Negative modes of Schwarzschild black hole in Einstein-Gauss-Bonnet Theory

对温斯顿·丘吉尔《关于希特勒入侵苏联的演讲》的文体分析6200字本文从分析文体的角度出发对温斯顿·丘吉尔的著名演讲《关于希特勒入侵苏联的演讲》中所使用的文体手段进行了详尽的分析,引导读者赏析中大量使用的平行、重复、押韵、比喻等修辞手法是如何使演讲稿达到音、形、义的完美结合,进而对演讲文体的风格进行了简要的论述。

文体学;音位学;词汇和语法;语义学The style of writing to Winston Churchill“Speech on Hitler’sInvasion of the U.S.S.R.” is analysedLi TingThe detailed analysis the main body of a book has been carried out from angle style of writing means used by the middle starting off giving a lecture “give a lecture about that Hitler invades Soviet” to Churchillian Winston ? q famousness analysing style of writing, guides reader to award Xi paralleling in the original Da Liang Shi Yong, repeats, is in rhyme, allegory waits rhetoric gimmick has been that draft reaches the tone, the shape, righteous consummation combines how to make to give a lecture, the style of writing style has carried out brief discussion on to giving a lecture then.Stylistics; Phonemics; Vocabulary and grammar; SemanticsG89C1673-8209(2009)02-0236-02Written by Winston Churchill, Speech on Hitler’s Invasion of the U.S.S.R. is full of powerful thoughts, magnificent expressions and well-structured. At that time, the U.S.S.R. was in danger and the disaster would quickly turn to the British. The goal of this speech was to arouse British people and win the cooperation of the USA, so, this speech is persuasive as its functional tenor is concerned; so far as the personal tenor is concerned, it is formal. General features of this speech can be summarized as: this persuasive speech is the speaking of what is written to be spoken to an audience on a formal occasion, so it presents features of formal written language while exhibiting characteristics of the written mode with its highly logical organization of thoughts, closely-knit structure and formal language. Just as language is multifunctional, so style is multileveled: Stylistic features can be formed at the three distinct levels: ① The level of Phonology; ② The level of Lexis and Grammar; ③ The level of Semantics. In order to find out the items of stylistic significance in this famous speech, this paper aims at analyzing it from one level of its language to another.1 Stylistic features at the first level―phonological features.Repetition of the previous initial sound may arouse the addressees’ sensitivity to the sound quality of a text, which includes Alliteration, as in “dull, drilled, docile”, “for his hearth and home”,Assonance, as in “clanking, heel-clicking”, “cowing and tying”. Besides, when the phoneticsymbolism is concerned, explosive sounds like[b],[p],[t],[d],[g],[k]usually gives a feeling of suddenness and strength while liquids and nasals a sense of peace and harmony. Winston used a series of explosives like “dandified, dull, docile, drilled, brutish, plodding, etc.” to impress his audience with the violent aggression of the Nazi army.2 Stylistic features at the second level―grammatical and lexical features.As public speeches are intended to inform, to persuade, and to appeal, most sentences are statements, as is shown in Churchill’s speech with only one question and two commands used in the whole speech. We know that casual speech tends to use short sentences. But a glance at Churchill’s speech reveals that its sentences are mostly long and complex ones. Though there can be several clauses in a longer sentence, they mainly come after the main clause, which presents no difficulty for the audience to comprehend. There are no sentences that are incomplete or elliptical. These characteristics show the double traits of public speaking: it is formal as is required of written language, and it is yet not difficult to understand at the time of the delivery as is required of the spoken mode. Stressing the idea by using synonyms or repeating the same word impinges on the audience’s ears, adding force, clearness or balance to a sentence. Examples for this can be found in Winston’s speech, e.g. “We have but one aim and one single, irrevocable purpose.” “…for we must speak out now at once, without a day’s delay.”3 Stylistic features at the third level―semantic features.The level of semantics concentrates on how the cohesive devices, paragraphing, discourse patterning, rhetorical devices and role structures contribute to the meaning of the text as a whole. Here, we will mainly focus on the rhetorical devices used in the speech. Clear organization is vital to speech making. Persuasive speeches, especially policy speeches often fall into problem-solution order. In Churchill’s speech, he showed how serious the problem was and the real intension of the Nazi regime. After having convinced the audience, Churchill called them to act with united strength. Besides, rhetorical devices are also employed by Churchill to make his speech vivid and forceful. Parallel structures are not only even in structure, but also coherent in meaning. They function not only as a kind of rhetoric device, but as a means of coherence as well. With its repetition of the sentence structure, the theme of the speech is closely connected and clearly illustrated. With so many parallel structures, the whole speech goes on smoothly and logically. The use of many parallel structures in this speech has filled the speech with strength and clear logic, e.g. “We will never parley, we will never negotiate…” “Behind all this glare, behind all this storm…” Repetition of the same word at the beginning or end of successive clauses or sentences helps create a strong emotional effect. For example: “ Any man or state who… Any man or state who…”In conclusion, Churchill’s Speech on Hitler’s Invasion of the U.S.S.R. is very typical of public persuasive speeches, with its stylistic characteristics on semantic, grammatical, lexical and phonological levels. As an excellent combination of sound, form and meaning, this speech has no doubt made significant contributions in the history and the present.参考文献[1] 张汉熙.《高级英语》上册[M],北京:外语教学与研究出版社,1995[2]徐有志.《英语文体学教程》[M].高等教育出版社,2005[3]徐有志.《现代英语文体学》[M].开封:河南大学出版社,1992[4]秦秋白.《英语文体和文体要略》[M]. 上海:上海外语教育出版社出,2006[5]霍晓军.《从文体学角度看FROST的诗》[J]. 太原:太原教育学院学报,2004。

