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Comment on `On the next-to-leading order gravitational spin(1)-spin(2) dynamics' by J. Stei

Comment on `On the next-to-leading order gravitational spin(1)-spin(2) dynamics' by J. Stei
Comment on `On the next-to-leading order gravitational spin(1)-spin(2) dynamics' by J. Stei

a r X i v :0712.2032v 1 [g r -q c ] 12 D e c 2007Comment on ‘On the next-to-leading order gravitational spin(1)-spin(2)dynamics’by

J.Steinho?et al.

Rafael A.Porto 1and Ira Z.Rothstein 2

1Physics Department,University of California,Santa Barbara

Santa Barbara,93106USA 2Carnegie Mellon University Dept.of Physics,

Pittsburgh PA 15213,USA

In this comment we explain the discrepancy found between the results in arXiv:0712.1716v1for the 3PN spin-spin potential and those previously derived in gr-qc/0604099.We point out that to compare one must include sub-leading lower order spin-orbit e?ects which contribute to the spin-spin potential once one transforms to the PN frame.When these e?ects are included the results in arXiv:0712.1716v1do indeed reproduce those found in gr-qc/0604099.In [1]we applied recently developed E?ective Field Theory (EFT)techniques [2,4]to compute the next to leading order (NLO)potential due to spin(1)-spin(2)interactions in the Newton-Wigner (NW)spin supplementarity condition (SSC).In [5]the NLO spin(1)-spin(2)was also calculated.Given our claim that,up to 4PN order the equations of motion in the spin(1)-spin(2)sector can be obtained via the traditional Hamiltonian approach [1,6],the authors then searched for the canonical transformation necessary to go between the results,?nding that no such transformation exists.However,in order to compare one must also consider spin-orbit e?ects,including a subleading one which becomes a spin(1)spin(2)-orbit term once written in terms of the coordinate velocity in the post-Newtonian one.In fact,once this new term is added,the result in [5]do indeed reproduce the potential calculated in [1].The spin-orbit 1.5PN potential is given by [4,6]V so 1.5P N =G N m 22ωab μS ab with ωab μthe Ricci rotation coe?cients.Expanding around ?at space we have e b μ=δb μ+h νμdt ,then reads S i 01=12S ij 1e j 0( x 1)+...=12r 2 ( n × S 2)× S 1 i +...,(2)where we have e j 0=H j 0/2=

G N 2h 0i,j S

ij [1,4],or by simple inspection of the Kerr metric in harmonic gauge.A similar term follows for particle 2.

In the EFT approach [1,2,4,6]each vertex scales with di?erent power of spin,and velocity,and therefore spin(1)-spin(2)interactions are those for which there is a spin tensor on each vertex.On the other hand spin-orbit terms are those for which the spin of one of the bodies interacts with the motion of the companion.As we can see from (1)and

(2),to obtain the full S 1S 2Hamiltonian up to 3PN we need to include a spin(1)spin(2)-orbit term given by

G 2

N

2r 4 ( S 1× n )·( n × S 2) =G 2N

M

2 and we thus get

V s1s2=?G N

2

v1· v2?3 v1· n v2· n? v21+ v22 ? S1· v1 S2· v2?3

2

( v1× S1)· n( v2× S2)· n

?6( v1× S2)· n( v2× S1)· n +G2N M r3 S1· S2?3 S1· n S2· n ,(4) where we also added last,the LO spin(1)-spin(2)potential at2PN order.

We can now show that the canonical transformation generated by(b=c=1/2in[5])

g=G N

m2

S

1· p2

S

2· n?

1

2r2 1m2 S2· S1 p2· n

,(5)

leads toδH NLO

SS =0at each order in G N.Thus the results in[5]do indeed reproduce those in[1].In a forthcoming

paper we will present full details of the spin-spin calculation.

[1]R.A.Porto and I.Z.Rothstein,Phys.Rev.Lett.97,021101(2006)

[2]W.Goldberger and I.Z Rothstein,Phys.Rev.D73,104029(2006).

[3]W.D.Goldberger and I.Z.Rothstein,Phys.Rev.D73,104030(2006)[arXiv:hep-th/0511133].“Les Houches lectures on

e?ective?eld theories and gravitational radiation,”arXiv:hep-ph/0701129.

[4]R.A.Porto,Phys.Rev.D73,104031(2006)

[5]J.Steinho?,S.Hergt and G.Schafer,arXiv:0712.1716v1[gr-qc].

