Schwarzschild black hole with global monopole charge

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《在月亮下面》-英汉对照14544

简介太阳是一个巨大的燃烧着的火球。

没有来自太阳的光和热，我们就不能在地球这颗行星上生存。

在这颗行星周围有臭氧层保护着我们免受太阳之火的伤害。

但是如果臭氧层破裂，将会怎样呢？现在臭氧层上已经有了小洞，有人说那些小洞会变得越来越大——直到有一天再没有任何东西挡在我们和太阳的火焰之间。

事情发生在2222年。

地球人制造了AOL——人工臭氧层。

美丽的地球又有了1000年的生命。

树又发芽了，下雨了，河里有了水……但现在是2522年，人工臭氧层已经破裂。

幼嫩的树林开场枯萎，河流也逐渐干涸。

凯和瑞拉从他们的宇宙飞船里看到了这危险的情景，可他们又能做些什么呢？月亮下面的部族里住着他们的朋友，可是路途遥远，而地球首领高格又不肯听从他们的指挥。

本书的作者罗维纳·阿金耶米是英国人。

曾在非洲居住和工作了多年。

现在她在剑桥工作和生活。

1 AOLFive hundred kilometres over Europe，ShipOM-45 moved north．In a room at the backof the ship，Kiah watched the numbers onthe computer in front of him．‘Time for dinner，’Rillasaid．The numbers changed quickly and Kiah's eyes didn't move．Rillawent across the room to his table．Shebegan to watch the numbers，too．‘What's wrong with thesatellite？’she asked．Shewas a beautiful girl，about twenty yearsold，with long black hair and big eyes．‘Nothing's wrong with thesatellite，’Kiah answered quietly．‘It'sthe AOL．’He began to write the numbersin the book on his table．Suddenly，the numbersstopped changing．Kiah looked at Rilla．‘OverEurope，’he said．‘It'shappening．The AOL is breaking up．Thereare big holes in the AOL and they're getting bigger．’‘You're right！Shallwe see Captain Seru now，before dinner？’Kiah stood up．He wasnearly two metres tall，with dark eyesand hair．‘Yes，comeon，’he said．Quickly，they went toCaptain Seru's room．They waited at thedoor．‘Come in！’captainSeru called．She was a little woman witha fat face．‘Come in！Wouldyou like a drink？’‘No，thankyou，’Kiah answered．‘I'dlike you to look at these numbers．’Kiahgave Captain Seru his book．1 人工臭氧层OM-45号宇宙飞船在欧洲上空500公里的高度向北飞行。

托福听力科学美国人60秒：黑洞三兄弟

托福考试(TOEFL)/备考辅导2017年托福听力科学美国人60秒：黑洞三兄弟【导语】为了帮助大家高效备考托福，熟悉托福听力，小编为大家带来2017年托福听力科学美国人60秒，希望对大家托福备考有所帮助。

更多精彩尽请关注小编！科学美国人60秒中英文翻译：黑洞三兄弟科学美国人60秒英文文本This is Scientific American 60 Secomds, Space. I'm Clara Moskowitz, got a minute?Inside most galaxies a supermassive black hole lurks. But one galaxy about 4 billion light-years from us was recently discovered to have not one, not even two, but three gigantic black holes at its center.Such triple systems appear to be extremely rare—only four are known. The newfound system includes two black holes orbiting each other very closely, about 450 light-years apart, with a third black hole a bit farther out. The central pair zoomaround each other at a fast clip, about 300 times the speed of sound on Earth. The hole trinity also represents the tightest trio of black holes known to date. It's described in thejournal Nature.As these objects continue to orbit at the center of their galaxy, gravity will eventuallypull them closerand closer together. Ultimately, they may even merge. Researchers hope this triple-black-hole system may be a good place to look for space-time ripples called gravitational waves. As their orbits shrink, the black holes should radiate away some of their orbital energy as the sought-after gravitational waves, predicted by Einstein a century ago.Thanks for the minute for Scientific American 60 Secomds, Tech. I'm Clara Moskowitz.2017年托福听力科学美国人60秒：黑洞三兄弟.doc [全文共767字] 编号：8535578。

史瓦西黑洞中的乌龟坐标的定义

史瓦西黑洞（Shwarzchild Black Hole）是一种理论天体，是由德国物理学家史瓦西（Karl Schwarzschild）在1916年根据爱因斯坦的广义相对论得到的一个解。

它描述了一个质量很大的天体，它的引力非常强大，甚至连光也无法从它的引力范围中逃逸，因此被称为“黑洞”。

在史瓦西黑洞的理论模型中，天体通常被描述为一个点状的引力源，也就是所谓的“奇点”。

奇点的周围有一个叫做“事件视界”的区域，从这个区域向外的任何物体都无法逃离黑洞的引力。

当物体越接近史瓦西黑洞的奇点时，它的引力场就越强大，以至于最终无法逃脱。

这种情况被称为“信息的丢失”，因为一旦物体穿过了事件视界，就再也无法传递任何信息到宇宙的其他部分。

史瓦西黑洞中的乌龟坐标系统是专门用来描述黑洞空间中物体的位置和运动的一种坐标系。

它采用了爱因斯坦引力场方程的球对称解，以便描述黑洞的引力场。

乌龟坐标系的引入使得我们可以更加方便地描述史瓦西黑洞内外的空间结构和物体的运动规律。

接下来，我们将详细介绍史瓦西黑洞中的乌龟坐标的定义和特点：1. 乌龟坐标系的引入史瓦西黑洞中的乌龟坐标系是通过对黑洞的度规进行适当的变换得到的。

在这个坐标系中，我们可以用一组坐标来描述出黑洞事件视界以内外的空间结构，并且最大限度地简化了爱因斯坦引力场方程的求解过程。

乌龟坐标系的引入使得我们可以更加清晰地看到黑洞内外的引力场结构，更加方便地描述物体的位置和运动状态。

2. 乌龟坐标系的特点乌龟坐标系最大的特点是它的“径向坐标”和“角向坐标”是独立的。

也就是说，我们可以将空间的度规化简为径向和角向两个部分，从而更加方便地描述空间的曲率和引力场的分布。

这使得乌龟坐标系成为了研究黑洞引力场结构的理想工具。

3. 乌龟坐标系的应用乌龟坐标系在研究黑洞引力场结构、分析黑洞内外物体的运动规律等方面都发挥了重要的作用。

通过乌龟坐标系，我们可以更加清晰地看到黑洞的引力场结构，预测物体穿越黑洞事件视界的情况，以及研究黑洞辐射等重要问题。

Black Hole(黑洞的科普,选修课)