黑羊效应英文文章The Black Sheep EffectIn the vast expanse of society, we often find ourselves drawn to the outliers, the individuals who defy the norm and stand out from the crowd. These "black sheep" of the herd, as they are sometimes referred to, captivate our attention and spark intrigue within us. The concept of the "black sheep effect" explores this phenomenon, delving into the psychological undercurrents that drive our fascination with those who deviate from the expected.At the heart of the black sheep effect lies the fundamental human need for belonging and social acceptance. We are inherently wired to seek out and conform to the norms of our respective groups, be it our families, our communities, or our broader social circles. When an individual within these groups exhibits behavior or characteristics that deviate from the accepted standards, it can create a sense of discomfort and unease among the other members.This discomfort arises from the fact that the presence of a "black sheep" challenges the cohesion and stability of the group. The individual's actions or beliefs may be perceived as a threat to thegroup's identity, values, or established ways of thinking. As a result, the group may feel compelled to distance itself from the deviant member, either through overt criticism, ostracization, or more subtle forms of social exclusion.However, it is precisely this sense of deviation that piques our curiosity and draws our attention to the black sheep. We are intrigued by their willingness to go against the grain, to challenge the status quo, and to forge their own path. In a world that often rewards conformity and discourages individuality, the black sheep's refusal to conform can be both captivating and unsettling.Moreover, the black sheep effect is not limited to our perception of others; it can also shape our own self-image and behavior. Individuals who perceive themselves as deviating from the group's norms may internalize a sense of being the "black sheep" of the herd. This self-perception can lead to a range of psychological consequences, including feelings of isolation, low self-esteem, and a heightened sensitivity to the judgments of others.The black sheep effect is not merely a social phenomenon; it also has deep-rooted evolutionary underpinnings. In our ancestral past, conformity and group cohesion were essential for survival. Those who strayed too far from the group faced increased risks of predation, starvation, and exclusion from vital resources. As a result,our brains have developed a predisposition to be attuned to and wary of individuals who deviate from the group.This evolutionary legacy continues to shape our modern social interactions and perceptions. When we encounter a black sheep, our brain's threat detection systems may be activated, triggering a heightened sense of vigilance and a desire to maintain the group's cohesion. This response can lead to a range of cognitive biases, such as the tendency to judge the black sheep more harshly or to attribute their behavior to internal flaws rather than external factors.However, it is important to recognize that the black sheep effect is not always a negative phenomenon. In some cases, the presence of a black sheep can serve as a catalyst for positive change within a group. By challenging the status quo and pushing the boundaries of accepted norms, the black sheep can inspire others to question their own assumptions, to think more critically, and to embrace new perspectives.Moreover, the black sheep's nonconformity can be a source of inspiration and empowerment for those who feel marginalized or underrepresented within their social circles. By witnessing the courage and resilience of the black sheep, individuals may be emboldened to embrace their own uniqueness and to assert their right to be different.In conclusion, the black sheep effect is a complex and multifaceted phenomenon that reflects the delicate balance between our need for social belonging and our innate desire for individuality. As we navigate the social landscape, it is important to be mindful of our own biases and to approach the black sheep with empathy, understanding, and an open mind. By embracing the diversity of human experience, we can foster a more inclusive and enriching social fabric, one that celebrates the unique contributions of all its members.。

