Generalized proof of the linearized second law in general quadric corrected
Einstein-Maxwell gravity

Xin-Yang Wang ${}^{\text{a}}$, Jie Jiang ${}^{\text{b}}$

College of Education for the Future, Beijing Normal University, Zhuhai 519087,
China

###### Abstract

Although the entropy of black holes in any diffeomorphism invariant theory of
gravity can be expressed as the Wald entropy, the issue of whether the entropy
always obeys the second law of black hole thermodynamics remains open. Since
the non-minimal coupling interaction between gravity and the electromagnetic
field in the general quadric corrected Einstein-Maxwell gravity can
sufficiently influence the expression of the Wald entropy, we would like to
check whether the Wald entropy of black holes in the quadric corrected gravity
still satisfies the second law. A quasistationary accreting process of black
holes is first considered, which describes that black holes is perturbed by
matter fields and eventually settle down to a stationary state. Two
assumptions that the matter fields should obey the null energy condition and
that a regular bifurcation surface exists on the background spacetime are
further proposed. According to the two assumptions and the Raychaudhuri
equation, we demonstrate that the Wald entropy monotonically increases along
the future event horizon under the linear order approximation of the
perturbation. This result indicates that the Wald entropy of black holes in
the quadric corrected gravity strictly obeys the linearized second law of
thermodynamics.

${}^{\text{a}}$<EMAIL_ADDRESS>
${}^{\text{b}}$ Corresponding author<EMAIL_ADDRESS>
##  1 Introduction

Black holes are special spacetime structures predicted by General Relativity.
The boundary of black holes is called the event horizon which is a null
hypersurface. Pioneering work by Hawking [1] demonstrated that the area of the
event horizon never decreases along the direction of the time evolution. From
this perspective, Bekenstein [2] first suggested that there may be a strong
connection between the area of the event horizon and the entropy of black
holes. Using the quantum field theory in curved spacetime, Hawking [3] showed
that the temperature and the entropy of black holes are defined by its surface
gravity and area of the event horizon. While the entropy of black holes is
expressed as $S_{BH}=A/4$, where $A$ is the area of the event horizon, and it
is called the Bekenstein-Hawking entropy. From the definition of the
temperature and the entropy of black holes, the four laws of thermodynamics of
black holes are constructed, and black holes can be identical with adiabatic
systems in the thermodynamics [4, 5, 6]. In the four laws of thermodynamics,
the two profound laws for black holes are the first and the second laws
respectively. If we recognize black holes as thermodynamic systems, the two
laws should be satisfied first. A natural question of whether the laws of
thermodynamics are the robust feature for black holes in any covariant
gravitational theory is proposed. Starting with this question, Iyer and Wald
[7, 8] established the first law of thermodynamics for any diffeomorphism
invariant gravitational theory, and the entropy of black holes in the
gravitational theory can be expressed as

$S_{\text{W}}=-2\pi\int_{s}d^{n}y\sqrt{\gamma}\frac{\partial\mathcal{L}}{\partial
R_{abcd}}\bm{\epsilon}_{ab}\bm{\epsilon}_{cd}\,,$ (1)

where $\mathcal{L}$ is the Lagrangian of the theory of gravity,
$\bm{\epsilon}_{ab}$ is the binormal for any slice of the event horizon,
$\sqrt{\gamma}$ is the volume element of the cross section of the horizon, and
$y$ is used to label the transverse coordinates on any slice. This entropy is
called the Wald entropy and it is no longer proportional to the area of the
event horizon. When a diffeomorphism invariant gravitational theory degraded
into the Einstein theory, the form of Wald entropy is identical with
Bekenstein-Hawking entropy as well.

In the study of the quantization of gravity, it still has not an appropriate
scheme to quantize the gravitational field until now. To study the interaction
between gravity and matter fields on the quantum scale, we should find the
corresponding low-energy effective field theory (EFT) first. The low-energy
EFT makes that some quantum corrected terms are added in the expression of the
Lagrangian. These terms contain the higher curvature terms and the non-minimal
coupling terms between gravity and matter fields. The existence of the quantum
correction terms will change the expression of the Wald entropy substantially.
Although it has been proved that the Wald entropy always obeys the first law
of thermodynamics for any diffeomorphism invariant theory [7, 8], whether the
Wald entropy in the theory of gravity with quantum correction still satisfies
the second law of thermodynamics has not been generally demonstrated. Along
this line of thought, the second law of black holes in the gravitational
theory with higher curvature correction has been investigated in previous
research works. According to the field redefinition, the Wald entropy of black
holes in $f(R)$ gravity has been proved it satisfies the second law of
thermodynamics [9, 10]. However, in the gravitational theory, the result shows
that the second law is violated when two black holes emerge [11].
Bhattacharjee et al. [12] suggest that it is enough for us to examine the
second law of black holes in quantum corrected gravitational theory under the
linear order approximation of the process considering a quasistationary
accretion process of black holes. Based on this perspective, the linearized
second law of black holes in Gauss-Bonnet and Lovelock theory has been
examined [13, 14]. Furthermore, Wall [15] proposes a general method to check
the linearized second law in arbitrary higher curvature gravity, while the
expression of the entropy obeys the linearized second law is also given.

Although the theory of gravity with any higher curvature correction has been
studied adequately, the situation that the Lagrangian contains the non-minimal
coupling interaction between gravity and matter fields is not involved. Since
the non-minimal coupling terms between the Riemann tensor and the scalar field
are contained in the Lagrangian of the Horndeski gravity, we have examined the
Wald entropy of black holes in the theory of gravity always satisfies the
linearized second law under the first-order approximation of the
quasistationary accretion process [16]. However, only studying the interaction
between gravity and the scalar field cannot sufficiently examine whether the
Wald entropy generally obeys the linearized second law in the theory of
gravity with any non-minimal coupling matter fields. We should continue to
consider the gravitational theory with other types of non-minimal coupling
matter fields to examine the second law of black holes. Therefore, in the
following, we would like to consider the most general quadric corrected
Einstein-Maxwell theory of gravity. In this theory, expect the pure Einstein-
Maxwell action, the quadric corrected terms also involved in the expression of
the Lagrangian. These quadric correction terms mainly contain the self-
interaction of the electromagnetic field, the non-minimal coupling interaction
between gravity and the electromagnetic field, and the quadratic curvature
correction. Similar to the case of higher curvature gravity, the non-minimal
coupling terms also affect the expression of Wald entropy in the same way.
Therefore, investigating whether the Wald entropy of black holes in the
general quadric corrected gravitational theory still satisfies the linearized
second law is also an important segment to examine the validity of the second
law of black hole thermodynamics at the first-order approximation of the
matter field perturbation.

The organization of the paper is as follows. In Sec. 2, the general quadric
corrected Einstein-Maxwell gravity is introduced, and the Wald entropy of
black holes in the theory of gravity is given. In Sec. 3, considering an
quasistationary accreting process of black holes and assuming that the matter
fields accreted by black holes should satisfy the null energy condition, we
examine whether the Wald entropy still satisfies the linearized second law of
black holes thermodynamics under the linear order approximation of the
perturbation. The paper ends with discussions and conclusions in Sec. 4.

##  2 General quadric corrected Einstein-Maxwell gravity and Wald entropy

We will consider the ($\text{n}+2$)-dimensional Einstein-Maxwell gravity with
general quadric correction terms. These correction terms mainly contain the
self-interaction terms of the electromagnetic field, the non-minimal coupling
interaction between gravity and the electromagnetic field, and the quadratic
curvature correction. The Lagrangian ($\text{n}+2$)-form of the gravitational
theory can be formally written as

$\bm{\mathcal{L}}=\bm{\mathcal{L}}_{\text{EM}}+\Delta\bm{\mathcal{L}}+\bm{\mathcal{L}}_{\text{mt}}\,.$
(2)

The first part of Eq. (2) represents the Lagrangian of the pure Einstein-
Maxwell gravitational theory,

$\bm{\mathcal{L}}_{\text{EM}}=\frac{1}{16\pi}\left(R-F_{ab}F^{ab}\right)\bm{\epsilon}\,,$
(3)

where $R$ is the Ricci scalar, $\bm{F}=d\bm{A}$ is the electromagnetic
strength, $\bm{A}$ is the vector potential of the electromagnetic field, and
$\bm{\epsilon}$ is the volume element of the spacetime. The second part of Eq.
(2) is the portion of the quadric correction, which can be decomposed into two
parts, i.e.,

$\Delta\bm{\mathcal{L}}=\Delta\bm{\mathcal{L}}_{1}+\Delta\bm{\mathcal{L}}_{\text{QC}}\,.$
(4)

The first part of Eq. (4) involves the non-minimal coupling between gravity
and the electromagnetic field as well as the self-interaction of the Maxwell
field, which can be specifically expressed as

$\begin{split}\Delta\bm{\mathcal{L}}_{1}=&\frac{1}{16\pi}\left(\alpha_{1}RF_{ab}F^{ab}+\alpha_{2}R_{ab}F^{ac}F^{b}_{\
c}+\alpha_{3}R_{abcd}F^{ab}F^{cd}\right.\\\
&\left.+\alpha_{4}F_{ab}F^{ab}F_{cd}F^{cd}+\alpha_{5}F_{ab}F^{bc}F_{cd}F^{da}\right)\bm{\epsilon}\,,\end{split}$
(5)

where the parameter $\alpha_{i}\ \left(i=1,2,3,4,5\right)$ is the coupling
constants of two types of the interaction, and the second part of Eq. (4) is
the quadratic curvature correction which is given as

$\Delta\bm{\mathcal{L}}_{\text{QC}}=\frac{1}{16\pi}\left(\beta_{1}R^{2}+\beta_{2}R_{ab}R^{ab}+\beta_{3}R_{abcd}R^{abcd}\right)\bm{\epsilon}\,,$
(6)

where $\beta_{i}\ \left(i=1,2,3\right)$ represent the coupling constant of the
gravitational self-interaction. The last part of Eq. (2) is the Lagrangian of
the additional matter fields in spacetime.

For stationary black holes in the general quadric correction theory of
gravity, the entropy of black holes can be described by the Wald entropy.
According to the definition of the Wald entropy and the form of the
Lagrangian, it can be expressed as

$\begin{split}S_{\text{W}}=S_{\text{EM}}+S_{\text{QC}}=\frac{1}{4}\int_{s}d^{n}y\sqrt{\gamma}&\left(\rho_{\text{EM}}+\rho_{\text{QC}}\right)\,.\end{split}$
(7)

Eq. (7) shows that the Wald entropy of black holes in the general quadric
corrected Einstein-Maxwell gravity can be divided into two parts. The first
part $S_{\text{EM}}$ is the entropy of black holes in the Einstein-Maxwell
gravity with the non-minimal coupling interaction between gravity and the
electromagnetic field as well as the self-interaction of the Maxwell field,
the second part $S_{\text{QC}}$ is the entropy that comes from the quadratic
curvature correction. While $\rho_{\text{EM}}$ and $\rho_{\text{QC}}$ are
densities of the two types of entropy. Since the entropy of black holes in
$F(\text{Riemann})$ gravity generally obeys the linearized second law shown in
Refs [13, 15], the second part of the Wald entropy obeys the linearized second
law naturally. To investigate the linearized second law of black holes in the
general quadric corrected Einstein-Maxwell gravity, it is not hard to think
that we only need to examine whether the first part of the Wald entropy
$S_{\text{EM}}$ still satisfies the linearized second law of thermodynamics.
In other words, the Lagrangian that is used to examine the linearized second
law is given as

$\bm{\mathcal{L}}=\bm{\mathcal{L}}_{\text{EM}}+\Delta\bm{\mathcal{L}}_{1}+\bm{\mathcal{L}}_{\text{mt}}\,.$
(8)

From the Lagrangian of Eq. (8), the equation of motion of the gravitational
part can be formally written as

$H_{ab}=8\pi T_{ab}\,.$ (9)

In which the specific expression of $H_{ab}$ is

$H_{ab}=G_{ab}+\frac{1}{2}\sum_{i=1}^{3}\alpha_{i}H_{ab}^{(i)}$ (10)

with

$\begin{split}H_{ab}^{(1)}=&2R_{ab}F_{cd}F^{cd}-4RF_{a}^{\
c}F_{bc}-g_{ab}RF_{cd}F^{cd}-2F^{cd}\nabla_{a}\nabla_{b}F_{cd}\\\
&-4\nabla_{a}F^{cd}\nabla_{b}F_{cd}-2F^{cd}\nabla_{b}\nabla_{a}F_{cd}+4g_{ab}F^{cd}\nabla_{e}\nabla^{e}F_{cd}\\\
&+4g_{ab}\nabla_{e}F_{cd}\nabla^{e}F^{cd}\,,\\\
H_{ab}^{(2)}=&\nabla_{a}F_{b}^{\ c}\nabla_{d}F_{c}^{\ d}-2R_{cd}F_{a}^{\
c}F_{b}^{\ d}-g_{ab}F^{cd}R_{de}F_{c}^{\ e}+\nabla_{b}F_{a}^{\
c}\nabla_{d}F_{c}^{\ d}\\\ &+F^{cd}\nabla_{d}\nabla_{a}F_{bc}+F_{b}^{\
c}\nabla_{d}\nabla_{a}F_{c}^{\ d}+F^{cd}\nabla_{d}\nabla_{b}F_{ac}+F_{a}^{\
c}\nabla_{d}\nabla_{b}F_{c}^{\ d}\\\ &+F_{b}^{\
c}\nabla_{d}\nabla^{d}F_{ac}+F_{a}^{\
c}\nabla_{d}\nabla^{d}F_{bc}+g_{ab}F^{cd}\nabla_{d}\nabla_{e}F_{c}^{\
e}+\nabla_{b}F_{cd}\nabla^{d}F^{\ c}_{a}\\\
&+\nabla_{a}F_{cd}\nabla^{d}F_{b}^{\
c}-g_{ab}\nabla_{c}F^{cd}\nabla_{e}F_{d}^{\
e}+g_{ab}F^{cd}\nabla_{e}\nabla_{d}F_{c}^{\
e}+g_{ab}\nabla_{d}F_{ce}\nabla^{e}F^{cd}\\\
&+2\nabla_{d}F_{bc}\nabla^{d}F_{a}^{\ c}\,,\\\ H_{ab}^{(3)}=&2F_{b}^{\
c}\nabla_{c}\nabla_{d}F_{a}^{\ d}-R_{acde}F_{b}^{\ c}F^{de}-F_{a}^{\
c}F^{de}R_{bcde}-g_{ab}R_{cdef}F^{cd}F^{ef}\\\ &+2F_{a}^{\
c}\nabla_{c}\nabla_{d}F_{b}^{\ d}+4\nabla_{c}F_{a}^{\ c}\nabla_{d}F_{b}^{\
d}+2F_{b}^{\ c}\nabla_{d}\nabla_{c}F_{a}^{\ d}+2F_{a}^{\
c}\nabla_{d}\nabla_{c}F_{b}^{\ d}\\\ &+4\nabla_{c}F_{bd}\nabla^{d}F_{a}^{\
c}\,,\end{split}$ (11)

and $G_{ab}=R_{ab}-1/2Rg_{ab}$ is the Einstein tensor. $T_{ab}$ in Eq. (9) is
the total stress-energy tenor of the theory, it can be decomposed as

$T_{ab}=T_{ab}^{\text{EM}}+T_{ab}^{\text{mt}}\,,$ (12)

where $T_{ab}^{\text{EM}}$ represents the stress-energy tenor of the
electromagnetic field, and $T_{ab}^{\text{mt}}$ is the stress-energy tensor of
extra matter fields.

