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On the Symmetries of the Edgar-Ludwig Metric

a r X i v :g r -q c /0109037v 1 11 S e p 2001On the Symmetries of the Edgar-Ludwig Metric

Alan Barnes

School of Engineering and Applied Science,Aston University,Aston

Triangle,Birmingham B47ET,UK ?

PACS number(s):04.20Submitted to Class.Quantum Grav.Date:7February 2008Abstract.The conformal Killing equations for the most general (non-plane wave)conformally ?at pure radiation ?eld are solved to ?nd the conformal Killing vectors.As expected ?fteen independent conformal Killing vectors exist,but in general the metric admits no Killing or homothetic vectors.However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case,overlooked by previous investigators,a two-dimensional homethety group exists.No higher dimensional groups of motions or homotheties are admitted by these metrics.

?E-Mail:barnesa @https://www.wendangku.net/doc/7411438671.html, Fax:+441213336215

1

2Alan Barnes

1.Introduction

The most general conformally?at pure radiation(or null?uid)?eld which is not a plane wave is given by(Ludwig and Edgar,2000)

ds2=(xV(x,y,u)?w2)du2+2xdudw?2wdudx?dx2?dy2(1)

with

V=N(u)(x2+y2)+2M(u)x+2F(u)y+2S(u)(2)

where M,N=0,F and S are arbitrary functions of the coordinate u.This metric generalises a solution found by Wils(1989).Only three of these functions are essential as the form of the metric is preserved by the coordinate transformations

du=d?u/α(?u)w=α?w+˙αx?x=x?y=y(3)

whereαis an arbitrary non-zero function of?u and a dot signi?es a partial derivative with respect to?u.Under this coordinate transformation V transforms as

?V(?x,?y,?u)=α?2(V(x,y,u)+2x(α¨α?˙α2))(4) In the original form of the metric(Edgar and Ludwig,1997),this transformation was(in e?ect)used to set M=0.However it is more convenient to use the coordinate freedom to set N=1as in Edgar and Vickers(1999).Thus,dropping tildes,the most general conformally?at pure radiation?eld is given by the metric(1)with

V=x2+y2+2M(u)x+2F(u)y+2S(u)(5)

where M,F and S are arbitrary functions of the coordinate u.

It is interesting to study the symmetries of the Edgar-Ludwig metric for a number of reasons.Firstly the metric is of interest in its own right as it has a physically realistic matter content,namely pure radiation.Being conformally?at the metric admits a ?fteen parameter group of conformal symmetries and it might be thought that,for special values of the functions M,F and S,some of these symmetries would reduce to homotheties or isometries as happens in the case of the conformally?at perfect ?uid solutions found by Stephani(1967).Although the most general Stephani metrics admit no isometries,there is a rich set of special cases which admit isometries;see Barnes and Rowlingson(1990)and Seixas(1992)for a treatment of the non-expanding case and Barnes(1998)for the expanding case.By way of contrast in the conformally ?at null?uid case,the special cases admitting homotheties and isometries are much more restricted except for the case of plane waves.In this paper only the non-plane wave case will be considered as the symmetry structure of plane waves solutions is well-known(Ehlers and Kundt,1962).

Secondly metrics such as the Stephani and Edgar-Ludwig solutions present a sti?test for implementations of the Karlhede algorithm(Karlhede,1980;Karlhede and

On the Symmetries of the Edgar-Ludwig Metric3 MacCallum,1982)as their classi?cation requires the calculation of the third and fourth derivatives of the curvature tensor respectively(Bradley,1986;Koutras A,1992; Skea,1997).In a recent paper Pollney et al.(2000)presented a classi?cation of the Edgar-Ludwig metric using the GRTensor implementation of the Karlhede algorithm. However their results are not consistent with those obtained by Skea(1997)using the implementation of the same algorithm in CLASSI(?A man,1987).GRTensor predicts a non-trivial isotropy group whereas CLASSI predicts no isometries at all.Moreover there appears to be a bug in the GRTensor output routines as evidenced by the form given for the real part of(D2Φ)4,4in Pollney et al.(2000).Given this uncertainty it is useful to investigate the symmetry structure by an alternative method,namely the direct integration of conformal Killing’s equations.Moreover this method unlike the standard version of the Karlhede algorithm,gives expressions for the conformal Killing vectors directly.The results on isometries obtained in this paper are consistent with those obtained by the CLASSI version of the Karlhede algorithm.

