text:
ICYMI: Reuters: Biden's child tax credit pays big in Republican states, popular with voters
**ICYMI**
Key Point: "The top 10 states by average monthly child tax credit payments in August – all from the West and Midwest – were: Utah, Idaho, South Dakota, Alaska, Nebraska, Wyoming, North Dakota, Iowa, Kansas and Montana, with monthly payments ranging from $515 to $456 in August. All voted last year for Trump over Biden and all have Republican governors."
Reuters: Biden's child tax credit pays big in Republican states, popular with voters
By Jason Lange and Chris Kahn
A one-year expansion of the U.S. child tax credit, a policy championed by President Joe Biden and his fellow Democrats over Republican opposition, has disproportionately benefited states that voted for former President Donald Trump in 2020, a Reuters review of Treasury Department data has found.
The current expanded tax credit has proven popular, a Reuters/Ipsos poll found, supported by 59% of U.S. adults including 75% of people who identified themselves as Democrats and 41% of people who identified as Republicans. The poll was conducted online Sept. 9-10, based on responses from 1,003 adults and with a credibility interval of 4 percentage points.
The policy's support among Republicans far outstripped their 11% backing for Biden's overall job performance in a separate Reuters/Ipsos poll.
The policy's popularity, experts said, might benefit Democrats in elections next year that will determine whether they retain control of Congress for the second half of Biden's term. Democrats are defending razor-thin majorities in the Senate and House of Representatives.
"That could make a difference in a whole lot of places where we have close Senate and House races," said Norman Ornstein, an expert on elections at the American Enterprise Institute.
The top 10 states by average monthly child tax credit payments in August – all from the West and Midwest – were: Utah, Idaho, South Dakota, Alaska, Nebraska, Wyoming, North Dakota, Iowa, Kansas and Montana, with monthly payments ranging from $515 to $456 in August. All voted last year for Trump over Biden and all have Republican governors.
In Wisconsin and Arizona – states that Biden narrowly won last year and are shaping up to have competitive Senate races next year – average payments in August were just under $450.
The policy gets cash to families even before they square annual tax bills. Republican-led states tend to have lower household incomes than states with Democratic leadership such as California and New York, thus benefiting more from the policy, which reduces tax credits to upper-income households.
The economic stimulus law, called the American Rescue Plan, raised the existing child tax credit from $2,000 per child to $3,000 per child for children over age 6 and from $2,000 to $3,600 for children under age 6, while upping the age limit from 16 to 17. Families benefit from the full credit if they earn up to $150,000 for a couple or $112,500 for a family with a single parent.
Since July, the U.S. Treasury has given more than 35 million households about $250 to $300 a month for each child under age 18, a policy some analysts say is already significantly reducing childhood poverty.
The tax credit already is being felt by Americans benefiting from it.
Lolitha Maria Scott, a 41-year old call center worker in Phoenix with five children, described the tax credit as a lifeline that she thinks should continue beyond this year because many working parents like her struggle to keep up with rising rent bills.
"I understand people say it's costly for the budget but it also helps the American people," Scott said in an interview.
Scott said the policy helped cement her plan to vote for Democratic Senator Mark Kelly, who is seeking re-election next year in Arizona.
The post ICYMI: Reuters: Biden's child tax credit pays big in Republican states, popular with voters appeared first on Democrats.


Acne scars can be classified as macular, atrophic, or hypertrophic. Macular scars are flat scars that are discolored. Hypertrophic scars are elevated and usually discolored. Atrophic scars are depressed scars that are further classified as ice pick, boxcar and rolling acne scars.
Microdermabrasion removes the outer layer of the epidermis and effectively improves mild acne scars. It is very safe and inexpensive.
Vistapen Micro-needling creates thousands of micropunctures in the skin, which stimulates collagen production. It is most helpful for rolling and boxcar scars. It is very safe and moderately expensive as it requires about 4 sessions.
Trichloroacetic acid (TCA) peel is a medium depth chemical peel that significantly reduces acne scars. It is inexpensive, but requires a week to recover.
TCA CROSS (Chemical Reconstruction Of Skin Scars) techniques involve carefully placing very small amounts of highly concentrated TCA to the bottom of individual ice pick and boxcar scars. Collagen formation occurs and reduces the scars. It is moderately expensive as it requires 2 to 3 sessions, but it has less down time compared to a TCA peel as only small areas are treated.
Subcision involves placing a needle under the acne scar and repeatedly moving it back and forth to free the skin from underlying scar tissue. It is inexpensive and safe and usually combined with other treatments for best results.
Dermal filler injections can be placed directly under the scar to elevate it. This can decrease acne scars for 1 to 2 years. They are safe and inexpensive as very small amounts are needed. Unfortunately, you have to purchase the entire syringe—most clients use the remainder to treat other areas of their face.
Lasers such as carbon dioxide (CO2), Erbium and Fraxel are very effective in improving acne scars. They are very expensive with a significant amount of downtime.
Minor surgical procedures including Punch excision techniques and fat transplantation can very successfully remove deep pitted scars.


The Latin name for this plant derives from the Latin verb urere, meaning "to burn." You'll understand why if you've ever encountered nettle leaves! Sometimes you will be walking in the country and feel like you've stung by a wasp or a bee, but there is no bug in site: you may instead have been "burned" by the stinging hairs of a nettle bush.
Native to Europe, Asia, and North America, the nettle is equipped with tiny hairs filled with stinging chemicals that are released when you touch them. Surprisingly, these compounds can actually bring relief to painful areas of the body. They have other promising medicinal uses, too, but in these cases you'll want to cook or dry the plant to remove the sting before you touch it!
Nettle juice was prescribed as an external treatment for snakebites and scorpion stings some 2,000 years ago. It has also been used as a kidney tonic, menstrual relief, a diuretic, and a treatment for certain respiratory disorders. Today it is generally used as a pain reliever for arthritis, for hay fever relief, and as a treatment for enlarged prostate.
A hundred or more chemicals have been identified in the nettle plant, including the neurotransmitters acetylcholine, serotonin and histamine. If you've got arthritis, slapping your joints with these tiny needles that inject these chemicals into your skin induces a positive healing response. Nettle leaf can also relieve allergies because it contains a variety of flavonoids that have antihistamine effects.
The root of stinging nettle contains lectins and sterols, which may help with benign prostate hypertrophy (BPH). If the enlarged prostate gland presses on the tube that empties urine from the bladder (the urethra), symptoms can include reduced urinary flow, incomplete emptying of the bladder, post-urination dripping, and the constant urge to urinate. Studies have shown stinging nettle is comparable to finasteride, a medication commonly prescribed for prostate enlargement. However, scientists aren't sure why nettle root reduces symptoms, since unlike finasteride, nettle does not decrease prostate size.


text:
Lane Anderson Carson (born August 21, 1947 in New Orleans, Louisiana, USA) is a licensed building contractor, real estate broker, and attorney in private practice who resides in Covington, the parish seat of St. Tammany Parish.

Career

Carson was 1st Platoon Leader, A Company, 1/11 Infantry, 5th Mech Infantry Division during the Vietnam War. He was wounded in operation Dewey Canyon II (Lam Son 719), discharged and returned home. He received the Purple Heart, Vietnam Campaign Medal, Vietnam Service Medal and Combat Infantryman Badge.

References

External links
http://www.reagan.utexas.edu/archives/speeches/1985/121185a.htm
http://pview.findlaw.com/view/2879092_1?noconfirm=0
https://web.archive.org/web/20061113185312/http://www.usna.com/News_Pubs/News/SearchNewsRead.asp?Article=4907
https://web.archive.org/web/20060821211016/http://www.rejuvenola.com/
https://www.senate.gov/comm/banking_housing_and_urban_affairs/general/noms/nibs/carson.htm
http://www.sos.louisiana.gov:8090/cgibin/?rqstyp=elcpr&rqsdta=03090452
https://web.archive.org/web/20060928231151/http://www.nibs.org/July03.pdf
http://www.thenewsstar.com/apps/pbcs.dll/article?AID=/20071206/NEWS01/712060313

1947 births
Living people
United States Army personnel of the Vietnam War
Baptists from Louisiana
Louisiana Democrats
Louisiana Republicans
Louisiana State University alumni
Members of the Louisiana House of Representatives
Politicians from New Orleans
Protestants from Louisiana
People from Covington, Louisiana
Tulane University Law School alumni
United States Army officers
Lawyers from New Orleans
State cabinet secretaries of Louisiana

redpajama_set_name: RedPajamaWikipedia


In Marriage: Communion of Life and Love, Bishop Galeone shares new insights on the Church's ancient and powerful wisdom. He deals extensively with questions of contraception and Natural Family Planning. Classic line: "Is it normal for a wife to wear ear plugs while listening to her husband?"
Click here to read Marriage: Communion of Life and Love in a printable form. In Spanish: Matrimonio: una Comunion de Vida y Amor.


How did you hear about Uloop News?
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Uloop News is one of the fastest growing news sites written by current college students for college students. Our college student writers cover campus and entertainment news, events, local businesses, sports and a variety of related topics. Many have also been featured on the Huffington Post, Buzzfeed and Business Insider thanks to our content partnership program.
Uloop has given me a platform to post thought-provoking articles and exercise my skills in writing to an audience. The flexible posting schedule has been a convenient way to put my love for writing to work as a science major. I would recommend writing consistently, whether for Uloop or on your own, to grow as a thinker, communicator, and human being.
Uloop gave me the instruction and exposure I needed to further my writing career and develop my editorial skill set. In writing as a National Columnist, I was able to have my articles featured on several news outlets, as well as journey deeper into the role of the journalist.
Writing for Uloop was the first job I've had in journalism and it provided me with skills and experiences which will help me for years to come. With Uloop, I've had the opportunity to get to know people around the country, and I've been introduced to a variety of people on my campus as well.


What sustainable grazing management means for the Northwest.
"Is meat sustainable?" It's a question too big to tackle in a single blog post, but the fact is that billions of people worldwide eat meat, in one form or another, and in the Northwest, farm sales of livestock represent nearly $3 billion in sales annually. It's useful, therefore, to consider whether some forms of meat production are better for natural systems than others.
One promising technique attempts to use livestock grazing to reverse desertification, restore grassland habitats, and even reduce global warming. It may hold promise for the Northwest's dry interior regions.
There's a fascinating back story to the so-called "brown revolution" for arid lands that starts with Allan Savory, who founded the Savory Institute based on the principles he discovered while ranching in Africa. One good account comes from Spencer Beebe, who profiled the work on Ecotrust's blog.
The guiding belief behind Savory's work is that while overgrazing can cause desertification, there's something to be learned from the fact that humans, grazing animals, and their habitats evolved together. For example, in arid environments today, sudden dry periods can halt the natural decay of plant material. But in prehistoric times, huge herds of grazing animals, such as buffalo or elephants, moved frequently across the landscape, in the process trampling organic material into the soil and breaking dry ground to allow water to seep in. The grazing herds also effectively "recycled" grassland forage by eating it, then returning it in the form of dung and urine. Herds moved on before they ran out of forage, which allowed the grazed area to "rest," recover, and re-grow, thanks to water infiltration and the chewed and trampled biological material, before grazers returned to repeat the process.
By Dr. Karen Launchbaugh, University of Idaho College of Natural Resources, via a Creative Commons license.
Domesticated livestock and fences changed the natural dynamic, allowing private herds to graze in a single location with no recovery for the rangelands. The result was often desertification, reduced productivity and biodiversity, and increased poverty for grazing communities. The solution, Savory believes, is "holistic grazing management" that moves herds in such a way that allows more "rest" of rangelands before the herd moves in again.
Rangeland—the extensive, unforested land dominating the western half of North America—constitutes a very large portion of Cascadia. In the Northwest, the most common form of rangeland is sagebrush country, sometimes called the "shrub-steppe," that provides habitat for sage grouse and many other creatures as well.
In Oregon, federal rangelands cover 14 million acres, fully 54 percent of the state's land. Idaho's rangeland covers 44 percent of the state, 9.5 million acres. By one estimate Washington's rangeland totals about 5 million acres, representing 27 percent of the state, and British Columbia contains more than 23 million acres of rangeland––about 25 percent of its total land — and a beef herd of about 215,000 head.
On both sides of the border, government has an outsize role in controlling rangelands. In British Columbia, over 90 percent of the land is Provincial Crown (that is, government-owned) land. The government leases rangeland, with most grazing occurring on Crown land between the Canadian Rockies and the Coast Mountains.
In the US, support for grazing has been a federal function, going back at least to the New Deal-Era Taylor Grazing Act of 1934, which directed the government to manage and protect federal rangelands, and to support grazing. That function now resides with the US Bureau of Land Management, which manages huge portions of Cascadia east of the Cascade Mountains.
Because rangeland management is largely the province of Cascadia's governments, public policy can play a leading role in catalyzing sustainable grazing practices. Before that happens, however, policymakers will need to better understand the benefits of Savory-style range management.
There may even be a climate benefit to better grazing practices. In a 2011 article in the Atlantic, Jim Howell, co-founder and CEO of Grasslands, addressed the connection between livestock and global warming, claiming that increasing the organic matter in the world's grasslands by just two percent would sequester 2880 billion tons of CO2.
For comparison, a February 2009 author in Scientific American estimated that the annual worldwide production of meat generated the equivalent of 6.5 billion tons CO2. (The analysis applied to all meat, not just livestock. However beef is one of the most greenhouse gas-intensive forms of meat.) For perspective then, if Jim Howell's estimate is right, better managed grasslands might be able to take up all the CO2 gases produced by 443 years of meat production at current rates.
It remains to be seen whether holistic grazing will take root with Cascadia, but the practice is already at the region's doorstep. Grasslands, LLC is applying holistic grazing practices in South Dakota and eastern Montana regions where buffalo once held sway.
It's important to acknowledge that some have objected to Savory's principles; see here and here, for example. In addition, it may take years before results from the Grasslands LLC operations in the Great Plains might can support the "brown revolution" hopes.
In the meantime, however, the Savory Institute has won support from organizations as diverse as Ecotrust in Portland and ranching advocates in Idaho. Here's hoping that the Institute's grand experiments in the Plains are successful, and can translate to more sustainable meat production and healthier Northwest rangelands.
John Abbotts is a former Sightline research consultant, who occasionally submits material that Sightline staff turn into blog posts.
John if I have an email address will send you a piece on the Science & Methodology behind the holistic planned grazing you wrote about.
You can send it to me and I'll make sure John gets it.
Of course, the world's shipping fleet emits 17 billion metric tons a year of CO2. That's four times the amount from meat, and four times the amount from all the cars combined. Care to give up your iPad? Didn't think so. Too cool.
Oops, three times the amount from meat. And thst's worldwide. Couldn't find the numbers for beef cattle. We know that the vegan "environmentalists" just hate steaks.
Thanks to Alan McCoy & Marie Burr for supporting a sustainable Northwest.


text:
\section{Introduction}

Metric perturbations of spacetimes are an important issue. The solution of the Einstein field equations (EFEs) for static, vacuum and spherically symmetric spacetime is the Schwarzschild spacetime. And the metric perturbations of the Schwarzschild black hole have been studied for a long time. To begin with, Regge, Wheeler and Vishveshwara et.al studied the odd-parity perturbation \cite{ReggeWheeler1957, Vishveshwara1970PRD}, while Zerilli and Moncrief investigated the even-parity perturbation \cite{Zerilli1970PRL,Zerilli1970PRD,Moncrief1974ext}. The perturbation theory of the Schwarzschild spacetime has been well summarized in Chandrasekhar's monograph \cite{chandrasekharbook}. After decades of research and development, this theory can be applied to a variety of different physical problems. A useful application is the quasi-normal modes of the perturbed black holes, which was initiated by Vishveshwara \cite{Vishveshwara1970Nature}, Chandrasekhar \cite{chandrasekhar1975} and Mashhoon \cite{Mashhoon1984} et.al., and the review articles of this topic can be found in \cite{Kokkotas1999,Nollert1999,Cardoso2004, Cardoso2009CQG,Konoplya2011,Pani2013IJMPA}. Another application is studying a particle moving around the Schwarzschild black hole. One can treat this point-particle as a perturbation of the Schwarzschild spacetime \cite{Price1971,Martel2004}. In addition, studying the metric perturbation can promote the analysis of the stability of the Schwarzschild spacetime \cite{Price1972,Dafermos2016,Wald2018}.