美国文学知到章节测试答案智慧树2023年最新泰山学院第一章测试1.The first settlers who became the founding fathers of the American nationwere quite a few of _______. （）参考答案:Puritans2.The common thread throughout American literature has been the emphasison the _______.（）参考答案:Individualism3.The ship “________”carried about one hundred Pilgrims and took 66 days tobeat its way across the Atlantic. In December of 1620, it put the Pilgrimsashore at Plymouth, Massachusetts.（）参考答案:Mayflower4.The English colonies in North America rose in arms against their parentcountry and the Continental Congress adopted ________.（）参考答案:the Declaration of Independence5.Which of the following works is not connected with Thomas Paine?（）参考答案:The Autobiography6.Which of the following works is written by Philip Freneau?（）参考答案:The Wild Honey Suckle7.The English began to settle down in the early 17th century.（）参考答案:对8.Poor Richard''s Almanac was written by Philip Freneau.（）参考答案:错9.The Autobiography of Benjamin Franklin can be divided into four parts.（）参考答案:对10.Philip Freneau is called the“Father of American Poetry”.（）参考答案:对第二章测试1.____________ was the most leading spirit of the Transcendental Club.（）参考答案:Emerson2.American literature produced only one female poet during the nineteenthcentury. This was ___________.（）参考答案:Emily Dickinson3.The House of Seven Gables is a famous mystery-haunted novel written by_____.（）参考答案:Nathaniel Hawthorne4.The following writers belong to the Romantic group in American literatureexcept _____.（）参考答案:William Blake5.There is a good reason to state that New England Transcendentalism wasactually _____ on the Puritan soil.（）参考答案:Unitarianism6.The poetic style Walt Whitman devised is now called blank verse, which ispoetry without a fixed beat or regular rhyme scheme.（）参考答案:错7.After his death, Henry Wadsworth Longfellow became the only American tobe honored with a bust in the Poet's Corner of Westminster Abbey.（）参考答案:对8. A superb book Nature came out of Henry David Thoreau's two-yearexperiment at Walden Pond.（）参考答案:错9.James Fenimore Cooper launched two kinds of immensely popular stories:the sea adventure tale, and the frontier saga.（）参考答案:对10.No other American poet ever surpassed Edgar Allan Poe's ability in the use ofEnglish as a medium of pure musical and rhythmic beauty.（）参考答案:对第三章测试1.Realism in American literature stretches from _____________ to the end of 19thCentury. （）参考答案:American Civil War2.Which ONE of the following concepts is related to the understanding ofliterary realism? （）参考答案:representation of characters, human nature and social actualities in a non-idealized way3.Where Mark Twain and William Dean Howells satirized European mannersat times, ________ was an admirer. （）参考答案:Henry James4.______________’s stories still had many unrealistic qualities: “tall tales” andunlikely coincidences. He is never a pure realist. （）参考答案:Mark Twain5.______________ is the novel into which Jack London put most of himself. （）参考答案:Martin Eden6.All the following concepts can be found in American naturalistic fictionEXCEPT ___________. （）参考答案:search for identity7.“_______________” was a term crea ted by the French novelist, Emile Zola. （）参考答案:naturalism8.Sister Carrie tells about a country girl coming to Chicago to look for a betterlife and to pursue the American Dream.（）参考答案:对9.The impact of Darwin’s evolutionary theory on the American thought and theinfluence of the nineteenth century French literature on the American men of letters gave rise to another school of realism: American naturalism.（）参考答案:对10.In The Octopus written by Stephen Crane, wheat farmers struggle to growcrops and send them to market for a profit, while being beleaguered by the inflated prices of the giant railroad conglomeration.（）参考答案:错第四章测试1.Who is considered to be the first Imagist theorist? （）参考答案:T. E. Hulme2.Which poem doesn’t belong to Imagist poems? （）参考答案:Had I Not Seen the Sun3.What are the artistic features of Modernism? （）参考答案:Fragmentation;Unusual typography;Allusive language;Stream-of-consciousness4.What are the masterpieces of Hemingway? （）.参考答案:The Old Man and the Sea;For Whom the Bell Tolls;A Farewell toArms;The Sun Also Rises5.Ezra pound laid down three Imagist poetic principles. （）参考答案:对6.The Road Not taken is written by T. S. Eliot. （）参考答案:错7.Fitzgerald was once praised “the poet laureate of the Jazz Age”. （）参考答案:对8.“The Sound and the Fury” was the masterpiece of ___.参考答案:null9.The jazz age refers to the decade of ___.参考答案:null10.What is the style of Hemingway’s novel?参考答案:null第五章测试1.Postmodernist fictions is a continuation of modernism's alienated mood,daring experimentation and disorienting techniques.（）参考答案:对2.Black humor rose in America in the1980s and 1990s.（）参考答案:错3.Vladimir Nabokov is a representative of Black humor.（）参考答案:对4.Catch-22, a novel accounting a frantic bombardier's desperate efforts tosurvive, was written by Joseph Heller.（）参考答案:对5.Avant-pop Art is an artistic genre based on the combination of avant-gardeand pop art.（）参考答案:对第六章测试1.Ethnic American literature can be divided into _________. （）参考答案:Native American literature;African American literature;Asian American literature;Jewish American literature2.N. Scott Momaday is a _______ American writer.（）参考答案:Native3.Louise Erdrich's Tetralogy include ________.（）参考答案:Tracks;The Beat Queen;The Bingo Palace;Love Medicine4.The author of The Color Purple is ________.（）参考答案:Alice Walker5.Which ONE of the following is known as the masterpiece of Richard Wright?（）参考答案:Native Son6.Toni Morrison is the first African American novelist who received the NobelPrize for Literature.（）参考答案:对ngston Hughes was one of the most talented and original black writers inthe 20th century.（）参考答案:对8.Celeste Ng's Everything I Never Told You is a novel beyond nation, race,gender and age.（）参考答案:对9.Toni Morrison won a Pulitzer Prize for fiction for her God Help the Child.（）参考答案:错10.The Color Purple is an epistolary novel that depicted rape, incest, bisexuality,and lesbian love among African Americans.（）参考答案:对。