Thirdly Ludwig and Edgar(2000)have developed a new formalism for the investigation of the existence of homotheties and isometries.They have used this formalism to investigate the symmetries of their metric and they plan to extend the method to handle conformal Killing vectors.Their formalism works well when only a single homothetic or Killing vector exists.However,when several such vectors exist, its use is somewhat controversial as evidenced by the discussion following Edgar’s talk on the method at the recent GR16meeting in Durban.Thus it is useful to have available results on the symmetry structure of this metric obtained by more tried and tested methods.

2.Conformal Killing Vectors

The conformal Killing equations areξi;j+ξj;i=2σg ij or equivalently

g ij,kξk+g ikξk,j+g kjξk,i=2σg ij(6)

A straightforward but somewhat lengthy calculation reveals that the components of the conformal Killing vectorξi and the associated conformal factor must have the form

ξu=(a(x2+y2)+by+dx+c)/x(7)

ξw=2aw2+w(2βy+γ+2a u x?d u)+H(u,x,y)(8)

ξx=(a(x2?y2)?by?c)w/x+2βxy+γx+a u(x2?y2)?b u y?c u(9)

ξy=(2ay+b)w+β(y2?x2)+γy+?+2a u xy+b u x(10)

σ=2aw+2a u x+2βy+γ(11)

where a,b,c,d,β,γand?are functions of the coordinate u and where u subscripts denote partial derivatives.

4Alan Barnes

Equations(7)-(11)could be used to investigate the conformal symmetries of any metric of the form(1)as their validity does not depend on the precise form of the function V in the metric.The xu,yu and uu components of the conformal Killing equations do depend on the precise form of V and remain to be satis?ed.For the Edgar-Ludwig metric where V is given by equation(5),integration of these three equations for H gives

H(x,y,w)=(a uu?2aM)(x2+y2)+(b uu?2bM)y+c uu?2cM

1

?

ay4+(aF+b/2)y3+(bF+aS+c/2)y2+(cF+bS)y+cS)

2

where k is a function of u only.The functionsβ,γ,?,a,b,c,d and k must also satisfy the following linear system of di?erential equations:

2βu=2aF?b(13)

γu=2aS?c(14)

?u=bS?cF(15)

2a uuu=2aM u+2βF+4a u M?2d u?γ(16)

b uuu=bM u+2βS?dF u+2b u M?2d u F??(17)

c uuu=cM u?dS u+2Mc u?2S

d u?F?+Sγ(18)

2d uu=?6aS+2bF?3c?2k(19)

2k u=?2aS u?2dM u+2a u S?2b u F+c u?4d u M(20) Using standard techniques this linear system may be reduced to a?rst order linear system for?fteen unknowns,namely a,b,c and their?rst and second derivatives, d and its?rst derivative together withβ,γ,?and k.The general solution of this system thus involves?fteen arbitrary constants of integration and so there is a?fteen-parameter family of conformal Killing vectors.This is to be expected as the metric is conformally?at and so admits a conformal symmetry group of maximal dimension. The algebra involved in deriving equations(12)-(20)is rather heavy and has been checked?using the computer algebra system Reduce(Hearn,1995).