In perturbation theory in general relativity, the redundant coordinate freedom can be eliminated by choosing specific gauges. The most familiar gauge in the Schwarzschild spacetime is the Regge-Wheeler (RW) gauge, which was first presented by Regge and Wheeler \cite{ReggeWheeler1957}. And where they also analysed the spherical harmonics and decomposed the general perturbation in the Schwarzschild spacetime into odd-parity and even-parity sectors. The RW gauge has the obvious advantage of algebraic simplicity, and it is widely used in the literature. Since then, the construction and the physical meaning of the gauge-invariant properties have attracted lots of attention. Using Lagrangian and Hamiltonian variational principles for the perturbation, Moncrief considered that the metric perturbations can be decomposed into the gauge invariant part and the gauge dependent part \cite{Moncrief1974ext,Moncrief1974int}. Gerlach and Sengupta discussed the construction of gauge invariant properties in general spherically symmetric spacetimes \cite{Gerlach1979,Gerlach1980}. Thorne reviewed and summarized various scalar, vector and tensor spherical harmonics with a uniform notation \cite{Thorne1980}. Martel and Poisson presented a gauge-invariant and covariant formalism, and also showed that the energy or angular-momentum radiation can be expressed in terms of gauge-invariant scalar functions \cite{Poisson2005}. Recently, Sopuerta considered that the master functions are linear combinations of the metric perturbations and their first-order derivatives, and discussed about the master equation for vacuum spherically symmetric spacetimes \cite{Sopuerta2021}. Besides the RW gauge, there exists a variety of gauge choices. For example, the light-cone gauge, which presented by Preston and Poisson \cite{Poisson2006}, can provide geometrical meaning to the coordinates in perturbed spacetimes. Another gauge choice is named as easy (EZ) gauge \cite{Thompson2016}, which was devised by Detweiler when he considered the gravitational self-force problem in the perturbed Schwarzschild spacetime \cite{Detweiler2008}. In the EZ gauge, the metric perturbation is singular on the black-hole horizon \cite{Poisson2018}.

Generally speaking, for metric perturbation of a spherically symmetric spacetime, the standard process is to decouple the even-parity and the odd-parity EFEs, obtain the wave equations and then solve the one dimensional Schr\"{o}dinger-like equation with an effective potential. One of the most important step is to construct the gauge-invariant variable and obtain the master equation. When the background metric takes the form
\begin{equation}
\label{ABeq1}
ds^2=-f(r)dt^2+f(r)^{-1}dr^2+ r^2(d\theta^2+\sin^2\theta d\varphi^2),
\end{equation}
the problems have been thoroughly studied for the theories of Einstein \cite{Kodama2003a,Kodama2003b}, Einstein-Maxwell  \cite{Kodama2004,Kodama2011}, and  Lovelock  \cite{TS09,TS10,Kodama2011} in higher dimensions. However, if one considers a non-vacuum spherical static black hole with hairs \cite{Heisenberg2018,Myung2019,Tomikawa2021}, the metric in general cannot be cast in the above form.  Instead,  the most general spherically symmetric static spacetimes should be described by the metric
\begin{equation}
\label{ABneq1}
ds^2=-A(r)dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2\theta d\varphi^2),
\end{equation}
where $A\cdot B\neq 1$, for which perturbations have not been studied in detail so far. In particular, based on the Post-Newtonian (PN) approximation, Darmour et.al investigated the gravitational radiation generated by inspiralling compact binary systems and presented a novel approach to map the two-body problem onto an effective-one-body (EOB) system \cite{Damour1999,Damour2000}. Recently, the discussions of self-consistent of radiation-reaction force in the EOB system shows that one should first solve the gravitational perturbation in the most general shyerically symmetric spacetimes \cite{Jing2022}. Therefore, a natural question is how to construct  gauge-invariant perturbation variables in the most general spherically symmetric spacetimes (\ref{ABneq1}), and then study the even-parity and odd-parity perturbations.

With the above considerations as our main motivations, in this paper we consider the most general spherically symmetric background spacetimes with metric perturbations, and the construction of gauge invariant variables. For even-parity perturbations, we find that there exist several gauge choices, including the EZ gauge and the RW gauge. Under the EZ gauge, we construct the gauge-invariant variables and obtain a third-order master equations. However, the third-order equation can be written as a second-order equation after the separation of radial and time variables. Under the RW gauge, a similar situation also occurs. For the odd-parity perturbations, the master equation remains a second-order wave equation as usual. It should be noted that such developed formulas are not only applicable to the most general EOB system, as pointed above  \cite{Jing2022}, but also to other modified theories of gravity, in which the background is described by the
most general metric (\ref{ABneq1}). These include  theories with high-order derivative terms \cite{Bert18a,Bert18b,SJ22}. In such theories, the field equations can be always written as $G_{\mu\mu} = \kappa T^{\text{eff.}}_{\mu\nu}$, where $T^{\text{eff.}}_{\mu\nu}$ represent the modifications to general relativity (GR). Certainly, in such theories extra fields are often introduced. In the latter, we need to consider
not only the effective Einstein field equations, but also the equations for matter fields. In this paper we shall mainly focus on the effective Einstein field equations [cf. Eq.(\ref{eq16}) to be given below and other components given in Appendix B], that is, the perturbations of the most general
spherically symmetric metric, and leave the  studies of perturbations for matter field equations to another occasion, as the latter will be involved with specific modified theories.

Considering the general metric perturbations in static spherically symmetric spacetimes, Throne showed how to construct a ten-spherical-harmonic basis \cite{Thorne1980}. Through this paper, we use the A-K notation \cite{Thompson2016}, which dealt only with the Schwarzschild spacetime as the background, and was first presented by Detweiler when he considered the self-force problem. The advantage of using this notation is that one can find the relation between the metric perturbation components and the gauge invariants. In this paper, through the gauge invariants representing different combinations of the metric components, we show that the gauge-invariant variables have the similar structure under the EZ and RW gauges.

The rest of this paper is organized as follows. In Sect. II, we discuss the basic framework. First, the ten orthogonal harmonics basis are introduced. Then the decomposition of non-vacuum Einstein equations and the A-K notation are reviewed. After the investigation of gauge freedom, we consider the EZ and RW gauges. In Sect. III, we first consider the gauge invariant properties. Then we focus on constructing the master equations for both even-parity and odd-parity perturbations. We also study the cases for $l=0,1$. In Sect. IV, an example that a small particle goes around a circular orbit in spherically symmetric spacetimes is investigated. Finally, we summarize our main results with some discussions.

Throughout this paper, we use the A-K notation similarly to \cite{Thompson2016}. Units will be chosen in which $c=G=1$. In Sect. III, the subscript, e.g., $\psi_0$ and $\psi_1$, always represents quantities for $l=0$ and $l=1$ cases, respectively. And the superscript with Roman letters, i.e. $\chi^\mathrm{I}$ and $\chi^\mathrm{II}$, represent the quantities under the EZ gauge or under the RW gauge, respectively.

\section{Basic Framework}

\subsection{Orthogonal Harmonics Basis}

Let us start with the most general spherically symmetric spacetimes
\begin{equation}
\label{SO2spacetime}
ds^2=g_{ab}^{(0)}dx^adx^b=-e^{2\Phi(r)}dt^2+e^{2\Lambda(r)}dr^2+r^2(d\theta^2+\sin^2\theta d\varphi^2) .
\end{equation}
To decompose tensor fields on the above background, we choose the orthogonal basis composed of scalar spherical harmonics, vector harmonics and tensor harmonics. First, we define two unnormalized and orthogonal co-vectors $v$ and $n$
\begin{equation}
v_a=(-1,0,0,0), \quad\quad n_a=(0,1,0,0),
\end{equation}
the projection operator onto the sphere surface
\begin{equation}
\Omega_{ab}=g_{ab}^{(0)}+e^{2\Phi}v_a v_b-e^{2\Lambda}n_a n_b=r^2 \text{diag}(0,0,1,\sin^2\theta),
\end{equation}
and the spatial Levi-Civita tensor, $\epsilon_{abc}\equiv v^d\epsilon_{dabc}$, where $\epsilon_{tr\theta\phi}=e^{\Phi+\Lambda}r^2\sin\theta$.

In general, the complete basis on the 2-sphere are constructed by 1-scalar spherical harmonic, $Y^{lm}=Y^{lm}(\theta,\varphi)$, 3 pure-spin vector harmonics and 6 tensor harmonics \cite{Thorne1980}. The pure-spin vector harmonics are given by
\begin{equation}
Y_a^{E,lm}=r\nabla_a Y^{lm},\quad Y_a^{B,lm}=r\epsilon_{ab}\hspace{0.1mm}^c n^b\nabla_c Y^{lm}, \quad Y_a^{R,lm}=n_aY^{lm}.
\end{equation}
And the pure-spin tensor harmonics are given by
\begin{align}
T_{ab}^{T0,lm}=\Omega_{ab}Y^{lm}, &\quad \quad T_{ab}^{L0,lm}=n_an_b Y^{lm}, \\
T_{ab}^{E1,lm}=rn_{(a}\nabla_{b)}Y^{lm}, &\quad \quad T_{ab}^{B1,lm}=rn_{(a}\epsilon_{b)c}\hspace{0.1mm}^d n^c\nabla_d Y^{lm}, \\
T_{ab}^{E2,lm}=r^2\left(\Omega_a\hspace{0.1mm}^c\Omega_b\hspace{0.1mm}^d-\frac{1}{2}\Omega_{ab}\Omega^{cd}\right)\nabla_c\nabla_dY^{lm}, &\quad \quad
T_{ab}^{B2,lm}=r^2\Omega_{(a}\hspace{0.1mm}^c\epsilon_{b)e}\hspace{0.1mm}^d n^e\nabla_c\nabla_d Y^{lm}.
\end{align}
Note that the vector harmonics are orthogonal to each other
\begin{equation}
\oint Y_a^{A,lm}(Y^a_{A',l'm'})^*d\Omega =N^{(vec)}(A,r,l)\delta_{AA'}\delta_{ll'}\delta_{mm'} ,
\end{equation}
with $\{A,A'\}=\{E,B,R\}$ and $N^{(vec)}(A,r,l)$ is the specific normalization factor for vector harmonics. The tensor harmonics are also orthogonal to each other
\begin{equation}
\oint T_{ab}^{A,lm}(T^{ab}_{A',l'm'})^*d\Omega =N^{(ten)}(A,r,l)\delta_{AA'}\delta_{ll'}\delta_{mm'} ,
\end{equation}
with $\{A,A'\}=\{T0, L0,E1,E2,B1,B2\}$ and $N^{(ten)}(A,r,l)$ is the specific normalization factor for tensor harmonics. The expression for these normalization functions $N^{(vec)}$ and $N^{(ten)}$ are given in Appendix A.

\subsection{Decomposition of Linearized Einstein Equations}
For perturbed spacetimes, we use $h_{ab}$ to represent the linear perturbation of the background spacetime $g_{ab}^{(0)}$, i.e., the metric of the perturbed spacetime can be written as
\begin{equation}
g_{ab}=g_{ab}^{(0)}+h_{ab}.
\end{equation}
The background metric and the perturbed metric satisfied the Einstein Field Equations (EFEs)
\begin{align}
 G_{ab}(g^{(0)}) &= 8\pi T_{ab}, \\
 G_{ab}(g^{(0)}+h) &= 8\pi (T_{ab}+\mathcal{T}_{ab}),
\end{align}
where $T_{ab}$ and $\mathcal{T}_{ab}$ denote the non-vacuum background and the perturbed energy-momentum tensor, respectively. Expanding the EFEs in terms of $h_{ab}$, we get
\begin{equation}
 G_{ab}(g^{(0)}+h)=G_{ab}(g^{(0)})-\frac{1}{2}E_{ab}(h),
\end{equation}
where $E_{ab}$ is the linearized Einstein operator
\begin{align}
\label{PerEFEs}
E_{ab}(h) =& \Box h_{ab}+\nabla_a\nabla_bh^{c}_{~c}-2\nabla_{(a}\nabla^ch_{b)c}+2R^{~c~d}_{a~b}h_{cd}
-(R_a\hspace{0.1mm}^c h_{bc}+R_b\hspace{0.1mm}^c h_{ac}) \nonumber \\
& +g_{ab}(\nabla^c\nabla^dh_{cd}-\Box h^{d}_{~d})-g_{ab}R^{cd}h_{cd}+R h_{ab} \nonumber \\
=& -16\pi\mathcal{T}_{ab}.
\end{align}
Note that now the Ricci curvature $R_{ab}$ and the scalar curvature $R$ of the background do not vanish in general. If $R_{ab}$ and $R$ vanish, then the background metric would reduce to the Schwarzschild metric, which is the situation discussed in \cite{Thompson2016}.

Detweiler decomposed the harmonic modes of the perturbed metric $h_{ab}$ as
\begin{align}
\label{eq17}
h_{ab}^{lm} =& \mathrm{A~}v_av_bY^{lm}+2\mathrm{B~}v_{(a}Y_{b)}^{E,lm}+2\mathrm{C~}v_{(a}Y_{b)}^{B,lm}+2\mathrm{D~}v_{(a}Y_{b)}^{R,lm}+\mathrm{E~}T^{T0,lm}_{ab} \nonumber \\
& +\mathrm{F~}T_{ab}^{E2,lm}+\mathrm{G~}T^{B2,lm}_{ab}+2\mathrm{H~}T^{E1,lm}_{ab}+2\mathrm{J~}T^{B1,lm}_{ab}+\mathrm{K~}T^{L0,lm}_{ab} ,
\end{align}
where all coefficients A through K are scalar functions of $(t,r)$, which were referred to as the A-K coefficients in \cite{Thompson2016}.