超大质量黑洞从吸积盘中吸积的概念图
Part.8 结语
Classification of black holes
分类质量大小
超重黑洞
~105–1010 M太阳
~0.001–400 AU
中介质量黑洞
~103 M太阳
~103 km ≈ R地球
恒星黑洞
~10 M太阳
~30 km
微型黑洞
up to ~M月球
up to ~0.1 mm
质量达到太阳十倍的黑洞
吸积accretion by black hole
黑洞通常是因为它们聚拢周围的气体产生辐射而被发现的，这一过程被称为吸积。
黑洞拉伸，撕裂并吞噬恒星恒星被黑洞吞噬
蒸发 Black hole evaporation
20世纪70年代，英国科学家霍金等人以量子力学为基础，对黑洞作了更缜密的考察，结果发现黑洞会像“蒸发”那样稳定地往外发射粒子。
Black Hole
Introduction to black hole
黑洞是时空曲率大到光都法从其视界逃脱的天体
Physicists
米切尔预言的黑洞
历史上，第一个意识到一个致密天体密度可以大到连光都无法逃逸的人是英国地理学家John Michell。他在1783年写给亨利· 卡文迪什一封信中提出这个想法的，他认为一个和太阳同等质量的天体，如果半径只有3公里，那么这个天体是不可见的，因为光无法逃离天体表面。
坍缩和爆发的压力进而使核心被压缩成致密的星核，而致密星核又分化为白矮星、中子星、黑洞。
视频：黑洞的产生
中子星的奥本海默极限
奥本海默（1904～1967）在1939年研究提出
如果中子星的质量超过3.2 m⊙(太阳质量的3.2倍)（精细的模型给出值在2～3 m⊙之间），则其的中子间的泡利斥力就再也阻挡不住星体引力坍缩，会进一步被压缩成一个体积很小而质量巨大的高密度引力源——黑洞。上述3.2 m⊙（常记作3）的中子星界限被称作“奥本海默极限”，任何超此极限的恒星都难以停留在中子星阶段。

高中语文人教版必修3 4.13黑洞简介

黑洞简介黑洞（Black hole）是现代广义相对论中，宇宙空间内存在的一种密度无限大，体积无限小的天体，所有的物理定理遇到黑洞都会失效。

1916年，德国天文学家卡尔·史瓦西（Karl Schwarzschild，1873～1916年）通过计算得到了爱因斯坦引力场方程的一个真空解，这个解表明，如果将大量物质集中于空间一点，其周围会产生奇异的现象，即在质点周围存在一个界面——“视界”一旦进入这个界面，即使光也无法逃脱。

这种“不可思议的天体”被美国物理学家约翰·阿奇巴德·惠勒（John Archibald Wheeler）命名为“黑洞”。

“黑洞是时空曲率大到光都无法从其视界逃脱的天体”。

黑洞是由质量足够大的恒星在核聚变反应的燃料耗尽而死亡后，发生引力坍缩产生的。

黑洞的质量极其巨大，而体积却十分微小，它产生的引力场极为强劲，以至于任何物质和辐射在进入到黑洞的一个事件视界（临界点）内，便再无法逃脱，甚至目前已知的传播速度最快的光（电磁波）也逃逸不出。