宇宙十大怪兽展开全文我们的宇宙实在是太奇怪了。

尽管诸如量子论、相对论和太阳中心说等前瞻性理论现已被普遍接受。

科学仍在继续向我们展示，宇宙中还存在许多令人费解的现象。

1、高德的不完备定理（G?del’s incompleteness theorems）高德的不完备定理更像是一组非常有趣的关于逻辑和哲学的数学定理，而不是严格意义上的科学。

但是整体上，这些逻辑和哲学与科学密切相关。

1931年，科特-高德证明了该定理：因为任何稍微复杂一点的逻辑体系都不可避免自我引用；所以对于给定的任意一组逻辑规则，除了最简单的之外，总会存在无法判定（证明或证伪）的命题。

这表明了世界上不存在能够证明或证伪所有命题的终极数学体系。

一个无法判定的命题可以被当成是“我总是说谎”的数学形式。

由于该命题引用了描述它的语言本身（译者注：“我总是说谎”有两方面的意思，一方面是指命题要表达的内容“我是一个爱说谎的人”，另一方面也可以指描述它的语言“我说‘我总是说谎’是说谎，其实我不是一个爱说谎的人”），所以永远无法知道这个命题的真假。

尽管如此，并不是只有自我引用的命题才是无法判定的。

高德的不完备定理的主要结论是，所有的逻辑体系都会存在无法证明或证伪的命题。

因此，所有的逻辑体系都不“完备”。

不完备定理的哲学含意广为传播。

由于没有一组规则能够解释所有可能的事件或结果，所以物理学上的“终极理论”是不存在的。

同时，这也说明了“证据”是比“真相”更不靠谱的概念。

这样的想法令科学家们惴惴不安。

因为这意味着世界上总会存在无法被“证据”证明的“真相”。

由于不完备定理对计算机也一样适用，这也意味着我们的想法是不完备的。

世界上有些想法我们永远无法明了，这包括我们的想法是否一致（比如说，我们的理性是否包含错误，自相矛盾）。

这是因为高德的不完备第二定理表示，没有一种一致的理论能够证明自己的一致性。

这意味着，没有任何理智的人能证明自己没有精神病。

同样，如果一个体系证明自己是一致的，那么这就不是一个一致的体系。

有关宇宙负片的作文英文回答：The concept of a negative film of the universe is both intriguing and thought-provoking. It raises questions about the nature of reality and our understanding of the cosmos.Imagine if we could see the universe in negative, with the colors inverted and the dark areas appearing bright. This would completely change our perception of the world around us. The stars that we are accustomed to seeing as bright points of light would now be dark voids against a bright background. The galaxies that we know to be spiral or elliptical in shape would appear as intricate patterns of light and dark.The negative film of the universe would not only alter our visual perception, but also challenge our understanding of the physics behind it. We would have to reevaluate our theories and models to account for the reversed colors andthe new patterns that emerge. This could potentially lead to new discoveries and a deeper understanding of the universe.Furthermore, the negative film of the universe could have practical applications. For example, it could be used to study dark matter and dark energy, which are still largely mysterious to us. By examining the negative images, scientists may be able to uncover hidden patterns or structures that were previously invisible to us. This could provide valuable insights into the nature of these elusive components of the universe.In addition, the negative film of the universe would have a profound impact on our perception of beauty and aesthetics. The inverted colors and patterns could create a whole new visual experience, challenging our preconceived notions of what is visually appealing. It would be like looking at a familiar painting in negative, discovering new details and nuances that were previously unnoticed.中文回答：宇宙负片的概念令人着迷，也引发了对现实本质和宇宙认知的思考。