For even-parity (polar part) perturbations with $l\geq 2$, the perturbed metric can be decomposed as
\begin{equation}
h_{ab}^{\text {even }}=\left(\begin{array}{cccc}
	\mathrm{A~}Y_{lm} & -\mathrm{D~}Y_{lm} & -r\mathrm{B~}\partial_{\theta} Y_{lm} & -r\mathrm{B~}\partial_{\phi} Y_{lm} \\
	\text {Sym} & \mathrm{K~}Y_{lm} & r\mathrm{H~}\partial_{\theta} Y_{lm} & r\mathrm{H~}\partial_{\phi} Y_{lm} \\
	\text {Sym} & \text {Sym} & r^{2}\left[\mathrm{E}+\mathrm{F}\left(\partial_{\theta}^2+\frac{1}{2}l(l+1)\right)\right]Y_{lm} & r^{2}\mathrm{F}\left[\partial_{\theta}\partial_{\phi}-\cot\theta\partial_{\phi}\right]Y_{lm} \\
	\text {Sym} & \text {Sym} & \text {Sym} & r^{2} \sin^2\theta\left[\mathrm{E}-\mathrm{F}\left(\partial_{\theta}^{2}+\frac{1}{2} l(l+1)\right)\right]Y_{lm}
\end{array}\right) .
\end{equation}
And for the odd-parity (axial part) perturbations with $l\geq 2$, the perturbed metric can be decomposed as
\begin{equation}
h_{a b}^{\mathrm{odd}}=e^{-\Phi-\Lambda}
\left(\begin{array}{cccc}
	0 & 0 &  r \csc\theta \mathrm{C~}\partial_{\phi} Y_{lm} & -r \sin \theta\mathrm{~C~}\partial_{\theta} Y_{lm} \\
	0 & 0 & -r \csc\theta \mathrm{J~}\partial_{\phi} Y_{lm} & r \sin \theta \mathrm{~J~}\partial_{\theta} Y_{lm} \\
	\mathrm{Sym} & \mathrm{Sym} & -r^2\csc\theta \mathrm{G}\left[\partial_{\theta} \partial_{\phi}-\cot \theta \partial_{\phi}\right] Y_{lm} & -\frac{r^{2}}{2}\mathrm{G}\left[\csc\theta \partial_{\phi}^{2}+\cos\theta\partial_{\theta}-\sin\theta\partial_{\theta}^2\right]Y_{lm} \\
	\mathrm{Sym} & \mathrm{Sym} & \mathrm{Sym} & r^{2}\mathrm{G}\left[\sin \theta \partial_{\theta} \partial_{\phi}-\cos \theta \partial_{\phi}\right] Y_{lm}
\end{array}\right) .
\end{equation}
These A-K notations could be linearly transformed into the notation taken by Regge and Wheeler \cite{Thompson2016}. To get the explicit expression of coefficients A-K, one can project the metric perturbation onto the tensor harmonic basis. We present these coefficients in Appendix A.

Now, we can get the A-K components of any rank-2 tensor in the spherically symmetric background. For example, from \eq{PerEFEs}, one can write
\begin{equation}
\label{eq16}
-16\pi\mathcal{T}_{\mathrm{A}}=E_{\mathrm{A}}=e^{4\Phi}\oint E_{ab}(h)(v^av^bY^{\ast}_{lm})d\Omega ,
\end{equation}
to represent the A-term of $\mathcal{T}_{ab}$ or $E_{ab}(h)$, which along the direction of $v^av^bY^{\ast}_{lm}$. The expression for $E_{\mathrm{A}}$-$E_{\mathrm{K}}$ are given in Appendix B.

\subsection{The Gauge Transformation}
Under a gauge transformation, $\tilde x^a=x^a+\xi^a$, the first-order metric perturbation $h_{ab}$ would be transformed as
\begin{equation}
\tilde h_{ab}=h_{ab}-2\nabla_{(a}\xi_{b)} .
\end{equation}
As a vector, $\xi^a$ can be projected onto the pure-spin harmonic basis as
\begin{equation}
\xi_a=\mathrm{P~}v_aY_{lm}+\mathrm{R~}Y_a^{R,lm}+\mathrm{S~}Y_a^{E,lm}+\mathrm{Q~}Y_a^{B,lm} ,
\end{equation}
where $\mathrm{P}$, $\mathrm{R}$, $\mathrm{S}$ and $\mathrm{Q}$ are scalar functions of $(t,r)$. The functions $\mathrm{P}$, $\mathrm{R}$ and $\mathrm{S}$ describe three degrees of gauge freedom for even-parity perturbations, while the function $\mathrm{Q}$ describes one degree of the gauge freedom for odd-parity perturbations. Next we use $\Delta$ to represent the A-K projections of $2\nabla_{(a}\xi_{b)}$. For example,
\begin{equation}
\Delta \mathrm{A} = \mathrm{A}-\mathrm{\tilde A} = 2e^{4\Phi}\oint \nabla_{(a}\xi_{b)}(v^av^bY^{\ast}_{lm})d\Omega ,
\end{equation}
here A and $\mathrm{\tilde A}$ correspond to the projections of $h_{ab}$ and $\tilde h_{ab}$, respectively. Projecting $2\nabla_{(a}\xi_{b)}$ onto the tensor harmonic basis, we obtain the A-K components of the term $2\nabla_{(a}\xi_{b)}$,
\begin{align}
\label{DeltaAK}
& \Delta \mathrm{A} = -2\frac{\partial \mathrm{P}}{\partial t}-2e^{-2\Lambda+2\Phi}\frac{\partial \Phi}{\partial r}\mathrm{R},
& \Delta \mathrm{B} &= -\frac{\partial \mathrm{S}}{\partial t}+\frac{1}{r}\mathrm{P}, \nonumber \\
& \Delta \mathrm{C} = -\frac{\partial }{\partial t}\mathrm{Q},
& \Delta \mathrm{D} &= \left(\frac{\partial }{\partial r}-2\frac{\partial \Phi}{\partial r} \right)\mathrm{P} -\frac{\partial \mathrm{R}}{\partial t},  \nonumber \\
& \Delta \mathrm{E} = 2\frac{e^{-2\Lambda}}{r}\mathrm{R}-\frac{l(l+1)}{r}\mathrm{S},
& \Delta \mathrm{F} &= \frac{2}{r}\mathrm{S}, \nonumber \\
& \Delta \mathrm{G} = \frac{2}{r}\mathrm{Q}, &\Delta \mathrm{H} &= \frac{1}{r}\mathrm{R}+\left(\frac{\partial}{\partial r}-\frac{1}{r}\right)\mathrm{S}, \nonumber \\
& \Delta \mathrm{J} = \left(\frac{\partial}{\partial r}-\frac{1}{r}-\frac{\partial \Phi}{\partial r}-\frac{\partial \Lambda}{\partial r}\right)\mathrm{Q},
& \Delta \mathrm{K} &=\left(2\frac{\partial }{\partial r}-2\frac{\partial \Lambda}{\partial r}\right)\mathrm{R}.
\end{align}

\subsection{Gauge Choices}
Generally speaking, $\xi^a$ has four independent functions, representing four degrees of freedom in spherically symmetric spacetimes. Hence by properly choosing these four functions we can work with different gauges. For example, under the gauge transformation, the scalar function $\mathrm{F}(t,r)$ would transform as
\begin{equation}
\mathrm{\tilde F} = \mathrm{F}-\Delta \mathrm{F} = \mathrm{F}-\frac{2}{r}\mathrm{S}.
\end{equation}
By setting $\mathrm{S}=r\mathrm{F}/2$, one degree of the gauge freedom is fixed, and $\mathrm{\tilde F}=0$. Then substituting this S back into \eq{DeltaAK}, one can move on to eliminate the next degrees of freedom. In even-parity perturbations, properly choosing the functions $\mathrm{P}$, $\mathrm{R}$ and $\mathrm{S}$ would fix three variables of the metric perturbation. In the odd-parity properly choosing the function $\mathrm{Q}$, one can fix one variable of the metric perturbation.

Note that in the odd-parity sector, $\Delta \mathrm{G}$ is proportional to $\mathrm{Q}$, but there exists some derivative relation between $\Delta \mathrm{C}$, $\Delta \mathrm{J}$ and Q. If we want to eliminate $\mathrm{\tilde G}$ under the gauge transformation, just set $Q=r\mathrm{G}/2$ then $\mathrm{\tilde G}=0$ and $\Delta \mathrm{C}$ and $\Delta \mathrm{J}$ are uniquely determined. If we eliminate $\mathrm{\tilde C}$ under the gauge transformation rather than $\mathrm{\tilde G}$, then $\mathrm{Q}$ could be an arbitrary function of $r$ with some integration constant, which could not determine $\Delta \mathrm{G}$ and $\Delta \mathrm{J}$ completely. Such a choice cannot completely fix the gauge freedom. Similarly, in the even-parity sector, in order to fix the gauge completely, one may first fix $\mathrm{S}$ from $\Delta \mathrm{F}$. Afterward, there are still several choices to fix $\mathrm{P}$ and $\mathrm{R}$ via $\Delta \mathrm{A}$, $\Delta \mathrm{B}$, $\Delta \mathrm{E}$ or $\Delta \mathrm{H}$. Below, we would discuss two useful gauge choices.

\textbf{Regge-Wheeler Gauge.} Regge and Wheeler first presented the RW gauge by setting certain RW variables to zero \cite{ReggeWheeler1957}. In the even-parity sector of the Schwarzschild spacetime, to eliminate three gauge freedom, RW set
\begin{equation}
h_0^{RW,\text{even}}=h_1^{RW,\text{even}}=G^{RW}=0 ,
\end{equation}
which corresponds to setting $\mathrm{\tilde B}=\mathrm{\tilde F}=\mathrm{\tilde H}=0$ in the A-K notation. And in the odd-parity sector, RW set $h_2^{RW}=0$, which corresponds to setting $\mathrm{\tilde G}=0$ \cite{Thompson2016}. For general spherically symmetric spacetimes, we set
\begin{align}
& \mathrm{S}^{RW} = \frac{r}{2}\mathrm{F}, & \mathrm{P}^{RW} &= r \mathrm{B}+\frac{r^2}{2}\frac{\partial \mathrm{F}}{\partial t}, \nonumber \\
& \mathrm{R}^{RW} = r \mathrm{H}-\frac{r}{2}\frac{\partial \mathrm{F}}{\partial r}, & \mathrm{Q}^{RW} &= \frac{r}{2}\mathrm{G},
\end{align}
which means
\begin{equation}
\xi_a^{RW}=\left(r \mathrm{B}+\frac{r^2}{2}\frac{\partial \mathrm{F}}{\partial t}\right)v_aY_{lm}
+\left(r \mathrm{H}-\frac{r}{2}\frac{\partial \mathrm{F}}{\partial r}\right)Y_a^{R,lm}+\frac{r}{2}\mathrm{F}Y_a^{E,lm}+\frac{r}{2}\mathrm{G}Y_a^{B,lm} .
\end{equation}
It should note that this gauge choice is exactly the same as the gauge choice with the A-K notation in the Schwarzschild spacetime, see, for example, Eq.(6.7) in \cite{Thompson2016}.

\textbf{Easy gauge.} The EZ gauge was first introduced by Detweiler when he considered the self-force problem, in which the following metric components are set to zero,
\begin{equation}
\mathrm{\tilde B}=\mathrm{\tilde E}=\mathrm{\tilde F}=\mathrm{\tilde G}=0 .
\end{equation}
To eliminate these metric components, the components of the gauge vector are chosen as
\begin{align}
& \mathrm{S}^{EZ} = \frac{r}{2}\mathrm{F}, & \mathrm{P}^{EZ} &= r \mathrm{B}+\frac{r^2}{2}\frac{\partial \mathrm{F}}{\partial t}, \nonumber \\
& \mathrm{R}^{EZ} = \frac{r}{4}l(l+1)e^{2\Lambda} \mathrm{F}+\frac{r}{2}e^{2\Lambda} \mathrm{E}, & \mathrm{Q}^{EZ} &= \frac{r}{2}\mathrm{G},
\end{align}
which means
\begin{equation}
\xi_a^{EZ}=\left(r \mathrm{B}+\frac{r^2}{2}\frac{\partial \mathrm{F}}{\partial t}\right)v_aY_{lm}
+\left(rl(l+1)e^{2\Lambda} \mathrm{F}+\frac{r}{2}e^{2\Lambda} \mathrm{E}\right)Y_a^{R,lm}+\frac{r}{2}\mathrm{F}Y_a^{E,lm}+\frac{r}{2}\mathrm{G}Y_a^{B,lm} .
\end{equation}

\section{Gauge Invariants and Master Equations}
In this section, we would first introduce a general set of gauge invariants in the spherically symmetric backgrounds. With these gauge invariants, we shall investigate how to construct the single master equation under certain gauge choices.

\subsection{Gauge Invariants}
Generally speaking, under any arbitrary gauge transformation, the gauge invariants can be constructed from \eq{DeltaAK}. For example,
\begin{align}
\Delta \mathrm{G} &= \frac{2}{r}\mathrm{Q}, \\
\Delta \mathrm{J} &= \left(\frac{\partial}{\partial r}-\frac{1}{r}-\frac{\partial \Phi}{\partial r}-\frac{\partial \Lambda}{\partial r}\right)\mathrm{Q} ,
\end{align}
from which, we obtain
\begin{equation}
\Delta \mathrm{J}+\frac{r}{2}\left(\frac{\partial \Phi}{\partial r}+\frac{\partial\Lambda}{\partial r}-\frac{\partial }{\partial r} \right)\Delta \mathrm{G} = 0 .
\end{equation}
The above equation indicates that one can define
\begin{equation}
\alpha = \mathrm{J}+\frac{r}{2}\left(\frac{\partial \Phi}{\partial r}+\frac{\partial\Lambda}{\partial r}-\frac{\partial }{\partial r} \right)\mathrm{G},
\end{equation}
and $\alpha$ is a gauge invariant quantity. Note that there are seven even-parity metric components and three odd-parity metric components in the metric perturbation, while there are three even-parity components and one odd-parity component in $\xi^a$, which tell us that we can construct four independent even-parity gauge invariants and two independent odd-parity gauge invariants for $l\geq 2$ cases. Following the construction of $\alpha$, we find that they can be constructed as
\begin{align}
\label{gaugeinv}
\alpha &= \mathrm{J}+\frac{r}{2}\left(\frac{\partial \Phi}{\partial r}+\frac{\partial\Lambda}{\partial r}-\frac{\partial }{\partial r} \right)\mathrm{G} , \nonumber \\
\beta &= -\mathrm{C}-\frac{r}{2}\frac{\partial }{\partial t}\mathrm{G} , \nonumber \\
\chi &= \mathrm{H}-\frac{1}{2}e^{2\Lambda}\mathrm{E}-\frac{l(l+1)}{4}e^{2\Lambda}\mathrm{F}-\frac{r}{2}\frac{\partial }{\partial r}\mathrm{F} , \nonumber \\
\psi &= \frac{1}{2}\mathrm{K} -\frac{r}{2}e^{2\Lambda}\frac{\partial \Lambda}{\partial r}\mathrm{E}-\frac{1}{2}e^{2\Lambda}\mathrm{E}
-\frac{r}{2}e^{2\Lambda}\frac{\partial }{\partial r}\mathrm{E}-\frac{r}{4}l(l+1)e^{2\Lambda}\frac{\partial \Lambda}{\partial r}\mathrm{F}-\frac{1}{4}l(l+1)e^{2\Lambda}\mathrm{F} -\frac{r}{4}l(l+1)e^{2\Lambda}\frac{\partial }{\partial r}\mathrm{F} , \nonumber \\
\delta &= \mathrm{D}+\frac{r}{2}e^{2\Lambda}\frac{\partial }{\partial t}\mathrm{E}+\left(2r\frac{\partial \Phi}{\partial r} -1\right) B-r\frac{\partial }{\partial r}\mathrm{B}
-\frac{r^2}{2}\frac{\partial^2}{\partial t\partial r}\mathrm{F}-\left[ r-r^2\frac{\partial \Phi}{\partial r}-\frac{r}{4}l(l+1)e^{2\Lambda}\right]\frac{\partial}{\partial t}\mathrm{F} , \nonumber \\
\epsilon &= -\frac{1}{2}\mathrm{A}-\frac{r}{2}e^{2\Phi}\frac{\partial \Phi}{\partial r}\mathrm{E}-r\frac{\partial }{\partial t}\mathrm{B}
-\frac{r}{4}l(l+1)e^{2\Phi}\frac{\partial \Phi}{\partial r}\mathrm{F}-\frac{r^2}{2}\frac{\partial^2}{\partial t^2}\mathrm{F} .
\end{align}
These relations are the same as Eq. (7.5) of \cite{Thompson2016} when the background is vacuum, which degenerates to the Schwarzschild spacetime.