[6]R.A.Porto,arXiv:gr-qc/0701106.

[7]A.Hanson and T.Regge,Ann.Phys.(N.Y.)87,498(1974).

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吞风吻雨葬落日未曾彷徨 欺山赶海践雪径也未绝望 拈花把酒偏折煞世人情狂 凭这两眼与百臂或千手不能防 天阔阔雪漫漫共谁同航 这沙滚滚水皱皱笑着浪荡 贪欢一饷偏教那女儿情长埋葬 吞风吻雨葬落日未曾彷徨 欺山赶海践雪径也未绝望 拈花把酒偏折煞世人情狂 凭这两眼与百臂或千手不能防 天阔阔雪漫漫共谁同航 这沙滚滚水皱皱笑着浪荡 贪欢一饷偏教那女儿情长埋葬 吞风吻雨葬落日未曾彷徨 欺山赶海践雪径也未绝望 拈花把酒偏折煞世人情狂 凭这两眼与百臂或千手不能防 天阔阔雪漫漫共谁同航 这沙滚滚水皱皱笑着浪荡 贪欢一饷偏教那女儿情长埋葬 二、《倚天屠龙记》——马景涛版片头曲:刀剑如梦——周华健 我剑何去何从 爱与恨情难独钟 我刀划破长空 是与非懂也不懂 我哭泪洒心中 悲与欢苍天捉弄 我笑我狂我疯 天与地风起云涌 我醉一片朦胧 恩和怨是幻是空 我醒一场春梦 生与死一切成空 来也匆匆去也匆匆 恨不能相逢 爱也匆匆恨也匆匆 一切都隋风

英语作文:关于共享单车的4篇范文

英语作文:关于共享单车的4篇范文 共享单车确实是今年最热的话题之一,很多小伙伴们留言问共享单车要怎么写英语作文,这一份写作指南想必大家都很需要。 词汇篇 bicycle-sharing /sharing bike /shared bikes 共享单车 bicycle /bike /cycle 自行车 ride a bicycle /take a ride /cycling /cycle /bicycle /bike 骑自行车 park 停车 Bluegogo 小蓝单车 Ofo bicycle 小黄车 Mobike 摩拜单车 APP 应用程序(全称为application) scan the QR code

扫描二维码 unlock the smart lock 解开智能锁 timer 计时器 user 用户 smartphone 智能手机 keep fit 健身 environmental protection/ environment friendly/ help the environment 环保的 In full swing 如火如荼 brightly coloured 色彩鲜艳 The combination of technology, creativity and market forces 科技、创意、市场力量的结合 has changed the image of the city 改变了城市形象

curb carbon dioxide emissions 减少二氧化碳排放 reduce road congestion 减少道路拥堵的情况 random parking 随机停车 illegal parking 乱停乱放 bicycle damage 车辆损坏 traffic violation 违章 social mind 公德心 quality 素质 范文篇 (一) 假定你是红星中学初三学生李华。你的美国朋友Jim在给你的邮件中提到他对中国新近出现的一种共享单车“mobike”很感兴趣,并请你做个简要介绍。请你给Jim回信,内容包括: 1. 这种单车的使用方法(如:APP查看车辆、扫码开锁等);

神话情话 歌词

神话情话歌词 神话情话 演唱:齐豫、周华健 合:爱是愉快是难过是陶醉是情绪或在日后视作传奇爱是盟约是习惯是时间是白发也叫你我乍惊乍喜完全遗忘自己 竟可相许生与死 来日谁来问起 天高风急双双远飞 爱是微笑是狂笑是傻笑是玩笑或是为着害怕寂寥爱是何价是何故在何世又何以对这世界雪中送火谁还祈求什么 可歌可泣的结果 谁能承受后果 翻天覆海不枉当初 啊……啊…… 合:有你有我雪中送火男:爱在迷迷糊糊盘古初开便开始 这浪浪漫漫旧故事 女:爱在朦朦胧胧前生今生和他生 怕错过了也不会知 男:跌落茫茫红尘南北西东亦相依 怕独自活着没意义 女:爱是来来回回情丝一丝又一丝 合:至你与我此生永惜别时