According to Eq. (8) and the definition of the Wald entropy, the first part of
the Wald entropy $S_{EM}$ is given as

$S_{EM}=\frac{1}{4}\int_{s}d^{n}y\sqrt{\gamma}\rho_{EM}\,,$ (13)

and the concrete expression of the density $\rho_{EM}$ can be written as

$\begin{split}\rho_{EM}=&1+\alpha_{1}\left(F^{\hat{a}\hat{b}}F_{\hat{a}\hat{b}}-2F_{ac}F_{bd}k^{a}k^{b}l^{c}l^{d}-4F_{a\hat{c}}F_{b}^{\
\hat{c}}k^{a}l^{b}\right)\\\
&-\alpha_{2}\left(F_{ac}F_{bd}k^{a}k^{b}l^{c}l^{d}+F_{a\hat{c}}F_{b}^{\
\hat{c}}k^{a}l^{b}\right)-2\alpha_{3}F_{ac}F_{bd}k^{a}k^{b}l^{c}l^{d}\,,\end{split}$
(14)

where we have used the symbol “ $\hat{}$ ” to label the spatial index, i.e.,
$\gamma_{a}^{\ c}\gamma_{b}^{\ d}X_{cd}=X_{\hat{a}\hat{b}}$, to simplify the
expressions. Comparing with the pure Einstein-Maxwell theory, we can see that
the non-minimal coupling terms directly affect the expression of the entropy,
and it is no longer proportional to the area of the cross section of the event
horizon. Therefore, to examine the linearized second law, we only need to
investigate whether the expression of Eq. (13) still obeys the linearized
second law of black hole thermodynamics under the first-order approximation of
the matter fields perturbation.

##  3 Examining the linearized second law of black holes in general quadric
corrected Einstein-Maxwell gravity

Since the non-minimal coupling terms in the Lagrangian of the general quadric
corrected Einstein-Maxwell gravity can directly affect the expression of the
Wald entropy of black holes, we would like to examine whether the Wald entropy
in the corrected gravitational theory still obeys the linearized second law of
black hole thermodynamics. To examine the linearized second law, a slow
physical accretion process should be introduced first as well. This process is
described as extra matter fields pass through the event horizon and fall into
the black hole during the quasistationary accretion process. It implies that
the process can be regarded as a perturbation for black holes by extra matter
fields. Furthermore, it should be required that after the perturbation
process, the spacetime geometry of black holes must finally settle down to a
stationary state.

In ($\text{n}+2$)-dimensional general quadric corrected Einstein-Maxwell
gravity, the event horizon of black holes is denoted as $\mathcal{H}$, which
is a ($\text{n}+1$)-dimensional null hypersurface. When a parameter $\lambda$
is chosen as an affine parameter on the event horizon, the event horizon can
be generated by the null vector field
$k^{a}=\left(\partial/\partial\lambda\right)^{a}$, while the null vector field
$k^{a}$ satisfies the geodesic equation $k^{b}\nabla_{b}k^{a}=0$. It can be
demonstrate that the choice of the null vector field does not influence on the
examination of the linearized second law [14, 17]. For any cross section on
the event horizon, a coordinates with two null fields
$\\{k^{a},l^{a},y^{a}\\}$ can be constructed, where $l^{a}$ is a second null
vector, and $y^{a}$ represents the transverse coordinates on any slice. The
relationships between two null vector fields $k^{a}$ and $l^{a}$ are

$k^{a}k_{a}=l^{a}l_{a}=0\,,\qquad k^{a}l_{a}=-1\,.$ (15)

Based on the two null vectors, the binormal of the cross section is defined as
$\epsilon_{ab}=2k_{[a}l_{b]}$, and the induced metric on any cross section of
the future event horizon is defined as

$\gamma_{ab}=g_{ab}+2k_{(a}l_{b)}\,.$ (16)

Since $\gamma_{ab}$ is the spatial metric, the relationship between the two
null vector and the induced metric can be expressed as
$k^{a}\gamma_{ab}=l^{a}\gamma_{ab}=0$. For any spatial tensor
$X_{a_{1}a_{2}\cdots}$, using the induced metric $\gamma_{ab}$, one can define
the spatial derivative operator $D_{a}$ as

$D_{a}X_{a_{1}a_{2}\cdots}=\gamma_{a}^{\ b}\gamma_{a_{1}\cdots}^{\
b_{1}\cdots}\nabla_{b}X_{b_{1}b_{2}\cdots}\,,$ (17)

and the spatial derivative operator is compatible to the induce metric,
$D_{c}\gamma_{ab}=0$.

The extrinsic curvature of the event horizon is defined as

$B_{ab}=\gamma_{a}^{\ c}\gamma_{b}^{\ d}\nabla_{c}k_{d}\,,$ (18)

and the evolution of the induced metric along the future event horizon can be
obtained as

$\gamma_{a}^{\ c}\gamma_{b}^{\
d}\mathcal{L}_{k}\gamma_{cd}=2\left(\sigma_{ab}+\frac{\theta}{n}\gamma_{ab}\right)=2B_{ab}\,,$
(19)

where $\sigma_{ab}$ and $\theta$ represents the shear and the expansion of the
event horizon respectively. The evolution of the extrinsic curvature along the
future event horizon can also be obtained as

$\gamma_{a}^{\ c}\gamma_{b}^{\ d}\mathcal{L}_{k}B_{cd}=B_{ac}B_{b}^{\
c}-\gamma_{a}^{\ c}\gamma_{b}^{\ d}R_{ecfd}k^{e}k^{f}\,.$ (20)

From this result, the Raychaudhuri equation is given as

$\frac{d\theta}{d\lambda}=-\frac{\theta^{2}}{n-2}-\sigma_{ab}\sigma^{ab}-R_{kk}\,,$
(21)

where we have used the convention $X_{kk}=X_{ab}k^{a}k^{b}$ for any tensor
$X_{ab}$.

To represent the perturbation that comes from black holes absorbing matter
fields, the sufficient small parameter $\epsilon$ is introduced to describe
the order of the approximation of the perturbation. According to the small
parameter, we assume that
$B_{ab}\sim\theta\sim\sigma_{ab}\sim\mathcal{O}\left(\epsilon\right)$. Since
we only would like to consider the linearized second law during the dynamical
process, the symbol “$\simeq$” will be used to represent the identity under
the linear order approximation of the matter fields perturbation in the
following calculation.

Considering the first-order approximation of the dynamical perturbation, the
linear version of the Raychaudhuri equation can be written as

$\frac{d\theta}{d\lambda}\simeq-R_{kk}\,,$ (22)

and Eq. (20) is simplified as

$\gamma_{a}^{\ c}\gamma_{b}^{\ d}\mathcal{L}_{k}B_{cd}\simeq-\gamma_{a}^{\
c}\gamma_{b}^{\ d}R_{ecfd}k^{e}k^{f}\,.$ (23)

From the physical perspective, the total stress-energy tensor should be
satisfied the null energy condition. Considering any null vector field
$n^{a}$, the null energy condition of $T_{ab}$ can be expressed as
$T_{ab}n^{a}n^{b}\geq 0$. It indicates that two parts of the total stress-
energy tensor are both obeying the null energy condition. An assumption should
be added in our demonstration that the matter fields always satisfy the null
energy condition. According to the calculation in Appendix A, it is shown that
in the coordinates with two null vectors $k^{a}$ and $l^{a}$, the total
stress-energy tensor under the first-order approximation of the perturbation
can be expressed as

$T_{ab}k^{a}k^{b}\simeq T_{ab}^{\text{mt}}k^{a}k^{b}\geq 0\,.$ (24)

It implies that when we only consider the linear order approximation, the
total stress-energy tensor obeys the requirement of the null energy condition
naturally.

From now on, we would like to examine whether the first part of the Wald
entropy still satisfies the linearized second law during the matter fields
perturbation. Following a similar train of thought of Ref. [14], when the
linearized second law is valid during the perturbation process while the total
stress-energy tensor obeys the null energy condition, the entropy of black
holes should be satisfied the following relationship under the first-order
approximation, i.e.,

$\begin{split}\mathcal{L}_{k}^{2}S\simeq-2\pi\int_{s}\tilde{\epsilon}T_{kk}=-\frac{1}{4}\int_{s}\tilde{\epsilon}H_{kk}<0\,.\end{split}$
(25)

Since we have assumed that the evolution of black holes eventually tends to a
stationary state after the dynamical process, it implies that the rate of
change of the entropy should vanish in the asymptotic future. In other words,
the second-order Lie derivative of the entropy is negative,
$\mathcal{L}_{k}^{2}S<0$, during the process. Meanwhile, the decrease of the
rate of change also implies that the value of the entropy of black holes
always increases with the perturbation process in the future null direction,
i.e., $\mathcal{L}_{k}S>0$. To investigate whether the first part of the Wald
entropy obeys the linearized second law, we only need to examine whether the
first part of the Wald entropy satisfies the relationship

$\mathcal{L}_{k}^{2}S_{\text{EM}}\simeq-\frac{1}{4}\int_{s}\tilde{\epsilon}H_{kk}<0$
(26)

under the linear order approximation. Therefore, in the following, we need to
obtain the expression of the integral of $H_{kk}$ on an arbitrary cross
section of the event horizon under the linear order approximation.

From Eq. (10), the specific expression of $H_{kk}$ is given as

$\begin{split}H_{kk}=&R_{kk}+\frac{1}{2}\sum_{i=1}^{3}\alpha_{i}H_{kk}^{(i)}\,,\end{split}$
(27)

where

$\begin{split}H_{kk}^{(1)}=&2R_{kk}F_{cd}F^{cd}-4Rk^{a}k^{b}F_{a}^{\
c}F_{bc}-2k^{a}k^{b}\nabla_{a}\nabla_{b}\left(F^{cd}F_{cd}\right)\,,\end{split}$
(28) $\begin{split}H_{kk}^{(2)}=&k^{a}k^{b}\nabla_{a}F_{b}^{\
c}\nabla_{d}F_{c}^{\ d}-2k^{a}k^{b}F_{a}^{\ c}F_{b}^{\
d}R_{cd}+k^{a}k^{b}\nabla_{b}F_{a}^{\ c}\nabla_{d}F_{c}^{\ d}\\\
&+k^{a}k^{b}F^{cd}\nabla_{d}\nabla_{a}F_{bc}+k^{a}k^{b}F_{b}^{\
c}\nabla_{d}\nabla_{a}F_{c}^{\
d}+k^{a}k^{b}F^{cd}\nabla_{d}\nabla_{b}F_{ac}\\\ &+k^{a}k^{b}F_{a}^{\
c}\nabla_{d}\nabla_{b}F_{c}^{\ d}+k^{a}k^{b}F_{b}^{\
c}\nabla_{d}\nabla^{d}F_{ac}+k^{a}k^{b}F_{a}^{\
c}\nabla_{d}\nabla^{d}F_{bc}\\\ &+k^{a}k^{b}\nabla_{b}F_{cd}\nabla^{d}F_{a}^{\
c}+k^{a}k^{b}\nabla_{a}F_{cd}\nabla^{d}F_{b}^{\
c}+2k^{a}k^{b}\nabla_{d}F_{bc}\nabla^{d}F_{a}^{\ c}\,,\end{split}$ (29)
$\begin{split}H_{kk}^{(3)}=&2k^{a}k^{b}F_{b}^{\ c}\nabla_{c}\nabla_{d}F_{a}^{\
d}-k^{a}k^{b}R_{acde}F_{b}^{\ c}F^{de}-k^{a}k^{b}R_{bcde}F_{a}^{\ c}F^{de}\\\
&+2k^{a}k^{b}F_{a}^{\ c}\nabla_{c}\nabla_{d}F_{b}^{\
d}+4k^{a}k^{b}\nabla_{c}F_{a}^{\ c}\nabla_{d}F_{b}^{\ d}+2k^{a}k^{b}F_{b}^{\
c}\nabla_{d}\nabla_{c}F_{a}^{\ d}\\\ &+2k^{a}k^{b}F_{a}^{\
c}\nabla_{d}\nabla_{c}F_{b}^{\
d}+4k^{a}k^{b}\nabla_{c}F_{bd}\nabla^{d}F_{a}^{\ c}\,.\end{split}$ (30)

For the integral form on any slice of the first term in Eq. (27), the
integrand under the first-order approximation is given as

$\begin{split}R_{kk}\sqrt{\gamma}=&-\left(\mathcal{L}_{k}\theta\right)\sqrt{\gamma}\\\
=&-\mathcal{L}_{k}\left(\theta\sqrt{\gamma}\right)+\theta\left(\mathcal{L}_{k}\sqrt{\gamma}\right)\\\
=&-\mathcal{L}_{k}^{2}\sqrt{\gamma}+\theta^{2}\sqrt{\gamma}\\\
\simeq&-\mathcal{L}_{k}^{2}\sqrt{\gamma}\,.\end{split}$ (31)

The integral form of the first term in Eq. (27) can be expressed as

$\begin{split}\int_{s}\tilde{\epsilon}R_{kk}\simeq\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left(-1\right)\,.\end{split}$
(32)

According to Appendix B, after retaining the linear order terms as well as
neglecting the high-order terms, the $H_{kk}^{(1)}$ can be finally written as

$\begin{split}H_{kk}^{(1)}=2R_{kk}F_{cd}F^{cd}-2\mathcal{L}_{k}^{2}\left(F^{cd}F_{cd}\right)\,.\end{split}$
(33)

To obtain the integral form of $H_{kk}^{(1)}$ on the cross-section, the
integrand should be simplified firstly. Combining with the induced volume
element and using the result of Eq. (31), the integrand in the integral form
of $H_{kk}^{(1)}$ can be calculated as

$\begin{split}H_{kk}^{(1)}\sqrt{\gamma}=&2R_{kk}F_{cd}F^{cd}\sqrt{\gamma}-2\mathcal{L}_{k}^{2}\left(F^{cd}F_{cd}\right)\sqrt{\gamma}\\\
\simeq&-2F_{cd}F^{cd}\left(\mathcal{L}_{k}^{2}\sqrt{\gamma}\right)-2\mathcal{L}_{k}^{2}\left(F^{cd}F_{cd}\right)\sqrt{\gamma}\\\
\simeq&-2\mathcal{L}_{k}^{2}\left[\left(F_{cd}F^{cd}\right)\sqrt{\gamma}\right]+2\mathcal{L}_{k}^{2}\left(F_{cd}F^{cd}\right)\sqrt{\gamma}\\\
&+4\mathcal{L}_{k}\left(F_{cd}F^{cd}\right)\left(\mathcal{L}_{k}\sqrt{\gamma}\right)-2\mathcal{L}_{k}^{2}\left(F_{cd}F^{cd}\right)\sqrt{\gamma}\\\
\simeq&\mathcal{L}_{k}^{2}\left[-2\left(F_{cd}F^{cd}\right)\sqrt{\gamma}\right]\,.\end{split}$
(34)

Expanding duplicated indexes in the tensors of electromagnetic field, the
integral form of $H_{kk}^{(1)}$ can be finally written as

$\begin{split}\int_{s}\tilde{\epsilon}H_{kk}^{(1)}\simeq\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left[4\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)-2F_{\hat{c}\hat{d}}F^{\hat{c}\hat{d}}+8\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\,.\end{split}$ (35)

According to Appendix C, the expression of $H_{kk}^{(2)}$ under the first-
order approximation of the dynamical process is given as

$\begin{split}H_{kk}^{(2)}\simeq&-2\left[\mathcal{L}_{k}\left(k^{b}F_{b}^{\
\hat{e}}\right)\right]\left[D_{\hat{e}}\left(k^{a}F_{ad}l^{d}\right)\right]-2\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\\\
&+2\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)D_{\hat{e}}\left(F^{\hat{c}\hat{e}}\right)+2\left(F^{\hat{e}\hat{d}}\right)D_{\hat{d}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]\\\
&-4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]+4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\\\
&-2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\left(R_{kk}\right)+4\left(k^{a}F_{ae}l^{e}\right)\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\\\
&+2\left(l^{d}F_{d}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(k^{a}F_{a\hat{e}}\right)\right]+2\left(k^{a}F_{a\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\\\
&+4\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]\left[\mathcal{L}_{k}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\,.\end{split}$ (36)