From now on, we shall work in the coordinates $\tilde x^a$. And for the save of simplicity, all the tildes will be dropped from now on. Note that we use the superscript $\mathrm{I}$ or $\mathrm{II}$ to denote the quantities or parameters under the EZ gauge or the RW gauge, respectively. Adopting the specific EZ gauge, we have
\begin{equation}
\mathrm{B}=\mathrm{E}=\mathrm{F}=\mathrm{G}=0 .
\end{equation}
Then, the gauge invariants become
\begin{align}
\label{gaugeinvtometricEZ}
\alpha &= \mathrm{J}, & \beta &=-\mathrm{C}, \nonumber \\
\chi^\mathrm{I} &= \mathrm{H}, & \psi^\mathrm{I} &= \frac{1}{2}\mathrm{K}, \nonumber \\
\delta^\mathrm{I} &= \mathrm{D}, & \epsilon^\mathrm{I} &= -\frac{1}{2}\mathrm{A} .
\end{align}
Here $\alpha$ and $\beta$ are not superscripted because they are the same under the EZ and RW gauges. Similarly, adopting the specific RW gauge, we have
\begin{equation}
\mathrm{B}=\mathrm{F}=\mathrm{H}=\mathrm{G}=0,
\end{equation}
and the even-parity gauge invariants become
\begin{align}
\label{gaugeinvtometricRW}
\chi^\mathrm{II} &= -\frac{1}{2}e^{2\Lambda}\mathrm{E} ,
& \psi^\mathrm{II} &= \frac{1}{2}\mathrm{K}-\frac{1}{2}\left(r\Lambda'+1\right)e^{2\Lambda}\mathrm{E}
-\frac{r}{2}e^{2\Lambda}\frac{\partial}{\partial r}\mathrm{E} , \nonumber \\
\delta^\mathrm{II} &= \mathrm{D}+\frac{r}{2}e^{2\Lambda}\frac{\partial}{\partial t}\mathrm{E} , &
\epsilon^\mathrm{II} &= -\frac{1}{2}\mathrm{A}-\frac{1}{2}r\Phi'e^{2\Phi}\mathrm{E} .
\end{align}

Using these gauges, each A-K projection of the linearized Einstein equations, i.e. \eqs{projEA}-\meq{projEJ}, can be rewritten as a combination of the gauge invariants listed in \eq{gaugeinv}. The results can be found in Appendix C. It is obvious that under the EZ gauge, the relationship between the gauge invariants and the perturbed metric components seems simpler, hence we first study the master equation under the EZ gauge.

\subsection{Master Equations for \texorpdfstring{$l\geq 2$}.}

Assuming that the perturbation of the stress-energy $\mathcal{T}_{ab}$ is known, i.e. $E_{\mathrm{A}}$-$E_\mathrm{K}$ are known quantities, now we look for the master equations in terms of gauge invariants.

\subsubsection{\textbf{Even-parity Perturbations and the EZ gauge}}

The Bianchi identities indicate that not all seven even-parity projection equations in Appendix C are independent. It has been shown that there are four independent gauge invariants. Noting the specific structure of the expressions of $E_{\mathrm{A}}^\mathrm{I}$, $E_{\mathrm{D}}^\mathrm{I}$, $E_{\mathrm{F}}^\mathrm{I}$, $E_{\mathrm{H}}^\mathrm{I}$ and $E_{\mathrm{K}}^\mathrm{I}$, we find that one can obtain
$\partial\delta^\mathrm{I}/\partial t$ from $\partial E_{\mathrm{D}}^\mathrm{I}/\partial t$, and $\epsilon^\mathrm{I}$ from $E_{\mathrm{A}}^\mathrm{I}$ or $E_{\mathrm{F}}^\mathrm{I}$. Substituting these relations into $2E_{\mathrm{H}}^\mathrm{I}+E_{\mathrm{K}}^\mathrm{I}$, after a large but tedious calculation, we find that the coupled partial differential equations for $\chi^\mathrm{I}$ and $\psi^\mathrm{I}$ can be constructed as

\begin{align}
\label{prchiEZ}
\frac{\partial}{\partial r}\chi^\mathrm{I} =& \frac{2e^{2\Lambda-2\Phi}r}{\tau^\mathrm{I} \eta^{\mathrm{I}}}
\left(\sigma^\mathrm{I}\frac{\partial^2}{\partial t^2}\chi^\mathrm{I}-2\frac{\partial^2}{\partial t^2}\psi^\mathrm{I}
+\frac{1}{2}e^{2\Lambda}r\frac{\partial }{\partial t}E_\mathrm{D}^\mathrm{I} \right)
+\frac{\gamma^\mathrm{I}}{r\tau^\mathrm{I}}\chi^\mathrm{I}+\frac{\rho^\mathrm{I}}{r\tau^\mathrm{I}}\psi^\mathrm{I}\nonumber \\
&-\frac{e^{2\Lambda}r}{2\tau^{\mathrm{I}}}\left(e^{2\Lambda}\lambda+2r\Phi'-2\right)E_\mathrm{F}^\mathrm{I}+\frac{e^{2\Lambda}r}{2\tau^\mathrm{I}}\left(E_\mathrm{K}^\mathrm{I}+2E_\mathrm{H}^\mathrm{I}\right) ,
\end{align}
\begin{align}
\label{prpsiEZ}
\frac{\partial}{\partial r}\psi^\mathrm{I} =& \frac{ e^{2\Lambda-2\Phi}r}{\tau^\mathrm{I}}
\left(\sigma^\mathrm{I}\frac{\partial^2}{\partial t^2}\chi^\mathrm{I}-2\frac{\partial^2}{\partial t^2}\psi^\mathrm{I}
+\frac{1}{2}e^{2\Lambda}r\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{I} \right)
+\frac{\mu^\mathrm{I}}{r \tau^\mathrm{I}}\chi^\mathrm{I}+\frac{\nu^\mathrm{I}}{r\tau^\mathrm{I}}\psi^\mathrm{I} \nonumber \\
& +\frac{ e^{2\Lambda}r}{4\tau^\mathrm{I}}\left[\kappa^{\mathrm{I}}E^\mathrm{I}_{\mathrm{F}}+\eta^{\mathrm{I}}\left(E_\mathrm{K}^\mathrm{I}+2E_\mathrm{H}^\mathrm{I}\right)\right]
-\frac{1}{4}e^{4\Lambda-2\Phi}rE_\mathrm{A}^\mathrm{I} ,
\end{align}
where $\lambda=l(l+1)$, and $\sigma^\mathrm{I}$, $\tau^\mathrm{I}$, $\eta^\mathrm{I}$, $\gamma^\mathrm{I}$, $\rho^\mathrm{I}$, $\mu^\mathrm{I}$, $\nu^\mathrm{I}$, $\kappa^\mathrm{I}$ are functions determined by the background spacetime, which can be find in Appendix D. Note that in these two equations, there are no spatial derivatives of the source terms. Now the goal becomes to decouple the gauge invariants $\chi^\mathrm{I}$ and $\psi^\mathrm{I}$, e.g. \eqs{prchiEZ} and \meq{prpsiEZ}. Introducing the gauge invariant
\begin{equation}
\label{evenmastervEZ}
Z^\mathrm{I}\hspace{0.1mm}^{(+)} = \sigma^\mathrm{I}\chi^\mathrm{I}-2\psi^\mathrm{I} ,
\end{equation}
we find that $Z^\mathrm{I}\hspace{0.1mm}^{(+)}$ satisfies the master equation
\begin{equation}
\label{evenmasterEZ}
\left[\frac{2e^{2\Lambda-2\Phi}r}{N^{\mathrm{I}}\eta^{\mathrm{I}}}
\left(2r(\eta^\mathrm{I}-\sigma^\mathrm{I})\sigma^{\mathrm{I}}\frac{\partial^3}{\partial t^2 \partial r}
+\frac{N_\mathrm{T}^\mathrm{I}}{N^\mathrm{I}\eta^{\mathrm{I}}\tau^\mathrm{I}}\frac{\partial^2}{\partial t^2}\right)
+\frac{2r\sigma^{\mathrm{I}}\tau^\mathrm{I}}{N^{\mathrm{I}}}\frac{\partial^2}{\partial r^2}+\frac{N_\mathrm{R}^\mathrm{I}}{\left({N^\mathrm{I}}\right)^2}\frac{\partial}{\partial r}
+\frac{N_\mathrm{Z}^\mathrm{I}}{r\left({N^\mathrm{I}}\right)^2\tau^\mathrm{I}}\right]Z^\mathrm{I}\hspace{0.1mm}^{(+)}
=S_{\text{even}}^\mathrm{I},
\end{equation}
with the source term given by
\begin{align}
\label{evensourceEZ}
S_{\text{even}}^\mathrm{I} =& \frac{e^{4\Lambda-2\Phi}r}{N^{\mathrm{I}}} \left[\frac{N_\mathrm{A}^\mathrm{I}}{2N^\mathrm{I}}E_\mathrm{A}^\mathrm{I}
+r\sigma^{\mathrm{I}}\tau^\mathrm{I}\frac{\partial}{\partial r}E_\mathrm{A}^\mathrm{I}
+\frac{rN_\mathrm{D}^\mathrm{I}}{N^\mathrm{I}\left(\eta^{\mathrm{I}}\right)^2\tau^\mathrm{I}}\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{I}
-\frac{2r^2(\eta^\mathrm{I}-\sigma^\mathrm{I})\sigma^{\mathrm{I}}}{\eta^{\mathrm{I}}}\frac{\partial^2}{\partial t\partial r}E_\mathrm{D}^\mathrm{I} \right]
-\frac{2e^{2\Lambda}r^2M^{\mathrm{I}}_{\mathrm{F}}\sigma^{\mathrm{I}}}{N^{\mathrm{I}}}\frac{\partial}{\partial r}E^{\mathrm{I}}_{\mathrm{F}} \nonumber \\
& +\frac{e^{2\Lambda}r}{\left(N^{\mathrm{I}}\right)^2\tau^{\mathrm{I}}} \left[\frac{1}{2}N_\mathrm{F}^\mathrm{I}E_\mathrm{F}^\mathrm{I}
+N_\mathrm{HK}^\mathrm{I}\left(E_\mathrm{H}^\mathrm{I}+\frac{1}{2}E_\mathrm{K}^\mathrm{I}\right)\right]
-\frac{e^{2\Lambda}r^2\left(\eta^{\mathrm{I}}-\sigma^{\mathrm{I}}\right)\sigma^{\mathrm{I}}}{N^{\mathrm{I}}}
\left(2\frac{\partial}{\partial r}E^{\mathrm{I}}_{\mathrm{H}}+\frac{\partial}{\partial r}E^{\mathrm{I}}_{\mathrm{K}}\right) .
\end{align}
In the above equation, $N^\mathrm{I}$, $N_\mathrm{T}^\mathrm{I}$, $N_\mathrm{R}^\mathrm{I}$, $N_\mathrm{Z}^\mathrm{I}$ and $N_\mathrm{A}^\mathrm{I}$, $N_\mathrm{D}^\mathrm{I}$, $M^{\mathrm{I}}_\mathrm{F}$, $N_\mathrm{F}^\mathrm{I}$, $N_\mathrm{HK}^\mathrm{I}$ are functions depending only on the background. The explicit expressions of them can be found in Appendix D. Unlike the well-known Zerilli equation, the master equation under the EZ gauge is a third-order equation. However, rewriting \eq{evenmasterEZ} in the form
\begin{equation}
\left( a\frac{\partial^3}{\partial t^2\partial r}+b\frac{\partial^2}{\partial t^2}+c\frac{\partial^2}{\partial r^2}+d\frac{\partial}{\partial r}+e \right)Z^\mathrm{I}\hspace{0.1mm}^{(+)}=S_{\text{even}}^\mathrm{I},
\end{equation}
setting $Z^\mathrm{I}\hspace{0.1mm}^{(+)}=Z^\mathrm{I}\hspace{0.1mm}^{(+)}(r)e^{i\omega t}$ and $S_{\text{even}}^\mathrm{I}=S_{\text{even}}^\mathrm{I}(r)e^{i\omega t}$, we obtain a second-order differential equation
\begin{equation}
\left[c\frac{\partial^2}{\partial r^2}+(d-a\omega^2)\frac{\partial}{\partial r}+(e-b\omega^2) \right]Z^\mathrm{I}\hspace{0.1mm}^{(+)}(r)=S_{\text{even}}^\mathrm{I}(r) .
\end{equation}

If the background becomes the Schwarzschild spacetime, the master equation \meq{evenmasterEZ} will reduce to the result given in \cite{Thompson2016}, in other words, the result of Zerilli \cite{Zerilli1970PRL}. Further discussions about the degeneration of our result could be found in Appendix E.