黑洞无法直接观测，但可以借由间接方式得知其存在与质量，并且观测到它对其他事物的影响。

借由物体被吸入之前的因高热而放出紫外线和X射线的“边缘讯息”，可以获取黑洞存在的讯息。

推测出黑洞的存在也可借由间接观测恒星或星际云气团绕行轨迹取得位置以及质量。

科学家最新研究理论显示，当黑洞死亡时可能会变成一个“白洞”，它不像黑洞吞噬邻近所有物质，而是喷射之前黑洞捕获的所有物质。

演化过程黑洞就是中心的一个密度无限大、时空曲率无限高、体积无限小的奇点和周围一部分空空如也的天区，这个天区范围之内不可见。

依据阿尔伯特-爱因斯坦的相对论，当一颗垂死恒星崩溃，它将聚集成一点，这里将成为黑洞，吞噬邻近宇宙区域的所有光线和任何物质。

黑洞的产生过程类似于中子星的产生过程：某一个恒星在准备灭亡，核心在自身重力的作用下迅速地收缩，塌陷，发生强力爆炸。

当核心中所有的物质都变成中子时收缩过程立即停止，被压缩成一个密实的星体，同时也压缩了内部的空间和时间。

Hydrodynamics and global structure of rotating Schwarzschild black holes

a r X i v :g r -q c /0503079v 1 18 M a r 2005Hydrodynamics and global structure of rotating Schwarzschild black holesSoon-Tae Hong ∗Department of Science Education,Ewha Womans University,Seoul 120-750KoreaSung-Won Kim †Department of Science Education,Ewha Womans University,Seoul 120-750Korea andAsia Paciﬁc Center for Theoretical Physics,Pohang 790-784Korea(Dated:February 7,2008)Exploiting a rotating Schwarzschild black hole metric,we study hydrodynamic properties of per-fect ﬂuid whirling inward toward the black holes along a conical surface.On the equatorial plane of the rotating Schwarzschild black hole,we derive radial equations of motion with eﬀective poten-tials and the Euler equation for steady state axisymmetric fuid.Moreover,numerical analysis is performed to ﬁgure out eﬀective potentials of particles on the rotating Schwarzschild manifolds in terms of angular velocity,total energy and angular momentum per unit rest mass.Higher dimen-sional global embeddings are also constructed inside and outside the event horizons of the rotating Schwarzschild black holes.PACS numbers:02.40.Ma,04.20.Dw,04.20.Jb,04.70,95.30.LKeywords:rotaing Schwarzschild black hole,hydrodynamics,Euler equation,global ﬂat embeddingI.INTRODUCTIONThe physics of a rotating charged sphere has long at-tracted the attention of physicists [1,2,3].From the ex-perimental viewpoint,the pulsars have given concrete ev-idence for the existence of rotating magnetized collapsed objects.From the theoretical viewpoint,the existing ex-act solutions of Einstein equations have shown that the most general stationary solution,which is asymptotically ﬂat with a regular horizon for a fully collapsed object,has to be rotating and endowed with a net charge.It is well known that the Kerr [4]family of solutions of the Ein-stein and Einstein-Maxwell equations is the general class of solutions which could represent the ﬁeld of a rotating neutral or electrically charged sphere in asymptotically ﬂat space.In the extended manifolds,all geodesics which do not reach the central ring singularities of the Kerr black hole are shown to be complete,and also those null or timelike geodesics which do reach the singularities are entirely conﬁned to the equator [5].Moreover,the Kerr metric has the region called the ergosphere where the asymptotic time translation Killing ﬁeld becomes space-like.In the ergosphere,all observers are forced to ro-tate in the direction of the rotation of the black hole.Recently,the rotating Schwarzschild wormhole metric was proposed to investigate classes of geodesics falling through it which do not encounter any energy condition violating matter [6].On the other hand,a familiar feature of exact solutions to the ﬁeld equations of general relativity is the presence of singularities.As novel ways of removing the coordi-nate singularities,the higher dimensional global ﬂatem-2 II.ROTATING SCHW ARZSCHILD BLACKHOLE WITH CONSTANTΩWe consider the rotating Schwarzschild black hole witha constant angular velocityΩwhose4-metric is describedasds2=−N2dt2+N−2dr2+r2dθ2+r2sin2θ(dφ−Ωdt)2.(2.1)Here in the units G=c=1the lapse function N2isdeﬁned asN2=1−2mr.(2.2)The event horizon r H is located at the pole of g rr,namely at the root of N2to yield r H=2m as in the static Schwarzschild black hole case.The four velocity is given byu a=dx adτ≈0.(2.4) As in the Schwarzschild black hole since the coordi-nates t andφare cyclic we have the timelike Killingﬁeld ξa and the axial Killingﬁeldψa.Corresponding to the Killingﬁeldsξa andψa we can thenﬁnd the conserved energy E and the angular momentum L per unit rest mass for geodesics given as followsE=−g abξa u b= r−r Hr−r H E−Ωrr−r H E+ 1r−r H L,(2.7)FIG.1:Eﬀective potentials V(x,y)with x=r/r H and y=Ωr H for null and timelike geodesics of particles with E=1and L=2r H.which are substituted into(2.6)to yield the radial equa-tion for the particle on the equatorial plane12u r u r+V(r,E,L),(2.8) with the eﬀective potentialV=−r H2r2+1r3 r H2L−Ωr3E .(2.9) Here theﬁrst and second terms denote the Newtonian and centrifugal barrier terms respectively,which are at-tainable from Newtonian mechanics,while the other terms are general relativistic corrections,including the black hole rotating eﬀects with the parameterΩ.If E<1 the orbit of the particle is bound so that it cannot reach inﬁnity,while if E>1the orbit is unbound except for a measure-zero set of unstable orbits[16].For the null geodesics withκ=0,weﬁnd the only ex-tremum of the eﬀective potential to be a maximum occur-ring at r=3r H/2as in the case of static Schwarzschild black hole.The eﬀective potential V(x,y)for the par-ticles with the total energy per unit rest mass E=1 and angular momentum per unit rest mass L=2r H is shown in the left graph in Fig.1where x and y denote the dimensionless variables x=r/r H and y=Ωr H,re-spectively.Here note that we have a maximal eﬀective potential at the position(x,y)=(3/2,1/2).Next,we consider the timelike geodesics.The eﬀective potential(2.9)withκ=1now should fulﬁl the conditiondVr H(2.11)where the upper(lower)sign refers to the stable(unsta-ble)orbit.In particular,for the case of L≫r H,we canﬁnd r s ≈2L 2/r H corresponding to the Newtonian radius of circular orbits of particles with angular momentum per mass L orbiting a spherical body of mass m .The energy per unit mass of the particle in the circular orbit of the radius r =r s is the value of the eﬀective potential V at that radius1r 1/2s (2r s −3r H )1/2+Ωr 1/2H r sr 3/2s (2r s −3r H )1/2.(2.14)On the other hand,the angular frequency ωφfor the circular orbit is found to beωφ=r 1/2H +21/2Ωr 3/2sr 1/2sinh k H t,z1=k −1Hr −r Hr 31/2≡f (r,r H ),(2.17)where the surface gravity k H is given byk H =1r 1/2sinh k H t,z1=k −1Hr −r Hr=r H −rr 1/2cosh k H t,z 1=k −1Hr H −rintroducing4-metricds2=−N2dt2+N−2dr2+r2dθ2+r2sin2θ dφ−2ar−4a2sin2θruφ,L=−2a sin2θ2r κ+L22κ−L2+2a22r H−6aEr s+r H(r4s−r H r3s−4a2)√−g nu a)=0,(3.7)where n is the proper number density of particles mea-sured in the rest frame of theﬂuid and∇a is the covariantderivative in the rotating Schwarzschild curved manifoldand g=det g ab.For steady state axisymmetricﬂow,the conservation of energy-momentumﬂuxes is similarlydescribed by the Einstein equation[18]∇b T b a=1−g∇b(√n −15where the continuity equation (3.7)hasbeenused.Theradial componentof(3.11)yieldsdρndnu r.(3.12)Here the energy loss Λand the energy gain Γare in-troduced to set the decrease in the entropy of inﬂowing gas equal to the diﬀerence Λ−Γ.Moreover,using the projection operator g ab +u a u b in the equation (3.10)we can obtain the general relativistic Euler equation on the direction perpendicular to the four velocity(P +ρ)u b ∇b u a +(g ab +u a u b )∇b P =0.(3.13)After some algebra,from (3.13)we obtain the radial com-ponent of the Euler equation for the steady state axisym-metric ﬂuiddr 2+2r dPr 1/2sinh k H t,z1=k −1Hr −r Hr 1/2sin θcos φ,z3=r 3+2ar 1/2cos θ,z 5=2a r 1/2sin θsin(φ+t ),z 7=2ar 4 1/2sin θcos t,z9=2a (r 3+2a )r 4 1/2cos θ,z11=2a (r 3+2a )r 1/2,z 13=dr2r H (r 2+rr H +r 2H )+2r 3+ar 3(r 3+2a )dr 2+2r H (r 2+rr H +r 2H )+2r 3+ar 3dt2,(3.19)6in terms of the positive deﬁnite lapse function (2.21)in-side the event horizon r H to yield the (8+6)GEMS struc-ture (3.15)with the coordinate transformationz=k −1Hr H −rr1/2sinh k H t,z 13=g (r ),(3.20)with (z 2,z 3,z 4,z 5,z 6,z 7,z 8,z 9,z 10,z 11,z 12)in (3.16).IV.CONCLUSIONSIn conclusion,taking an ansatz for a rotating Schwarzschild black hole analogous to the rotating Schwarzschild wormhole [6]we have investigated hydro-dynamic properties of the perfect ﬂuid spiraling inward toward the black holes along a conical surface.Here we have exploited the fact that the coordinates t and φarecyclic in the rotating Schwarzschild metric to ﬁnd the timelike Killing ﬁeld and the axial Killing ﬁeld,to which we could obtain the conserved energy and the angular momentum per unit rest mass for geodesics.On the equatorial plane of the rotating Schwarzschild black hole,we have derived the radial equations of mo-tion with the eﬀective potential.We have also performed numerical analysis of the eﬀective potentials of particles on the rotating Schwarzschild manifolds in terms of an-gular velocity,total energy and angular momentum per unit rest mass.Finally,we have studied the rotating Schwarzschild black hole manifolds to construct (8+6)higher dimensional global embeddings inside and outside the event horizons.AcknowledgmentsWe would like to acknowledge ﬁnancial support in part from the Korea Science and Engineering Founda-tion Grant R01-2000-00015.[1]L.I.Shiﬀ,Proc.Nat.Acad.Sci.25,391(1939).[2]P.M.S.Blackett,Phil.Trans.Roy.Soc.Lon.245,309(1952).[3]R.Ruﬃni and A.Treves,Astrophy.Lett.13,109(1972).[4]R.P.Kerr,Phys.Rev.Lett.11,237(1963).[5]B.Carter,Phys.Rev.174,1559(1968).[6]E.Teo,Phys.Rev.D 58,024014(1998).[7]K.Schwarzschild,Sitzber.Deut.Akad.Wiss.Berlin,KI.Math.-Phys.Tech.pp.189-196(1916).[8]M.Spivak,Diﬀerential Geometry (Publish or Perish,Berkeley 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Space-Time Non-Commutativity near Horizon of a Black Hole