The first four terms of Eq. (36) can be simplified as

$\begin{split}&-2\left[\mathcal{L}_{k}\left(k^{b}F_{b}^{\
\hat{e}}\right)\right]\left[D_{\hat{e}}\left(k^{a}F_{ad}l^{d}\right)\right]-2\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\\\
&+2\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)D_{\hat{e}}\left(F^{\hat{c}\hat{e}}\right)+2\left(F^{\hat{e}\hat{d}}\right)D_{\hat{d}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]\\\
=&-2D_{\hat{e}}\left[\left(k^{a}F_{ad}l^{d}\right)\mathcal{L}_{k}\left(k^{b}F_{b}^{\
\hat{e}}\right)\right]+2\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\\\
&-2\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]+2D_{\hat{e}}\left[\left(F^{\hat{c}\hat{e}}\right)\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)\right]\\\
&-2\left(F^{\hat{c}\hat{e}}\right)D_{\hat{e}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)\right]+2\left(F^{\hat{c}\hat{e}}\right)D_{\hat{e}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)\right]\\\
=&-2D_{\hat{e}}\left[\left(k^{a}F_{ad}l^{d}\right)\mathcal{L}_{k}\left(k^{b}F_{b}^{\
\hat{e}}\right)\right]+2D_{\hat{e}}\left[\left(F^{\hat{c}\hat{e}}\right)\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)\right]\,.\end{split}$
(37)

Based on the assumption that the event horizon is compact, the surface term in
the integral on the cross-section of the event horizon can be ignored
directly. It means that the first four terms of Eq. (36) do not contributed to
the final result. The fifth and sixth terms of Eq. (36) can be canceled with
each other. Utilizing the result of Eq. (31) again, the seventh and eighth
terms of Eq. (36) with the induced volume element can be calculated as

$\begin{split}&-2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\left(R_{kk}\right)\sqrt{\gamma}+4\left(k^{a}F_{ae}l^{e}\right)\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\sqrt{\gamma}\\\
\simeq&2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\mathcal{L}_{k}^{2}\sqrt{\gamma}+2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\right]\sqrt{\gamma}\\\
\simeq&2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\sqrt{\gamma}\right]-2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\right]\sqrt{\gamma}\\\
&-4\mathcal{L}_{k}\left[\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\right]\theta\sqrt{\gamma}+2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\right]\sqrt{\gamma}\\\
\simeq&2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\sqrt{\gamma}\right]\,.\end{split}$
(38)

Therefore, the integral form of the seventh and eighth terms in Eq. (36) is
obtained as

$\begin{split}&\int_{s}\tilde{\epsilon}\left[-2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\left(R_{kk}\right)+4\left(k^{a}F_{ae}l^{e}\right)\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\\\
=&\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left[2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\right]\,.\end{split}$
(39)

For the last three terms of Eq. (36), the integrand of these terms under the
first-order approximation can be simplified as

$\begin{split}&2\left(l^{d}F_{d}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(k^{a}F_{a\hat{e}}\right)\right]\sqrt{\gamma}+2\left(k^{a}F_{a\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\sqrt{\gamma}\\\
&+4\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]\left[\mathcal{L}_{k}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\sqrt{\gamma}\\\
=&2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\sqrt{\gamma}\\\
=&2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\sqrt{\gamma}\right]-4\mathcal{L}_{k}\left[\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\theta\sqrt{\gamma}\\\
&-2\left[\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\left(\mathcal{L}_{k}\theta\right)\sqrt{\gamma}-2\left[\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\theta^{2}\sqrt{\gamma}\\\
\simeq&2\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\sqrt{\gamma}\right]\,.\end{split}$ (40)

The integral form of the last three terms can be further written as

$\begin{split}&\int_{s}\tilde{\epsilon}\left[2\left(l^{d}F_{d}^{\
\hat{e}}\right)\mathcal{L}_{k}^{2}\left(k^{a}F_{a\hat{e}}\right)+2\left(k^{a}F_{a\hat{e}}\right)\mathcal{L}_{k}^{2}\left(l^{d}F_{d}^{\
\hat{e}}\right)+4\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\mathcal{L}_{k}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\\\
=&\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left[2\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\,.\end{split}$ (41)

Therefore, the integral form of $H_{kk}^{(2)}$ can be finally expressed as

$\begin{split}\int_{s}\tilde{\epsilon}H_{kk}^{(2)}=\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left[2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)+2\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\,.\end{split}$ (42)

The result of $H_{kk}^{(3)}$ under the first-order approximation is given by
Appendix D, the specific expression can be expressed as

$\begin{split}H_{kk}^{(3)}\simeq&-8\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]-8\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)\\\
&-4\left(k^{c}F_{ce}l^{e}\right)\left(k^{d}F_{df}l^{f}\right)\left(R_{kk}\right)+8\left(k^{a}F_{ae}l^{e}\right)\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\,.\end{split}$
(43)

According to the compactness of the event horizon, the integral form of the
first two terms in Eq. (43) can be simplified as

$\begin{split}&\int_{s}\tilde{\epsilon}\left[-8\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]-8\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)\right]\\\
=&-8\int_{s}\tilde{\epsilon}D^{\hat{c}}\left[\left(k^{a}F_{ad}l^{d}\right)\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)\right]+8\int_{s}\tilde{\epsilon}D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)\\\
&-8\int_{s}\tilde{\epsilon}\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)\\\
=&0\,,\end{split}$ (44)

The integral of $H_{kk}^{(3)}$ is further given as

$\begin{split}\int_{s}\tilde{\epsilon}H_{kk}^{(3)}=&\int_{s}\tilde{\epsilon}\left[-4\left(k^{c}F_{ce}l^{e}\right)\left(k^{d}F_{df}l^{f}\right)\left(R_{kk}\right)+8\left(k^{a}F_{ae}l^{e}\right)\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\right]\,.\end{split}$
(45)

Using the result of Eq. (31), the integrand in Eq. (45) can be simplified as

$\begin{split}H_{kk}^{(3)}\sqrt{\gamma}=&-4\left(k^{c}F_{ce}l^{e}\right)\left(k^{d}F_{df}l^{f}\right)\left(R_{kk}\right)\sqrt{\gamma}+8\left(k^{a}F_{ae}l^{e}\right)\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\sqrt{\gamma}\\\
\simeq&4\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\mathcal{L}_{k}^{2}\sqrt{\gamma}+4\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\right]\sqrt{\gamma}\\\
\simeq&4\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\sqrt{\gamma}\right]-4\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\right]\sqrt{\gamma}\\\
&-4\mathcal{L}_{k}\left[\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\right]\mathcal{L}_{k}\sqrt{\gamma}+4\mathcal{L}_{k}^{2}\left[\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\right]\sqrt{\gamma}\\\
\simeq&\mathcal{L}_{k}^{2}\left[4\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\sqrt{\gamma}\right]\,.\end{split}$
(46)

Therefore, the integral form of $H_{kk}^{(3)}$ can be finally obtained as

$\begin{split}\int_{s}\tilde{\epsilon}H_{kk}^{(3)}=\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left[4\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\right]\,.\end{split}$
(47)

Combining with Eq. (32), Eq. (35), Eq. (42), and Eq. (47), the integral form
of $H_{kk}$ can be given as

$\begin{split}\int_{s}\tilde{\epsilon}H_{kk}=&\int_{s}\tilde{\epsilon}\left(R_{kk}+\frac{1}{2}\sum_{i=1}^{3}\alpha_{i}H_{kk}^{(i)}\right)\\\
\simeq&-\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\left[1+\alpha_{1}F_{\hat{c}\hat{d}}F^{\hat{c}\hat{d}}-2\alpha_{1}\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)-4\alpha_{1}\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)\right.\\\
&\left.-\alpha_{2}\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)-\alpha_{2}\left(k^{a}F_{a\hat{e}}\right)\left(l^{d}F_{d}^{\
\hat{e}}\right)-2\alpha_{3}\left(k^{a}F_{ab}l^{b}\right)\left(k^{c}F_{cd}l^{d}\right)\right]\\\
=&-\mathcal{L}_{k}^{2}\int_{s}\tilde{\epsilon}\
\rho_{\text{EM}}\,,\end{split}$ (48)

where we have used Eq. (14) in the last step. After adding coefficient $1/4$
to both sides of Eq. (48), we have

$\mathcal{L}_{k}^{2}S_{\text{EM}}=-\frac{1}{4}\int_{s}\tilde{\epsilon}H_{kk}<0\,.$
(49)

This result implies that the value of $S_{\text{EM}}$ always increases, while
its growth rate gradually decreases until the end of the perturbation process.
In other words, the value of the first part of the Wald entropy increases with
the dynamical process, and black holes finally settle down to a stationary
state after the perturbation. Since the second part of the Wald entropy
$S_{\text{QC}}$ has been demonstrated that it always obeys the linearized
second law, the Wald entropy of black holes in general quadric corrected
Einstein-Maxwell gravity still satisfies the linearized second law under the
first-order approximation of the matter fields perturbation.

##  4 Discussions and Conclusions

Although the Wald entropy of black holes in general quadric corrected
Einstein-Maxwell gravity has been generally demonstrated that it always
satisfies the linearized second law, there are still some key issues that
should be discussed in further detail. In our demonstration, In our
demonstration, the Wald entropy has been used to represent the entropy of
black holes in the corrected gravitational theory. However, the Wald entropy
is only suitable for describing the entropy of black holes at the initial and
final states because the two states are both stationary, while the expression
of the Wald entropy is independent of the selection of the cross-section on
the event horizon. Since the states of black holes are no longer stationary
during the perturbation process, the cross-section is evolving with the
process all the time. And the expression of the Wald entropy also changes with
the evolution of the cross-section according to the definition. This situation
makes it impossible to distinguish which expression of the Wald entropy
corresponds to the entropy of black holes during the perturbation. The
definition of the cross-section comes from the choice of the null generator
$k^{a}$. It means that the evolution of the cross-section is caused by the
change of $k^{a}$ with the perturbation. Fortunately, the proof of the
linearized second law is independent of the choice of the null generator as
mentioned above. It indicates that our demonstration is valid for any cross-
section on the event horizon whether or not the area of the cross-section
corresponds to the entropy of black holes. Besides, for the issue of which
area of cross-section corresponds to the entropy of black holes during the
perturbation process, we cannot solve it only relying on the first and the
second laws of thermodynamics unless involving some other constraint
conditions.

The entropy of black holes in any diffeomorphism invariant theory of gravity
is generally given by the Wald entropy, while it has been shown that the Wald
entropy always obeys the first law of black hole thermodynamics. However,
whether the Wald entropy obeys the second law of thermodynamics still requires
further study. When we consider a gravitational theory that contains the non-
minimal coupling interaction between gravity and matter fields, the expression
of the Wald entropy can be sufficiently affected by the non-minimal coupling
terms in the Lagrangian. Therefore, examining whether the Wald entropy of
black holes in the gravitational theory with the non-minimum coupling matter
fields satisfies the second law is an important prerequisite for examining
whether it generally satisfies the second law in any diffeomorphism invariant
gravity. The general quadric corrected Einstein-Maxwell gravity mainly
includes three types of correction terms, which describe self-interaction of
the electromagnetic field, the non-minimal coupling interaction between
gravity and the Maxwell field, and the quadratic interaction of gravity.
According to the definition of the Wald entropy, the non-minimal coupling
terms and the quadratic curvature terms will sufficiently affect the form of
the Wald entropy, while the expression of the Wald entropy can be divided into
the two parts. The first part is the entropy of black holes in the Einstein-
Maxwell gravity with non-minimal coupling interaction and the self-interaction
of the electromagnetic field, the second part of the Wald entropy comes from
the quadratic curvature correction. Since we only want to investigate the
linearized second law, the two parts of the Wald entropy can be examined
respectively whether it obeys the linearized second law. However, it has shown
that the second part of the Wald entropy always obeys the linearized second
law. So to investigate the linearized second law of black holes in the general
quadric corrected Einstein-Maxwell gravity, we only need to examine whether
the first part of the Wald entropy still obeys the linearized second law. To
study this issue, a quasistationary accreting process that matter fields pass
through the event horizon and perturb the geometry of the black hole is
considered, while matter fields are required to satisfy the null energy
condition. Two assumptions that black holes will settle down to a stationary
state after the perturbation process and that there is a regular bifurcation
surface in the background spacetime are further introduced. According to the
Raychaudhuri equation, we demonstrate that the second-order Lie derivative of
the first part of the Wald entropy is negative under the linear order
approximation of the dynamical accreting process. It implies that the value of
the entropy must increase along the direction of the future event horizon
during the process, while the magnitude of its increase gradually decreases.
The result indicates that black holes can eventually tend to a stationary
state, and the first part of the Wald entropy always satisfies the requirement
of the linearized second law during the perturbation process. Since the two
parts of the Wald entropy are obeying the linearized second law, the Wald
entropy of black holes in the general quadric correction Einstein-Maxwell
gravity always satisfies the linearized second law.

## Acknowledgement

X.-Y. W. is supported by the Talents Introduction Foundation of Beijing Normal
University with Grant No. 111032109. J. J. is supported by the National
Natural Science Foundation of China (NSFC) with Grants No. 11775022 and
11873044.

##  Appendix A Examining the null energy condition of the total stress-energy
tensor

We would like to demonstrate that the total stress-energy tensor in Eq. (9)
should satisfy the null energy tensor in this appendix. The stress energy
tensor can be decomposed into the two parts, which can be formally written as

$T_{ab}=T_{ab}^{\text{EM}}+T_{ab}^{\text{mt}}\,.$ (50)

We have assumed that additional matter fields should always satisfy the null
energy condition. It means that for any null vector $n^{a}$, the null energy
condition of the stress-energy tensor of matter fields can be expressed as

$T_{ab}^{\text{mt}}n^{a}n^{b}\geq 0\,.$ (51)

Hence, to prove the total stress-energy tensor $T_{ab}$ obeying the null
energy condition, we only need to demonstrate the electromagnetic field part
of stress-energy tensor $T_{ab}^{\text{EM}}$ satisfies the null energy
condition. From the equation of motion, the specific expression of
$T_{ab}^{\text{EM}}$ is given by

$\begin{split}T_{ab}^{\text{EM}}=&\alpha_{4}\left(4F_{a}^{\
c}F_{bc}F_{de}F^{de}-\frac{1}{2}g_{ab}F_{cd}F^{cd}F_{ef}F^{ef}\right)\\\
&+\alpha_{5}\left(4F_{a}^{\ c}F_{b}^{\ d}F_{c}^{\
e}F_{de}-\frac{1}{2}g_{ab}F_{c}^{\ e}F^{cd}F_{d}^{\
f}F_{ef}\right)\,.\end{split}$ (52)

According to the coordinates defined in Sec. 3, the electromagnetic field part
of stress-energy tensor $T_{ab}^{\text{EM}}$ contracting with two null vectors
$k^{a}$ can be further expressed as

$T_{kk}^{\text{EM}}=k^{a}k^{b}T_{ab}^{\text{EM}}=4\alpha_{4}k^{a}k^{b}F_{a}^{\
c}F_{bc}F_{de}F^{de}+4\alpha_{5}F_{a}^{\ c}F_{b}^{\ d}F_{c}^{\ e}F_{de}\,.$
(53)

Before investigating the null energy condition of $T_{ab}^{\text{EM}}$, an
convention should be introduced first. In the following calculation, we will
use the parenthesis or the square brackets with a subscript to represents the
order of the expression, i.e., $(\ )_{n}$ or $[\ ]_{n}$, $n=0,1$. The
expression is a zero-order or a background quantity when $n=0$, and the
expression is a first-order quantity when $n=1$. According to the convention
to represent the order of the expression, we further examine whether the
stress-energy tensor $T_{ab}^{\text{EM}}$ satisfies the null energy condition.
Using the definition of the induced metric $\gamma^{ab}$ to expand the index
$c$ in the expression, the first term of Eq. (53) can be written as

$\begin{split}4\alpha_{4}k^{a}k^{b}F_{a}^{\
c}F_{bc}F_{de}F^{de}=4\alpha_{4}\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(k^{b}F_{b\hat{c}}\right)_{1}\left(F_{ef}F^{ef}\right)_{0}\simeq
0\,.\end{split}$ (54)

Since the result is the multiplication of two first-order terms, the first
term of Eq. (53) is the second-order term. The first term of Eq. (53) should
be dropped out because only the first-order approximation of the matter fields
perturbation is considered in our demonstration. Similarly, the second term in
Eq. (53) can be calculated as

$\begin{split}&4\alpha_{5}F_{a}^{\ c}F_{b}^{\ d}F_{c}^{\ e}F_{de}\\\
=&4\alpha_{5}\left(k^{a}F_{a\hat{c}}\right)_{1}\left(k^{b}F_{b\hat{d}}\right)_{1}F^{\hat{c}\hat{e}}F^{\hat{d}}_{\
\hat{e}}-4\alpha_{5}\left(k^{a}F_{ae}l^{e}\right)_{0}\left(k^{b}F_{bf}l^{f}\right)_{0}\left(k^{c}F_{c\hat{g}}\right)_{1}\left(k^{d}F_{d}^{\
\hat{g}}\right)_{1}\\\
&-8\alpha_{5}\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{b}F_{b}^{\
\hat{e}}\right)_{1}\left(k^{c}F_{c\hat{f}}\right)_{1}\left(l^{d}F_{d}^{\
\hat{f}}\right)_{0}\\\ \simeq&0\,.\end{split}$ (55)

We can easily see that at least two first-order terms exist in each term of
the above equation. It means that the second term in Eq. (53) is also a
second-order term. Since the first and the second terms of Eq. (53) are both
proportional to the second-order approximation, the electromagnetic part of
the stress-energy tensor can be neglected directly when only considering the
first-order approximation of the matter fields perturbation. Therefore,
utilizing the null energy condition of extra matter fields, the null energy
condition of the total stress-energy tensor can be simplified expressed as

$T_{kk}\simeq T_{kk}^{\text{mt}}\geq 0$ (56)

under the first-order approximation. It is shown that the total stress-energy
tensor always satisfies the null energy condition.