Once \eq{evenmasterEZ} is solved, together with the definition of the master variable given by \eq{evenmastervEZ}, the gauge invariants $\chi^\mathrm{I}$ and $\psi^\mathrm{I}$ can be solved from
\begin{align}
\chi^\mathrm{I} =& \frac{1}{N^\mathrm{I}}
\left[\frac{4e^{2\Lambda-2\Phi}r^2\left(\eta^\mathrm{I}-\sigma^\mathrm{I}\right)}
{\eta^\mathrm{I}}\frac{\partial^2}{\partial t^2}Z^\mathrm{I}\hspace{0.1mm}^{(+)}
+2r\tau^\mathrm{I}\frac{\partial }{\partial r}Z^\mathrm{I}\hspace{0.1mm}^{(+)} +M_1^\mathrm{I}Z^\mathrm{I}\hspace{0.1mm}^{(+)} -e^{4\Lambda-2\Phi}r^2\tau^\mathrm{I} E_\mathrm{A}^\mathrm{I} \right. \nonumber \\
& \left. +\frac{2e^{4\Lambda-2\Phi}r^3\left(\eta^\mathrm{I}-\sigma^\mathrm{I}\right)}
{\eta^\mathrm{I}}\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{I}
+2e^{2\Lambda}r^2M^\mathrm{I}_{\mathrm{F}} E_\mathrm{F}^\mathrm{I}
+e^{2\Lambda}r^2\left(\eta^\mathrm{I}-\sigma^\mathrm{I}\right)\left(E_\mathrm{K}^\mathrm{I}+2E_\mathrm{H}^\mathrm{I}\right) \right] ,
\end{align}
\begin{align}
\psi^\mathrm{I} =& \frac{\sigma^\mathrm{I}}{2N^\mathrm{I}}
\left[\frac{4e^{2\Lambda-2\Phi}r^2\left(\eta^\mathrm{I}-\sigma^\mathrm{I}\right)}
{\eta^\mathrm{I}}\frac{\partial^2}{\partial t^2}Z^\mathrm{I}\hspace{0.1mm}^{(+)}
+2r\tau^\mathrm{I}\frac{\partial }{\partial r}Z^\mathrm{I}\hspace{0.1mm}^{(+)} -M_2^\mathrm{I}Z^\mathrm{I}\hspace{0.1mm}^{(+)} -e^{4\Lambda-2\Phi}r^2\tau^\mathrm{I} E_\mathrm{A}^\mathrm{I} \right. \nonumber \\
& \left. +\frac{2e^{4\Lambda-2\Phi}r^3\left(\eta^\mathrm{I}-\sigma^\mathrm{I}\right)}
{\eta^\mathrm{I}}\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{I}
+2e^{2\Lambda}r^2M^\mathrm{I}_{\mathrm{F}} E_\mathrm{F}^\mathrm{I}
+e^{2\Lambda}r^2\left(\eta^\mathrm{I}-\sigma^\mathrm{I}\right)\left(E_\mathrm{K}^\mathrm{I}+2E_\mathrm{H}^\mathrm{I}\right) \right],
\end{align}
where $M_1^\mathrm{I}$ and $M_2^\mathrm{I}$ are given in Appendix D. Then, the remaining two even-parity gauge invariants $\delta^\mathrm{I}$ and $\epsilon^\mathrm{I}$ can be obtained from \eqs{projEDgaugeinEZ} and \meq{projEFgaugeinEZ}, given respectively by
\begin{equation}
\delta^\mathrm{I} = \frac{1}{-2-e^{2\Lambda}\left(-2+\lambda\right)+4r\Lambda'}
\left(e^{2\Lambda}r^2E_\mathrm{D}^\mathrm{I}+e^{2\Lambda}\lambda r\frac{\partial}{\partial t}\chi^\mathrm{I}-4r\frac{\partial }{\partial t}\psi^\mathrm{I} \right) ,
\end{equation}
\begin{equation}
\epsilon^\mathrm{I} =e^{-2\Lambda+2\Phi}\left[\left(1-r\Lambda'+r\Phi'\right)\chi^\mathrm{I}
-\psi^\mathrm{I}+r\frac{\partial }{\partial r}\chi^\mathrm{I} +\frac{1}{2}e^{2\Lambda}r^2E_\mathrm{F}^\mathrm{I} \right] .
\end{equation}
From \eq{gaugeinvtometricEZ}, the even-parity metric perturbation components $\mathrm{A}$, $\mathrm{D}$, $\mathrm{H}$, $\mathrm{K}$ could be read out directly.

\subsubsection{\textbf{Even-parity Perturbations and the RW gauge}}

Similar to the development provided in the previous subsection, using the field equations $E_\mathrm{A}^\mathrm{II}$ to $E_\mathrm{K}^\mathrm{II}$ in Appendix C, one can eliminate the gauge invariants $\delta^\mathrm{II}$ and $\epsilon^\mathrm{II}$, and then obtain the following coupled equations
\begin{align}
\label{prchiRW}
\frac{\partial }{\partial r}\chi^\mathrm{II} =& \frac{2e^{2\Lambda-2\Phi}r}{\tau^\mathrm{II}\eta^{\mathrm{II}}}
\left(\sigma^\mathrm{II}\frac{\partial^2}{\partial t^2}\chi^\mathrm{II}-2\frac{\partial^2}{\partial t^2}\psi^\mathrm{II}
+\frac{1}{2}e^{2\Lambda}r\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{II}\right)
+\frac{\gamma^\mathrm{II}}{r \tau^\mathrm{II}}\chi^\mathrm{II}+\frac{\rho^\mathrm{II}}{r \tau^\mathrm{II}}\psi^\mathrm{II} \nonumber \\
& -\frac{e^{2\Lambda}r}{2\tau^\mathrm{II}}(e^{2\Lambda}\lambda+2r\Phi'-2)E_\mathrm{F}^\mathrm{II}
+\frac{e^{2\Lambda}r}{2\tau^\mathrm{II}}\left(2E_\mathrm{H}^\mathrm{II}+E_\mathrm{K}^\mathrm{II}\right) ,
\end{align}
\begin{align}
\label{prpsiRW}
\frac{\partial }{\partial r}\psi^\mathrm{II} =& \frac{e^{2\Lambda-2\Phi}r}{\tau^\mathrm{II}}
\left(\sigma^\mathrm{II}\frac{\partial^2}{\partial t^2}\chi^\mathrm{II}-2\frac{\partial^2}{\partial t^2}\psi^\mathrm{II}
+\frac{1}{2}e^{2\Lambda}r\frac{\partial }{\partial t}E_\mathrm{D}^\mathrm{II}\right)
+\frac{\mu^\mathrm{II}}{r \tau^\mathrm{II}}\chi^\mathrm{II}+\frac{\nu^\mathrm{II}}{r \tau^\mathrm{II}}\psi \nonumber \\
& +\frac{e^{2\Lambda}r}{4\tau^\mathrm{II}}\left(\kappa^\mathrm{II}E_\mathrm{F}^\mathrm{II}
+2\eta^\mathrm{II}E_\mathrm{H}^\mathrm{II}+\eta^\mathrm{II}E_\mathrm{K}^\mathrm{II}\right)-\frac{1}{4}e^{4\Lambda-2\Phi}rE_\mathrm{A}^\mathrm{II} ,
\end{align}
where the parameters $\sigma^\mathrm{II}$, $\tau^\mathrm{II}$, $\eta^\mathrm{II}$,  $\kappa^\mathrm{II}$, $\rho^\mathrm{II}$, $\mu^\mathrm{II}$, $\nu^\mathrm{II}$ depend only on the background, and the explicit expressions of them can be found in Appendix D.

We find that the master variable can be similarly constructed as
\begin{eqnarray}
\label{evenmastervRW}
Z^\mathrm{II}\hspace{0.1mm}^{(+)}=\sigma^\mathrm{II}\chi^\mathrm{II}-2\psi^\mathrm{II} ,
\end{eqnarray}
and then the master equation under the RW gauge is given by
\begin{equation}
\label{evenmasterRW}
\left[\frac{2e^{2\Lambda-2\Phi}r}{N^{\mathrm{II}}\eta^{\mathrm{II}}}
\left(4r^2(\Lambda'+\Phi')\sigma^{\mathrm{II}}\frac{\partial^3}{\partial t^2 \partial r}
+\frac{N_\mathrm{T}^\mathrm{II}}{N^\mathrm{II}\eta^{\mathrm{II}}\tau^\mathrm{II}}\frac{\partial^2}{\partial t^2}\right)
-\frac{2r\sigma^{\mathrm{II}}\tau^\mathrm{II}}{N^{\mathrm{II}}}\frac{\partial^2}{\partial r^2}+\frac{N_\mathrm{R}^\mathrm{II}}{\left({N^\mathrm{II}}\right)^2}\frac{\partial}{\partial r}
+\frac{N_\mathrm{Z}^\mathrm{II}}{r\left({N^\mathrm{II}}\right)^2\tau^\mathrm{II}}\right]Z^\mathrm{II}\hspace{0.1mm}^{(+)}
=S_{\text{even}}^\mathrm{II},
\end{equation}
with the source term given by
\begin{align}
\label{evensourceRW}
S_{\text{even}}^\mathrm{II} =& \frac{e^{4\Lambda-2\Phi}r}{N^{\mathrm{II}}} \left[\frac{N_\mathrm{A}^\mathrm{II}}{2N^\mathrm{II}}E_\mathrm{A}^\mathrm{II}
-r\sigma^{\mathrm{II}}\tau^\mathrm{II}\frac{\partial}{\partial r}E_\mathrm{A}^\mathrm{II}
+\frac{rN_\mathrm{D}^\mathrm{II}}{N^\mathrm{II}\left(\eta^{\mathrm{II}}\right)^2\tau^\mathrm{II}}\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{II}
-\frac{4r^3(\Lambda'+\Phi')\sigma^{\mathrm{II}}}{\eta^{\mathrm{II}}}\frac{\partial^2}{\partial t\partial r}E_\mathrm{D}^\mathrm{II} \right]
-\frac{2e^{2\Lambda}r^2 M^{\mathrm{II}}_\mathrm{F}\sigma^{\mathrm{II}}}{N^{\mathrm{II}}}\frac{\partial}{\partial r}
E^{\mathrm{II}}_{\mathrm{F}}\nonumber \\
& +\frac{e^{2\Lambda}r}{\left(N^{\mathrm{II}}\right)^2\tau^{\mathrm{II}}} \left[\frac{1}{2}N_\mathrm{F}^\mathrm{II}E_\mathrm{F}^\mathrm{II}
+N_\mathrm{HK}^\mathrm{II}\left(E_\mathrm{H}^\mathrm{II}+\frac{1}{2}E_\mathrm{K}^\mathrm{II}\right) \right]-\frac{2e^{2\Lambda}r^3\left(\Lambda
'+\Phi'\right)\sigma^{\mathrm{II}}}{N^{\mathrm{II}}}\left(2\frac{\partial}{\partial r}E^{\mathrm{II}}_{\mathrm{H}}+\frac{\partial}{\partial r}E^{\mathrm{II}}_{\mathrm{K}}\right) ,
\end{align}
where $N^\mathrm{II}$, $N_\mathrm{T}^\mathrm{II}$, $N_\mathrm{R}^\mathrm{II}$, $N_\mathrm{Z}^\mathrm{II}$, $N_\mathrm{A}^\mathrm{II}$, $N_\mathrm{D}^\mathrm{II}$, $M^{\mathrm{II}}_\mathrm{F}$, $N_\mathrm{F}^\mathrm{II}$ all depend on the background, which can be found in Appendix D. Similarly to the case under the EZ gauge, the master equation is a third-order equation, and which can be transformed as a second-order differential equation. Note that when the background metric takes the form
\begin{equation}
ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2
\end{equation}
The relation $\Lambda'+\Phi'=0$ makes the third-order terms in \eq{evenmasterRW} vanishes. In the Schwarzschild case, the master equation \eq{evenmasterRW} will also reduce to the result of Zerilli, see Appendix E. Once \eq{evenmasterRW} is solved, one can get $\chi^\mathrm{II}$ and $\psi^\mathrm{II}$ from
\begin{align}
\chi^\mathrm{II} =& \frac{1}{N^\mathrm{II}}
\left[\frac{8e^{2\Lambda-2\Phi}r^3\left(\Lambda'+\Phi'\right)}
{\eta^\mathrm{II}}\frac{\partial^2}{\partial t^2}Z^\mathrm{II}\hspace{0.1mm}^{(+)}
-2r\tau^\mathrm{II}\frac{\partial }{\partial r}Z^\mathrm{II}\hspace{0.1mm}^{(+)} +M_1^\mathrm{II}Z^\mathrm{II}\hspace{0.1mm}^{(+)} +e^{4\Lambda-2\Phi}r^2\tau^\mathrm{II} E_\mathrm{A}^\mathrm{II} \right. \nonumber \\
& \left. +\frac{4e^{4\Lambda-2\Phi}r^4\left(\Lambda'+\Phi'\right)}
{\eta^\mathrm{II}}\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{II}
+2e^{2\Lambda}r^2M^\mathrm{II}_{\mathrm{F}} E_\mathrm{F}^\mathrm{II}
+2e^{2\Lambda}r^3\left(\Lambda'+\Phi'\right)\left(E_\mathrm{K}^\mathrm{II}+2E_\mathrm{H}^\mathrm{II}\right) \right] ,
\end{align}
\begin{align}
\psi^\mathrm{II} =& \frac{\sigma^\mathrm{II}}{2N^\mathrm{II}}
\left[\frac{8e^{2\Lambda-2\Phi}r^3\left(\Lambda'+\Phi'\right)}
{\eta^\mathrm{II}}\frac{\partial^2}{\partial t^2}Z^\mathrm{II}\hspace{0.1mm}^{(+)}
-2r\tau^\mathrm{II}\frac{\partial }{\partial r}Z^\mathrm{II}\hspace{0.1mm}^{(+)} +\frac{M_2^\mathrm{II}}{\sigma^{\mathrm{II}}}Z^\mathrm{II}\hspace{0.1mm}^{(+)} +e^{4\Lambda-2\Phi}r^2\tau^\mathrm{II} E_\mathrm{A}^\mathrm{II} \right. \nonumber \\
& \left. +\frac{4e^{4\Lambda-2\Phi}r^4\left(\Lambda'+\Phi'\right)}
{\eta^\mathrm{II}}\frac{\partial}{\partial t}E_\mathrm{D}^\mathrm{II}
+2e^{2\Lambda}r^2M^\mathrm{II}_{\mathrm{F}} E_\mathrm{F}^\mathrm{II}
+2e^{2\Lambda}r^3\left(\Lambda'+\Phi'\right)\left(E_\mathrm{K}^\mathrm{II}+2E_\mathrm{H}^\mathrm{II}\right) \right],
\end{align}
and the remaining two gauge invariants $\delta^\mathrm{II}$ and $\epsilon^\mathrm{II}$ can be obtained from $E_\mathrm{D}^\mathrm{II}$ and $E_\mathrm{F}^\mathrm{II}$, which are given respectively by
\begin{equation}
\delta^\mathrm{II} = \frac{1}
{-2-e^{2\Lambda}\left(\lambda-2\right)+4r\Lambda'}\left[e^{2\Lambda}r^2E^{\mathrm{II}}_\mathrm{D}+r\left(2+e^{2\Lambda}\left(\lambda-2\right)+4r\Phi'\right)\frac{\partial}{\partial t}\chi^\mathrm{II}-4r\frac{\partial }{\partial t}\psi^\mathrm{II}\right] ,
\end{equation}
\begin{equation}
\epsilon^\mathrm{II} =e^{-2\Lambda+2\Phi}\left[\left(1-r\Lambda'+r\Phi'\right)\chi^\mathrm{II}
-\psi^\mathrm{II}+r\frac{\partial }{\partial r}\chi^\mathrm{II} +\frac{1}{2}e^{2\Lambda}r^2E_\mathrm{F}^\mathrm{II} \right] .
\end{equation}
When $\chi^\mathrm{II}$, $\psi^\mathrm{II}$, $\delta^\mathrm{II}$ and $\epsilon^\mathrm{II}$ are solved, the even-parity metric perturbation components $\mathrm{A}$, $\mathrm{D}$, $\mathrm{E}$, $\mathrm{K}$ are given by
\begin{align}
\mathrm{A} &= 2e^{-2\Lambda+2\Phi}r\Phi'\chi^\mathrm{II}-2\epsilon^\mathrm{II}, & \mathrm{D} &= \delta^\mathrm{II} +r\frac{\partial }{\partial t}\chi^\mathrm{II}, \nonumber \\
\mathrm{E} &= -2e^{-2\Lambda}\chi^\mathrm{II}, & \mathrm{K} &= 2\psi^\mathrm{II}+2(r\Lambda'-1)\chi^\mathrm{II}-2r\frac{\partial }{\partial r}\chi^\mathrm{II}.
\end{align}

Under the RW gauge, the master variable can be written as the combination of the gauge invariants $\chi^\mathrm{II}$ and $\psi^\mathrm{II}$. Note that Zerilli constructed the master variable as the combination of $K_{LM}$ and $R_{LM}$ \cite{Zerilli1970PRL}, which correspond to a combination of $\chi^\mathrm{II}$ and $\delta^\mathrm{II}$ in our paper.