3.Homotheties and Isometries

We now consider whether the metric can admit any homothetic motions or isometries. For a proper homothetic motion,the conformal factorσappearing in equation(6) must be a non-zero constant and for an isometry we haveσ=0.Hence from equation (11)it follows immediately that a=β=0andγ=σ.Sinceσis constant,equations (13)-(15)imply that b=c=?u=0.Furthermore from(16)and(17)it follows ?For full details see the?les https://www.wendangku.net/doc/7411438671.html,/~barnesa/el.red and el.log.

On the Symmetries of the Edgar-Ludwig Metric5 that d=d0?1

?2u2?f0?u(24)

2

where m0,f0and s0are arbitrary constants.Without loss of generality we may set f0=0by means of the coordinate transformation?u=u?f0/?if?=0or?y=y+f0 if?=0.Thus the metric admits a Killing vector if and only if the function V in the metric(1)can be written as

V=x2+y2+2m0x?2?uy+2s0+?2u2(25) The Killing vector has the form

ξ=?u+??y(26) This result agrees with that obtained by Ludwig and Edgar(2000)and is consistent with that of Skea(1997)who worked in a coordinate system in which M=0rather than N=1in equation(2).

For an homothety we may scaleξi so thatσ=1and then,by the coordinate transformation?u=u?2d0,we may set d0=0.The solution of equations(21)-(23) in this case is

M=m1u?2F=f1u?2+?S=s1u?4+f1?u?2+?2/2(27) where f1,m1and s1are arbitrary constants.Without loss of generality we may set ?=0by means of the coordinate transformation?y=y+?.Thus the metric admits a homothetic vector if and only if the metric function V in the metric(1)can be written as

V=x2+y2+2m1u?2x+2f1u?2y+2s1u?4(28)

6Alan Barnes

The homothetic vector has the form

ξ=?1

2

w?w+x?x+y?y(29)

This result is also consistent with that obtained by Ludwig and Edgar(2000)although they used a di?erent coordinate system from the one they used to investigate Killing vectors.In this coordinate system the function N appearing in V in equation(2)is not normalised to unity.The use of di?erent coordinate systems for the investigation of homotheties and isometries makes it di?cult to determine whether the spacetime can admit two(or more)Killing and homothetic vectors.However the analysis is straightforward if the same coordinate system is used throughout;equations(25)and (28)are satis?ed simultaneously if and only if

V=x2+y2(30) and in this case the metric(1)admits a Killing vectorξ0and a homothetic vectorξ1 given by

ξ0=?uξ1=?1

2

w?w+x?x+y?y(31)

In this case the(maximal)homothety group is two-dimensional.The group is non-Abelian as the commutator of its generators is

[ξ0ξ1]=?

1

On the Symmetries of the Edgar-Ludwig Metric7 Bradley M1986Class.Quantum Grav.3317–34

Edgar S B and Ludwig G1997Class.Quantum Grav.14L65–8

Edgar S B and Vickers J A1999Class.Quantum Grav.16589–604

Ehlers J and Kundt W1962Gravitation:an introduction to current research,ed.L Witten Wiley,New York

Hearn A C1995Reduce User’s Manual,Version3.6Rand,Santa Monica,CA

Karlhede A1980Gen.Rel.Grav.12693–707

Karlhede A and MacCallum M A H1982Gen.Rel.Grav.14673–82

Koutras A1992Class.Quantum Grav.9L143–5

Ludwig G and Edgar S B2000Class.Quantum Grav.171683–1705

Pollney D,Skea J E F and d’Inverno R A2000Class.Quantum Grav.172885–2902

Seixas W1992Class.Quantum Grav.9225–38

Skea J E F1997Class.Quantum Grav.142393-2404

Stephani H1967Commun.Math.Phys.4137–42

Wils P1989Class.Quantum Grav.61243–51

?A man J E1987Manual for CLASSI:classi?cation program for geometries in general relativity University of Stockholm,Institute of Theoretical Physics technical report