##  Appendix B Simplification of $H_{kk}^{(1)}$ under the linear order
approximation

In this appendix, we want to obtain the expression of Eq. (28) under the
first-order approximation of the quasistationary accreting process. Before
analyzing the expression of $H_{kk}^{(i)},i=1,2,3$, four important identities
that will be used heavily in the following calculations to simplify the
expression of the equation should be introduced. Firstly, we should prove that
two null vectors $k^{a}$ and $l^{a}$ on the background spacetime will satisfy
the identity

$k^{a}\nabla_{a}l_{b}=0\,.$ (57)

In order to demonstrate this identity, one can define the Killing vector
$\xi^{a}=\left(\partial/\partial v\right)^{a}$ which generates the event
horizon of the black holes on the background spacetime. Since the parameter
$v$ is not the affine parameter, the geodesic equation for the Killing vector
$\xi^{a}$ should be expressed as $\xi^{a}\nabla_{a}\xi^{b}=\kappa\xi^{b}$,
where $\kappa$ represents the surface gravity of black holes. From the
definition of $\xi^{a}$, the null vector $k^{a}$ can be obtained as
$k^{a}=e^{-\kappa v}\xi^{a}$. Using the identity $k^{a}l_{a}=-1$, the second
null vector $l^{a}$ can be naturally written as $l^{a}=e^{\kappa v}s^{a}$,
where $s^{a}$ is also a Killing vector. Furthermore, two relationships between
the two null Killing vectors can be given as

$\xi^{a}s_{a}=-1,\qquad\mathcal{L}_{\xi}s^{a}=\xi^{b}\nabla_{b}s^{a}-s^{b}\nabla_{b}\xi^{a}=0\,.$
(58)

From the definitions of two Killing vectors, $k^{a}\nabla_{a}l^{b}$ on the
background spacetime can be calculated as

$\begin{split}k^{a}\nabla_{a}l^{b}=&e^{-\kappa
v}\xi^{a}\nabla_{a}\left(e^{\kappa v}s^{b}\right)\\\
=&\xi^{a}\nabla_{a}s^{b}+\kappa s^{b}\\\ =&s^{a}\nabla_{a}\xi^{b}+\kappa
s^{b}\\\ =&\kappa s^{a}\hat{\epsilon}_{a}^{\ b}+\kappa s^{b}\\\
=&0\,,\end{split}$ (59)

where we have used the identity [8]

$\nabla_{a}\xi_{b}=\kappa\hat{\epsilon}_{ab}$ (60)

with the definition of the binormal $\hat{\epsilon}_{ab}=k_{a}\wedge
l_{b}=\xi_{a}\wedge s_{b}$ in the last step. Furthermore, using the definition
of $\xi^{a}$ and Eq. (60), the second and third identities on the background
spacetime can be respectively calculated as

$\begin{split}\gamma^{ab}\nabla_{b}k_{c}&=\gamma^{ab}\nabla_{b}\left(e^{-\kappa
v}\xi_{c}\right)\\\ &=\gamma^{ab}\xi_{c}\nabla_{b}e^{-\kappa
v}+\gamma^{ab}e^{-\kappa v}\nabla_{b}\xi_{c}\\\ &=\kappa\gamma^{ab}e^{-\kappa
v}\hat{\epsilon}_{bc}\\\ &=0\,,\end{split}$ (61)

and

$\begin{split}\gamma^{ab}\nabla_{c}k_{b}&=\gamma^{ab}\nabla_{c}\left(e^{-\kappa
v}\xi_{b}\right)\\\ =&\gamma^{ab}\xi_{b}\nabla_{c}e^{-\kappa
v}+\gamma^{ab}e^{-\kappa v}\nabla_{c}\xi_{b}\\\ =&\gamma^{ab}e^{-\kappa
v}\kappa\hat{\epsilon}_{cb}\\\ =&0\,.\end{split}$ (62)

From the results, one can see that two quantities $\gamma^{ab}\nabla_{b}k_{c}$
and $\gamma^{ab}\nabla_{c}k_{b}$ are both vanishing on the background
spacetime. In other words, two quantities are both the first-order terms. The
fourth identity on the background spacetime is

$\begin{split}\mathcal{L}_{k}\gamma_{ab}=&k^{c}\nabla_{c}\gamma_{ab}+\gamma_{cb}\nabla_{a}k^{c}+\gamma_{ac}\nabla_{b}k^{c}\\\
=&k^{c}\nabla_{c}\left(g_{ab}+k_{a}l_{b}+k_{b}l_{a}\right)\\\
=&l_{b}k^{c}\nabla_{c}k_{a}+k_{a}k^{c}\nabla_{c}l_{b}+l_{a}k^{c}\nabla_{c}k_{b}+k_{b}k^{c}\nabla_{c}l_{a}\\\
=&0\,,\end{split}$ (63)

where we have used the definition of the induced metric and the results of Eq.
(61) and Eq. (62) in the second step, the geodesic equation
$k^{b}\nabla_{b}k_{a}=0$ and the result of Eq. (59) is used in the last step.
Next, we will directly use the results of Eq. (57), Eq. (61), Eq. (62), and
Eq. (63) to simplify the specific expression of $H_{kk}^{(i)},i=1,2,3$.

After expanding duplicated indexes of the second term in Eq. (28), the
specific expression of $H_{kk}^{(1)}$ can be written as

$\begin{split}H_{kk}^{(1)}=2R_{kk}F_{cd}F^{cd}-4F_{a}^{\ c}F_{b}^{\
d}k^{a}k^{b}R\gamma_{cd}-2k^{a}k^{b}\nabla_{a}\nabla_{b}\left(F^{cd}F_{cd}\right)\,.\end{split}$
(64)

Under the linear order approximation, the first term of Eq. (64) is

$2R_{kk}\left(F_{cd}F^{cd}\right)=2\left(R_{kk}\right)_{1}\left(F_{cd}F^{cd}\right)_{0}\sim\mathcal{O}\left(\epsilon\right)\,.$
(65)

The second term of Eq. (64) is

$-4F_{a}^{\ c}F_{b}^{\ d}k^{a}k^{b}R\gamma_{cd}=-4\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(k^{b}F_{b\hat{c}}\right)_{1}\left(R\right)_{0}\simeq
0\,.$ (66)

The third of Eq. (64) can be simplified as

$-2k^{a}k^{b}\nabla_{a}\nabla_{b}\left(F^{cd}F_{cd}\right)=-2\left[\mathcal{L}_{k}^{2}\left(F^{cd}F_{cd}\right)\right]_{1}\sim\mathcal{O}\left(\epsilon\right)\,.$
(67)

Therefore, the expression of $H_{kk}^{(1)}$ under the first-order
approximation can be simplified as

$\begin{split}H_{kk}^{(1)}=2R_{kk}F_{cd}F^{cd}-2\mathcal{L}_{k}^{2}\left(F^{cd}F_{cd}\right)\,.\end{split}$
(68)

##  Appendix C Simplification of $H_{kk}^{(2)}$ under the linear order
approximation

We will continue to calculate Eq. (29) in the third appendix. Expanding
duplicated indexes in the expression again, Eq. (29) can be further written as

$\begin{split}H_{kk}^{(2)}=&-2F_{ae}F_{bf}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}R_{cd}-2F_{a}^{\
c}F_{b}^{\ d}k^{a}k^{b}R^{ef}\gamma_{ce}\gamma_{df}\\\ &+4F_{ad}F_{b}^{\
e}k^{a}k^{b}k^{c}l^{d}R_{c}^{\
f}\gamma_{ef}+4k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{b}F_{ae}\nabla_{c}F_{df}\\\
&+4F_{ad}F_{b}^{\ e}k^{a}k^{b}k^{c}l^{d}R_{c}^{\ f}\gamma_{ef}+2F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\ f}\nabla_{c}\nabla_{b}F_{df}\\\
&+4k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{b}F_{ae}\nabla_{d}F_{cf}+4F_{ae}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{d}\nabla_{c}F_{bf}\\\
&+4F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{f}\nabla_{d}\nabla_{c}F_{bf}+2k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{b}F_{df}\nabla_{e}F_{ac}\\\
&+4F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{f}\nabla_{d}\nabla_{c}F_{bf}+2F_{a}^{\
c}k^{a}k^{b}\gamma_{c}^{\ d}\nabla_{e}\nabla^{e}F_{bd}\\\
&+2k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{e}F_{ac}\nabla_{f}F_{bd}-2k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{b}F_{ad}\nabla_{f}F_{ce}\\\
&+2k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{b}F_{ac}\nabla_{f}F_{de}+2F^{cd}k^{a}k^{b}\gamma_{c}^{\
e}\gamma_{d}^{\ f}\nabla_{f}\nabla_{b}F_{ae}\\\ &+2F_{a}^{\
c}k^{a}k^{b}\gamma_{c}^{\ d}\gamma^{ef}\nabla_{f}\nabla_{b}F_{de}-2F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\ f}\nabla_{f}\nabla_{c}F_{bd}\\\
&-2F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\,.\end{split}$
(69)

For the first term of Eq. (69), it can be given as

$\begin{split}-2F_{ae}F_{bf}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}R_{cd}=-2\left(k^{a}F_{ae}l^{e}\right)_{0}\left(k^{b}F_{bf}l^{f}\right)_{0}\left(R_{kk}\right)_{1}\sim\mathcal{O}\left(\epsilon\right)\,.\end{split}$
(70)

The second term of Eq. (69) is

$\begin{split}-2F_{a}^{\ c}F_{b}^{\
d}k^{a}k^{b}R^{ef}\gamma_{ce}\gamma_{df}=-2\left(k^{a}F_{a\hat{c}}\right)_{1}\left(k^{b}F_{b\hat{d}}\right)_{1}\left(R^{\hat{c}\hat{d}}\right)_{0}\simeq
0\,.\end{split}$ (71)

The third term of Eq. (69) is given as

$\begin{split}4F_{ad}F_{b}^{\ e}k^{a}k^{b}k^{c}l^{d}R_{c}^{\
f}\gamma_{ef}=4\left(k^{a}F_{ad}l^{d}\right)_{0}\left(k^{b}F_{b\hat{e}}\right)_{1}\left(k^{c}R_{c}^{\
\hat{e}}\right)_{1}\simeq 0\,.\end{split}$ (72)

The fourth term of Eq. (69) is

$\begin{split}&4k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{b}F_{ae}\nabla_{c}F_{df}\\\
=&4k^{b}\nabla_{b}\left(k^{a}F_{ae}\gamma^{e}_{\
g}\right)k^{c}\nabla_{c}\left(l^{d}F_{df}\gamma^{fg}\right)\\\
=&4\left[\left[\mathcal{L}_{k}\left(k^{a}F_{ae}\gamma^{e}_{\
g}\right)\right]_{1}-\left(k^{a}F_{ae}\gamma^{e}_{\
m}\right)_{1}\left(\gamma^{m}_{\ b}\nabla_{g}k^{b}\right)_{1}\right]\\\
&\times\left[\left[\mathcal{L}_{k}\left(l^{d}F_{df}\gamma^{fg}\right)\right]_{0}+\left(l^{d}F_{df}\gamma^{f}_{\
n}\right)_{0}\left(\gamma^{nc}\nabla_{c}k^{g}\right)_{1}\right]\\\
\simeq&4\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]_{1}\left[\mathcal{L}_{k}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (73)

The fifth term of Eq. (69) can be written as

$\begin{split}&2F_{d}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{c}\nabla_{b}F_{af}\\\ =&2\left(l^{d}F_{d}^{\
\hat{g}}\right)_{0}k^{c}\nabla_{c}\left[k^{b}\nabla_{b}\left(k^{a}F_{af}\gamma^{f}_{\
g}\right)\right]\\\ =&2\left(l^{d}F_{d}^{\
\hat{g}}\right)_{0}k^{c}\nabla_{c}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{g}}\right)-\left(k^{a}F_{af}\gamma^{f}_{\
h}\right)_{1}\left(\gamma^{h}_{\ b}\nabla_{\hat{g}}k^{b}\right)_{1}\right]\\\
\simeq&2\left(l^{d}F_{d}^{\
\hat{g}}\right)_{0}\left[\mathcal{L}_{k}^{2}\left(k^{a}F_{a\hat{g}}\right)-\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{b}}\right)\right]_{1}\left(\gamma^{\hat{b}}_{\
c}\nabla_{\hat{g}}k^{c}\right)_{1}\right]\\\ \simeq&2\left(l^{d}F_{d}^{\
\hat{e}}\right)_{0}\left[\mathcal{L}_{k}^{2}\left(k^{a}F_{a\hat{e}}\right)\right]_{1}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (74)

The sixth term of Eq. (69) is given as

$\begin{split}&2F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{c}\nabla_{b}F_{df}\\\ =&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left[k^{b}\nabla_{b}\left(l^{d}F_{df}\gamma^{f}_{\
g}\right)\right]\\\ =&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left[\mathcal{L}_{k}\left(l^{d}F_{d\hat{g}}\right)-\left(l^{d}F_{d\hat{h}}\right)_{0}\left(\gamma^{\hat{h}}_{\
b}\nabla_{g}k^{b}\right)_{1}\right]\\\ \simeq&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}\left[\mathcal{L}_{k}^{2}\left(l^{d}F_{d\hat{g}}\right)-\left[\mathcal{L}_{k}\left(l^{d}F_{d\hat{b}}\right)\right]_{0}\left(\gamma^{\hat{b}}_{\
c}\nabla_{g}k^{c}\right)_{1}\right]\\\
\simeq&2\left(k^{a}F_{a\hat{e}}\right)_{1}\left[\mathcal{L}_{k}^{2}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (75)

The seventh term of Eq. (69) can be expressed as

$\begin{split}4k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{b}F_{ae}\nabla_{d}F_{cf}=4\left[\mathcal{L}_{k}\left(k^{a}F_{ae}l^{e}\right)\right]_{1}\left[\mathcal{L}_{k}\left(k^{c}F_{cf}l^{f}\right)\right]_{1}\simeq
0\,.\end{split}$ (76)