合:爱是微笑是狂笑是傻笑是玩笑或是为着害怕寂寥爱是何价是何故在何世又何以对这世界雪中送火谁还祈求什么 可歌可泣的结果 谁能承受后果 翻天覆海不枉当初 啊……啊…… 合:有你有我雪中送火男:爱在迷迷糊糊盘古初开便开始 这浪浪漫漫旧故事 女:爱在朦朦胧胧前生今生和他生 怕错过了也不会知 男:跌落茫茫红尘南北西东亦相依 怕独自活着没意义 女:爱是来来回回情丝一丝又一丝 合:至你与我此生永不问别时 男:爱在迷迷糊糊盘古初开便开始 这浪浪漫漫旧故事 女:爱在朦朦胧胧前生今生和他生 怕错过了也不会知 男:跌落茫茫红尘南北西东亦相依 怕独自活着没意义 女:爱是来来回回情丝一丝又一丝 合:至你与我此生永不惜别时 女):这次是我真的决定离开,远离那些许久不懂的悲哀,想让你忘却愁绪,忘记关怀,放开这纷纷扰扰自由自在。

(男):那次是你不经意的离开,成为我这许久不变的悲哀,于是淡漠了繁华,无法再开怀,于是我守着寂寞不能归来。 (合):啊.......拥起落落余晖任你采摘; 啊.......留住刹那永远为你开。 (女):这次是我真的决定离开,远离那些许久不懂的悲哀,想让你忘却愁绪,忘记关怀,放开这纷纷扰扰自由自在。 (男):那次是你不经意的离开,成为我这许久不变的悲哀,于是淡漠了繁华,无法再开怀,要陪你远离寂寞自由自在。 (合):啊.......拥起落落余晖任你采摘; 啊.......留住刹那永远为你开。

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Moreover, there will be far more pollution resulted from cars. Once a car started, it release huge amount of dirty gas and heat, which lead to great pollution and result in green house effect. 再者,由于私家车将会有更多的污染。一旦汽车起动,它释放大量的脏气和热量,这会导致很大的污染和温室效应。 All in all,Chinesepeople should take public transpotation into consideration first。 总之,中国人应该以公共交通工具为第一选择。 Too many cars have caused a lot of serious problems in our world. Besides congestion, aidents and fast fuel consumption, cars are responsible for a good part of air pollution in big cities. One possible solution is to design and develop clean cars and clean fuels. In Shanghai, some of the public buses begin to run on natural gas, which does not give off as much carbon dioxide as petrol. Another solution is to develop modern public transportation systems and restrict the use of private cars. If the price of

歌词总汇

叶倩文.潇洒走一回 天地悠悠 过客匆匆 潮起又潮落 恩恩怨怨 生死白头 几人能看透 红尘啊滚滚 痴痴啊情深 聚散终有时 留一半清醒留一半醉至少梦里有你追随我拿青春赌明天 你用真情换此生 岁月不知人间 多少的忧伤 何不潇洒走一回 **** 天地悠悠 过客匆匆 潮起又潮落 恩恩怨怨 生死白头 几人能看透 红尘啊滚滚 痴痴啊情深 聚散终有时 留一半清醒留一半醉至少梦里有你追随我拿青春赌明天 你用真情换此生 岁月不知人间 多少的忧伤 何不潇洒走一回 红尘啊滚滚 痴痴啊情深 聚散终有时 留一半清醒留一半醉至少梦里有你追随我拿青春赌明天你用真情换此生 岁月不知人间 多少的忧伤 何不潇洒走一回 岁月不知人间 多少的忧伤 何不潇洒走一回 啦……啦…… 啦……啦…… 踏浪 小小的一片云呀慢慢地走过来 请你们歇歇脚呀暂时停下来 山上的山花儿开呀我才到山上来 原来嘛你也是上山看那山花儿开啦……啦…… 啦……啦…… 小小的一阵风呀慢慢的走过来 请你们歇歇脚呀暂时停下来 海上的浪花儿开呀我才到海边来 原来嘛你也爱浪花才到海边来啦……啦…… 啦……啦…… 小小的一片云呀慢慢地走过来 请你们歇歇脚呀暂时停下来 山上的山花儿开呀我才到山上来 原来嘛你也是上山看那山花儿开啦……啦…… [00:01.00]一笑中*高正洋 [00:01.49]蝶儿闯入我梦我在蝶梦之中[00:06.12]是梦是醒有什么不同[00:10.58]辗转滚滚红尘流动朗朗时空[00:17.66]谁是谁非由谁来搬弄[00:23.69]相思纠缠不停终究是空[00:26.12]痴痴恋恋哭哭笑笑 [00:28.99]爱情一声叹息盼一次相逢[00:32.01]聚聚散散浮云浮萍浮生如梦[00:39.18]虚虚幻幻人世人间人生几何[00:44.97]看风云过尽人生起落天地有情 [00:50.57]何必问恩怨对与错爱恨情仇都付于