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英语造句大全English sentence 在句子中,更好的记忆单词! 1、(1)、able adj. 能 句子:We are able to live under the sea in the future. (2)、ability n. 能力 句子:Most school care for children of different abilities. (3)、enable v. 使。。。能句子:This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的句子:We must have the accurate calculation. (2)、accurately adv. 精确地 句子:His calculation is accurately. 3、(1)、act v. 扮演 句子:He act the interesting character. (2)、actor n. 演员 句子:He was a famous actor. (3)、actress n. 女演员 句子:She was a famous actress. (4)、active adj. 积极的 句子:He is an active boy. 4、add v. 加 句子:He adds a little sugar in the milk. 5、advantage n. 优势 句子:His advantage is fight. 6、age 年龄n. 句子:His age is 15. 7、amusing 娱人的adj. 句子:This story is amusing. 8、angry 生气的adj. 句子:He is angry. 9、America 美国n.

(完整版)主谓造句

主语+谓语 1. 理解主谓结构 1) The students arrived. The students arrived at the park. 2) They are listening. They are listening to the music. 3) The disaster happened. 2.体会状语的位置 1) Tom always works hard. 2) Sometimes I go to the park at weekends.. 3) The girl cries very often. 4) We seldom come here. The disaster happened to the poor family. 3. 多个状语的排列次序 1) He works. 2) He works hard. 3) He always works hard. 4) He always works hard in the company. 5) He always works hard in the company recently. 6) He always works hard in the company recently because he wants to get promoted. 4. 写作常用不及物动词 1. ache My head aches. I’m aching all over. 2. agree agree with sb. about sth. agree to do sth. 3. apologize to sb. for sth. 4. appear (at the meeting, on the screen) 5. arrive at / in 6. belong to 7. chat with sb. about sth. 8. come (to …) 9. cry 10. dance 11. depend on /upon 12. die 13. fall 14. go to … 15. graduate from 16. … happen 17. laugh 18. listen to... 19. live 20. rise 21. sit 22. smile 23. swim 24. stay (at home / in a hotel) 25. work 26. wait for 汉译英: 1.昨天我去了电影院。 2.我能用英语跟外国人自由交谈。 3.晚上7点我们到达了机场。 4.暑假就要到了。 5.现在很多老人独自居住。 6.老师同意了。 7.刚才发生了一场车祸。 8.课上我们应该认真听讲。9. 我们的态度很重要。 10. 能否成功取决于你的态度。 11. 能取得多大进步取决于你付出多少努力。 12. 这个木桶能盛多少水取决于最短的一块板子的长度。

初中英语造句

【it's time to和it's time for】 ——————这其实是一个句型,只不过后面要跟不同的东西. ——————It's time to跟的是不定式(to do).也就是说,要跟一个动词,意思是“到做某事的时候了”.如: It's time to go home. It's time to tell him the truth. ——————It's time for 跟的是名词.也就是说,不能跟动词.如: It's time for lunch.(没必要说It's time to have lunch) It's time for class.(没必要说It's time to begin the class.) They can't wait to see you Please ask liming to study tonight. Please ask liming not to play computer games tonight. Don’t make/let me to smoke I can hear/see you dance at the stage You had better go to bed early. You had better not watch tv It’s better to go to bed early It’s best to run in the morning I am enjoy running with music. With 表伴随听音乐 I already finish studying You should keep working. You should keep on studying English Keep calm and carry on 保持冷静继续前行二战开始前英国皇家政府制造的海报名字 I have to go on studying I feel like I am flying I have to stop playing computer games and stop to go home now I forget/remember to finish my homework. I forget/remember cleaning the classroom We keep/percent/stop him from eating more chips I prefer orange to apple I prefer to walk rather than run I used to sing when I was young What’s wrong with you There have nothing to do with you I am so busy studying You are too young to na?ve I am so tired that I have to go to bed early