The eighth term of Eq. (69) is written as

$\begin{split}4F_{ae}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{d}\nabla_{c}F_{bf}=4\left(k^{a}F_{ae}l^{e}\right)_{0}\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]_{1}\sim\mathcal{O}\left(\epsilon\right)\,.\end{split}$
(77)

Utilizing the following relation

$\left(\nabla_{d}\nabla_{c}-\nabla_{c}\nabla_{d}\right)F_{bf}=R_{dcb}^{\ \ \
g}F_{gf}+R_{dcf}^{\ \ \ g}F_{bg}\,,$ (78)

one can write the ninth term of Eq. (69) as

$\begin{split}&4F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{f}\nabla_{d}\nabla_{c}F_{bf}\\\
=&4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{d}F_{bf}+4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcb}^{\
\ \ g}F_{gf}\\\ &+4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcf}^{\ \ \
g}F_{bg}\,.\end{split}$ (79)

According to the Bianchi identity of the Maxwell field $\nabla_{[d}F_{bf]}=0$,
the ninth term can further be written as

$\begin{split}&4F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{f}\nabla_{d}\nabla_{c}F_{bf}\\\
=&-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{f}F_{db}-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{b}F_{fd}\\\
&+4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcb}^{\ \ \
g}F_{gf}+4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcf}^{\ \ \
g}F_{bg}\,.\end{split}$ (80)

The first term of Eq. (80) is given as

$\begin{split}&-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{f}F_{db}\\\
=&-4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\left(k^{b}l^{d}\nabla_{f}F_{db}\right)\\\
=&-4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\left[\nabla_{f}\left(k^{b}F_{db}l^{d}\right)-\left(\nabla_{f}k^{b}\right)l^{d}F_{db}-k^{b}\left(\nabla_{f}l^{d}\right)F_{db}\right]\\\
=&-4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\nabla_{f}\left(k^{b}F_{db}l^{d}\right)+4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\left(l^{d}F_{db}\right)\nabla_{f}k^{b}\\\
&+4k^{a}F_{ae}\gamma^{ef}l^{d}F_{db}k^{c}\nabla_{c}\left(\nabla_{f}k^{b}\right)+4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\left(F_{db}k^{b}\right)\nabla_{f}l^{d}\\\
&+4k^{a}F_{ae}\gamma^{ef}F_{db}k^{b}k^{c}\nabla_{c}\left(\nabla_{f}l^{d}\right)\,.\end{split}$
(81)

For the first term of Eq. (81), it can be calculated as

$\begin{split}&-4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\nabla_{f}\left(k^{b}F_{db}l^{d}\right)\\\
=&-4\left(k^{a}F_{ae}\gamma^{eg}\right)\gamma_{g}^{\
f}k^{c}\nabla_{f}\nabla_{c}\left(k^{b}F_{db}l^{d}\right)\\\
=&-4\left(k^{a}F_{a\hat{e}}\right)_{1}D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{db}l^{d}\right)\right]_{1}-4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(\gamma_{\hat{e}}^{\
f}\nabla_{f}k^{c}\right)_{1}\nabla_{c}\left(k^{b}F_{db}l^{d}\right)\\\
\simeq&0\,.\end{split}$ (82)

The second of Eq. (81) is

$\begin{split}4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\left(l^{d}F_{db}\right)\nabla_{f}k^{b}=4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(\gamma_{\hat{e}}^{\
f}\nabla_{f}k^{b}\right)_{1}k^{c}\nabla_{c}\left(l^{d}F_{db}\right)\simeq
0\,.\end{split}$ (83)

The third term of Eq. (81) can be evaluated as

$\begin{split}4k^{a}F_{ae}\gamma^{ef}l^{d}F_{db}k^{c}\nabla_{c}\left(\nabla_{f}k^{b}\right)\simeq
4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left[\mathcal{L}_{k}\left(\gamma_{\hat{e}}^{\
f}\nabla_{f}k^{b}\right)\right]_{1}\left(l^{d}F_{db}\right)\simeq
0\,.\end{split}$ (84)

For the fourth term of Eq. (81), it can be calculated as

$\begin{split}&4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\left(F_{db}k^{b}\right)\nabla_{f}l^{d}\\\
=&4\left(k^{a}F_{a}^{\ \hat{h}}\right)k^{c}\nabla_{c}\left(\gamma_{d}^{\
g}F_{gb}k^{b}-k_{d}l^{g}F_{gb}k^{b}-k^{(g}k^{d)}F_{[gb]}l_{d}\right)\left(\nabla_{\hat{h}}l^{d}\right)\\\
=&4\left(k^{a}F_{a}^{\
\hat{h}}\right)_{1}\left[\mathcal{L}_{k}\left(F_{\hat{d}b}k^{b}\right)\right]_{1}\left(\nabla_{\hat{h}}l^{\hat{d}}\right)-4\left(k^{a}F_{a}^{\
\hat{h}}\right)_{1}\left(F_{\hat{m}b}k^{b}\right)_{1}\left(\gamma^{\hat{m}}_{\
c}\nabla_{d}k^{c}\right)_{1}\left(\nabla_{\hat{h}}l^{d}\right)\\\
&-4\left(k^{a}F_{a}^{\
\hat{h}}\right)_{1}\left[\mathcal{L}_{k}\left(l^{g}F_{gb}k^{b}\right)\right]_{1}\left(k_{d}\nabla_{\hat{h}}l^{d}\right)\\\
\simeq&0\,.\end{split}$ (85)

For the fifth term of Eq. (81), we have

$\begin{split}&4k^{a}F_{ae}\gamma^{ef}F_{db}k^{b}k^{c}\nabla_{c}\left(\nabla_{f}l^{d}\right)\\\
=&4\left(k^{a}F_{a\hat{e}}\right)_{1}\left(\gamma^{g}_{\
d}F_{gb}k^{b}\right)_{1}k^{c}\nabla_{c}\left(\nabla^{\hat{e}}l^{d}\right)-4\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{(g}k^{b)}F_{[gb]}\right)k^{c}\nabla_{c}\left(l_{d}\nabla^{\hat{e}}l^{d}\right)\\\
&+4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(l^{g}F_{gb}k^{b}\right)_{0}k^{c}\nabla_{c}\left[l^{d}\left(\gamma_{\hat{e}}^{\
f}\nabla_{f}k_{d}\right)_{1}\right]\\\ \simeq&0\,.\end{split}$ (86)

The result of the first term of Eq. (80) under the first-order approximation
is

$-4k^{a}F_{ae}\gamma^{ef}k^{c}\nabla_{c}\nabla_{f}\left(k^{b}F_{db}l^{d}\right)\simeq
0\,.$ (87)

The second term of Eq. (80) is

$\begin{split}&-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{b}F_{fd}\\\
=&-4\left(k^{a}F_{ae}\gamma^{eg}\right)k^{c}\nabla_{c}\left[k^{b}\nabla_{b}\left(\gamma_{g}^{\
f}F_{fd}l^{d}\right)\right]\\\ \simeq&-4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (88)

The third term of Eq. (80) is calculated as

$\begin{split}&4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcb}^{\ \ \
g}F_{gf}\\\
=&4\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{b}k^{c}l^{d}R_{acb\hat{g}}\right)_{1}\left(F^{\hat{g}\hat{e}}\right)_{0}-4\left(k^{a}F_{a\hat{e}}\right)\left(k^{c}l^{d}k^{(b}k^{h)}R_{dc[bh]}\right)\left(l^{g}F_{g}^{\
\hat{e}}\right)\\\
&-4\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{b}k^{c}l^{d}l^{h}R_{dcbh}\right)_{0}\left(k^{g}F_{g}^{\
\hat{e}}\right)_{1}\\\ \simeq&0\,.\end{split}$ (89)

The fourth term of Eq. (80) is written as

$\begin{split}&4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcf}^{\ \ \
g}F_{bg}\\\ =&4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{c}l^{d}R_{dc\hat{e}\hat{g}}\right)_{0}\left(k^{b}F_{b}^{\
\hat{g}}\right)_{1}-4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{c}k^{h}l^{d}R_{dc\hat{e}h}\right)_{1}\left(k^{b}F_{bg}l^{g}\right)_{0}\\\
&-4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left(k^{c}l^{d}l^{h}R_{dc\hat{e}h}\right)\left(k^{(b}k^{g)}F_{[bg]}\right)\\\
\simeq&0\,.\end{split}$ (90)

So the ninth term of Eq. (69) under the first-order approximation can be
expressed as

$4F_{a}^{\
e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{f}\nabla_{d}\nabla_{c}F_{bf}\simeq-4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\,.$
(91)

The tenth term of Eq. (69) is

$\begin{split}&2k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{b}F_{df}\nabla_{e}F_{ac}\\\
=&2\left(k^{b}\nabla_{b}F_{df}\right)\gamma^{cd}\gamma^{ef}\nabla_{e}\left(k^{a}F_{ac}\right)-2\left(k^{b}\nabla_{b}F_{df}\right)\gamma^{cd}F_{ac}\left(\gamma^{ef}\nabla_{e}k^{a}\right)\\\
=&2\left(k^{b}\nabla_{b}F_{df}\right)\gamma^{cd}\gamma^{ef}\nabla_{e}\left(k^{a}F_{ag}\gamma^{g}_{\
c}-k^{(a}k^{g)}F_{[ag]}l_{c}-k_{c}k^{a}F_{ag}l^{g}\right)\\\
&-2\left[\mathcal{L}_{k}\left(F_{\hat{m}\hat{n}}\right)\right]_{1}\left(F_{a}^{\
\hat{m}}\right)\left(\gamma^{e\hat{n}}\nabla_{e}k^{a}\right)_{1}+2\left(F_{d\hat{n}}\right)\left(\gamma_{b}^{\
d}\nabla_{\hat{m}}k^{b}\right)_{1}\left(F_{a}^{\
\hat{m}}\right)\left(\gamma^{e\hat{n}}\nabla_{e}k^{a}\right)_{1}\\\
&+2\left(F_{\hat{m}f}\right)\left(\gamma_{b}^{\
f}\nabla_{n}k^{b}\right)_{1}\left(F_{a}^{\
\hat{m}}\right)\left(\gamma^{en}\nabla_{e}k^{a}\right)_{1}\\\
\simeq&2\left[\mathcal{L}_{k}\left(F^{\hat{c}\hat{e}}\right)\right]_{1}D_{\hat{e}}\left(k^{a}F_{a\hat{c}}\right)_{1}-2\left[\mathcal{L}_{k}\left(F_{\hat{m}\hat{n}}\right)\right]_{1}\left(k^{a}F_{ag}l^{g}\right)_{0}\left(\gamma^{c\hat{m}}\gamma^{e\hat{n}}\nabla_{e}k_{c}\right)_{1}\\\
&-2\left(k^{b}\nabla_{b}F_{df}\right)\left(\gamma^{cd}k_{c}\right)D^{f}\left(k^{a}F_{ag}l^{g}\right)\\\
\simeq&0\,.\end{split}$ (92)

The eleventh term of Eq. (69) is further calculated as

$\begin{split}&-2k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}F_{bf}\nabla_{e}F_{ad}\\\
=&-2\left[k^{c}\nabla_{c}\left(k^{b}F_{bf}\gamma^{f}_{\
g}\right)\right]\left(k^{a}l^{d}\gamma^{ge}\nabla_{e}F_{ad}\right)\\\
=&-2\left[\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}-\left(k^{b}F_{b\hat{f}}\right)_{1}\left(\gamma^{\hat{f}}_{\
c}\nabla_{\hat{g}}k^{c}\right)_{1}\right]\\\
&\times\left[D^{\hat{g}}\left(k^{a}F_{ad}l^{d}\right)-\left(F_{ad}l^{d}\right)\left(\gamma^{\hat{g}e}\nabla_{e}k^{a}\right)_{1}-\left(k^{a}F_{ad}\right)\left(\gamma^{\hat{g}e}\nabla_{e}l^{d}\right)\right]\\\
\simeq&-2\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]_{1}\left[D^{\hat{e}}\left(k^{a}F_{ad}l^{d}\right)\right]_{0}+2\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]_{1}\left(k^{a}F_{a\hat{d}}\right)_{1}\left(\nabla^{\hat{e}}l^{\hat{d}}\right)\\\
&-2\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\left(k^{(a}k^{c)}F_{[ac]}\right)\left(l_{d}\nabla^{\hat{e}}l^{d}\right)+2\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]_{1}\left(k^{a}F_{ac}l^{c}\right)_{0}\left(l^{d}\nabla^{\hat{e}}k_{d}\right)_{1}\\\
\simeq&-2\left[\mathcal{L}_{k}\left(k^{b}F_{b}^{\
\hat{e}}\right)\right]_{1}\left[D_{\hat{e}}\left(k^{a}F_{ad}l^{d}\right)\right]_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (93)

The twelfth term of Eq. (69) is given as

$\begin{split}&2F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\nabla_{e}\nabla^{e}F_{bd}\\\
=&-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{d}F_{bf}-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{d}\nabla_{c}F_{bf}\\\
&+2F_{ac}k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{f}\nabla_{e}F_{bd}\,.\end{split}$
(94)

Using the Bianchi identity again, the first term of Eq. (94) can be expressed
as

$\begin{split}&-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{d}F_{bf}\\\
=&2k^{a}F_{ae}\gamma^{ef}k^{c}k^{b}l^{d}\nabla_{c}\nabla_{f}F_{db}+2k^{a}F_{ae}\gamma^{ef}k^{c}k^{b}l^{d}\nabla_{c}\nabla_{b}F_{fd}\,.\end{split}$
(95)

For the first term of Eq. (95), it can be calculated as

$\begin{split}&2k^{a}F_{ae}\gamma^{ef}k^{c}k^{b}l^{d}\nabla_{c}\nabla_{f}F_{db}\\\
=&2\left(k^{a}F_{ae}\gamma^{ef}\right)k^{c}\nabla_{c}\left[\nabla_{f}\left(k^{b}F_{db}l^{d}\right)-l^{d}F_{db}\nabla_{f}k^{b}-k^{b}F_{db}\nabla_{f}l^{d}\right]\\\
=&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left[\gamma_{\hat{g}}^{\
f}\nabla_{f}\left(k^{b}F_{db}l^{d}\right)-l^{d}F_{db}\left(\gamma_{\hat{g}}^{\
f}\nabla_{f}k^{b}\right)_{1}-k^{b}F_{db}\gamma_{\hat{g}}^{\
f}\nabla_{f}l^{d}\right]\\\ \simeq&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left[\gamma_{\hat{g}}^{\
f}\nabla_{f}\left(k^{b}F_{db}l^{d}\right)\right]-2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left[\left(k^{b}F_{\hat{d}b}\right)_{1}\left(\nabla_{\hat{g}}l^{\hat{d}}\right)\right]\\\
&+2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left(k^{(b}k^{h)}F_{[hb]}l_{d}\gamma_{\hat{g}}^{\
f}\nabla_{f}l^{d}\right)-2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}k^{c}\nabla_{c}\left[\left(k^{b}F_{hb}l^{h}\right)\left(l^{d}\nabla_{\hat{g}}k_{d}\right)_{1}\right]\\\
\simeq&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}\left[\mathcal{L}_{k}\left[D_{\hat{g}}\left(k^{b}F_{db}l^{d}\right)\right]-\left(\gamma_{c}^{\
f}\nabla_{\hat{g}}k^{c}\right)_{1}\nabla_{f}\left(k^{b}F_{db}l^{d}\right)\right]\\\
\simeq&2\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}D_{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{db}l^{d}\right)\right]_{1}\\\
\simeq&0\,.\end{split}$ (96)

The second term of Eq. (95) is

$\begin{split}&2k^{a}F_{ae}\gamma^{ef}k^{c}k^{b}l^{d}\nabla_{c}\nabla_{b}F_{fd}\\\
=&2\left(k^{a}F_{ae}\gamma^{eg}\right)_{1}k^{c}\nabla_{c}\left[k^{b}\nabla_{b}\left(\gamma_{g}^{\
f}F_{fd}l^{d}\right)\right]\\\ \simeq&2\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (97)

