{{1}}$
$\displaystyle+{\frac{225}{16}}{\Lambda}^{4}{\mathcal{A}}^{2}(\mathcal{A}-3M)(-9M+\mathcal{A}),$
$\displaystyle\mathcal{N}$
$\displaystyle=-{\frac{16}{25}}\mathcal{A}(\mathcal{K}{\Lambda}^{3}+{\frac{3375}{256}})\mathcal{K}{f_{{1}}}^{10}+(-{\frac{32}{25}}\mathcal{K}^{2}{\Lambda}^{3}+{\frac{63}{20}}\Lambda\mathcal{A}^{2}\mathcal{K}-{\frac{27}{2}}\mathcal{K}){f_{{1}}}^{9}-{\frac{39}{40}}\mathcal{A}\Lambda\mathcal{K}(-{\frac{420}{13}}+\mathcal{A}^{2}\Lambda)f_{1}^{8}$
$\displaystyle+((\mathcal{K}{\Lambda}^{3}+{\frac{135}{8}})\mathcal{A}^{4}-{\frac{39}{4}}\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}+{\frac{513}{20}}\mathcal{AK}{\Lambda}^{2}M+{\frac{216}{5}}\mathcal{K}\Lambda)f_{1}^{7}+(({\frac{18}{5}}\mathcal{K}{\Lambda}^{3}+{\frac{135}{8}})\mathcal{A}^{3}+(-{\frac{48}{5}}\mathcal{K}{\Lambda}^{3}M$
$\displaystyle-{\frac{135}{2}}M)\mathcal{A}^{2}-39\mathcal{AK}{\Lambda}^{2}+{\frac{234}{5}}\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{6}+({\frac{99}{4}}\Lambda\mathcal{A}^{3}M-{\frac{192}{5}}\mathcal{AK}{\Lambda}^{3}M+8\mathcal{A}^{2}{\Lambda}^{3}\mathcal{K}-{\frac{225}{4}}\mathcal{A}^{4}\Lambda-$
$\displaystyle{\frac{216}{5}}\mathcal{K}{\Lambda}^{2}){f_{{1}}}^{5}-48\Lambda(6/5\mathcal{K}{\Lambda}^{2}M-1/3\mathcal{AK}{\Lambda}^{2}+\mathcal{A}^{4}M\Lambda-{\frac{69}{16}}M\mathcal{A}^{2}+{\frac{9}{8}}\mathcal{A}^{3}){f_{{1}}}^{4}+60{\Lambda}^{2}({\frac{16}{75}}\mathcal{K}\Lambda+\mathcal{A}^{4}$
$\displaystyle-2\mathcal{A}^{3}M+{\frac{33}{10}}\mathcal{A}^{2}{M}^{2}){f_{{1}}}^{3}+(60\mathcal{A}^{4}{\Lambda}^{3}M+(-36{\Lambda}^{3}{M}^{2}+54{\Lambda}^{2})\mathcal{A}^{3}-192\mathcal{A}^{2}M{\Lambda}^{2}){f_{{1}}}^{2}-20\mathcal{A}^{2}{\Lambda}^{3}(-{\frac{27}{5}}\mathcal{A}M$
$\displaystyle+\mathcal{A}^{2}+{\frac{54}{5}}{M}^{2})f_{{1}}-16\mathcal{A}^{2}{\Lambda}^{3}(\mathcal{A}-3M),$
$\displaystyle\mathcal{P}$
$\displaystyle={\frac{9}{16}}{f_{{1}}}^{8}\mathcal{AK}+(\mathcal{K}-{\frac{17}{120}}\Lambda\mathcal{A}^{2}\mathcal{K}){f_{{1}}}^{7}+1/45\mathcal{A}\Lambda\mathcal{K}(-{\frac{153}{2}}+\mathcal{A}^{2}\Lambda){f_{{1}}}^{6}+({\frac{4}{15}}\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}-2/3\mathcal{AK}{\Lambda}^{2}M-8/3\mathcal{K}\Lambda$
$\displaystyle-{\frac{15}{16}}\mathcal{A}^{4}){f_{{1}}}^{5}+(-5/4\mathcal{A}^{3}+{\frac{35}{8}}M\mathcal{A}^{2}+4/3\mathcal{AK}{\Lambda}^{2}-4/3\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{4}+5/2\Lambda({\frac{32}{45}}\mathcal{K}\Lambda-{\frac{13}{30}}\mathcal{A}^{3}M+\mathcal{A}^{4}){f_{{1}}}^{3}$
$\displaystyle+(\mathcal{A}^{2}\Lambda