\subsubsection{\textbf{Odd-parity Perturbations}}

For the odd-parity perturbations, the EZ gauge and the RW gauge are identical. So, in the following we shall not distinguish them. We construct the odd-parity master variable as
\begin{equation}
\label{oddvariablel2}
Z^{(-)}=\left(r+r^2\Lambda'+r^2\Phi'\right)\beta-r^2\frac{\partial}{\partial r}\beta+r^2\frac{\partial}{\partial t}\alpha ,
\end{equation}
then the master equation is given by
\begin{equation}
\label{oddmasterl2}
\left\{ -\frac{\partial^2}{\partial t^2}+e^{-2\Lambda+2\Phi}
\left[\frac{\partial^2}{\partial r^2}-\left(\frac{X'}{X}+3\Lambda'+3\Phi'\right)\frac{\partial}{\partial r}+\frac{N_{\text{odd}}}{r^2} \right]\right\} Z^{(-)} = S_{\text{odd}} ,
\end{equation}
with the source term given by
\begin{eqnarray}
\label{oddsourcel2}
S_{\text{odd}} = e^{2\Phi}r^2\left[\left( \frac{1}{r}+\Lambda'-\Phi'-\frac{X'}{X}\right)E_\mathrm{C}
+\frac{\partial }{\partial r}E_\mathrm{C}+\frac{\partial }{\partial t}E_\mathrm{J}\right] ,
\end{eqnarray}
where $X$ and $N_{\text{odd}}$ are functions only depending on the background, and the explicit expressions of which can be found in Appendix D. Once \eq{oddmasterl2} is solved, together with the structure of the odd-parity master variable \eq{oddvariablel2} and the perturbed field equations \eqs{projECgaugein} and \meq{projEJgaugein}, we have
\begin{equation}
\label{reconsalpha}
\alpha = -\frac{e^{2\Lambda-2\Phi}}{X} \left(e^{2\Phi}r^2E_\mathrm{J}+\frac{\partial }{\partial t}Z^{(-)} \right),
\end{equation}
and
\begin{equation}
\label{reconsbeta}
\beta = \frac{1}{X}\left[ e^{2\Lambda}r^2E_\mathrm{C}-\left(\frac{1}{r}-2\Lambda'-2\Phi' \right)Z^{(-)}-\frac{\partial }{\partial r}Z^{(-)} \right].
\end{equation}
Then, the odd-parity metric perturbation components $\mathrm{C}$ and $\mathrm{J}$ can be read off from these expressions.

\subsection{Specific Cases for \texorpdfstring{$l=0,1$}.}

In the previous subsection, we have discussed the construction of master variables and master equations for even-parity and odd-parity perturbations for the $l\geq 2$ cases. However, the decomposition of the metric perturbation would take some other forms for the specific cases of $l=0, 1$. In this section, we investigate how to solve the metric perturbation for $l=0,1$. Note that in this subsection, we use, e.g. $\mathrm{A}_0$ and $\mathrm{A}_1$, to represent the scalar functions in the metric perturbation and the gauge vector for $l=0$ and $l=1$ cases, respectively. And we also use, e.g. $E_{\mathrm{A}}^{(l=0)}$ and $E_{\mathrm{A}}^{(l=1)}$ to represent the projection function $E_{\mathrm{A}}$ for $l=0$ and $l=1$ cases, respectively.

\subsubsection{\textbf{\texorpdfstring{$l=0$}. case}}

For the special case $l=0$, there remains one scalar spherical harmonic function $Y^{00}=\frac{1}{2\sqrt{\pi}}$, which leads the metric perturbation to
\begin{equation}
h^{(l=0)}_{ab} = \frac{1}{2\sqrt{\pi}}(\mathrm{A}_0v_av_b+2\mathrm{D}_0v_{(a}n_{b)}+\mathrm{E}_0\sigma_{ab}+\mathrm{K}_0n_ab_b),
\end{equation}
and the gauge vector takes the form
\begin{equation}
\xi^{(l=0)}_a = \frac{1}{2\sqrt{\pi}}(\mathrm{P}_0v_a+\mathrm{R}_0n_a).
\end{equation}
In the $l=0$ case, we only have four metric perturbation components, and all of them are even-parity. Under the gauge transformation, the metric perturbation will be transformed as
\begin{align}
\label{DeltaAKl0}
\Delta \mathrm{A}_0 &= -2\frac{\partial }{\partial t}\mathrm{P}_0-2e^{-2\Lambda+2\Phi}\Phi'\mathrm{R}_0,
& \Delta \mathrm{D}_0 &= \left(\frac{\partial }{\partial r}-2\Phi' \right) \mathrm{P}_0 -\frac{\partial }{\partial t}\mathrm{R}_0, \nonumber \\
\Delta \mathrm{E}_0 &= 2\frac{e^{-2\Lambda}}{r}\mathrm{R}_0,
& \Delta \mathrm{K}_0 &= \left(2\frac{\partial }{\partial r}-2\Lambda' \right)\mathrm{R}_0 .
\end{align}
The structure of $\Delta \mathrm{A}_0$, $\Delta \mathrm{D}_0$ and $\Delta \mathrm{K}_0$ are the same as the $l\geq 2$ cases, but $\Delta \mathrm{E}_0$ is different from the expression given in \eq{DeltaAK} since $\mathrm{S}$ does not exist for $l=0$. From \eq{DeltaAKl0}, we can construct two gauge invariants
\begin{align}
\psi_0 &= \frac{1}{2}\mathrm{K}_0-\frac{r}{2}e^{2\Lambda}\Lambda'\mathrm{E}_0-\frac{1}{2}e^{2\Lambda}\mathrm{E}_0-\frac{r}{2}e^{2\Lambda}\frac{\partial}{\partial r}\mathrm{E}_0 , \nonumber \\
o_0 &= \frac{1}{2}\frac{\partial }{\partial r}\mathrm{A}_0-\Phi'\mathrm{A}_0+\frac{\partial }{\partial t}\mathrm{D}_0+\frac{1}{2}e^{2\Phi}(\Phi'+r\Phi'')\mathrm{E}_0+\frac{1}{2}e^{2\Phi}r\Phi'\frac{\partial}{\partial r}\mathrm{E}_0
+\frac{1}{2}e^{2\Lambda}r\frac{\partial^2}{\partial t^2}\mathrm{E}_0 .
\end{align}

Choosing the gauge $\mathrm{D}_0=\mathrm{E}_0=0$, we find that $E_\mathrm{A}^{(l=0)}$, $E_\mathrm{D}^{(l=0)}$, $E_\mathrm{E}^{(l=0)}$ and $E_\mathrm{K}^{(l=0)}$ are given by
\begin{align}
E_\mathrm{A}^{(l=0)} =& 2r^{-2}e^{-2\Lambda}(-1+e^{2\Lambda}+2r\Lambda')\mathrm{A}_0  +2r^{-2}e^{-4\Lambda+2\Phi}(4r\Lambda'-1)\mathrm{K}_0-2r^{-1}e^{-4\Lambda+2\Phi}\frac{\partial }{\partial r}\mathrm{K}_0 , \nonumber \\
E_\mathrm{D}^{(l=0)} =& 2r^{-1}e^{-2\Lambda}\frac{\partial }{\partial t}\mathrm{K}_0 , \nonumber \\
E_\mathrm{E}^{(l=0)} =& 2r^{-1}e^{-2\Lambda-2\Phi}(r\Lambda'\Phi'-r\Phi''-\Phi')\mathrm{A}_0-r^{-1}e^{-2\Lambda-2\Phi}(r\Lambda'+2r\Phi'-1)\frac{\partial }{\partial r}\mathrm{A}_0+e^{-2\Lambda-2\Phi}\frac{\partial^2}{\partial r^2}\mathrm{A}_0 \nonumber \\
& +2r^{-1}e^{-4\Lambda}(\Phi'+r{\Phi'}^2+r\Phi''-2r\Lambda'\Phi'-2\Lambda')\mathrm{K}_0+r^{-1}e^{-4\Lambda}(r\Phi'+1)\frac{\partial }{\partial r}\mathrm{K}_0
+e^{-2\Lambda-2\Phi}\frac{\partial^2}{\partial t^2}\mathrm{K}_0 , \nonumber \\
E_\mathrm{K}^{(l=0)} =& -4r^{-1}e^{-2\Phi}\Phi'\mathrm{A}_0+2r^{-1}e^{-2\Phi}\frac{\partial }{\partial r}\mathrm{A}_0
+2r^{-2}\mathrm{K}_0 .
\end{align}
The gauge invariants $\psi_0$ and $o_0$ can be constructed as
\begin{equation}
\label{psil0}
\psi_0 = \frac{1}{2}\mathrm{K}_0, \quad\quad o_0= \frac{1}{2}\frac{\partial }{\partial r}\mathrm{A}_0-\Phi'\mathrm{A}_0.
\end{equation}
And then
$E_\mathrm{D}^{(l=0)}$, $E_\mathrm{E}^{(l=0)}$ and $E_\mathrm{K}^{(l=0)}$ can be written as
\begin{align}
E_\mathrm{D}^{(l=0)} =& 4r^{-1}e^{-2\Lambda}\frac{\partial}{\partial t}\psi_0, \label{EDl0} \\
E_\mathrm{E}^{(l=0)} =& -2r^{-1}e^{-2\Lambda-2\Phi}(r\Lambda'-1)o_0+2e^{-2\Lambda-2\Phi}\frac{\partial}{\partial r}o_0+4r^{-1}e^{-4\Lambda}(\Phi'-2\Lambda'+r{\Phi'}^2+r\Phi''-2r\Lambda'\Phi')\psi_0 \nonumber \\
& +2r^{-1}e^{-4\Lambda}(r\Phi'+1)\frac{\partial }{\partial r}\psi_0+2e^{-2\Lambda-2\Phi}\frac{\partial^2}{\partial t^2}\psi_0, \label{EEl0}  \\
E_\mathrm{K}^{(l=0)} =& 4r^{-1}e^{-2\Phi}o_0+4r^{-2}\psi_0. \label{EKl0}
\end{align}

From \eq{EKl0}, we have
\begin{equation}
\label{o0l0}
o_0=\frac{1}{4}e^{2\Phi}\left(rE_\mathrm{K}^{(l=0)}-\frac{4}{r}\psi_0\right) .
\end{equation}
Substituting \eqs{EDl0} and \meq{o0l0} into \eq{EEl0}, we find that $\psi_0$ satisfied
\begin{equation}
\label{psi02}
\left(\iota_0 \frac{\partial }{\partial r}-\sigma_0\right)\psi_0=\frac{r}{2}e^{4\Lambda}S^{(l=0)},
\end{equation}
where $\iota_0$ and $\sigma_0$ are functions depend on background, which can be found in Appendix D, and $S^{(l=0)}$ is the source term,
\begin{equation}
S^{(l=0)}=\frac{1}{2}e^{-2\Phi}r\frac{\partial }{\partial t}E^{(l=0)}_{\mathrm{D}}-E^{(l=0)}_{\mathrm{E}}
+\frac{1}{2}e^{-2\Lambda}\left[r\frac{\partial }{\partial r}E_{\mathrm{K}}^{(l=0)}+\left(2-r\Lambda'+2r\Phi'\right)E_{\mathrm{K}}^{(l=0)}\right].
\end{equation}

\subsubsection{\textbf{\texorpdfstring{$l=1$}. case}}

For the case $l=1$, the tensor harmonic basis $T_{ab}^{E2,1m}$ and $T_{ab}^{B2,1m}$ vanish, hence the scalar functions $\mathrm{F}_1$ and $\mathrm{G}_1$ would no longer exist. The four components of the gauge vector $\xi_a$ imply that there are only three gauge invariants for even-parity and one gauge invariant for odd-parity. The remaining eight projections of $2\nabla_{(a}\xi_{b)}$ are the same as for $l\geq2$ cases.

First, we investigate the even-parity sector. As we explain for $l\geq2$ cases, one should first determine the function $\mathrm{S}$ of the gauge vector $\xi_a$ to make $\mathrm{F}$ vanish under the gauge transformation. However, when $l=1$, the lack of $\mathrm{F}_1$ prevents us from constructing the gauge invariants as the cases for $l\geq2$. The even-parity $l=1$ metric perturbations are related to the linear momentum of the system \cite{Zerilli1970PRD}.
Note that if we take the gauge choice $\mathrm{B}_1=\mathrm{E}_1=\mathrm{H}_1=0$, which was introduced by Zerilli in the Schwarzschild spacetime, the relation between $E_\mathrm{A}$ and metric components would no longer be a simple relationship. However, taking the gauge
\begin{equation}
\mathrm{A}_1=\mathrm{E}_1=\mathrm{H}_1=0,
\end{equation}
we find that $E_\mathrm{A}^{(l=1)}$ and the metric perturbed component $\mathrm{K_1}$ have a simple relation. In particular, we find
\begin{align}
E_\mathrm{A}^{(l=1)} =& -2r^{-1}e^{-4\Lambda+2\Phi}\left(\frac{\partial }{\partial r}\mathrm{K}_1+r^{-1}(1+e^{2\Lambda}-4r\Lambda')\mathrm{K}_1 \right) , \label{EAl1} \\
E_\mathrm{D}^{(l=1)}  =& 2r^{-2}(1-2r\Phi')\mathrm{B}_1+2r^{-1}\frac{\partial}{\partial r}\mathrm{B}_1+ 2r^{-1}e^{-2\Lambda} \frac{\partial }{\partial t}\mathrm{K}_1-2r^{-2}e^{-2\Lambda}(1-2r\Lambda')\mathrm{D}_1, \label{EDl1} \\
E_\mathrm{K}^{(l=1)} +4E_\mathrm{H}^{(l=1)}
=& 2r^{-2}e^{-2\Lambda}(-2+e^{2\Lambda}-2r\Phi')\mathrm{K}_1  +4r^{-1}e^{-2\Phi}(1-e^{2\Lambda})\frac{\partial}{\partial t}\mathrm{B}_1-4e^{-2\Phi}\frac{\partial^2}{\partial t\partial r}\mathrm{B}_1. \label{EKl1}
\end{align}
From \eq{EAl1}, the metric perturbation function $\mathrm{K}_1$ can be found. This solution can be used in \eq{EKl1} to solve $\mathrm{B}_1$, and then from \eq{EDl1}, $\mathrm{D}_1$ could also be solved. The remaining quantities, such as $E_\mathrm{B}$ or $E_\mathrm{E}$, can be used to check the consistency of the solutions.