The result of the first term of Eq. (94) is

$-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{d}F_{bf}\simeq
2\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\,.$
(98)

According to the definition of the Riemann tensor

$\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}\right)X_{cd}=R_{abc}^{\ \ \
e}X_{ed}+R_{abd}^{\ \ \ e}X_{ce}\,,$ (99)

the second term of Eq. (94) is calculated as

$\begin{split}&-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{d}\nabla_{c}F_{bf}\\\
=&-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\left(\nabla_{c}\nabla_{d}F_{bf}+R_{dcb}^{\
\ \ g}F_{gf}+R_{dcf}^{\ \ \ g}F_{bg}\right)\\\
=&-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{d}F_{bf}-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcb}^{\
\ \ g}F_{gf}\\\ &-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcf}^{\ \ \
g}F_{bg}\,.\end{split}$ (100)

Since the first term of Eq. (LABEL:secondtwelfthhkk2) is identical with the
first term of Eq. (95), the result of this term can be directly written as

$\begin{split}-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{d}F_{bf}\simeq
2\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\,.\end{split}$
(101)

The second and the third term of Eq. (LABEL:secondtwelfthhkk2) can be further
calculated as

$\begin{split}&-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcb}^{\ \ \
g}F_{gf}-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{dcf}^{\ \ \ g}F_{bg}\\\
=&-2\left(k^{a}F_{a\hat{e}}\right)_{1}\left(F^{\hat{f}\hat{e}}\right)_{0}\left(k^{b}k^{c}l^{d}R_{dcb\hat{f}}\right)_{1}-2\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{g}F_{g}^{\
\hat{e}}\right)_{1}\left(k^{b}k^{c}l^{d}l^{h}R_{dcbh}\right)_{0}\\\
&-2\left(k^{a}F_{a}^{\ \hat{e}}\right)_{1}\left(k^{b}F_{b}^{\
\hat{f}}\right)_{1}\left(k^{c}l^{d}R_{dc\hat{e}\hat{f}}\right)_{0}-2\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{b}l^{g}F_{bg}\right)_{0}\left(k^{c}l^{d}k^{h}R_{dc\hat{e}h}\right)_{1}\\\
&-2\left(k^{a}F_{a\hat{e}}\right)\left(l^{g}F_{g}^{\
\hat{e}}\right)\left(k^{c}l^{d}k^{(b}k^{h)}R_{dc[bh]}\right)-2\left(k^{a}F_{a}^{\
\hat{e}}\right)\left(k^{(b}k^{g)}F_{[bg]}\right)\left(k^{c}l^{d}l^{h}R_{dc\hat{e}h}\right)\\\
\simeq&0\,.\end{split}$ (102)

Therefore, the second term of Eq. (LABEL:secondtwelfthhkk2) under the linear
order approximation is

$-2F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{d}\nabla_{c}F_{bf}\simeq
2\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\sim\mathcal{O}\left(\epsilon\right)\,.$
(103)

The third term of Eq. (94) is further written as

$\begin{split}&2F_{ac}k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{f}\nabla_{e}F_{bd}\\\
=&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}\gamma^{hf}\nabla_{f}\left(\gamma_{\hat{g}}^{\
d}\gamma_{h}^{\ e}k^{b}\nabla_{e}F_{bd}\right)-2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}\left(\gamma_{\hat{g}}^{\ d}\gamma_{h}^{\
e}\nabla_{e}F_{bd}\right)\left(\gamma^{hf}\nabla_{f}k^{b}\right)_{1}\\\
\simeq&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}D_{f}\left[D^{f}\left(k^{b}F_{b\hat{g}}\right)_{1}-\gamma^{d}_{\
\hat{g}}F_{bd}\left(\gamma^{fe}\nabla_{e}k^{b}\right)_{1}\right]\\\
\simeq&0\,.\end{split}$ (104)

The result of twelfth term of Eq. (69) under the first-order approximation is
given as

$\begin{split}2F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\nabla_{e}\nabla^{e}F_{bd}\simeq 4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\,.\end{split}$
(105)

The thirteenth term of Eq. (69) can be calculated as

$\begin{split}&2k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{e}F_{ac}\nabla_{f}F_{bd}\\\
=&2\left[D^{h}\left(k^{a}F_{a}^{\ \hat{g}}\right)_{1}-F_{a}^{\
\hat{g}}\left(\gamma^{eh}\nabla_{e}k^{a}\right)_{1}\right]\left[D_{h}\left(k^{b}F_{b\hat{g}}\right)_{1}-F_{b\hat{g}}\left(\gamma_{h}^{f}\nabla_{f}k^{b}\right)_{1}\right]\\\
\simeq&0\,.\end{split}$ (106)

The fourteenth term of Eq. (69) can be expressed as

$\begin{split}&-2k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{b}F_{ad}\nabla_{f}F_{ce}\\\
=&-2\left[\mathcal{L}_{k}\left(k^{a}F_{ad}l^{d}\right)\right]_{1}\left[D^{\hat{g}}\left(k^{c}F_{c\hat{g}}\right)_{1}-F_{c\hat{g}}\left(\gamma^{\hat{g}f}\nabla_{f}k^{c}\right)_{1}\right]\\\
\simeq&0\,.\end{split}$ (107)

The fifteenth term of Eq. (69) is given as

$\begin{split}&2k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{b}F_{ac}\nabla_{f}F_{de}\\\
=&2k^{b}\nabla_{b}\left(k^{a}F_{ac}\gamma^{c}_{\
g}\right)\gamma^{gd}\gamma^{e}_{\ h}\gamma^{hf}\nabla_{f}F_{de}\\\
=&2\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{g}}\right)-\left(k^{a}F_{a\hat{m}}\right)_{1}\left(\gamma^{\hat{m}}_{\
b}\nabla_{\hat{g}}k^{b}\right)_{1}\right]\left[\gamma^{hf}\nabla_{f}\left(F^{\hat{g}}_{\
h}\right)\right]\\\
\simeq&2\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)\right]_{1}\left[D_{e}\left(F^{\hat{c}\hat{e}}\right)\right]_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (108)

The sixteenth term of Eq. (69) is

$\begin{split}&2F^{cd}k^{a}k^{b}\gamma_{c}^{\ e}\gamma_{d}^{\
f}\nabla_{f}\nabla_{b}F_{ae}\\\ =&2\left(\gamma^{c}_{\ g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{ge}\gamma^{hf}k^{a}k^{b}\nabla_{f}\nabla_{b}F_{ae}\\\
=&2\left(\gamma^{c}_{\ g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{hf}\nabla_{f}\left(\gamma^{ge}k^{a}k^{b}\nabla_{b}F_{ae}\right)-2\left(\gamma^{c}_{\
g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{hf}\gamma^{ge}k^{b}\nabla_{b}F_{ae}\nabla_{f}k^{a}\\\
&-2\left(\gamma^{c}_{\ g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{hf}\gamma^{ge}k^{a}\nabla_{b}F_{ae}\nabla_{f}k^{b}\,.\end{split}$
(109)

For the first term of Eq. (109), we have

$\begin{split}&2\left(\gamma^{c}_{\ g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{hf}\nabla_{f}\left(\gamma^{ge}k^{a}k^{b}\nabla_{b}F_{ae}\right)\\\
=&2\left(F_{\hat{g}\hat{h}}\right)D^{\hat{h}}\left[k^{b}\nabla_{b}\left(\gamma^{\hat{g}e}k^{a}F_{ae}\right)\right]\\\
=&2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}D^{\hat{h}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{g}}\right)\right]_{1}-2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}D^{\hat{h}}\left[\left(k^{a}F_{a\hat{f}}\right)_{1}\left(\gamma^{f}_{\
b}\nabla_{\hat{g}}k^{b}\right)_{1}\right]\\\
\simeq&2\left(F^{\hat{e}\hat{d}}\right)_{0}D_{\hat{d}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]_{1}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (110)

The second term of Eq. (109) can be simplified as

$\begin{split}&-2\left(\gamma^{c}_{\ g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{hf}\gamma^{ge}k^{b}\nabla_{b}F_{ae}\nabla_{f}k^{a}\\\
=&-2\left(F^{\hat{g}}_{\
\hat{h}}\right)k^{b}\nabla_{b}\left(F_{\hat{m}\hat{g}}\right)\left(\gamma^{\hat{h}f}\nabla_{f}k^{\hat{m}}\right)+2\left(F^{\hat{g}}_{\
\hat{h}}\right)k^{b}\nabla_{b}\left(k^{a}F_{a\hat{g}}\right)\left(l_{m}\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)\\\
&+2\left(F^{\hat{g}}_{\
\hat{h}}\right)k^{b}\nabla_{b}\left(l^{a}F_{a\hat{g}}\right)\left(\gamma^{\hat{h}f}k_{m}\nabla_{f}k^{m}\right)\\\
=&-2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left[\mathcal{L}_{k}\left(F_{\hat{m}\hat{g}}\right)\right]_{1}\left(\gamma^{\hat{h}f}\nabla_{f}k^{\hat{m}}\right)_{1}+2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left(F_{\hat{m}\hat{n}}\right)_{0}\left(\gamma^{\hat{n}}_{\
b}\nabla_{\hat{g}}k^{b}\right)_{1}\left(\gamma^{\hat{h}f}\nabla_{f}k^{\hat{m}}\right)_{1}\\\
&+2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left(F_{\hat{n}\hat{g}}\right)_{0}\left(\gamma^{\hat{n}}_{\
b}\nabla_{m}k^{b}\right)_{1}\left(\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}+2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{g}}\right)\right]_{1}\left(l_{m}\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}\\\
&-2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left(k^{a}F_{a\hat{n}}\right)_{1}\left(\gamma^{\hat{n}}_{\
b}\nabla_{\hat{g}}k^{b}\right)_{1}\left(l_{m}\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}\\\
\simeq&0\,.\end{split}$ (111)

The third term of Eq. (109) can be expressed as

$\begin{split}&-2\left(\gamma^{c}_{\ g}\gamma^{d}_{\
h}F_{cd}\right)\gamma^{hf}\gamma^{ge}k^{a}\nabla_{b}F_{ae}\nabla_{f}k^{b}\\\
=&-2\left(F^{\hat{g}}_{\ \hat{h}}\right)\left(k^{a}\gamma_{\hat{g}}^{\
e}\gamma^{b}_{\
m}\nabla_{b}F_{ae}\right)\left(\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)+2\left(F^{\hat{g}}_{\
\hat{h}}\right)k^{b}\nabla_{b}\left(k^{a}F_{a\hat{g}}\right)\left(l_{m}\gamma^{hf}\nabla_{f}k^{m}\right)\\\
&+2\left(F^{\hat{g}}_{\ \hat{h}}\right)\left(\gamma_{\hat{g}}^{\
e}k^{a}l^{b}\nabla_{b}F_{ae}\right)\left(\gamma^{\hat{h}f}k_{m}\nabla_{f}k^{m}\right)\\\
=&-2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left[D_{m}\left(k^{a}\gamma_{\hat{g}}^{\
e}F_{ae}\right)\right]_{1}\left(\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}+2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left(\gamma_{\hat{g}}^{\
e}F_{ae}\right)\left(\gamma^{b}_{\
m}\nabla_{b}k^{a}\right)_{1}\left(\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}\\\
&+2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left[\mathcal{L}_{k}\left(k^{a}\gamma_{\hat{g}}^{\
e}F_{ae}\right)\right]_{1}\left(l_{m}\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}-2\left(F^{\hat{g}}_{\
\hat{h}}\right)_{0}\left(\gamma_{n}^{\
e}k^{a}F_{ae}\right)_{1}\left(\gamma^{n}_{\
b}\nabla_{\hat{g}}k^{b}\right)_{1}\left(l_{m}\gamma^{\hat{h}f}\nabla_{f}k^{m}\right)_{1}\\\
\simeq&0\,.\end{split}$ (112)

So the sixteenth term of Eq. (69) under the first-order approximation can be
obtained as

$2F^{cd}k^{a}k^{b}\gamma_{c}^{\ e}\gamma_{d}^{\
f}\nabla_{f}\nabla_{b}F_{ae}\simeq
2\left(F^{\hat{e}\hat{d}}\right)D_{\hat{d}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]$
(113)

The seventeenth term of Eq. (69) can be further calculated as

$\begin{split}&2F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{f}\nabla_{b}F_{de}\\\
=&2\left(k^{a}F_{ac}\gamma^{cg}\right)\gamma_{h}^{\
f}\nabla_{f}\left(\gamma_{g}^{\
d}\gamma^{eh}k^{b}\nabla_{b}F_{de}\right)-2\left(k^{a}F_{ac}\gamma^{cg}\right)_{1}\left(\gamma_{g}^{\
d}\gamma^{eh}F_{de}\right)_{0}\left(\gamma_{h}^{\
f}\nabla_{f}k^{b}\right)_{1}\\\ \simeq&2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}D^{\hat{h}}\left[\mathcal{L}_{k}\left(F_{\hat{g}\hat{h}}\right)\right]_{1}-2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}D^{\hat{h}}\left[\left(F_{\hat{f}\hat{h}}\right)_{0}\left(\gamma^{\hat{f}}_{\
b}\nabla_{\hat{g}}k^{b}\right)_{1}\right]\\\ &-2\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}D^{\hat{h}}\left[\left(F_{\hat{g}\hat{f}}\right)_{0}\left(\gamma^{\hat{f}}_{\
b}\nabla_{\hat{h}}k^{b}\right)_{1}\right]\\\ \simeq&0\,.\end{split}$ (114)

The eighteenth term of Eq. (69) can be written as

$\begin{split}&-2F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{f}\nabla_{c}F_{bd}\\\
=&-2\left(k^{a}F_{ae}\right)\gamma^{ef}\nabla_{f}\left(k^{b}l^{d}k^{c}\nabla_{c}F_{bd}\right)+2\left(k^{a}F_{a\hat{e}}\right)_{1}\left(\gamma^{\hat{e}f}\nabla_{f}k^{b}\right)_{1}\left(l^{d}k^{c}\nabla_{c}F_{bd}\right)\\\
&+2\left(k^{a}F_{ae}\right)\left(\gamma^{ef}\nabla_{f}l^{d}\right)\left(k^{b}k^{c}\nabla_{c}F_{bd}\right)+2\left(k^{a}F_{a\hat{e}}\right)_{1}\left(\gamma^{\hat{e}f}\nabla_{f}k^{c}\right)_{1}\left(k^{b}l^{d}\nabla_{c}F_{bd}\right)\\\
\simeq&-2\left(k^{a}F_{a\hat{e}}\right)_{1}D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{bd}l^{d}\right)\right]_{1}+2\left(k^{a}F_{a\hat{e}}\right)k^{c}\nabla_{c}\left(k^{b}F_{b\hat{h}}\right)\left(\nabla^{\hat{e}}l^{\hat{h}}\right)\\\
&-2\left(k^{a}F_{a\hat{e}}\right)k^{c}\nabla_{c}\left(k^{(b}k^{d)}F_{[bd]}\right)\left(l^{h}\nabla^{\hat{e}}l_{h}\right)+2\left(k^{a}F_{a\hat{e}}\right)_{1}\left[\mathcal{L}_{k}\left(k^{b}F_{bd}l^{d}\right)\right]_{1}\left(l_{h}\gamma^{\hat{e}f}\nabla_{f}k^{h}\right)_{1}\\\
\simeq&2\left(k^{a}F_{a\hat{e}}\right)_{1}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{h}}\right)\right]_{1}\left(\nabla^{\hat{e}}l^{\hat{h}}\right)-2\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{b}F_{b\hat{c}}\right)_{1}\left(\nabla^{\hat{e}}l^{h}\right)\left(\nabla_{h}k^{\hat{c}}\right)\\\
\simeq&0\,.\end{split}$ (115)