M-11M+10/3\mathcal{A})\mathcal{A}^{2}\Lambda{f_{{1}}}^{2}-5/3\mathcal{A}^{2}{\Lambda}^{2}(\mathcal{A}^{2}+3{M}^{2}-2\mathcal{A}M)f_{{1}}-{\frac{20}{9}}\mathcal{A}^{2}{\Lambda}^{2}(\mathcal{A}-3M),$
$\displaystyle\mathcal{S}$
$\displaystyle={\frac{9}{20}}{f_{{1}}}^{6}\mathcal{AK}-{\frac{2}{35}}\mathcal{K}(\mathcal{A}^{2}\Lambda-15){f_{{1}}}^{5}-4/5\mathcal{AK}\Lambda{f_{{1}}}^{4}+(-{\frac{9}{14}}\mathcal{A}^{4}-{\frac{48}{35}}\mathcal{K}\Lambda){f_{{1}}}^{3}-{\frac{15}{14}}\mathcal{A}^{2}(-{\frac{16}{5}}M+\mathcal{A}){f_{{1}}}^{2}$
$\displaystyle+\mathcal{A}^{3}\Lambda(\mathcal{A}-3/7M)f_{{1}}+{\frac{12}{7}}\mathcal{A}^{2}\Lambda(\mathcal{A}-3M),$
$\displaystyle\mathcal{T}$
$\displaystyle=-{\frac{27}{8}}{f_{{1}}}^{7}\mathcal{K}+9/4{f_{{1}}}^{6}\mathcal{AK}\Lambda-3/2{f_{{1}}}^{5}\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}+((\mathcal{K}{\Lambda}^{3}+{\frac{135}{8}})\mathcal{A}^{3}+3\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{4}-{\frac{135}{4}}\mathcal{A}M{f_{{1}}}^{3}$
$\displaystyle+{\frac{45}{2}}{f_{{1}}}^{2}\mathcal{A}^{2}\Lambda
M-30f_{{1}}{\Lambda}^{2}\mathcal{A}^{3}M+30\mathcal{A}{M}^{2}{\Lambda}^{2},$
$\displaystyle\mathcal{U}$
$\displaystyle={\frac{189}{16}}\mathcal{AK}{f_{{1}}}^{7}-{\frac{27}{4}}\mathcal{K}(-2+{A}^{2}\Lambda){f_{{1}}}^{6}+{\frac{15}{4}}\mathcal{A}\Lambda(\mathcal{A}^{2}\Lambda-{\frac{18}{5}})\mathcal{K}{f_{{1}}}^{5}+((-{\frac{135}{4}}-2\mathcal{K}{\Lambda}^{3})\mathcal{A}^{4}+15/2\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}$
$\displaystyle-21/2\mathcal{AK}{\Lambda}^{2}M){f_{{1}}}^{4}+((-{\frac{135}{8}}-4\mathcal{K}{\Lambda}^{3})\mathcal{A}^{3}+{\frac{405}{4}}M\mathcal{A}^{2}-12\mathcal{K}{\Lambda}^{2}M){f_{{1}}}^{3}+{\frac{135}{4}}\mathcal{A}^{3}\Lambda(-5/3M+\mathcal{A}){f_{{1}}}^{2}$
$\displaystyle+(60\mathcal{A}^{4}{\Lambda}^{2}M-45\mathcal{A}^{2}\Lambda\,M)f_{{1}}+30\mathcal{A}^{2}{\Lambda}^{2}M(\mathcal{A}-3\,M),$
(34) $\displaystyle\mathcal{V}$
$\displaystyle=-{\frac{405}{32}}\mathcal{AK}{f_{{1}}}^{7}+{\frac{81}{16}}\mathcal{K}(\mathcal{A}^{2}\Lambda-4){f_{{1}}}^{6}-3/2\mathcal{A}\Lambda(\mathcal{A}^{2}\Lambda-{\frac{27}{2}})\mathcal{K}{f_{{1}}}^{5}+(18\mathcal{K}\Lambda+9\mathcal{AK}{\Lambda}^{2}M+{\frac{405}{16}}\mathcal{A}^{4}$
$\displaystyle-6\mathcal{A}^{2}{\Lambda}^{2}\mathcal{K}){f_{{1}}}^{4}+(-{\frac{405}{4}}\,M\mathcal{A}^{2}+{\frac{405}{16}}\mathcal{A}^{3}+18\mathcal{K}{\Lambda}^{2}M-6\mathcal{AK}{\Lambda}^{2}){f_{{1}}}^{3}-{\frac{135}{4}}\mathcal{A}^{3}\Lambda(\mathcal{A}-7/6\,M){f_{{1}}}^{2}$
$\displaystyle-{\frac{45}{2}}\mathcal{A}^{2}\Lambda(\mathcal{A}^{2}\Lambda\,M-4\,M+\mathcal{A})f_{{1}}+15/2\mathcal{A}^{2}{\Lambda}^{2}(\mathcal{A}-3\,M)^{2},$