a r X i v :g r -q c /0502111v 1 25 F eb 2005Space-Time Non-Commutativity near Horizonof a Black HoleM.Martinis and V.Mikuta-MartinisTheoretical Physics Division,Rudjer Boˇs kovi´c Institute,10001Zagreb,Croatia E-mail:martinis@irb.hr,vmikuta@irb.hr1IntroductionEinstein’s general theory of relativity[1]describes gravity as a manifestation of the curvature of spacetime.A fundamental instability against collapse implies the existence of black holes as stable solutions of Einstein’s equations.A black hole is formed if a massive object(e.g.a star)collapses into an inﬁnitely dense state known as a singularity.In this picture the curvature of spacetime becomes extreme and prevents any particle even light from escaping to inﬁnity.A black hole may have several horizons that fully characterize its structure.The simplest three-dimensional geometry for a black hole is a sphere(known as a Schwarzschild sphere),its surface deﬁnes the event horizon.In the case of a spherical black hole,with Rµν=0,all horizons coincide at the Schwarzschild’s critical radius r s=2GMc−2.The Quantum Field Theory(QFT)in curved spacetime with classical event horizon is,however,troubled by the singularity at the horizon[2].This problem may be solved by treating the black hole as a quantum state which implies that the energy of the black hole and its corresponding time do not commute at the horizon[3].In this picture we study the dynamics of a scalarﬁeld in the near-horizon region described by a static Klein-Gordon(KG)operator which in this case becomes the Hamiltonian of the system.The dynamics of a scalarﬁeld in the near-horizon region,and its associated SO(2,1)conformal symmetry have been studied in many papers[4],[5],[6],[7],[8]in which a complete treatments of conformal quantum mechanics and of near-horizon symmetry were made. In this letter,we present the explicit construction of the time operator in the near-horizon region in terms of the generators of the aﬃne group,and discuss its self-adjointness[9].2Scalar Field in the Near-Horizon RegionThe Schwarzschild geometry of a static spherical black hole is described by the metric(c=¯h=G=1)ds2=−f(r)dt2+[f(r)]−1dr2+r2dΩ2,(1) where rf(r)=r−r s,and dΩ2=dθ2+sin2θdϕ2.Near horizon(r∼r s),f(r) behaves as f(r)∼2κ(r−r s),whereκ=1/2r s denotes the surface gravity.The equation of motion of a free scalarﬁeldΦ(x)in this background metric is −1fω2φ+fφrr+(f′+2f/r)φr−l(l+1)xφ,the KG equation in the near-horizon region,for small x,reduces to a scale invariant Schroedinger equation[d24+Θ2)2(p2+gSinceˆH is x-coordinate dependent,we expect spacetime noncommutativity, [ˆt,x]=0.What isˆt as an operator?Due to Pauli theorem[15]no such self-adjoint operator should exist if the spectrum of the self-adjoint Hamiltonian is semibounded or discrete.In quantum theoryˆH is essentially self-adjoint only for g>3/4in the domainD0={ψ∈L2(R+,dx),ψ(0)=ψ′(0)=0}(8) In this domain it has a continuous spectrum for g≥3/4with E>0but no ground state at E=0[4].For g≤3/4,the Hamiltonian is not essentially self-adjoint[6],[8],but it admits a one-parameter family of self-adjoint extension labeled by a U(1)parameter e iz,where z is a real number,which labels the domains D z of the extended Hamiltonian.The set D z contains all the vectors in D0,and vectors of the formψz=ψ++e izψ−.For g=−1xK02cotzE zandψz exhibits a scale behavior of the typeψz∼√This algebra can be easily extended to the full SO(2,1)conformal algebra by adding to the set(ˆH,ˆD)the conformal generatorˆK=x2/2.In this case the g-dependent constant quadratic Casimir operator is obtainedC2=14−32(ˆDˆH−1+ˆH−1ˆD),(13)which obeys the required commutation relation,[ˆH,ˆt]=i.Although,bothˆH andˆD separately can be made self-adjoint operators in the domain L2(R+,dE) it is not true for aˆt-operator which containsˆH−1[17].It is clear thatˆt is not a self-adjoint operator in the domain L2(R+,dE)whereˆH−→EˆD−→−i(E d2)(14)ˆt−→−i d2=α>0.This allow usto calculate all the matrix elements involving time operator and even to study their classical limit as¯h→0at the horizon.In the limit g−→0,we haveˆH−→H0=p2/2andˆt−→t0whereˆt 0=−1The Aharonov-Bohn time operatorˆt0is not self-adjoint and its eigenfunctions are not orthonormal.4ConclusionIn this paper,we have studied the properties of a scalarﬁeld in the near-horizon region of a massive Schwarzschild black hole.The quantum Hamilto-nian governing the near-horizon dynamics is found to be scale invariant and has the full conformal group as a dynamical symmetry ing only the generators of the aﬃne group,we constructed the time operator near-horizon. The self-adjointness ofˆH andˆt is also discussed.5AcknowledgmentThis work was supported by the Ministry of Science and Technology of the Republic of Croatia under Contract No.0098004.References[1]Nicolson I1981Gravity,Black Holes,and the Universe(David and Charles,London1981)[2]’t Hooft G2004Horizons,gr-qc/0401027.[3]Bai H and Yan Mu-Lin Quantum Horizon,gr-qc/0403064.[4]de Alfaro V et al1976Nuovo Cimento A34569Camblong H E and Ord´o nez C R2003Phys.Rev.D6*******[5]Strominger A1998JHEP9802009Claus P et al1998Phys.Rev.Lett.814553[6]Govindarajam T R et al2000Nucl.Phys.B583291[7]Gibbons G W and Townsand P K1999Phys.Lett.B454187[8]Birmingham D et al2001Phys.Lett.B505191Gupta K S and Sen S2002Phys.Lett.B526121[9]Martinis M and Mikuta V2003Fizika B12285[10]’t Hooft G1996Int.Jour.Mod.Phys.A114623[11]Gupta K S and Sen S2003Hidden Degeneracy in the Brick Wall Model of BlackHoles hep-th/0302183[12]Camblong H E and Ord´o nez C R2004Black Hole Thermodynamics from Near-Horizon Conformal Quantum Mechanics hep-th/0411008[13]’t Hooft G2000Holographic Principle hep-th/0003004[14]Camblong H E et al2000Phys.Rev.Lett.851590Camblong H E et al2001Ann.Phys.(NY)28714and57[15]Pauli W1958”Die allgemeinen Prinzipien der Wellenmechanik”inEncyclopedia of Physics,ed.Fluegge S(Springer-Verlag,Berlin1958)1-168[16]Moretti L and Pinamonti N2002Nucl.Phys.B647131[17]Klauder J R1999Noncanonical Quantization of Gravity I.Foundation of AﬃneQuantum Gravity gr-qc/9906013[18]Aharonov Y and Bohm D1961Phys.Rev.1221649。