For the nineteenth term of Eq. (69), it can be expressed as

$\begin{split}&-2F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\\\
=&-2\left(k^{a}F_{ad}l^{d}\right)\gamma^{gf}\nabla_{f}\left(k^{b}k^{c}\gamma_{g}^{\
e}\nabla_{c}F_{be}\right)+2\left(k^{a}F_{ad}l^{d}\right)k^{c}\nabla_{c}\left(\gamma_{g}^{\
e}F_{be}\right)\left(\gamma^{gf}\nabla_{f}k^{b}\right)\\\
&+2\left(k^{a}F_{ad}l^{d}\right)\left(k^{b}\gamma_{g}^{\
e}\nabla_{c}F_{be}\right)\left(\gamma^{gf}\nabla_{f}k^{c}\right)\,.\end{split}$
(116)

For the first term of Eq. (116), it can be calculated as

$\begin{split}&-2\left(k^{a}F_{ad}l^{d}\right)\gamma^{gf}\nabla_{f}\left(k^{b}k^{c}\gamma_{g}^{\
e}\nabla_{c}F_{be}\right)\\\
=&-2\left(k^{a}F_{ad}l^{d}\right)_{0}D^{g}\left[\mathcal{L}_{k}\left(k^{b}F_{be}\gamma^{e}_{\
g}\right)-\left(k^{b}F_{b\hat{e}}\right)_{1}\left(\gamma^{\hat{e}}_{\
c}\nabla_{g}k^{c}\right)_{1}\right]\\\
\simeq&-2\left(k^{a}F_{ad}l^{d}\right)_{0}D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]_{1}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (117)

The second term of Eq. (116) is

$\begin{split}&2\left(k^{a}F_{ad}l^{d}\right)k^{c}\nabla_{c}\left(\gamma_{g}^{\
e}F_{be}\right)\left(\gamma^{gf}\nabla_{f}k^{b}\right)\\\
=&2\left(k^{a}F_{ad}l^{d}\right)_{0}\left[\mathcal{L}_{k}\left(F_{\hat{h}\hat{g}}\right)\right]_{1}\left(\gamma^{\hat{g}f}\nabla_{f}k^{\hat{h}}\right)_{1}-2\left(k^{a}F_{ad}l^{d}\right)_{0}\left(F_{\hat{h}\hat{m}}\right)_{0}\left(\gamma^{\hat{m}}_{\
c}\nabla_{g}k^{c}\right)_{1}\left(\gamma^{gf}\nabla_{f}k^{\hat{h}}\right)_{1}\\\
&-2\left(k^{a}F_{ad}l^{d}\right)_{0}\left(F_{\hat{m}\hat{g}}\right)_{0}\left(\gamma^{\hat{m}}_{\
c}\nabla_{h}k^{c}\right)_{1}\left(\gamma^{\hat{g}f}\nabla_{f}k^{h}\right)_{1}-2\left(k^{a}F_{ad}l^{d}\right)_{0}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]\left(l_{h}\gamma^{\hat{g}f}\nabla_{f}k^{h}\right)_{1}\\\
&+2\left(k^{a}F_{ad}l^{d}\right)_{0}\left(k^{b}F_{b\hat{m}}\right)_{1}\left(\gamma^{\hat{m}}_{\
c}\nabla_{g}k^{c}\right)_{1}\left(l_{h}\gamma^{gf}\nabla_{f}k^{h}\right)_{1}\\\
\simeq&0\,.\end{split}$ (118)

The third term of Eq. (116) can be expressed as

$\begin{split}&2\left(k^{a}F_{ad}l^{d}\right)\left(k^{b}\gamma_{g}^{\
e}\nabla_{c}F_{be}\right)\left(\gamma^{gf}\nabla_{f}k^{c}\right)\\\
=&2\left(k^{a}F_{ad}l^{d}\right)_{0}D_{h}\left(k^{b}F_{b\hat{g}}\right)_{1}\left(\gamma^{\hat{g}f}\nabla_{f}k^{h}\right)_{1}-2\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{c}_{\
h}\nabla_{c}k^{b}\right)_{1}\left(F_{b\hat{g}}\right)\left(\gamma^{\hat{g}f}\nabla_{f}k^{h}\right)_{1}\\\
&-2\left(k^{a}F_{ad}l^{d}\right)_{0}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}\left(l_{h}\gamma^{gf}\nabla_{f}k^{h}\right)_{1}\\\
&+2\left(k^{a}F_{ad}l^{d}\right)_{0}\left(k^{b}F_{b\hat{m}}\right)_{1}\left(\gamma^{\hat{m}}_{\
c}\nabla_{g}k^{c}\right)_{1}\left(l_{h}\gamma^{gf}\nabla_{f}k^{h}\right)_{1}\\\
\simeq&0\,.\end{split}$ (119)

Therefore, the result of the nineteenth term of Eq. (69) is

$\begin{split}-2F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\simeq-2\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\,.\end{split}$
(120)

So the expression of $H_{kk}^{(2)}$ under the linear order approximation can
be written as

$\begin{split}H_{kk}^{(2)}\simeq&-2\left[\mathcal{L}_{k}\left(k^{b}F_{b}^{\
\hat{e}}\right)\right]\left[D_{\hat{e}}\left(k^{a}F_{ad}l^{d}\right)\right]-2\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\\\
&+2\mathcal{L}_{k}\left(k^{a}F_{a\hat{c}}\right)D_{\hat{e}}\left(F^{\hat{c}\hat{e}}\right)+2\left(F^{\hat{e}\hat{d}}\right)D_{\hat{d}}\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]\\\
&-4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]+4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(F_{\hat{e}d}l^{d}\right)\right]\\\
&-2\left(k^{a}F_{ae}l^{e}\right)\left(k^{b}F_{bf}l^{f}\right)\left(R_{kk}\right)+4\left(k^{a}F_{ae}l^{e}\right)\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\\\
&+2\left(l^{d}F_{d}^{\
\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(k^{a}F_{a\hat{e}}\right)\right]+2\left(k^{a}F_{a\hat{e}}\right)\left[\mathcal{L}_{k}^{2}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\\\
&+4\left[\mathcal{L}_{k}\left(k^{a}F_{a\hat{e}}\right)\right]\left[\mathcal{L}_{k}\left(l^{d}F_{d}^{\
\hat{e}}\right)\right]\,.\end{split}$ (121)

##  Appendix D Simplification of $H_{kk}^{(3)}$ under the linear order
approximation

In the final appendix, we will calculate Eq. (30) under the first-order
approximation. After duplicated indexes in every term of the equation as well,
Eq. (30) can be further written as

$\begin{split}H_{kk}^{(3)}=&-2F_{a}^{\ c}F^{de}k^{a}k^{b}R_{bfgh}\gamma_{c}^{\
f}\gamma_{d}^{\ g}\gamma_{e}^{\ h}+4F_{a}^{\ e}F_{d}^{\
f}k^{a}k^{b}k^{c}l^{d}R_{bgch}\gamma_{e}^{\ g}\gamma_{f}^{\ h}\\\ &+4F_{a}^{\
e}F_{b}^{\ f}k^{a}k^{b}k^{c}l^{d}R_{cgdh}\gamma_{e}^{\ g}\gamma_{f}^{\
h}+4F_{ae}F_{b}^{\ g}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}R_{cfdh}\gamma_{g}^{\ h}\\\
&-4F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{c}\nabla_{f}F_{bd}-4F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{f}\nabla_{c}F_{bd}\\\
&-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{f}F_{be}-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\\\
&+8k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{b}F_{ae}\nabla_{d}F_{cf}+8F_{ae}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{d}\nabla_{c}F_{bf}\\\
&+4F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{d}\nabla_{f}F_{be}+4F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{f}\nabla_{d}F_{be}\\\
&+4k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{d}F_{bf}\nabla_{e}F_{ac}-8k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}F_{bf}\nabla_{e}F_{ad}\\\
&+4k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{d}F_{ac}\nabla_{f}F_{be}-8k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{b}F_{ad}\nabla_{f}F_{ce}\,.\end{split}$
(122)

The first term of Eq. (122) is

$\begin{split}-2F_{a}^{\ c}F^{de}k^{a}k^{b}R_{bfgh}\gamma_{c}^{\
f}\gamma_{d}^{\ g}\gamma_{e}^{\ h}=-2\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(F^{\hat{d}\hat{e}}\right)_{0}\left(k^{b}R_{b\hat{c}\hat{d}\hat{e}}\right)_{1}\simeq
0\,.\end{split}$ (123)

The second term of Eq. (122) can be calculated as

$\begin{split}4F_{a}^{\ e}F_{d}^{\ f}k^{a}k^{b}k^{c}l^{d}R_{bgch}\gamma_{e}^{\
g}\gamma_{f}^{\ h}=4\left(k^{a}F_{a}^{\ \hat{e}}\right)_{1}\left(l^{d}F_{d}^{\
\hat{f}}\right)_{0}\left(k^{b}k^{c}R_{b\hat{e}c\hat{f}}\right)_{1}\simeq
0\,.\end{split}$ (124)

The third term of Eq. (122) can be written as

$\begin{split}4F_{a}^{\ e}F_{b}^{\ f}k^{a}k^{b}k^{c}l^{d}R_{cgdh}\gamma_{e}^{\
g}\gamma_{f}^{\ h}=\left(k^{a}F_{a}^{\ \hat{e}}\right)_{1}\left(k^{b}F_{b}^{\
\hat{f}}\right)_{1}\left(k^{c}l^{d}R_{c\hat{e}d\hat{f}}\right)_{0}\simeq
0\,.\end{split}$ (125)

The fourth term of Eq. (122) is expressed as

$\begin{split}4F_{ae}F_{b}^{\
g}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}R_{cfdh}\gamma_{g}^{\
h}=\left(k^{a}F_{ae}l^{e}\right)_{0}\left(k^{b}F_{b}^{\
\hat{g}}\right)_{1}\left(k^{c}l^{f}k^{d}R_{cfd\hat{g}}\right)_{1}\simeq
0\,.\end{split}$ (126)

According to the relation (99), the fifth and sixth terms of Eq. (122) can be
written as

$\begin{split}&-4F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{c}\nabla_{f}F_{bd}-4F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{f}\nabla_{c}F_{bd}\\\
=&-8F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{bd}-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfb}^{\
\ \ g}F_{gd}\\\ &-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfd}^{\ \ \
g}F_{bg}\,.\end{split}$ (127)

For the first term of Eq. (127), it is further calculated as

$\begin{split}&-8F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{bd}\\\
=&-8\left(k^{a}F_{a\hat{e}}\right)_{1}D^{e}\left[\mathcal{L}_{k}\left(k^{b}l^{d}F_{bd}\right)\right]_{1}+8\left(k^{a}F_{a\hat{e}}\right)_{1}\left(\gamma^{\hat{e}f}\nabla_{f}k^{c}\right)_{1}\left(k^{b}l^{d}\nabla_{c}F_{bd}\right)\\\
&+8\left(k^{a}F_{a\hat{e}}\right)_{1}\left(\gamma^{\hat{e}f}\nabla_{f}k^{b}\right)_{1}\left(l^{d}k^{c}\nabla_{c}F_{bd}\right)+8\left(k^{a}F_{a\hat{e}}\right)\left(k^{c}k^{b}\nabla_{c}F_{bd}\right)\left(\nabla^{\hat{e}}l^{d}\right)\\\
\simeq&8\left(k^{a}F_{a\hat{e}}\right)_{1}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}\left(\nabla^{\hat{e}}l^{\hat{g}}\right)-8\left(k^{a}F_{a\hat{e}}\right)_{1}\left(k^{b}F_{b\hat{d}}\right)_{1}\left(\gamma^{\hat{d}}_{\
c}\nabla_{g}k^{c}\right)_{1}\left(\nabla^{\hat{e}}l^{g}\right)\\\
&-8\left(k^{a}F_{a\hat{e}}\right)k^{c}\nabla_{c}\left(k^{(b}k^{d)}F_{[bd]}\right)\left(l^{g}\nabla^{\hat{e}}l_{g}\right)-8\left(k^{a}F_{a\hat{e}}\right)_{1}\left[\mathcal{L}_{k}\left(k^{b}F_{bd}l^{d}\right)\right]_{1}\left(k^{g}\nabla^{\hat{e}}l_{g}\right)\\\
\simeq&0\,.\end{split}$ (128)

The second term of Eq. (127) is

$\begin{split}&-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfb}^{\ \ \
g}F_{gd}\\\ =&-4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{b}k^{c}R_{c\hat{e}b\hat{f}}\right)_{1}\left(F^{\hat{f}}_{\
d}l^{d}\right)_{0}+4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{c}k^{b}l^{h}R_{c\hat{e}bh}\right)_{1}\left(k^{g}F_{gd}l^{d}\right)_{0}\\\
&+4\left(k^{a}F_{a}^{\
\hat{e}}\right)\left(k^{c}k^{(b}k^{h)}R_{c\hat{e}[bh]}\right)\left(l^{(g}l^{d)}F_{[gd]}\right)\\\
\simeq&0\,.\end{split}$ (129)

The third term of Eq. (127) can be calculated as

$\begin{split}&-4F_{ae}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfd}^{\ \ \
g}F_{bg}\\\ =&-4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{c}l^{d}R_{c\hat{e}d\hat{f}}\right)_{0}\left(k^{b}F_{b}^{\
\hat{f}}\right)_{1}+4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{c}l^{d}l^{h}R_{c\hat{e}dh}\right)_{0}\left(k^{(b}k^{g)}F_{[bg]}\right)\\\
&+4\left(k^{a}F_{a}^{\
\hat{e}}\right)_{1}\left(k^{c}l^{d}k^{h}R_{c\hat{e}dh}\right)_{1}\left(k^{b}l^{g}F_{bg}\right)_{0}\\\
\simeq&0\,.\end{split}$ (130)

So the result of the fifth and sixth terms of Eq. (122) under the first-order
approximation is

$\begin{split}&-4F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{c}\nabla_{f}F_{bd}-4F_{a}^{\ e}k^{a}k^{b}k^{c}l^{d}\gamma_{e}^{\
f}\nabla_{f}\nabla_{c}F_{bd}\simeq 0\,.\end{split}$ (131)

Utilizing the relation (99), the seventh and eighth terms of Eq. (122) can be
combined as

$\begin{split}&-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{f}F_{be}-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\\\
=&-8F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfb}^{\
\ \ g}F_{ge}\\\ &-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfe}^{\ \ \
g}F_{bg}\,.\end{split}$ (132)

The first term of Eq. (132) is given as

$\begin{split}&-8F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\\\
=&-8\left(k^{a}F_{ad}l^{d}\right)\gamma^{gf}\nabla_{f}\left(\gamma_{g}^{\
e}k^{b}k^{c}\nabla_{c}F_{be}\right)+8\left(k^{a}F_{ad}l^{d}\right)\left(\gamma^{ef}\nabla_{f}k^{b}\right)k^{c}\nabla_{c}F_{be}\\\
&+8\left(k^{a}F_{ad}l^{d}\right)\left(\gamma^{ef}\nabla_{f}k^{c}\right)k^{b}\nabla_{c}F_{be}\,.\end{split}$
(133)

The first term of Eq. (133) is

$\begin{split}&-8\left(k^{a}F_{ad}l^{d}\right)\gamma^{gf}\nabla_{f}\left(\gamma_{g}^{\
e}k^{b}k^{c}\nabla_{c}F_{be}\right)\\\
=&-8\left(k^{a}F_{ad}l^{d}\right)_{0}D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]_{1}+8\left(k^{a}F_{ad}l^{d}\right)_{0}\gamma^{gf}\nabla_{f}\left[\left(k^{b}F_{be}\gamma_{h}^{\
e}\right)_{1}\left(\gamma^{h}_{\ c}\nabla_{g}k^{c}\right)_{1}\right]\\\
\simeq&-8\left(k^{a}F_{ad}l^{d}\right)_{0}D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]_{1}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (134)