Then we investigate the odd-parity sector. For $l\geq2$ cases, one should determine the function $\mathrm{Q}$ of the gauge vector $\xi_a$ to make $\mathrm{G}$ vanish under the gauge transformation. For $l=1$ case, $\mathrm{G}_1$ no longer exists, which means that we can not construct the gauge invariants $\alpha$ and $\beta$ as in the $l\geq 2$ cases. However, from the components of the projection of $2\nabla_{(a}\xi_{b)}$
\begin{align}
\Delta \mathrm{C}_1 =& -\frac{\partial }{\partial t}\mathrm{Q},\\
\Delta \mathrm{J}_1 =& \left(\frac{\partial}{\partial r}-\frac{1}{r}-\Phi'-\Lambda'\right)\mathrm{Q},
\end{align}
we can construct a gauge invariant property $\alpha_1$ as
\begin{equation}
\label{mastervoddl1}
\alpha_1 =r^2\frac{\partial }{\partial t}\mathrm{J}_1+r^2\frac{\partial }{\partial r}\mathrm{C}_1-r\left(1+r\Phi'+r\Lambda' \right)\mathrm{C}_1 .
\end{equation}
Then, for $l=1$, the odd-parity of the projection of EFEs are
\begin{align}
E_\mathrm{C}^{(l=1)}  =
& -r^{-1}e^{-2\Lambda}(-2+3r\Lambda'+3r\Phi')\frac{\partial }{\partial r}\mathrm{C}_1+e^{-2\Lambda}\frac{\partial^2 }{\partial r^2}\mathrm{C}_1
-r^{-1}e^{-2\Lambda}(-3+2r\Lambda'+2r\Phi')\frac{\partial }{\partial t}\mathrm{J}_1+e^{-2\Lambda}\frac{\partial^2 }{\partial t\partial r}\mathrm{J}_1 , \label{ECl1}\nonumber \\
&-r^{-2}e^{-2\Lambda}\left(
-2+2r^2{\Lambda'}^2-3r\Phi'+r\Lambda\left(1+6r\Phi' \right)-r^2\Lambda''-3r^2\Phi''\right)\mathrm{C}_1 \\
E_\mathrm{J}^{(l=1)}  =& -r^{-1}e^{-2\Lambda}\left(
2\Lambda'\left(1+r\Phi'\right)-2(\Phi'+r{\Phi'}^2+r\Phi'')
\right)\mathrm{J}_1  -e^{-2\Phi}\frac{\partial^2}{\partial t^2}\mathrm{J}_1\nonumber \\
&+ r^{-1}e^{-2\Phi}(1+r\Lambda'+r\Phi')\frac{\partial }{\partial t}\mathrm{C}_1-e^{-2\Phi}\frac{\partial^2}{\partial t\partial r}\mathrm{C}_1 \label{EJl1}.
\end{align}

Together with \eq{mastervoddl1}, \eq{ECl1} and \eq{EJl1} can be decoupled, and the master equation for the odd-parity perturbation is given by
\begin{equation}
\label{mastereqoddl1}
\left\{ -\frac{\partial^2}{\partial t^2}+e^{-2\Lambda+2\Phi}
\left[\frac{\partial^2}{\partial r^2}-\left(\frac{X_1'}{X_1}+3\Lambda'+3\Phi'\right)\frac{\partial}{\partial r}
+\frac{N_{\text{odd}}^{(l=1)}}{r^2} \right] \right\} \alpha_1=S_{\text{odd}}^{(l=1)} ,
\end{equation}
where $S_{\text{odd}}^{(l=1)}$ is the source term,
\begin{equation}
S_{\text{odd}}^{(l=1)}=e^{2\Phi}r^2\left[\left(\frac{1}{r}+\Lambda'-\Phi'-\frac{X_1'}{X_1}\right)E_\mathrm{C}^{(l=1)}
+\frac{\partial}{\partial r}E_\mathrm{C}^{(l=1)}  +\frac{\partial}{\partial t}E_\mathrm{J}^{(l=1)}   \right].
\end{equation}
In the above equations, $X_1$ and $N_{\text{odd}}^{(l=1)}$ are all depend only on the background, which can be found in Appendix D. Once \eq{mastereqoddl1} is solved, the metric perturbation components $\mathrm{C}_1$ and $\mathrm{J}_1$ can be determined by
\begin{align}
\mathrm{C}_1 =& -\frac{1}{X_1}\left[e^{2\Lambda}r^2E_\mathrm{C}^{(l=1)} -\left(\frac{1}{r}-2\Lambda'-2\Phi'\right)\alpha_1-\frac{\partial }{\partial r}\alpha_1 \right], \label{C1} \\
\mathrm{J}_1 =& -\frac{e^{2\Lambda-2\Phi}}{X_1}\left(e^{2\Phi}r^2E_\mathrm{J}^{(l=1)} +\frac{\partial }{\partial t}\alpha_1 \right). \label{J1}
\end{align}

\section{A Point Particle as the Source}

In this section, we present a simple example that a small object moves along a circular orbit around the center of a spherically symmetric spacetime. We analyse the solutions for $l=0,1$ in this section. For $l\geq 2$ cases, we only provide a general outline.

Assuming that the small object moves along the worldline $z(\tau)$ with mass $\mu$ and four velocity $u_a$, the stress-energy tensor of this point particle takes the form \cite{Poisson2011}
\begin{equation}
T_{ab}=\int\frac{\mu u_au_b}{\sqrt{-g^{(0)}}}\delta^4[x-z(\tau)]d\tau ,
\end{equation}
where $\tau$ is the proper time, $g^{(0)}$ and $\delta^4$ are the determinant of the background and the four-dimensional Dirac delta function, respectively. Generalizing the standard analysis from the textbook \cite{Waldbook}, the time-like geodesics in general spherically symmetric spacetime is
\begin{equation}
-1 =  g_{ab}u^au^b = -e^{2\Phi}{\dot t}^2+e^{2\Lambda}{\dot r}^2+r^2{\dot\varphi}^2.
\end{equation}
Using the static Killing field $\xi^a=(\partial/\partial t)^a$ and the rotational Killing field $\psi^a=(\partial/\partial\varphi)^a$, two conserved quantities $\mathcal{E}$ and $L$ can be defined as
\begin{equation}
\mathcal{E} = -g_{ab}\xi^a u^b = e^{2\Phi}\dot t, ~~~~
L = g_{ab}\psi^au^b = r^2 \dot\varphi .
\end{equation}
Then the geodesic equation reads
\begin{equation}
\frac{1}{2}{\dot r}^2+\frac{1}{2}e^{-2\Lambda}\left(\frac{L^2}{r^2}+1\right)=\frac{1}{2}e^{-2\Lambda-2\Phi}\mathcal{E}^2 .
\end{equation}

Considering that the particle moves along a circular orbit with radio $R$, which is determined by $\partial V/\partial r=0$, the four velocity of the particle can be written as
\begin{equation}
u_a= (-\mathcal{E},0,0,L),
\end{equation}
and now $\mathcal{E}$ and $L$ represent the energy and angular momentum of the particle, given respectively by,
\begin{equation}
\label{exampleEL}
\mathcal{E} = \frac{e^{\Phi|_R}}{\sqrt{R\Lambda'|_R+1}},  ~~~~ L = R\sqrt{\frac{-R\Lambda'|_R}{R\Lambda'|_R+1}}.
\end{equation}
where $|_R$ denotes the corresponding function taking its value along the orbit. The orbital frequency can also be defined as
\begin{equation}
\Omega^2=\left(\frac{u^\varphi}{u^t}\right)^2 = -\frac{e^{2\Phi}\Lambda'|_R}{R} ,
\end{equation}
which gives two useful relations
\begin{equation}
\label{energyangular}
\mathcal{E} = \frac{e^{2\Phi|_R}}{\Omega R^2}L, ~~~~~~~~ \Omega L = -R\Lambda'|_R\cdot\mathcal{E}.
\end{equation}

Projecting the stress-energy tensor $T_{ab}$ to the harmonic basis, one can obtain $E_\mathrm{A}$-$E_\mathrm{K}$. The results reveal that $E_\mathrm{D}$, $E_\mathrm{H}$, $E_\mathrm{J}$ and $E_\mathrm{K}$ vanish automatically, and the non-vanishing projections of the stress-energy tensor are given as follows. For $l\geq 0$, we have
\begin{align}
E_\mathrm{A}^{lm} =& -16\pi e^{-\Lambda+\Phi}\frac{\mu \mathcal{E}}{r^2}\delta(r-R)Y_{lm}^*\left(\frac{\pi}{2},\Omega t\right),  \label{exampleEA}\\
E_\mathrm{E}^{lm} =& -8\pi e^{-\Lambda-\Phi}\frac{\mu L\Omega R^2}{r^4}\delta (r-R)Y_{lm}^*\left(\frac{\pi}{2},\Omega t\right), \label{exampleEE}
\end{align}
where $E_\mathrm{A}$ and $E_\mathrm{E}$ satisfy the relation $E_\mathrm{A}=-\frac{2e^{2\Phi}r^2}{R^3\Lambda'}E_\mathrm{E}$. For $l\geq 1$, we find
\begin{align}
E_\mathrm{B}^{lm} =& -\frac{16\pi}{l(l+1)}e^{-\Lambda-\Phi}\frac{\mu \mathcal{E}\Omega R^2}{r^3}\delta (r-R)
\left(\frac{\partial }{\partial \varphi}Y_{lm}^*(\theta,\varphi)\right)\Big|_{\theta=\frac{\pi}{2},\varphi=\Omega t}, \label{exampleEB}\\
E_\mathrm{C}^{lm} =& -\frac{16\pi}{l(l+1)}\frac{\mu \mathcal{E} \Omega R^2}{r^3}\delta(r-R)
\left(\frac{\partial }{\partial \theta}Y^*_{lm}(\theta,\varphi)\right)\Big|_{\theta=\frac{\pi}{2},\varphi=\Omega t} , \label{exampleEC}
\end{align}
and for $l\geq 2$,
\begin{align}
E_\mathrm{F}^{lm} =& \frac{16\pi(l-2)!}{(l+2)!}e^{-\Lambda-\Phi}\frac{\mu L \Omega R^2}{r^4}\delta(r-R)
\left[2\frac{\partial^2}{\partial \theta^2} Y^*_{lm}(\theta,\varphi) +l(l+1)Y_{lm}^*(\theta,\varphi)\right]\Big|_{\theta=\frac{\pi}{2},\varphi=\Omega t}, \label{exampleEF} \\
E_\mathrm{G}^{lm} =& -\frac{32\pi (l-2)!}{(l+2)!}\frac{\mu L \Omega R^2}{r^4}\delta (r-R)
\left(\frac{\partial^2}{\partial \theta \partial \varphi}Y^*_{l,m}(\theta,\varphi)\right)\Big|_{\theta=\frac{\pi}{2},\varphi=\Omega t}. \label{exampleEG}
\end{align}

\subsection{Perturbations for \texorpdfstring{$l=0$}.}

For $l=0$, substituting \eqs{exampleEE} into \eq{psi02}, we have
\begin{equation}
\left(\iota_0 \frac{\partial }{\partial r}-\sigma_0\right)\psi_0= -\frac{r}{2}e^{4\Lambda}E_\mathrm{E}^{00}.
\end{equation}
Using the standard method to solve the above equation, one can obtain
\begin{align}
\psi_0 =& -\exp\left(\int^r\frac{\sigma_0(r)}{\iota_0(r)}dr\right)
\int^{r}\left[\frac{r'e^{4\Lambda(r')}}{2\iota_0(r')}\text{exp}\left(-\int^{r'}\frac{\sigma_0(r'')}{\iota_0(r'')}dr'' \right) E^{00}_\mathrm{E}(r') \right]dr'  \nonumber \\
=& -\exp\left({\int^r\frac{\sigma_0}{\iota_0}dr-\int^R\frac{\sigma_0}{\iota_0}dr }\right)
\frac{R e^{4\Lambda|_R}}{2\iota_0|_R}\bar{E}^{00}_\mathrm{E}\Theta (r-R) \nonumber \\
=& -\exp\left(\int^r_R\frac{\sigma_0}{\iota_0}dr\right)
\frac{R e^{4\Lambda|_R}}{2\iota_0|_R}\bar{E}^{00}_\mathrm{E}\Theta (r-R) ,
\end{align}
where $\Theta(r-R)$ is the unit step function, and $\bar{E}_\mathrm{E}^{00}$ denote the evaluated coefficient given by
\begin{equation}
\bar{E}^{00}_{\mathrm{E}}=-4\sqrt{\pi}e^{-\Lambda|_R-\Phi|_R}\frac{\mu L \Omega}{R^2}= 4\sqrt{\pi}e^{-\Lambda|_R-\Phi|_R}\frac{\mu \Lambda'|_R \cdot\mathcal{E}}{R}.
\end{equation}
From \eq{psil0}, we have
\begin{equation}
\mathrm{K}_0=2\psi_0=-4\sqrt{\pi}\frac{\Lambda'|_R \cdot \mu\mathcal{E}}{\iota_0|_R}
\exp\left(\int_R^r\frac{\sigma_0}{\iota_0}dr+3\Lambda|_R-\Phi|_R\right)\Theta(r-R).
\end{equation}
Using \eq{o0l0}, we can get $o_0$, and then solve \eq{psil0} to obtain $\mathrm{A}_0$
\begin{equation}
\mathrm{A}_0 = 4\sqrt{\pi}e^{2\Phi}\frac{\Lambda'|_R \cdot \mu \mathcal{E}}{\iota_0|_R}\cdot \mathcal{A}\cdot
\exp\left[-\int^R\frac{\sigma_0}{\iota_0}dr+3\Lambda|_R-\Phi|_R\right]\Theta(r-R) ,
\end{equation}
where
\begin{equation}
\mathcal{A}=\int r^{-1}
\exp\left[ \int^r \frac{\sigma_0(r')}{\iota_0(r')}dr'\right]dr.
\end{equation}

With $\mathrm{A}_0$ and $\mathrm{K}_0$ in hand, the metric perturbation can be directly written out. Since both $\mathrm{A}_0$ and $\mathrm{K}_0$ contain step function, it is obvious that inside the orbit, i.e. $r<R$, the perturbation would vanishes. While outside the orbit, i.e. $r>R$, the perturbed metric can be given by
\begin{align}
h_{ab}^{00} =& \frac{1}{2\sqrt\pi}(\mathrm{A}_0 v_av_b+\mathrm{K}_0n_an_b) \nonumber \\
=& 2e^{2\Phi}\frac{\Lambda'|_R \cdot \mu\mathcal{E}}{\iota_0|_R} \cdot \mathcal{A}\cdot
\exp\left[ -\int^R\frac{\sigma_0}{\iota_0}dr+3\Lambda|_R-\Phi|_R\right] v_av_b \nonumber \\
& -2\frac{ \Lambda'|_R \cdot \mu\mathcal{E}}{ \iota_0|_R}
\exp\left[\int^r_R\frac{\sigma_0}{\iota_0}dr+3\Lambda|_R-\Phi|_R \right]n_an_b.
\end{align}
Unlike the Schwarzschild spacetime, there is no clear definition of the mass $M$ for general spherically symmetric spacetimes. It is obvious that the perturbation for $l=0$ only affects $g_{tt}$ and $g_{rr}$. In the Schwarzschild spacetime, this perturbation is equivalent to adding an extra mass $\delta m$ to the system \cite{Zerilli1970PRD}.