宇宙十大怪兽

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。

很全面的资料-中子星与黑洞 Neutron Stars and Black Holes

If you could compress Earth to a smaller radius => higher escape velocity from the surface.
vesc
vesc
The Schwarzschild Radius
=> There is a limiting radius where the escape velocity reaches the speed of light, c:
Black Holes
Just like white dwarfs (Chandrasekhar limit: 1.4 Msun), there is a mass limit for neutron stars:
Neutron stars can not exist with masses > 3 Msun
We know of no mechanism to halt the collapse of a compact object with > 3 Msun. It will collapse into a single point – a singularity:
=> A Black Hole!
Escape Velocity
Pulsars / Neutron Stars
Neutron star surface has a temperature of ~ 1 million K.
Cas A in X-rays
Wien’s displacement law,
lmax = 3,000,000 nm / T[K] gives a maximum wavelength of lmax = 3 nm, which corresponds to X-rays.

Strong gravitational lensing in a noncommutative black-hole spacetime

a r X i v :1012.1670v 3 [g r -q c ] 19 F eb 2011Strong gravitational lensing in a noncommutative black-hole spacetimeChikun Ding,∗Shuai Kang,and Chang-Yong ChenDepartment of Physics and Information Engineering,Hunan Institute of Humanities Science and Technology,Loudi,Hunan 417000,P.R.ChinaSongbai Chen †and Jiliang Jing ‡Institute of Physics and Department of Physics,Hunan Normal University,Changsha,Hunan 410081,P.R.China Key Laboratory of Low Dimensional Quantum Structures and Quantum Control (Hunan Normal University),Ministry of Education,P.R.China.AbstractNoncommutative geometry may be a starting point to a quantum gravity.We study the inﬂuence of the spacetime noncommutative parameter on the strong ﬁeld gravitational lensing in the non-commutative Schwarzschild black-hole spacetime and obtain the angular position and magniﬁcation of the relativistic images.Supposing that the gravitational ﬁeld of the supermassive central object of the galaxy described by this metric,we estimate the numerical values of the coeﬃcients and ob-servables for strong gravitational paring to the Reissner-Norstr¨o m black hole,we ﬁnd that the inﬂuences of the spacetime noncommutative parameter is similar to those of the charge,just these inﬂuences are much smaller.This may oﬀer a way to distinguish a noncommutative black hole from a Reissner-Norstr¨o m black hole,and may probe the spacetime noncommutative constant ϑ[1]by the astronomical instruments in the future.PACS numbers:04.70.-s,95.30.Sf,97.60.LfI.INTRODUCTIONThe theoretical discovery of radiating black holes disclosed theﬁrst window on the mysteries of quantum gravity.Though after thirty years of intensive research,the full quantum gravity is still unknown.However there are two candidates for quantum gravity,which are the string theory and the loop quantum gravity.By the string/black hole correspondence principle[2],stringy eﬀects cannot be neglected in the late stage of a black hole.In the string theory,coordinates of the target spacetime become noncommutating operators on a D-brane as[3][ˆxµ,ˆxν]=iϑµν,(1.1) whereϑµνis a real,anti-symmetric and constant tensor which determines the fundamental cell discretization of spacetime much in the same way as the Planck constant discretizes the phase space,[ˆx i,ˆp j]=i δij. Motivated by string theory arguments,noncommutative spacetime has been reconsidered again and is believed to aﬀord a starting point to quantum gravity.Noncommutative spacetime is not a new conception,and coordinate noncommutativity also appears in anotherﬁelds,such as in quantum Hall eﬀect[4],cosmology[5],the model of a very slowly moving charged particle on a constant magneticﬁeld[6],the Chern-Simon’s theory[7],and so on.The idea of noncommutative spacetime dates back to Snyder[8]who used the noncommutative structure of spacetime to introduce a small length scale cut-oﬀinﬁeld theory without breaking Lorentz invariance and Yang[9]who extended Snyder’s work to quantize spacetime in1947before the renormalization theory.Noncommutative geometry[10]is a branch of mathematics that has many applications in physics,a good review of the noncommutative spacetime is in[11,12].The fundamental notion of the noncommutative geometry is that the picture of spacetime as a manifold of points breaks down at distance scales of the order of the Planck length:Spacetime events cannot be localized with an accuracy given by Planck length[12]as well as particles do in the quantum phase space.So that the points on the classical commutative manifold should then be replaced by states on a noncommutative algebra and the point-like object is replaced by a smeared object[13]to cure the singularity problems at the terminal stage of black hole evaporation[14].The approach to noncommutative quantumﬁeld theory follows two paths:one is based on the Weyl-Wigner-Moyal*-product and the other on coordinate coherent state formalism[13].In a recent paper,following the coherent state approach,it has been shown that Lorentz invariance and unitary,which are controversial questions raised in the*-product approach[15],can be achieved by assumingϑµν=ϑdiag(ǫ1,...,ǫD/2),(1.2) whereϑ[1]is a constant which has the dimension of length2,D is the dimension of spacetime[16]and,there isn’t any UV/IR mixing.Inspire by these results,various black hole solutions of noncommutative spacetime have been found[17];thermodynamic properties of the noncommutative black hole were studied in[18];the evaporation of the noncommutative black hole was studied in[19];quantized entropy was studied in[20],and so on.It is interesting that the noncommutative spacetime coordinates introduce a new fundamental natural length √scalebe described by this metric and then obtain the numerical results for the observational gravitational lensing parameters deﬁned in Sec.II.Then,we make a comparison between the properties of gravitational lensing in the noncommutative Schwarzschild and Reissner-Norstr¨o m metrics.In Sec.IV,we present a summary.II.DEFLECTION ANGLE IN THE NONCOMMUTATIVE SCHW ARZSCHILD BLACK HOLESPACETIMEThe line element of the noncommutative Schwarzschild black hole reads[14]ds2=−f(r)dt2+dr2r√ϑ→∞.And Eq.