歌词)

动力火车-当 当还珠格格主题曲(动力火车) 喔...喔..喔.喔.喔 喔...喔..喔.喔.喔 喔...喔..喔.喔.喔 喔...喔..喔 当山峰没有棱角的时候 当河水不再流 当时间停住日夜不分 当天地万物化为虚有 我还是不能和你分手 不能和你分手 你的温柔是我今生最大的守候当太阳不再上升的时候 当地球不再转动 当春夏秋冬不再变化 当花草树木全部凋残 我还是不能和你分散 不能和你分散 你的笑容是我今生最大的眷恋让我们红尘作伴活的潇潇洒洒策马奔腾共享人世繁华 对酒当歌唱出心中喜悦 轰轰烈烈把握青春年华 让我们红尘作伴活的潇潇洒洒策马奔腾共享人世繁华 对酒当歌唱出心中喜悦 轰轰烈烈把握青春年华

大约在冬季 轻轻地我将离开你 请将眼角的泪拭去 漫漫长夜里未来日子里亲爱的你别为我哭泣 前方的路虽然太凄迷 请在笑容里为我祝福 虽然迎着风虽然下着雨我在风雨之中念着你 没有你的日子里 我会更加珍惜自己 没有我的岁月里 你要保重你自己 你问我何时归故里 我也轻声地问自己 不是在此时不知在何时我想大约会是在冬季 不是在此时不知在何时我想大约会是在冬季

恋曲1980 你曾经对我说你永远爱着我 爱情这东西我明白但永远是什么姑娘你别哭泣我俩还在一起 今天的欢乐将是明天创痛的回忆啦啦啦啦啦啦啦啦啦啦啦啦 今天的欢乐将是明天创痛的回忆什么都可以抛弃什么也不能忘记现在你说的话都只是你的勇气春天刮着风秋天下着雨 春风秋雨多少海誓山盟随风远去啦啦啦啦啦啦啦啦啦啦啦啦 亲爱的莫再说你我永远不分离 你不属于我我也不拥有你 姑娘世上没有人有占有的权利 或许我们分手就这么不回头 至少不用编织一些美丽的藉口 啦啦啦啦啦啦啦啦啦啦啦啦 亲爱的或许明天我们要分离 啦啦啦啦啦啦啦啦啦啦啦啦 亲爱的莫再说你我永远不分离 啦啦啦啦啦啦啦啦啦啦啦啦 亲爱的莫再说你我明天要分离 啦啦啦啦啦啦啦啦啦啦啦啦 亲爱的莫再说你我永远不分离 恋曲1990罗大佑 乌溜溜的黑眼珠和你的笑脸 怎么也难忘记你容颜的转变 轻飘飘的旧时光就这么溜走 转头回去看看时已匆匆数年 苍茫茫的天涯路是你的飘泊 寻寻觅觅长相守是我的脚步 黑漆漆的孤枕边是你的温柔 醒来时的清晨里是我的哀愁 或许明日太阳西下倦鸟已归时 你将已经踏上旧时的归途 人生难得再次寻觅相知的伴侣 生命终究难舍蓝蓝的白云天 轰隆隆的雷雨声在我的窗前 怎么也难忘记你离去的转变 孤单单的身影后寂寥的心情 永远无怨的是我的双眼