\subsection{Perturbations for \texorpdfstring{$l=1$}.}

\subsubsection{\textbf{Even-parity Perturbations}}

For the even-parity perturbations with $l=1$, we find that we can set directly $\mathrm{A}_1=\mathrm{E}_1=\mathrm{H}_1=0$. From \eq{EAl1}, we find that $\mathrm{K}_1$ can be determined by $E_\mathrm{A}$. Note that $Y_{1m}^*(\frac{\pi}{2},\varphi)$ would vanish for $m=0$, hence
\begin{equation}
\mathrm{K}_1= -8\pi R\frac{\mu\mathcal{E}}{r^2}Y_{1,\pm1}^*\left(\frac{\pi}{2},\Omega t\right)
\exp\left({-\int_R^r\frac{1+e^{2\Lambda}-4r\Lambda'}{r}dr+3\Lambda|_R-\Phi|_R}\right)\Theta(r-R).
\end{equation}
Substituting $\mathrm{K}_1$ into \eq{EKl1}, together with $E_\mathrm{H}=E_\mathrm{K}=0$, we have
\begin{equation}
\label{exampleKl1}
\frac{\partial}{\partial r}\left(\frac{\partial}{\partial t}\mathrm{B}_1\right)-r^{-1}(1-e^{2\Lambda})\frac{\partial}{\partial t}\mathrm{B}_1
= \mathcal{K}\mathrm{K}_1 ,
\end{equation}
where
\begin{equation}
\mathcal{K}=\frac{1}{2}r^{-2}e^{2\Phi-2\Lambda}(-2+e^{2\Lambda}-2r\Phi') .
\end{equation}
Hence, $\mathrm{B}_1$ can be solved by \eq{exampleKl1}. Then, considering \eq{EDl1} with $E_\mathrm{D}=0$, in principle, the metric perturbed component $\mathrm{D}_1$ can also be obtained.

\subsubsection{\textbf{Odd-parity Perturbations}}

To study the metric perturbation components $\mathrm{C}_1$ and $\mathrm{J}_1$, we should first solve the master variable $\alpha_1$ from \eq{mastereqoddl1}. Then using \eqs{C1} and \meq{J1}, $\mathrm{C}_1$ and $\mathrm{J}_1$ can be obtained. However, $\mathrm{G}$ automatically vanish for the $l=1$ case, there exists one freedom for the odd-parity. If we impose the gauge $\mathrm{C}_1=0$, the master variable becomes
\begin{equation}
\alpha_1=r^2\frac{\partial}{\partial t}\mathrm{J}_1 .
\end{equation}
Then, \eq{C1} becomes
\begin{equation}
\frac{\partial }{\partial r}\alpha_1+\left(\frac{1}{r}-2\Lambda'-2\Phi' \right)\alpha_1=e^{2\Lambda}r^2E_{\mathrm{C}} .
\end{equation}
The solution of this equation is
\begin{align}
\alpha_1 =& -\frac{16\pi}{l(l+1)}\cdot\frac{\mu\mathcal{E}\Omega R^2}{r} \exp(2\Lambda+2\Phi-2\Phi|_R)
\cdot\left(\frac{\partial }{\partial \theta}Y^*_{lm}(\theta,\varphi)\right)|_{\theta=\frac{\pi}{2},\varphi=\Omega t} \nonumber \\
=& -8\pi \frac{\mu L}{r}e^{2\Lambda+2\Phi}\left(\frac{\partial }{\partial \theta}Y^*_{lm}(\theta,\varphi)\right)|_{\theta=\frac{\pi}{2},\varphi=\Omega t} ,
\end{align}
where we have usedamour the relation \eq{energyangular} and the fact $l=1$. From $\alpha_1$, one can solve $\mathrm{J}_1$ with a undetermined function that only depend on $r$. However, the metric perturbation component $\mathrm{J}_1$ yields a contribution to $h^{1m}_{r\theta}$ and $h^{1m}_{r\varphi}$.

Next, we impose the gauge $\mathrm{J}_1=0$, and now the gauge invariant $\alpha_1$ becomes
\begin{equation}
\alpha_1=r^2\frac{\partial}{\partial r}\mathrm{C}_1-r\left(1+r\Lambda'+r\Phi'\right)\mathrm{C}_1 .
\end{equation}
Substituting this relation into \eqs{C1} and \meq{J1}, we know that $\alpha_1$ should only be a function of $r$, and $\mathrm{C}_1$ satisfies the second-order differential equation
\begin{align}
& r^2\frac{\partial^2 }{\partial r^2}\mathrm{C}_1+r\left(2-3r\Lambda'-3r\Phi'\right)\frac{\partial }{\partial r}\mathrm{C}_1 - \left(
2-2r^2{\Lambda'}^2+3r\Phi'-r\Lambda'(1+6r\Phi')+r^2\Lambda''+3r^2\Phi''
\right)\mathrm{C}_1 =e^{2\Lambda}r^2E_\mathrm{C} ,
\end{align}
which is \eq{ECl1} with $\mathrm{J}_1=0$. Considering that $E_\mathrm{C}$ contains the delta function, we predict that the solution of $\mathrm{C}_1$ contain the step function $\Theta(r-R)$, with two arbitrary constants of integration. One can be determined by considering that the background spacetime without
intrinsic angular momentum, and the other can be determined by the fact that perturbation $\mathrm{C}_1$ would be convergent at infinity \cite{Thompson2016}. If the background metric degenerates to the Schwarzschild spacetime, our equation would become Eq. (10.21) of \cite{Thompson2016}, which gives a non-vanishing $h_{t\varphi}^{01}$ outside the circular orbit to describe the adding angular momentum of the system.

\subsection{Perturbations For \texorpdfstring{$l\geq 2$}.}

Generally speaking, for $l\geq 2$ the problem is much more mathematically involved, and normally we have to seek help from numerical computations. From the source term, \eqs{evensourceEZ}, \meq{evensourceRW} or \meq{oddsourcel2}, we know that they depend only on \eqs{exampleEA}-\meq{exampleEG}. All these terms are proportional to $e^{-im\Omega t}$, which implies that our master equations, \eqs{evenmasterEZ}, \meq{evenmasterRW} and \meq{oddmasterl2}, can be written as the second-order ordinary differential equations in $r$ for each $l$ and $m$. And the distributional sources of these differential equations would be vanish everywhere except at the circular orbit radius $r=R$.

An basic outline is as follows. To solve the second-order equations, one can study two regions separately. One region is from the circular orbit radius to infinity, i.e. $r\in(R,\infty)$. In this region, a naturally boundary condition is considering an appropriate radiation at spatial infinity. In the other region there are several different situations. For example, if the background spacetime has an event horizon, then one should determine this region from the event horizon to the circular orbit radius, i.e., $r\in (r_+,R)$. Then solve the differential equation with the boundary condition at the event horizon $r_+$. Another example is that the background spacetime is a perfect fluid star without event horizons, then one should determine the other region from the center of the star to the circular orbit radius, i.e., $r\in (0,R)$. Then solve the differential equation with some boundary conditions at the center of the star. Finally, matching the solutions obtained in the two separate regions properly across the boundary $r=R$, we obtain the perturbations valid over the whole spacetime.

\section{Conclusions and Discussions}

In this paper, we systematically study the gauge invariant perturbations of a general spherically symmetric background. First, we find that, in general spherically symmetric spacetimes, for even-parity, there are several gauge choices. One is the well-known RW gauge, and the other is the EZ gauge. For odd-parity perturbation, only one gauge choice exists, that is, by setting $\mathrm{G}=0$. Then, we mainly focus on the construction of master equations for $l\geq 2$. For even-parity perturbation,  under the EZ gauge or the RW gauge, the master equation \meq{evenmasterEZ} or \meq{evenmasterRW} are the third-order equations. However, after the separation of variables, the equations all reduce to a second-order equation. For odd-parity perturbation, the master equation is also constructed as \eq{oddmasterl2}. Next, the cases for $l=0$ and $l=1$ are discussed. For the $l=0$ case, we present the equations that the gauge invariants satisfy. For even-parity perturbations with $l=1$, we find that the metric perturbation components are still determined when the source is specified. And for odd-parity perturbations with $l=1$, the master equation \meq{mastereqoddl1} is still a wave-like equation. Finally, using our general results, we investigate a point particle moving along a circular orbit in general spherically symmetric spacetimes. In particular, the form of the solutions for $l=0$ and $l=1$ are carefully discussed.

Our results can be applied to various modified theories of gravity, in which the background is described by the general static metric (\ref{ABneq1}), instead of
the particular one (\ref{ABeq1}). In particular, it can be applied to  the EOB system. Jing et. al. pointed out that one should consider the  Hamilton equations for an EOB system self-consistently \cite{Jing2022}. Specifically, considering that the EOB system takes the spinless effective metric $g_{\mu\nu}^{eff}$ \cite{Damour1999, Damour2016, Damour2018}, the Hamiltonian $H[g_{\mu\nu}^{eff}]$ and the radiation-reaction force $\mathcal{F}^{circ}_\varphi[g_{\mu\nu}^{eff}]$ should be both based on the same effective metric. Under the quasi-circular approximation, the radiation-reaction force can be obtained by the energy-loss rate,
\begin{equation}
\mathcal{F}^{circ}_\varphi[g_{\mu\nu}^{eff}]\simeq \frac{1}{\dot{\varphi}}\frac{dE[g_{\mu\nu}^{eff}]}{dt}.
\end{equation}
Considering that the perturbed Weyl tensor $\psi_4^B$ can be divided into the even-parity $\psi_4^{BE}$ and the odd-parity $\psi_4^{BO}$ parts \cite{Jing2022}, then the energy-loss rate can be calculated from $\psi_4^{BE}$ and $\psi_4^{BO}$ via the relation,
\begin{equation}
\frac{dE[g_{\mu\nu}^{eff}]}{dt}=\frac{c^3}{16\pi G\omega^2}\int\left\{\left[Re(\psi_4^{BE}+\psi_4^{BO})\right]^2+\left[Im(\psi_4^{BE}+\psi_4^{BO})\right]^2\right\}r^2d\Omega^2.
\end{equation}
So, the key step to obtain a self-consistent radiation-reaction force $\mathcal{F}^{circ}_\varphi[g_{\mu\nu}^{eff}]$ is to solve the solution of $\psi_4^{BE}$ and $\psi_4^{BO}$. In \cite{Jing2022}, the authors constructed the decoupled equations for both even-parity $\psi_4^{BE}$ and odd-parity $\psi_4^{BO}$ in the effective metric spacetime rather than in the Schwarzschild spacetime. Generally speaking, $\psi_4^{BO}$ is only related to the odd-parity perturbation, i.e. the $\mathrm{C}$ and $\mathrm{J}$ terms defined in Eq.(\ref{eq17}). When \eq{oddmasterl2} is solved, one can determine $\mathrm{C}$ and $\mathrm{J}$ by \eqs{reconsalpha} and \meq{reconsbeta}, and then $\psi_4^{BO}$ can be calculated. Similar processing can be done for the even-parity perturbation. However, Ref. \cite{Jing2022} only considered the background effective metric that takes the form as \eq{ABeq1}, which can be applied to the Post-Minkowskian (PM) approximation \cite{Damour2016,Damour2018,He2021}, if the undeterminated parameters $d_i$ are well constrained. While in this paper we consider the most general spherical symmetric metric, which can be applied to the EOB theory with either the PN approximation or the PM approximation. We wish to come back to this important issue soon.

\acknowledgments
We would like very much to thank Xiaokai He and Tao Zhu for valuable discussions, and also thank Zhucun Li. This work was partially supported by the National Key Research and Development Program of China Grant No.2020YFC2201503, the National Natural Science Foundation of China under Grants No. 11705053, No. 11975203 and No. 12035005, and the Hunan Provincial Natural Science Foundation of China under Grant No. 2022JJ40262. The work of Fang was supported in part by China Scholarship Council for the visiting post-doc program at Baylor University.

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text:
Remember Gaza
26.12.2009 18:06 | Anti-militarism | Palestine | Birmingham | Liverpool | Sheffield | South Coast | World
From December 27th 2008 to January 18th 2009 Israeli bombers pounded Gaza. The Israeli assault was the most violent since the occupation began in 1967. On the eighth day of the attack Israeli Occupation Forces (IOF) entered Gaza and remained there for two weeks. Throughout the 22 days of the assault civilians were delierately targeted, hospitals were attacked, schools were bombed and 1417 people were killed, including at least 318 children.
The international Palestine solidarity movement is calling for January 2010, the first anniversary of the massacre, to be a month of solidarity actions in support for the campaign for boycott, divestment and sanctions against Israel. One such call has come from the Free Gaza movement, whose volunteers broke the siege in order to stand in solidarity with the people of Gaza during the massacre.
On December 27th demonstrations, commemorating the first day of the massacre, were held in Brighton, Sheffield, Liverpool and Birmingham while a thousand strong demonstration was held outside the Israeli embassy in London. Elsewhere Manchester's Target Brimar campaign has called for 22 days of waging peace, incuding several actions against Manchester based arms manufacturer, Brimar, who supply equipment used in Israeli fighter aircraft, the Palestine Solidarity Campaign has called for a week of action, from the 9th to the 17th of January, against supermarkets selling Israeli goods, a mass 'die in' is planned in Trafalgar Square on the 16th and Smash EDO have called for a mass demonstration against EDO MBM/ITT, now ITT Integrated Structures, who supply components for F-15s and F-16s used by the Israeli military.
The Viva Palestina convoy was supposed to arrive in Gaza today, breaking the siege and bringing much needed medicine and rebuilding materials. However, the convoy has not been allowed to enter Egypt and has been in Aqabah, Jordan, since December 25th. On the 27th the convoy members began a hunger strike calling for the Egyptian government to allow them to enter. Activists are also arriving in Egypt to join the Gaza Freedom March, a march from Egypt into Gaza and to the Erez crossing. Freedom marchers have already suffered repression from the Egyptian authorities with dozens arrested (and then released) and a Gaza massacre memorial in Cairo shut down.
Links: Palestine Solidarity Campaign|Boycott Israeli Goods Campaign|Smash EDO|Target Brimar|Palestinian Return Centre|Gaza Freedom March|Viva Palestina|Free Gaza Movement|Palestinian BDS Campaign|
During the massacre people around the world took action in solidarity with Gaza and for a boycott of Israel. In London nightly demonstrations were held outside the Israeli embassy [1|2|3], local demonstrations took place across the UK and on the 10th December 100 000 people marched on the Israeli embassy. Actions took place against companies profiting from Israel's occupation: supermarkets selling Israeli goods, Starbucks, Ahava and Agrexco. Six campaigners decommissioned EDO MBM/ITT in Brighton, causing an alleged £500 000 worth of damage and a wave of resistance swept across university campuses.
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The numbers speak for themselves — underclassrising.net
The Occupation Began In 1948 — Zionism Is Not Judaism

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