(2.1)leads to the mass distribution m(r)=2Mγ 3/2,r2/4ϑ /√ϑ,the event horizons are given byr±=4Mπγ 3/2,r2±/4ϑ ,(2.4)which behaviors as that of Reissner-Norstr¨o m black hole.The line element(2.1)describes the geometry of a noncommutative black hole and should give us useful insights about possible spacetime noncommutative eﬀects on strong gravitational lensing.As in[27,28,30],we set2M=1and rewrite the metric(2.1)asds2=−A(r)dt2+B(r)dr2+C(r) dθ2+sin2θdφ2 ,(2.5) withA(r)=f(r),B(r)=1/f(r),C(r)=r2.(2.6) The deﬂection angle for the photon coming from inﬁnite can be expressed asα(r0)=I(r0)−π,(2.7)where r 0is the closest approach distance and I (r 0)is [27,28]I(r 0)=2∞r 0C (r )A (r 0)C (r )=A ′(r )2−r 3psπϑe−r 2ps√2,r 2psϑ.ϑ0.2540.2420.230r ps 1.494051.497211.49890√0.2180.2060.1940.1821.499621.499891.499981.50000ϑ→0,it can recovers that in the commutative Schwarzschild black hole spacetime whichr ps =1.5.Fig.1shows that the relation between the photon sphere radius and the spacetime noncommutative parameter ϑis very coincident to the functionr ps =1.5−7.8×107√ϑ∈(0,19−32q 2)/4,which implies that there exist some distinct eﬀects of the noncommutative parameterϑon gravitational lensing in the strong ﬁeld limit.FIG.1:Theﬁgure is for the radius of the photon sphere in the noncommutative Schwarzschild black hole spacetime √with diﬀerentϑ17.Following the method developed by Bozza[30,37],we deﬁne a variabler0z=1−A(r)B(r)C(r0)wherep(r0)=2−3√2,r202ϑ√4ϑ,q(r0)=3√2,r204ϑ√4ϑ 2+r20u ps−1+¯b+O(u−u ps),(2.19) where¯a=R(0,r ps)q(r ps)= 1−r4psπϑe−r2ps2,¯b=−π+bR +¯a log4q2(r ps) 2A(r ps)−r2ps A′′(r ps)A3(r ps),b R=I R(r ps),p′(r ps)=dpϑas in[30].Because the values of various low derivative of integrand ofI R(r ps)atϑ→0is zero,we can getb R=2log[6(2−√ϑ).(2.21) Then we can obtain the¯a,¯b and u ps,and describe them in Fig(2).Figures(2)tell us that with the increase ofϑthe coeﬃcient¯a increase,the¯b slowly increases atﬁrst,then decrease quickly when it arrives at a peak, and the minimum impact parameter u ps decreases,which is similar to that in the Reissner-Norstr¨o m black hole metric.However,as shown in Fig.(2),in the noncommutative Schwarzschild black hole,¯a increases more slowly,both of¯b and u ps decrease more slowly.In a word,comparing to the Reissner-Nordstrom black hole,the inﬂuences of the spacetime noncommutative parameter on the strong gravitational lensing is similar to those of the charge,merely they are much smaller.On the other side,in principle we can distinguish a noncommutative Schwarzschild black hole from the Reissner-Nordstrom black hole and,may be probe the value of the spacetime noncommutative constant by using strongﬁeld gravitational lensing.0.100.120.140.160.180.200.220.241.0001.0011.0021.0031.0041.005a0.100.120.140.160.180.200.220.240.400280.400260.400240.400220.40020b0.100.120.140.160.180.200.220.242.597942.597962.597982.598002.598022.598042.59806u p sq a0.100.150.200.250.300.350.400.4040.4020.4000.3980.396qbqu p sFIG.2:Variation of the coeﬃcients of the strong ﬁeld limit ¯a ,¯b and the minimum impact parameter u ps with the spacetime noncommutative parameter√ϑ.Considering the source,lens and observer are highly aligned,the lens equation in strong gravitational lensing can be written as [39]β=θ−D LSbetween the source and the lens,θis the angular separation between the image and the lens,∆αn=α−2nπis the oﬀset of deﬂection angle and n is an integer.The position of the n-th relativistic image can be approximated asu ps e n(β−θ0n)D OSθn=θ0n+¯a,(2.24)θ0n are the image positions corresponding toα=2nπ.The magniﬁcation of n-th relativistic image is given byu2ps e n(1+e n)D OSµn=∞n=2µn.(2.28) For highly aligned source,lens and observer geometry,these observable can be simpliﬁed ass=θ∞e¯b−2π¯a.(2.29) The strong deﬂection limit coeﬃcients¯a,¯b and the minimum impact parameter u ps can be obtain through measuring s,R andθ∞.Then,comparing their values with those predicted by the theoretical models,we can identify the nature of the black hole lens.III.NUMERICAL ESTIMATION OF OBSER V ATIONAL GRA VITATIONAL LENSINGPARAMETERSIn this section,supposing that the gravitational ﬁeld of the supermassive black hole at the galactic center of Milk Way can be described by the noncommutative Schwarzschild black hole metric,we estimate the numerical values for the coeﬃcients and observables of the strong gravitational lensing,and then we study the eﬀect of the spacetime noncommutative parameter ϑon the gravitational lensing.The mass of the central object of our Galaxy is estimated to be 2.8×106M ⊙and its distance is around 8.5kpc.For diﬀerent ϑ,the numerical value of the minimum impact parameter u ps ,the angular position of the asymptotic relativistic images θ∞,the angular separation s and the relative magniﬁcation of the outermost relativistic image with the other relativistic images r m are listed in the table (II).It is easy to obtain thatTABLE II:Numerical estimation for main observables and the strong ﬁeld limit coeﬃcients for black hole at the center of our galaxy,which is supposed to be described by the noncommutative Schwarzschild black hole metric.R s is Schwarzschild radius.r m =2.5log R .ϑs (µarcsecs)u ps /R S¯b16.8706.82191.0000.160.021092.59808−0.4002316.86996.821701.000030.200.021162.59807−0.4001916.86936.800521.003140.240.023042.59752−0.4005816.85506.547741.041870.100.120.140.160.180.200.220.2416.869016.869216.869416.869616.8698Θ0.100.120.140.160.180.200.220.246.7856.7906.7956.8006.8056.8106.8156.820r m0.100.120.140.160.180.200.220.240.02110.02120.02130.02140.02150.02160.02170.0218 s0.100.150.200.250.300.350.4014.515.015.516.016.5qΘ0.100.150.200.250.300.350.406.06.26.46.66.8qr m0.100.150.200.250.300.350.400.0250.0300.035qsFIG.4:Strong gravitational lensing by the Galactic center black hole.Variation of the values of the angular positionθ∞,the relative magnitudes r m and the angular separation s with parameter√the table(II),we alsoﬁnd that as the parameterϑincreases,the minimum impact parameter u ps,the angular position of the relativistic imagesθ∞and the relative magnitudes r m decrease,but the angular separation s increases.From Fig.