歌词

什么时候谁答应过,天亮会否定所有黑夜,我们拥有的,多不过付出的一切。王菲不爱我的我不爱 情字何解怎落笔都不对,而我独缺你一生的了解。周杰伦兰亭序 我们今生注定是沧桑,哭着来要笑着走过呀。朋友啊让我们一起牢牢铭记呀,我们今生兄弟情谊长。旭日阳刚今生缘 只因为在人群中多看了你一眼,再也没能忘掉你的容颜。王菲传奇 宁愿用这一生等你发现,我一直在你身旁,从未走远。王菲传奇 天空越蔚蓝越怕抬头看,电影越圆满就越觉得伤感。刘若英一辈子的孤单 要不是沧海桑田真爱怎么会浮现。待渡过斜风冷雨春暖在眼前。SHE 一眼万年 你的泪水化为漫天飞舞的彩蝶,爱是翼下之风两心相随自在飞。孙楠韩红美丽的神话 我们的故事并不算美丽,却如此难以忘记。金莎星月神话 如果当初勇敢地在一起,会不会不同结局,你会不会也有千言万语,埋在沉默的梦里。金莎星月神话 愿心中,永远留着我的笑容,伴你走过每一个春夏秋冬。张学友祝福 人生几何,能够得到知己,失去生命的力量也不可惜。邓丽君我只在乎你 人有悲欢离合月有阴晴圆缺,此事古难全。但愿人长久千里共婵娟。王菲但愿人长久 给我一万年一两岁也都无所谓,有些爱逃不出天网恢恢。周迅爱恨恢恢 听海哭的声音,叹息著谁又被伤了心却还不清醒。张惠妹听海 我又回头去追去醉,就算我追到最后只剩冰雪,天都为我伤悲冷的爱快枯萎,任漫天风雪覆盖我的心碎。熊天平雪候鸟 只能用笑容,期待着雨过天晴的彩虹。蔡依林天空 谁能告诉我,有没有这样的笔,能画出一双双不流泪的眼睛。孙楠只要有你 追寻,我生命的那份纯真,心中抹不去的那一片云彩,追寻那永远属于我们的那份无悔的忠贞。我苦苦追寻,那人世间的大爱无疆,大道无垠。孙楠追寻 这是一个多美丽又遗憾的世界,我们就这样抱着笑着还流着泪。朴树生如夏花 一路春光啊,一路荆棘呀,惊鸿一般短暂,如夏花一样绚烂。朴树生如夏花 最懂我的人,谢谢一路默默的陪着我,让我拥有好故事可以说。信乐团海阔天空 若生命直到这里,从此没有我,我会找个天使替我去爱你。安琥天使的翅膀 盼了好久终於盼到今天,忍了好久终於把梦实现,那些不变的风霜早就无所谓,我累也不说累。刘德华今天 凡人却视而不见规矩定方圆,悟性悟觉悟空心甘情愿。简简单单陪你走好每一天。刘德华悟如果这是通往爱的旅途,也许过程注定要荆棘密布,但我不后悔选择这条路,你的爱让我深深体会活着的感触。梁静茹路 穿越千年的哀愁,是你在尽头等我,最美丽的感动,会值得用一生守候。飞儿乐团千年之恋 爱是一步一步坚强,奋不顾身的绽放。飞儿乐团荆棘里的花 我穿越伤害,最美的答案,是你的爱。飞儿乐团刺鸟 死了都要爱,不哭到微笑不痛快,宇宙毁灭心还在。信乐团死了都要爱 想留不能留才最寂寞,没说完温柔只剩离歌。信乐团离歌 我愿意为你我愿意为你,我愿意为你被放逐天际,只要你真心拿爱与我回应,什么都愿意,什么都愿意为你。王菲我愿意 等到风景都看透,也许你会陪我看细水长流。王菲红豆 如果说分手是苦痛的起点,那在终点之前我愿意再爱一遍,想要对你说的不敢说的爱,会不会有人可以明白。————周杰伦轨迹

智能汽车的英语作文(3篇)

智能汽车的英语作文(3篇) 篇三:智能汽车的英语作文 In 2080, cars are going to be different. The car of new type is made from China. The motor is quicker than other cars. The car can be droved 500 kilometers per hour. Computers are going to be inside every car. It is going to control the car. You don’t need to be worried about the red light. The car is going to go on special tracks above the city. It would be not so crowded and noisy. There is a special air conditioner in the car. In winter you can be warm there. In summer you can be pleasantly cool. If you have this car, you will be very fashionable. 篇二:智能汽车 Smart carsMeters, Leo the library cardEven the past 70 years has changed little basically the car industry, will also feel thein... zhi neng qi che ZHI NENG car Intelligent Automobile.Intelligent Car is a computer, such as the latest scientific and technologicalachievements and modern product of the combination of the auto industry, which "understanding." Usually on autopilot, automatic transmission, and ev... The petroleum is also useful one day, sooner or later must by other energy substitution, the electric car not have the air pollution, the noise quite to be also small, the operation is quite convenient, moreover, the electric c... Fly think of Carle Cup National University intelligent car contest. 篇一:智能汽车的英语作文 Intelligent Car is a computer, such as the latest scientific and technological achievements and modern product of the combination of the auto industry, which "understanding." Usually on autopilot, automatic transmission, and even with the automatic identification of the function of the road. In addition, the car has a range of ancillary facilities should be computerized, and often gives novelty.

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