General_Knowledge_about_American_Literature_2_1_34. The Age of Realism in the literary history of the United States refers to the period from to .A. 1861/1920B. 1865/1920C. 1861/1914D. 1865/191435. Mark Twain is well-known for his .A. frontier themeB. symbolismC. local colorD. international theme36. Theodore Dreiser and Jack London are among the best representative writers of literary , which is greatly influenced by Darwin.A. naturalismB. sentimentalismC. romanticismD. transcendentalism37. The major trend in American literature in the seventies and eighties of the 19th century is .A. romanticismB. realismC. sentimentalismD. naturalism38. The three dominant figures of the American Realistic Period are the following EXCEPT .A. Jack LondonB. William Dean HowellsC. Mark TwainD. Henry James39. Whose fame largely rested on his handling of the international theme?A. Mark TwainB. Stephen CraneC. Theodore DreiserD. Henry James40. The book that gives fairly accurate picture of southern plantation life is .A. An American TragedyB. The Call of the WildC. Uncle Tom’s CabinD. A Hazard of New Fortunes41. expresses Jack London’s view that success means an adaptation to circumstances—a coordination of inner energy and external force.A. Martin EdenB. The Call of the WildC. White FangD. The Sea-Wolf42. Jack London wrote as “an attack on individualism”.A. Martin EdenB. The Call of the WildC. White FangD. The Sea-Wolf43. Which of the following statement is NOT true of the American naturalists?A. They stressed the possible triumph of human will.B. They ventured the forbidden subjects such as sex, death,and violence.C. They wrote in a daring, open, and direct manner.D. They see human beings no more than a physical object under the control of biological and environmental forces.Reference Answers: DCABA; DCBAA44. is regarded as one of the finest “psychological”fiction writers in America because he explored the motivations and frustration of his fictional characters in terms of Freudianism.A. Jack LondonB. Mark TwainC. Stephen CraneD. Sherwood Anderson45. In 1900, Jack London published his first collection of short stories, named .A. The Son of the WolfB. The Sea WolfC. The Law of LifeD. White Fang46. With Howells, James, and Mark Twain active on the scene, became the major trend in the seventies and eighties of the nineteenth century.A. sentimentalismB. romanticismC. realismD. naturalism47. Which book is NOT written by Mark Twain?A. The Gilded AgeB. The Prince and the PauperC. Life on the MississippiD. The Portrait of a Lady48. The novel Sister Carrie was written by .A. Henry JamesB. Theodore DreiserC. Jack LondonD. Frank Norris49. The novel that “touched off the American Civil War” refers to .A. Gone with the WindB. Uncle Tom’s CabinC. Farewell to ArmsD. The Grapes of Wrath50. Maggie: A Girl of the Streets is an example of naturalist fiction, written by .A. Stephen CraneB. Edith WhartonC. Edgar Allan PoeD. Eugene Debs51. is considered to be Theodore Dreiser’s greatest work.A. The American TragedyB. Sister CarrieC. The FinancierD. Trilogy of Desire52. The book from which “all modern American literature comes” refers to .A. The Great GatsbyB. The Sun Also RisesC. Moby DickD. The Adventures of Huckleberry Finn53. Frank Norris’s novel has been viewed as “the first full-bodied naturalistic American novel”.A. Vandorer and the BruteB. McTeagueC. Sister CarrieD. The Red Badge of CourageReference Answers: DACDB; BAADB54. Jack London’s masterwork is somewhat autobiographical.A. The Sea WolfB. The People of the AbyssC. Martin EdenD. The Star Rover55. The short stories often with surprise ending such as The Gift to the Magi and The Last Leaf are written by .A. O. HenryB. Theodore DreiserC. William FaulknerD. Jack London56. was an American writer, journalist and humorist, who wona world-wide reputation for his stories of the youthful adventures of Tom Sawyer and Huckleberry Finn.A. Ernest HemingwayB. Mark TwainC. Norman MailerD. J.D. Salinger57. made his name as a leading naturalistic writer with his masterwork, Winesburg, Ohio, a picture of life in a typically small Midwestern town.A. Frank