(4),weﬁnd that in the noncommutative Schwarzschild black hole with the increase of parameter ϑ,the angular positionθ∞and magnitudes r m decreases more slowly,angular separation s increases more slowly than those in the Reissner-Norstr¨o m black hole spacetime.This means that the bending angle is smaller and the relative magniﬁcation of the outermost relativistic image with the other relativistic images is bigger in the noncommutative Schwarzschild black hole spacetime.In order to identify the nature of these two compact objects lensing,it is necessary for us to measure angular separation s and the relative magniﬁcation r m in the astronomical observations.Tables(II)tell us that the resolution of the extremely faint image is∼0.03µarc sec,which is too small.However,with the development of technology,the eﬀects of the spacetime noncommutative constantϑon gravitational lensing may be detected in the future.IV.SUMMARYNoncommutative geometry may be a starting point to a quantum gravity.Spacetime noncommutative constant would be a new fundamental natural constant which can aﬀect the classical gravitational eﬀect such as gravitational lensing.Studying the strong gravitational lensing can help us to probe the spacetime noncommutative constant and the noncommutative gravity.In this paper we have investigated strongﬁeld lensing in the noncommutative Schwarzschild black hole spacetime to study the inﬂuence of the spacetime noncommutative parameter on the strong gravitational lensing.The model was applied to the supermassive black hole in the Galactic center.Our results show that with the increase of the parameterϑthe minimum impact parameter u ps,the angular position of the relativistic imagesθ∞and the relative magnitudes r m decrease,and the angular separation s paring to the Reissner-Norstr¨o m black hole,weﬁnd that the angular positionθ∞and magnitude r m decrease more slowly,angular separation s increases more slowly.In a word,the inﬂuences of spacetime noncommutative parameter are similar to those of the charge, just they are much smaller.This may oﬀer a way to distinguish a noncommutative Schwarzschild black hole from a Reissner-Norstr¨o m black hole by the astronomical instruments in the future.AcknowledgmentsThis work was partially supported by the Scientiﬁc Research Foundation for the introduced talents of Hunan Institute of Humanities Science and Technology.S.Kang’s work was supported by the National Natural Science Foundation of China(NNSFC)No.10947101;C.-Y.Chen’s work was supported by the NNSFC No.11074070; J.Jing’s work was supported by the NNSFC No.10675045,No.10875040and No.10935013,973Program No. 2010CB833004and the HPNSFC No.08JJ3010S.Chen’s work was supported by the NNSFC No.10875041, the PCSIRT No.IRT0964and the construct program of key disciplines in Hunan Province.[1]The notationϑused here is a constant as well as Plank constant ,but we still call it as a spacetime noncommutativeparameter since 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a rXiv:g r-qc/97334v 113Mar1997Schwarzschild black hole with global monopole charge Naresh Dadhich ∗Inter-University Centre for Astronomy &Astrophysics,Post Bag 4,Ganeshkhind,Pune -411007,India.K.Narayan †and U.A.Yajnik ‡Department of Physics,Indian Institute of Technology Bombay,Mumbai 400076.Abstract We derive the metric for a Schwarzschild black hole with global monopole charge by relaxing asymptotic ﬂatness of the Schwarzschild ﬁeld.We then study the eﬀect of global monopole charge on particle orbits and the Hawking radiation.It turns out that existence,bound-edness and stability of circular orbits scale up by (1−8πη2)−1,andthe perihelion shift and the light bending by (1−8πη2)−3/2,while the Hawking temperature scales down by (1−8πη2)2the Schwarzschild values.Here ηis the global charge.PACS numbers:04.20.Jb,14.80.Hv,04.50.+h IITB-TP-971011IntroductionThe Schwarzschild solution is the unique spherically symmetric solution of the vacuum Einstein equation.It represents a static black hole.It turns out[1]that a global monopole charge accompanies spontaneous breaking of global0(3)symmetry into U(1)in phase transitions in the Universe.Putting on global charge to the Schwarzschild black hole will amount to breaking the vacuum and asymptoticﬂatness of the Schwarzschild spacetime.It still rep-resents a localized object with a horizon.Our main aim in this paper is to study the eﬀect of global monopole charge on particle orbits and the Hawking radiation.The spacetime of pure global monopole charge,when the Schwarzschild mass vanishes,can be regarded in some sense to be“minimally”curved for gravitational charge density,4πρc=R ik u i u k,u i u i=1,vanishes.That means the relativistic(active gravitational)mass is zero for the spacetime[2].All radial trajectories will always remain parallel.Thus introduction of global monopole charge does not signiﬁcantly alter the nature of the Schwarzschild ﬁeld.We wish to investigate what eﬀect does its introduction entail on physi-cally measurable quantities.Recently a similar investigation has been carried out for a global monopole in the Kaluza-Klein spacetime[3].In Sec.2we shall brieﬂy outline the derivation of the metric for a static black hole with a global monopole charge followed in Sec.3byﬁeld theoretic considerations.The eﬀect of global charge on particle orbits and the Hawk-ing radiation process will be investigated in Secs.4and5.We conclude with a discussion in Sec.6.2The MetricLet us write the general spherically symmetric metricds2=Bdt2−Adr2−r2(dθ2+sin2θdϕ2)(2.1) where A and B are functions of r and t.2Now the equations R01=0and R00=R11will imply B=A−1=1+2φ, whereφ=φ(r)[4].Note that no boundary condition has been used.With thisR00=R11=−▽2φ(2.2) and2R22=R33=−.