The second term of Eq. (133) is given as

$\begin{split}&8\left(k^{a}F_{ad}l^{d}\right)\left(\gamma^{ef}\nabla_{f}k^{b}\right)k^{c}\nabla_{c}F_{be}\\\
=&8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{\hat{m}g}\nabla^{\hat{e}}k_{g}\right)_{1}\left[\mathcal{L}_{k}\left(F_{\hat{m}\hat{e}}\right)\right]_{1}-8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{\hat{n}g}\nabla_{f}k_{g}\right)_{1}\left(F_{\hat{n}\hat{c}}\right)_{0}\left(\gamma^{mf}\nabla_{m}k^{\hat{c}}\right)_{1}\\\
&-8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{\hat{m}f}\nabla_{f}k_{g}\right)_{1}\left(\gamma^{ng}\nabla_{n}k^{\hat{c}}\right)_{1}\left(F_{\hat{c}\hat{m}}\right)_{0}-8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(l^{g}\gamma^{\hat{h}f}\nabla_{f}k_{g}\right)_{1}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{h}}\right)\right]_{1}\\\
&+8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{hf}l^{g}\nabla_{f}k_{g}\right)_{1}\left(\gamma^{\hat{m}}_{\
c}\nabla_{h}k^{c}\right)_{1}\left(k^{b}F_{b\hat{m}}\right)_{1}\\\
\simeq&0\,.\end{split}$ (135)

The third term of Eq. (133) is calculated as

$\begin{split}&8\left(k^{a}F_{ad}l^{d}\right)\left(\gamma^{ef}\nabla_{f}k^{c}\right)k^{b}\nabla_{c}F_{be}\\\
=&8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{\hat{h}f}\nabla_{f}k^{g}\right)_{1}D_{g}\left(k^{b}F_{b\hat{h}}\right)_{1}-8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(\gamma^{hf}\nabla_{f}k^{g}\right)_{1}\left(F_{b\hat{h}}\right)\left(\gamma_{g}^{\
c}\nabla_{c}k^{b}\right)_{1}\\\
&-8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(l_{g}\gamma^{\hat{h}f}\nabla_{f}k^{g}\right)_{1}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{h}}\right)\right]_{1}+8\left(k^{a}F_{ad}l^{d}\right)_{0}\left(l_{g}\gamma^{hf}\nabla_{f}k^{g}\right)_{1}\left(k^{b}F_{b\hat{e}}\right)_{1}\left(\gamma_{c}^{\
\hat{e}}\nabla_{h}k^{c}\right)_{1}\\\ \simeq&0\,.\end{split}$ (136)

So the first term of Eq. (132) under the linear order approximation can be
finally expressed as

$\begin{split}-8F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\simeq-8\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]\,.\end{split}$
(137)

The second term of Eq. (132) is

$\begin{split}&-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfb}^{\ \ \
g}F_{ge}\\\
=&-4\left(k^{c}F_{cd}l^{d}\right)_{0}\left(F^{[\hat{e}\hat{f}]}\right)_{0}\left(k^{a}k^{b}R_{a(\hat{e}\hat{f})b}\right)_{1}+4\left(k^{d}F_{de}l^{e}\right)\left(l^{f}F_{f}^{\
\hat{g}}\right)\left(k^{a}k^{(b}k^{c)}R_{a\hat{g}[bc]}\right)\\\
&+4\left(k^{c}F_{cf}l^{f}\right)_{0}\left(k^{d}F_{d}^{\
\hat{g}}\right)_{1}\left(k^{a}k^{b}l^{e}R_{a\hat{g}be}\right)_{1}\\\
\simeq&0\,.\end{split}$ (138)

The third term of Eq. (132) can be written as

$\begin{split}&-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}R_{cfe}^{\ \ \
g}F_{bg}\\\
=&-4\left(k^{c}F_{ce}l^{e}\right)_{0}\left(k^{d}F_{df}l^{f}\right)_{0}\left(k^{a}k^{b}R_{ab}\right)_{1}+4\left(k^{b}F_{bd}l^{d}\right)_{0}\left(k^{c}F_{c}^{\
\hat{g}}\right)_{1}\left(k^{a}R_{a\hat{g}}\right)_{1}\\\
\simeq&-4\left(k^{c}F_{ce}l^{e}\right)_{0}\left(k^{d}F_{df}l^{f}\right)_{0}\left(R_{kk}\right)_{1}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (139)

The result of the seventh and eighth terms of Eq. (122) is

$\begin{split}&-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}\nabla_{f}F_{be}-4F_{ad}k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{f}\nabla_{c}F_{be}\\\
\simeq&-8\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]-4\left(k^{c}F_{ce}l^{e}\right)\left(k^{d}F_{df}l^{f}\right)\left(R_{kk}\right)\,.\end{split}$
(140)

The ninth term of Eq. (122) can be written as

$\begin{split}8k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{b}F_{ae}\nabla_{d}F_{cf}=8\left[\mathcal{L}_{k}\left(k^{a}F_{ae}l^{e}\right)\right]_{1}\left[\mathcal{L}_{k}\left(k^{c}F_{cf}l^{f}\right)\right]_{1}\simeq
0\,.\end{split}$ (141)

For the tenth term of Eq. (122), we have

$\begin{split}8F_{ae}k^{a}k^{b}k^{c}k^{d}l^{e}l^{f}\nabla_{d}\nabla_{c}F_{bf}=8\left(k^{a}F_{ae}l^{e}\right)_{0}\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]_{1}\sim\mathcal{O}\left(\epsilon\right)\,.\end{split}$
(142)

Using the relation (99) again, the eleventh and twelfth terms of Eq. (122) can
be expressed as

$\begin{split}&4F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{d}\nabla_{f}F_{be}+4F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{f}\nabla_{d}F_{be}\\\
=&8k^{a}F_{ac}\gamma^{cd}\gamma^{ef}k^{b}\nabla_{f}\nabla_{d}F_{be}+4F_{ac}k^{a}k^{b}\gamma^{cd}\gamma^{ef}R_{dfb}^{\
\ \ g}F_{ge}\\\ &+4F_{ac}k^{a}k^{b}\gamma^{cd}\gamma^{ef}R_{dfe}^{\ \ \
g}F_{bg}\,.\end{split}$ (143)

For the first term of Eq. (143), we have

$\begin{split}&8k^{a}F_{ac}\gamma^{cd}\gamma^{ef}k^{b}\nabla_{f}\nabla_{d}F_{be}\\\
=&8\left(k^{a}F_{a}^{\ \hat{g}}\right)\gamma_{h}^{\
f}\nabla_{f}\left(\gamma_{\hat{g}}^{\
d}\gamma^{eh}k^{b}\nabla_{d}F_{be}\right)-8\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}\left(\gamma_{h}^{\
f}\nabla_{f}k^{b}\right)_{1}\left(\gamma_{\hat{g}}^{\
d}\gamma^{eh}\nabla_{d}F_{be}\right)\\\ \simeq&8\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}D_{\hat{h}}\left[D_{\hat{g}}\left(k^{b}F_{b}^{\
\hat{h}}\right)_{1}\right]-8\left(k^{a}F_{a}^{\
\hat{g}}\right)_{1}D_{\hat{h}}\left[\left(F_{b}^{\
\hat{h}}\right)\left(\gamma_{\hat{g}}^{\
d}\nabla_{d}k^{b}\right)_{1}\right]\\\ \simeq&0\,.\end{split}$ (144)

The second term of Eq. (143) is

$\begin{split}&4F_{ac}k^{a}k^{b}\gamma^{cd}\gamma^{ef}R_{dfb}^{\ \ \
g}F_{ge}\\\ =&4\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(k^{b}R_{\hat{c}\hat{e}b\hat{g}}\right)_{1}\left(F^{\hat{g}\hat{e}}\right)_{0}-4\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(k^{b}l^{h}R_{\hat{c}\hat{e}bh}\right)_{0}\left(k^{g}F_{g}^{\
\hat{e}}\right)_{1}\\\ &-4\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(k^{(b}k^{h)}R_{\hat{c}\hat{e}[bh]}\right)\left(l^{g}F_{g}^{\
\hat{e}}\right)\\\ \simeq&0\,.\end{split}$ (145)

The third term of Eq. (143) can be calculated as

$\begin{split}&4F_{ac}k^{a}k^{b}\gamma^{cd}\gamma^{ef}R_{dfe}^{\ \ \
g}F_{bg}\\\ =&-4\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(R_{\hat{c}\hat{g}}\right)_{0}\left(k^{b}F_{b}^{\
\hat{g}}\right)_{1}+4\left(k^{a}F_{a}^{\
\hat{c}}\right)\left(l^{h}R_{\hat{c}h}\right)\left(k^{(b}k^{g)}F_{[bg]}\right)\\\
&+4\left(k^{a}F_{a}^{\
\hat{c}}\right)_{1}\left(k^{h}R_{\hat{c}h}\right)_{1}\left(k^{b}F_{bg}l^{g}\right)_{0}\\\
\simeq&0\,.\end{split}$ (146)

After combining the eleventh and twelfth terms of Eq. (122), the result under
the first-order approximation can be obtained as

$\begin{split}4F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{d}\nabla_{f}F_{be}+4F_{a}^{\ c}k^{a}k^{b}\gamma_{c}^{\
d}\gamma^{ef}\nabla_{f}\nabla_{d}F_{be}\simeq 0\,.\end{split}$ (147)

The thirteenth term of Eq. (122) can be expressed as

$\begin{split}&4k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{d}F_{bf}\nabla_{e}F_{ac}\\\
=&4\left[D^{\hat{g}}\left(k^{b}F_{b\hat{h}}\right)_{1}-\left(\gamma^{\hat{g}d}\nabla_{d}k^{b}\right)_{1}\left(F_{b\hat{h}}\right)\right]\left[D^{\hat{h}}\left(k^{a}F_{a\hat{g}}\right)_{1}-\left(\gamma^{e\hat{h}}\nabla_{e}k^{a}\right)_{1}\left(F_{a\hat{g}}\right)\right]\\\
\simeq&0\,.\end{split}$ (148)

The fourteenth term of Eq. (122) is

$\begin{split}&-8k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}F_{bf}\nabla_{e}F_{ad}\\\
=&-8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\gamma^{ge}\nabla_{e}\left(k^{a}F_{ad}l^{d}\right)+8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\gamma^{ge}l^{d}F_{ad}\nabla_{e}k^{a}\\\
&+8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\gamma^{ge}k^{a}F_{ad}\nabla_{e}l^{d}\,.\end{split}$ (149)

The first term of Eq. (149) can be expressed as

$\begin{split}&-8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\gamma^{ge}\nabla_{e}\left(k^{a}F_{ad}l^{d}\right)\\\
=&-8\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}D^{\hat{g}}\left(k^{a}F_{ad}l^{d}\right)_{0}+8\left(k^{b}F_{b\hat{h}}\right)_{1}\left(\gamma^{\hat{h}}_{\
c}\nabla_{g}k^{c}\right)_{1}D^{g}\left(k^{a}F_{ad}l^{d}\right)_{0}\\\
=&-8\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)\right]_{1}D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)_{0}\\\
\sim&\mathcal{O}\left(\epsilon\right)\,.\end{split}$ (150)

The second term of Eq. (149) is

$\begin{split}&8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\gamma^{ge}l^{d}F_{ad}\nabla_{e}k^{a}\\\
=&8\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}\left(\gamma^{\hat{g}e}\nabla_{e}k^{a}\right)_{1}\left(l^{d}F_{ad}\right)-8\left(k^{b}F_{b\hat{h}}\right)_{1}\left(\gamma^{\hat{h}}_{\
c}\nabla_{g}k^{c}\right)_{1}\left(\gamma^{ge}\nabla_{e}k^{a}\right)_{1}\left(l^{d}F_{ad}\right)\\\
\simeq&0\,.\end{split}$ (151)

The third term of Eq. (149) is

$\begin{split}&8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\gamma^{ge}k^{a}F_{ad}\nabla_{e}l^{d}\\\
=&8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\left(k^{a}F_{ad}\gamma^{hd}\right)\left(\gamma^{ge}\nabla_{e}l_{h}\right)+8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\left(k^{a}F_{ad}l^{d}\right)\left(\gamma^{ge}l_{h}\nabla_{e}k^{h}\right)\\\
&-8k^{c}\nabla_{c}\left(\gamma_{g}^{\
f}k^{b}F_{bf}\right)\left(k^{(a}k^{d)}F_{[ad]}\right)\left(\gamma^{ge}l^{h}\nabla_{e}l_{h}\right)\\\
=&8\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}\left(k^{a}F_{a}^{\
\hat{h}}\right)_{1}\left(\gamma^{\hat{g}e}\nabla_{e}l_{\hat{h}}\right)-8\left(k^{b}F_{b\hat{f}}\right)\left(\gamma_{c}^{\
\hat{f}}\nabla_{g}k^{c}\right)_{1}\left(k^{a}F_{a}^{\
\hat{h}}\right)_{1}\left(\gamma^{ge}\nabla_{e}l_{\hat{h}}\right)\\\
&+8\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{g}}\right)\right]_{1}\left(k^{a}F_{ad}l^{d}\right)_{0}\left(l_{h}\gamma^{\hat{g}e}\nabla_{e}k^{h}\right)_{1}-8\left(k^{b}F_{b\hat{f}}\right)_{1}\left(\gamma_{c}^{\
\hat{f}}\nabla_{g}k^{c}\right)_{1}\left(k^{a}F_{ad}l^{d}\right)_{0}\left(l_{h}\gamma^{ge}\nabla_{e}k^{h}\right)_{1}\\\
\simeq&0\,.\end{split}$ (152)

So the result of the fourteenth term of Eq. (122) can be obtained as

$-8k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{c}F_{bf}\nabla_{e}F_{ad}\simeq-8\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)\right]D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)\,.$
(153)

The fifteenth term of Eq. (122) is further calculated as

$\begin{split}&4k^{a}k^{b}\gamma^{cd}\gamma^{ef}\nabla_{d}F_{ac}\nabla_{f}F_{be}\\\
=&4\left[D^{\hat{g}}\left(k^{a}F_{a\hat{g}}\right)_{1}-\left(\gamma^{\hat{g}d}\nabla_{d}k^{a}\right)_{1}\left(F_{a\hat{g}}\right)\right]\left[D^{\hat{h}}\left(k^{b}F_{b\hat{h}}\right)_{1}-\left(\gamma^{\hat{h}f}\nabla_{f}k^{b}\right)_{1}\left(F_{b\hat{h}}\right)\right]\\\
\simeq&0\,.\end{split}$ (154)

The sixteenth term of Eq. (122) is

$\begin{split}&-8k^{a}k^{b}k^{c}l^{d}\gamma^{ef}\nabla_{b}F_{ad}\nabla_{f}F_{ce}\\\
=&-8\left[\mathcal{L}_{k}\left(k^{a}F_{ad}l^{d}\right)\right]_{1}\left[D^{\hat{g}}\left(k^{c}F_{c\hat{g}}\right)_{1}-\left(\gamma^{\hat{g}f}\nabla_{f}k^{c}\right)_{1}\left(F_{c\hat{g}}\right)\right]\\\
\simeq&0\,.\end{split}$ (155)

After retaining the first-order terms and ignoring the high-order term as
well, the result of $H_{kk}^{(3)}$ under the first-order approximation can be
expressed as

$\begin{split}H_{kk}^{(3)}\simeq&-8\left(k^{a}F_{ad}l^{d}\right)D^{\hat{e}}\left[\mathcal{L}_{k}\left(k^{b}F_{b\hat{e}}\right)\right]-8\mathcal{L}_{k}\left(k^{b}F_{b\hat{c}}\right)D^{\hat{c}}\left(k^{a}F_{ad}l^{d}\right)\\\
&-4\left(k^{c}F_{ce}l^{e}\right)\left(k^{d}F_{df}l^{f}\right)\left(R_{kk}\right)+8\left(k^{a}F_{ae}l^{e}\right)\left[\mathcal{L}_{k}^{2}\left(k^{b}F_{bf}l^{f}\right)\right]\,.\end{split}$
(156)

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