NorrisB. Upton SinclairC. Sherwood AndersonD. John Steinbeck58. American author book The Red Badge of Courage published in 1895 brought him international fame.A. Stephen Crane’sB. Alex Haley’sC. Norman Mailer’sD. John Steinbeck’s59. American author Theodore Dreiser was generally regarded as an outstanding representative of , whose novels like Sister Carrie and An American Tragedy depict real-life subjects in a different harsh light.A. naturalism and realismB. romanticism and naturalismC. modernism and naturalismD. colonialism and modernism60. is a prolific American novelist and short story writer, whose works such as The Call of the Wild and Martin Eden deal romantically with the overwhelming power of nature and the struggle for survival.A. Northrop FryeB. Jack LondonC. Richard WrightD. Alice Walker61. Which of the following is NOT written by Henry James?A. Daisy MillerB. The Portrait of a LadyC. The BostoniansD. The Genius62. American writer Harriet Stowe was best-known for the anti-slavery novel , which was written in reaction to the FugitiveSlave Act of 1850.A. A Portrait of the Artist as a Young ManB. After Many a SummerC. Uncle Tom’s CabinD. Gulliver’s Travels63. At the end of the 1860s, the two representatives of American Realism were Mark Twain and .A. Stephen CraneB. Ernest HemingwayC. William Dean HowellsD. John SteinbeckReference Answers: CABCA; ABDCC64. The leader of the imagist movement in American literature is .A. Wallace StevensB. Ezra PoundC. Robert FrostD. Thomas Stearns Eliot65. The great dividing line between the nineteenth century and the contemporary American Literature is .A. the Civil WarB. The First World WarC. the Second World WarD. the Great Depression66. wrote about the disintegration of the old social system in the American Southern States.A. Jerome David SalingerB. John UpdikeC. William FaulknerD. F. Scott Fitzgerald67. All the following writers belong to “The Beat Generation” EXCEPT .A. Allen GinsbergB. John UpdikeC. Jack KerouacD. Lawrence Ferlinghetti68. The Naked Lunch is written by .A. William Seward BurroughsB. Lawrence FerlinghettiC. Allen GinsbergD. Jack Kerouac69. showed great interest in Chinese literature and translated the poetry of Li Bai into English.A. Ezra PoundB. Robert FrostC. T.S. EliotD. Emily Dickinson70. is often acclaimed as the literary spokesman of the Jazz Age.A. Ernest HemingwayB. F. Scott FitzgeraldC. William FaulknerD. Ezra Pound71. Which of the following is NOT a black author?A. Richard WrightB. Allen GinsbergC. James BaldwinD. Ralph Ellison72. In which period did Canada’s most famous and successful writers appear?A. Post-war periodB. Period of nation-buildingC. Early colonial periodD. Pre-European period73. called for an American renaissance in culture and collaborated in fashioning a new kind of poetry called Imagism.A. Carl SandburgB. Ezra PoundC. Dos PassosD. Scott FitzgeraldReference Answers: BBCBA; ABBAB74. Robert Frost is a regional poet in the sense that his poems depict mostly .A. the frontier lifeB. the sea adventureC. the puritan communityD. New England landscape75. Which of the following authors is NOT the Nobel Prize winner?A. Ernest HemingwayB. Eugene O’NeillC. William FaulknerD. F. Scott Fitzgerald76. is John Steinbeck’s masterpiece.A. An American TragedyB. The Adventures of Huckleberry FinnC. A Farewell to ArmsD. The Grapes of Wrath77. is Hemingway’s first true novel, which portrays “the lost generation”.A. For Whom the Bell TollsB. The Old Man and the SeaC. The Sun Also RisesD. A Farewell to Arms78.William Faulkner was the foremost American writer of the 20th century.A. New EnglandB. WesternC. SouthernD. Black79. Who wrote the famous two-line poetry In a Station of the Metro?A. Ezra PoundB. W.B. YeatsC. T.S. EliotD. D.M Thomas80. Ezra Pound’s major work of poetry is the long poem called .A. Hign Seluyn MauberleyB. Make It NewC. Pisan CantoD. Polite Essays81. In the novel The Old Man and the Sea, Hemingway portrayed an old fisherman named , who shows triumphant even in defeat.A. SantiagoB. Jack BarnesC. Martin EdenD. Tom Sawyer82. The Great Gatsby is written by .A. Ernest HemingwayB. Sinclair