$\displaystyle\mathcal{W}$
$\displaystyle={\frac{13}{20}}{f_{{1}}}^{6}\mathcal{AK}-2/15\mathcal{K}(-9+\mathcal{A}^{2}\Lambda){f_{{1}}}^{5}-4/5\mathcal{AK}\Lambda\,{f_{{1}}}^{4}+(-{\frac{16}{15}}\mathcal{K}\Lambda-\mathcal{A}^{4}){f_{{1}}}^{3}+(5\,M\mathcal{A}^{2}-3/2\mathcal{A}^{3}){f_{{1}}}^{2}$
$\displaystyle+\mathcal{A}^{3}\Lambda(\mathcal{A}-M)f_{{1}}+4/3\mathcal{A}^{2}\Lambda(\mathcal{A}-3\,M),$
$\displaystyle\mathcal{X}$
$\displaystyle=-4/5\,{f_{{1}}}^{4}\mathcal{AK}-8/5\mathcal{K}{f_{{1}}}^{3}+f_{{1}}\mathcal{A}^{4}+2\mathcal{A}^{2}\left(\mathcal{A}-3\,M\right).$
(35)

##  Appendix B Near Horizon Solutions

Here, we present additional terms that appear in the near horizon solution
(11):

$\displaystyle f_{3}$
$\displaystyle=-\dfrac{1}{72f_{1}^{2}\mathcal{K}r_{+}^{2}(2+r_{+}f_{1})}(-20r_{+}^{5}f_{2}-20\,\Lambda\,r_{+}^{5}-20\,r_{+}^{4}f_{1}+96\,\mathcal{K}f_{1}^{2}r_{+}^{3}f_{2}^{2}+96\,f_{1}^{3}\mathcal{K}+$
$\displaystyle
96\,\mathcal{K}f_{1}r_{+}^{2}f_{2}^{2}-152\,\mathcal{K}f_{1}^{3}r_{+}^{2}f_{2}+20\,r_{+}\,Q^{2}+58\,r_{+}f_{1}^{4}\mathcal{K}-240\mathcal{K}f_{1}^{2}r_{+}f_{2}),$
(36) $\displaystyle f_{4}$
$\displaystyle=\dfrac{1}{3456r_{+}^{3}f_{1}^{4}\mathcal{K}^{2}(2+r_{+}\,f_{1})^{2}}(-6912\,f_{1}^{5}\mathcal{K}^{2}-9744\,f_{1}^{6}\mathcal{K}^{2}r_{+}+1910{r_{+}}^{5}f_{1}^{4}\mathcal{K}-3464\,{r_{+}}^{2}f_{1}^{7}\mathcal{K}^{2}$
$\displaystyle+100\,r_{+}^{9}f_{2}+23040\,\mathcal{K}^{2}r_{+}^{4}f_{2}^{3}f_{1}^{3}+11520\,\mathcal{K}^{2}r_{+}^{5}f_{2}^{3}f_{1}^{4}-3840\,r_{+}^{6}f_{2}^{2}f_{1}\mathcal{K}+13824\,\mathcal{K}^{2}r_{+}^{3}f_{2}^{3}f_{1}^{2}+$
$\displaystyle
1680\,r_{+}^{5}f_{2}f_{1}^{2}\mathcal{K}-3360\,r_{+}^{6}f_{2}f_{1}\mathcal{K}\Lambda+27648\,r_{+}\,f_{2}f_{1}^{4}\mathcal{K}^{2}-55680\,r_{+}^{3}f_{2}^{2}f_{1}^{4}\mathcal{K}^{2}+40560\,r_{+}^{2}f_{2}f_{1}^{5}\mathcal{K}^{2}$
$\displaystyle-34560\,r_{+}^{2}f_{2}^{2}f_{1}^{3}\mathcal{K}^{2}-3600\,\mathcal{K}f_{1}^{2}r_{+}^{7}f_{2}^{2}+200\,\mathcal{K}f_{1}^{3}r_{+}^{6}f_{2}-23040\,\mathcal{K}^{2}f_{1}^{5}r_{+}^{4}f_{2}^{2}+15112\,\mathcal{K}^{2}f_{1}^{6}r_{+}^{3}f_{2}$
$\displaystyle+2320\,r_{+}^{6}f_{1}^{3}\mathcal{K}\Lambda+3360\,f_{1}^{2}\mathcal{K}\Lambda\,r_{+}^{5}-3120\,\mathcal{K}f_{1}^{2}r_{+}^{7}f_{2}\Lambda+100\,r_{+}^{9}\Lambda+100\,r_{+}^{8}f_{1}-1840\,r_{+}^{2}{Q}^{2}f_{1}^{3}\mathcal{K}$
$\displaystyle-2400\,r_{+}\,{Q}^{2}f_{1}^{2}\mathcal{K}+3360\,r_{+}^{2}f_{2}f_{1}\mathcal{K}{Q}^{2}+3120\,\mathcal{K}f_{1}^{2}r_{+}^{3}f_{2}{Q}^{2}-100\,r_{+}^{5}{Q}^{2}+2640\,f_{1}^{3}\mathcal{K}r_{+}^{4}).$
(37)

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