The Kite Runner-美句摘抄及造句

《The Kite Runner》追风筝的人--------------------------------美句摘抄 1.I can still see Hassan up on that tree, sunlight flickering through the leaves on his almost perfectly round face, a face like a Chinese doll chiseled from hardwood: his flat, broad nose and slanting, narrow eyes like bamboo leaves, eyes that looked, depending on the light, gold, green even sapphire 翻译:我依然能记得哈桑坐在树上的样子,阳光穿过叶子,照着他那浑圆的脸庞。他的脸很像木头刻成的中国娃娃,鼻子大而扁平,双眼眯斜如同竹叶,在不同光线下会显现出金色、绿色,甚至是宝石蓝。 E.g.: A shadow of disquiet flickering over his face. 2.Never told that the mirror, like shooting walnuts at the neighbor's dog, was always my idea. 翻译:从来不提镜子、用胡桃射狗其实都是我的鬼主意。E.g.:His secret died with him, for he never told anyone. 3.We would sit across from each other on a pair of high

翻译加造句

一、翻译 1. The idea of consciously seeking out a special title was new to me., but not without appeal. 让我自己挑选自己最喜欢的书籍这个有意思的想法真的对我具有吸引力。 2.I was plunged into the aching tragedy of the Holocaust, the extraordinary clash of good, represented by the one decent man, and evil. 我陷入到大屠杀悲剧的痛苦之中,一个体面的人所代表的善与恶的猛烈冲击之中。 3.I was astonished by the the great power a novel could contain. I lacked the vocabulary to translate my feelings into words. 我被这部小说所包含的巨大能量感到震惊。我无法用语言来表达我的感情(心情)。 4,make sth. long to short长话短说 5.I learned that summer that reading was not the innocent(简单的) pastime(消遣) I have assumed it to be., not a breezy, instantly forgettable escape in the hammock(吊床),( though I’ ve enjoyed many of those too ). I discovered that a book, if it arrives at the right moment, in the proper season, will change the course of all that follows. 那年夏天,我懂得了读书不是我认为的简单的娱乐消遣,也不只是躺在吊床上,一阵风吹过就忘记的消遣。我发现如果在适宜的时间、合适的季节读一本书的话,他将能改变一个人以后的人生道路。 二、词组造句 1. on purpose 特意,故意 This is especially true here, and it was ~. (这一点在这里尤其准确,并且他是故意的) 2.think up 虚构,编造,想出 She has thought up a good idea. 她想出了一个好的主意。 His story was thought up. 他的故事是编出来的。 3. in the meantime 与此同时 助记:in advance 事前in the meantime 与此同时in place 适当地... In the meantime, what can you do? 在这期间您能做什么呢? In the meantime, we may not know how it works, but we know that it works. 在此期间,我们不知道它是如何工作的,但我们知道,它的确在发挥作用。 4.as though 好像,仿佛 It sounds as though you enjoyed Great wall. 这听起来好像你喜欢长城。 5. plunge into 使陷入 He plunged the room into darkness by switching off the light. 他把灯一关,房

改写句子练习2标准答案

The effective sentences:(improve the sentences!) 1.She hopes to spend this holiday either in Shanghai or in Suzhou. 2.Showing/to show sincerity and to keep/keeping promises are the basic requirements of a real friend. 3.I want to know the space of this house and when it was built. I want to know how big this house is and when it was built. I want to know the space of this house and the building time of the house. 4.In the past ten years,Mr.Smith has been a waiter,a tour guide,and taught English. In the past ten years,Mr.Smith has been a waiter,a tour guide,and an English teacher. 5.They are sweeping the floor wearing masks. They are sweeping the floor by wearing masks. wearing masks,They are sweeping the floor. 6.the drivers are told to drive carefully on the radio. the drivers are told on the radio to drive carefully 7.I almost spent two hours on this exercises. I spent almost two hours on this exercises. 8.Checking carefully,a serious mistake was found in the design. Checking carefully,I found a serious mistake in the design.