LewisC. John SteinbeckD. Francis Scott Fitzgerald83. In 1962, John Steinbeck was awarded the for literature.A. Pulitzer PrizeB. Nobel PrizeC. Bolligen PrizeD. National PrizeReference Answers: DDDCC; ACADB84. In 1951, Jerome David Salinger published his first and only novel .A. The Catcher in the RyeB. Breakfast at Tiffa ny’sC. Bullet ParkD. Let Your Mind Alone85. The Rabbit series were written by .A. John UpdikeB. Jerome David SalingerC. Mark TwainD. Ezra Pound86. The poem Howl is written by .A. Allen GinsbergB. John UpdikeC. Lawrence FerlinghettiD. Jack Kerouac87. was Jack Kerouac’s best novel.A. Mexico City BluesB. Howl and the PoemsC. Visions of CodyD. On the Road/doc/2b16407284.html,wrence Ferlinghetti is a/an .A. playwrightB. poetC. novelistD. essayist89. , born in America, considered as one of the great poets in both American literature and British literature, got British nationality in 1927.A. John UpdikeB. T.S. EliotC. Mark TwainD. Ezra Pound90. Ralph Ellison was an African-American writer, whose novel published in 1952 gained a wide critical success, and he has been compared to such writers as Melville and Hawthorne.A. The Grapes of WrathB. Invisible ManC. Native SonD. The Sound and the Fury91. William Faulkner, author of , was awarded the Noble Prize for Literature in 1949.A. The Grapes of WrathB. Invisible ManC. To Kill a MockingbirdD. The Sound and the Fury92. Alex Haley became famous with the publication of the novel , which woke up an interest in genealogy, particularly among African-Americans.A. Native SonB. RootsC. The Age of InnocenceD. Sophie’s Choice93. Catch-22 was written by American writer and the phrase “catch-22” has entered the English language to signify a no-win situation, particularly one created by a law, regulation or circumstance.A. Joseph ConradB. Upton SinclairC. Joseph HellerD. Mark TwainReference Answers: AAADB; BBDBC94. Ernest Hemingway is one of the most famous American novelist, short-story writer and essayist, whose simple prose style in the works like have influenced wide range of writers.A. All the King’s MenB. The Sun Also RisesC. The Color PurpleD. The Adventures of Huckleberry Finn95. Sinclair Lewis, the author of , is an American novelist, playwright, and social critic who has gained popularity with satirical novels.A. Tom JonesB. Main StreetC. Moby DickD. The Bluest Eye96. is an American poet whose works are often set in rural New England, exploring the relationships between individuals and between people and nature.A. Ernest HemingwayB. Robert FrostC. F. Scott FitzgeraldD. John Steinbeck97. American writer Harper Lee is famous for her race relations novel which was awarded the Pulitzer Prize in 1961.A. The Scarlet LetterB. To Kill a MockingbirdC. The Great GatsbyD. Naked Lunch98. E.E. Cummings is best known for his lyrical and typographically eccentric .A. novelsB. poetryC. proseD. drama99. American author Margaret Mitchell wrote an enormously popular novel , which tells a story about the Civil War and Reconstruction as seen from the southern point of view.A. Winesburg, OhioB. Go Tell It on the MountainC. Gone with the WindD. Invisible Man100. best-known novel is The Catcher in the Rye, a story about a rebellious teenage schoolboy and his quixotic experiences in New York.A. Jero me David Salinger’sB. Robert Frost’sC.F. Scott FitzgeraldD. John Steinbeck101. received the Nobel Prize for Literature in 1962, and he is best remembered for The Grapes of Wrath, a novel widely considered to be a 20th century classic.A. Sinclair LewisB. Jerome SalingerC.John SteinbeckD. F. Scott Fitzgerald102. Pearl S. Buck, who won the Nobel Prize in 1938, wrote many novels about China, and the best known is .A. The Good EarthB. The Caine MutinyC. A New LifeD. The Portrait of a Lady103. , one of America’s greatest playwrights, won the Nobel Prize in 1936, the first American playwright to receive the honor. Some of his most famous works include Anna Christie, the Hairy Ape, The Iceman Cometh, Long Day’s Journey into Night and so on.A. Arthur MillerB. Tennessee WilliamsC. Bernard MalamudD. Eugene Galdstone O’NeillReference Answers: BBBBB; CACAD。