用以下短语造句

M1 U1 一. 把下列短语填入每个句子的空白处(注意所填短语的形式变化): add up (to) be concerned about go through set down a series of on purpose in order to according to get along with fall in love (with) join in have got to hide away face to face 1 We’ve chatted online for some time but we have never met ___________. 2 It is nearly 11 o’clock yet he is not back. His mother ____________ him. 3 The Lius ___________ hard times before liberation. 4 ____________ get a good mark I worked very hard before the exam. 5 I think the window was broken ___________ by someone. 6 You should ___________ the language points on the blackboard. They are useful. 7 They met at Tom’s party and later on ____________ with each other. 8 You can find ____________ English reading materials in the school library. 9 I am easy to be with and _____________my classmates pretty well. 10 They __________ in a small village so that they might not be found. 11 Which of the following statements is not right ____________ the above passage? 12 It’s getting dark. I ___________ be off now. 13 More than 1,000 workers ___________ the general strike last week. 14 All her earnings _____________ about 3,000 yuan per month. 二.用以下短语造句: 1.go through 2. no longer/ not… any longer 3. on purpose 4. calm… down 5. happen to 6. set down 7. wonder if 三. 翻译: 1.曾经有段时间,我对学习丧失了兴趣。(there was a time when…) 2. 这是我第一次和她交流。(It is/was the first time that …注意时态) 3.他昨天公园里遇到的是他的一个老朋友。(强调句) 4. 他是在知道真相之后才意识到错怪女儿了。(强调句) M 1 U 2 一. 把下列短语填入每个句子的空白处(注意所填短语的形式变化): play a …role (in) because of come up such as even if play a …part (in) 1 Dujiangyan(都江堰) is still ___________in irrigation(灌溉) today. 2 That question ___________ at yesterday’s meeting. 3 Karl Marx could speak a few foreign languages, _________Russian and English. 4 You must ask for leave first __________ you have something very important. 5 The media _________ major ________ in influencing people’s opinion s. 6 _________ years of hard work she looked like a woman in her fifties. 二.用以下短语造句: 1.make (good/full) use of 2. play a(n) important role in 3. even if 4. believe it or not 5. such as 6. because of

英语造句

English sentence 1、(1)、able adj. 能 句子:We are able to live under the sea in the future. (2)、ability n. 能力 句子:Most school care for children of different abilities. (3)、enable v. 使。。。能 句子:This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的 句子:We must have the accurate calculation. (2)、accurately adv. 精确地 句子:His calculation is accurately. 3、(1)、act v. 扮演 句子:He act the interesting character.(2)、actor n. 演员 句子:He was a famous actor. (3)、actress n. 女演员 句子:She was a famous actress. (4)、active adj. 积极的 句子:He is an active boy. 4、add v. 加 句子:He adds a little sugar in the milk. 5、advantage n. 优势 句子:His advantage is fight. 6、age 年龄n. 句子:His age is 15. 7、amusing 娱人的adj. 句子:This story is amusing. 8、angry 生气的adj. 句子:He is angry. 9、America 美国n. 句子:He is in America. 10、appear 出现v. He appears in this place. 11. artist 艺术家n. He is an artist. 12. attract 吸引 He attracts the dog. 13. Australia 澳大利亚 He is in Australia. 14.base 基地 She is in the base now. 15.basket 篮子 His basket is nice. 16.beautiful 美丽的 She is very beautiful. 17.begin 开始 He begins writing. 18.black 黑色的 He is black. 19.bright 明亮的 His eyes are bright. 20.good 好的 He is good at basketball. 21.British 英国人 He is British. 22.building 建造物 The building is highest in this city 23.busy 忙的 He is busy now. 24.calculate 计算 He calculates this test well. 25.Canada 加拿大 He borns in Canada. 26.care 照顾 He cared she yesterday. 27.certain 无疑的 They are certain to succeed. 28.change 改变 He changes the system. 29.chemical 化学药品

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