Hawthorne’s “black vision" of life and human beings09 英4As we can see， Hawthorne’s literary world turns out to be a most disturbed， torme ntedand problematical one possible to imagine. This has much to do with his “black vision” of life and human beings，especially in The Scarlet Letter，this kind of writing characteristic can be tracked most adequately。

So ,what is black vision exactly？Section1．The Hawthorne’s black vision of life and human beings According to Hawthorne，“There is evil in every human heart， which may re main latent， perhaps， through the whole life, but circumstances may rouse it to acti vity ”。

One source of evil that Hawthorne is concerned most is overreaching intellect， whi ch usually refers to someone who is too proud， too sure of himself. He believed tha t “the wrong doing of one generation lives into the successive ones,” Hawthorne looks around the world with black glasses，the more sins they saw，and the darker the world is。

Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identifies the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great difficulty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR werefirst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs werefirst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will find this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the field.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a fit to the fundamental and first overtone QNM.In the next section we attempt to give a mathematical definition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In thefifth section we discuss the excitation and observation of QNMs andfinally in the sixth section we will mention the more significant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asfinite strings,mem-branes,or cavitiesfilled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedfield.) There is generally an infinite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isfixed at its ends.All such systems can be described by systems of partial differential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of thefiniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for differential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the differential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The definition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling thefinite string has solutions of various degrees of differentiability. To describe all“realistic situations”,clearly C∞functions should be sufficient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to infinity.The simplest case are waves on an infinite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice differentiable solution.No boundary condition is needed at infinity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of thefields are spatially compact, or–the other extreme–solutions with infinite total energy for which thefields grow at spatial infinity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near infinity! Again it turns out that mathematically it is more convenient to study all solu-tions offinite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that wefind a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for infinite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually different to normal modes.To define quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisfies the ordinary differential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and defineG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The definition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which defines a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be defined as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical significance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in finite parts of the space by a finite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions defined by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On finite intervals and for late times the solution is approximated by a finite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near infinity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Thefirst new point concerns the definitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,define the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at infinity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an infinite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largefinite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for sufficiently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniquefingerprint which would lead to the direct identification of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The difference between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisfies the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate defined byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the effective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testfields as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingfield.For“polar”perturbations the effective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type offluid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving differential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential difficulty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes differentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because infinity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the defining integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an infinite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。