(2.4)rNow R22=0will determine k=0and we obtain the Schwarzschild solution. Retaining k will make R22=0and will give rise to stresses,kT00=T11=−L=14(ψaψa−η2)2.(3.1) Topologically non-trivial self-supporting solutions to this system can be found. The ansatz describing a monopole isx aψa(¯x)=ηf(r).(3.5)r2The stress tensor of the system outside the core can be approximated asη2T00=T11=)dt2−(1+2k−2Mris usually negligible.However if we consider the spherically symmetric gravi-tational collapse of the matter around such a monopole,weﬁnd that a black hole is formed.The parameter M then corresponds to the mass of the black hole.Even when M=0,this spacetime has non-zero curvature:R00=R11=R01=0(3.8) but8πη2R22=of O(3)(more generally,Spin(N)instead of O(N))then an isospinor species χmay exist.Now a2πspace rotation will induce a2πO(3)rotation which will change the sign of the wavefunction which therefore can not be globally deﬁned.Note that the fate of the global charge S z of the O(3)group will be the same as that of any other global charge such as the baryon number.4Particle orbitsIn this Section we wish to investigate the eﬀect of global charge on the particle orbits;existence,boundedness and stability of circular orbits,light deﬂection and perihelion-shift.4.1Circular orbits:For the metric(3.7),we have△˙t=E,r2˙ϕ=l(4.1) where△=1+2k−2M/r,˙t=dt/ds;E and l are speciﬁc energy and angular momentum of a test particle.Substituting(4.1)in the metric(we have set θ=π/2as usual),l2˙r2=E2−△(4.3)1+2k√where equality refers to the photon orbit.Of course l≥2(4.4)1+2k6while stability of circular orbits will further require V′′≥0,givingr s≥6Mr0)2△(r0)r(4.6) where r0is the radius vector of the closest approach to the gravitating body. The deﬂection angle will be given byδϕ=2|ϕ(r0)−ϕ(∞)|−π1+2k.(4.7)Note that when M=0,ϕhas the period2π/√r0(1+2k)3/2≈4Mr2+r2−(△−1(r+)−△−1(r−))−1r2dr(4.9) where r±refer to aphelion and perihelion points.The perihelion shift is given by7δϕ=2|ϕ(r+)−ϕ(r−)|−2π1+2k(4.10)and we thus obtainδϕ=6πML(1+12πη2)(4.11)whereL=1r++1 1+2k−2MRωl(r)dr∗2+(ω2−[l(l+1)r−2+2Mr−3][1−2Mr−1])Rωl=0.(5.6)Because of the‘potential’term in square brackets,the standard incoming waves will partially scatter back oﬀthe gravitationalﬁeld to become a su-perposition of incoming and outgoing waves.However,the eﬀective potential vanishes as r∗→±∞(i.e.for r→2M,∞).Thus,in those regions the mode functions will be of the form(1re−iωv Y lm).(5.7)The surface gravity of the black hole formed as the result of gravitational collapse isκ=(1+2k)24M.(5.8)This redeﬁnition of the surface gravity reduces the calculation to that of the Schwarzschild case.With this redeﬁnition,the Penrose diagram of the present system is identical to that of the Schwarzschild case.As in the Schwarzschild case,assuming there are no bound states at late times,the ingoing and outgoing modes form a complete basis in terms of which theﬁeld at late times can be expanded.Then an outgoing mode of the form r−1e−iωu Y lm at late times can be traced back to early times in the same fashion as in the Schwarzschild case.At early times,the ray,moving along constant phase v lines will have the form e−iωu(v).The aﬃne parameter on the past horizon can be chosen asλ=−ce−κu.(5.9) Then the functionu(v)=−1const].(5.10) 9This can be Bogolubov transformed in the same manner as the Schwarzschild case to give the Planckian spectrum1Nβ=e8πMω/(1+2k)2−1.(5.12) Thus the thermal spectrum is recovered but the temperature is now(1−8πη2)2/8πM.Consider next the quanta of theψﬁeld responsible for the monopole.The equations satisﬁed by˜ψa=ψa−ψa bg(5.13) are,after linearisation,2˜ψa+2λη2ψa bg( bψb bg˜ψb)=0.(5.14) Asymptotically the equation becomesx a2˜ψa+2λη2Thus the fate of our black hole is similar to that of the magnetically charged black holes[6].While the question of its ultimate fate cannot be settled within the semiclassical approximation,the answer will be the same as for a Schwarzschild black hole.If it can in fact evaporate completely,and does so,then it leaves behind a monopole identical to the original one before collapse.In concluding this section,we note that the persistence of the thermal radiation is not surprising.It is known from work in axiomaticﬁeld theory [7]that when horizons exist,a unique nontrivial automorphism exists for the ground state.The automorphism works out to be periodicity in time,leading to the Kubo-Martin-Schwinger boundary condition on the Green function. 6DiscussionThe main purpose of this investigation was to study the eﬀect of global monopole charge on particle orbits and the Hawking radiation.It turns out that existence,boundedness and stability threshold for circular orbits scale up by(1−8πη2)−1,perihelion shift and light bending by(1−8πη2)−3/2,while the Hawking temperature scales down by(1−8πη2)2the Schwarzschild val-ues.This is how the global monopole chargeηaﬀects the particle orbits and the Hawking radiation thermal spectrum.It may be noted that the event horizon is given by r=2M(1−8πη2)−1,while the red-shifted proper accel-eration remains unchanged as M/r2.It is the scaling up of the horizon that leads to scaling down of the surface gravity(temperature)of the hole.The spacetime(3.7)with M=0can be looked upon as“minimally”curved for it has zero gravitational charge(R ik u i u k=0)which leads to vanishing of acceleration as well as tidal acceleration for radial motion.Its curvature only manifests in producing tidal acceleration for transverse mo-tion.Acknowledgement:KN thanks Jawaharlal Nehru Centre for Advanced Research,Bangalore for award of a summer fellowship and IUCAA for pro-viding the facilities.11References[1]M Barriola and A Vilenkin(1989)Phys.Rev.Lett.63,341.[2]N.Dadhich(1995)GR-14Abstracts,A.98.[3]A.Banerjee,S.Chatterjee and A.A.Sen(1996)Class.Quantum Grav.13,3141.[4]N.Dadhich,On the Schwarzschildﬁeld,to be published.[5]S.Weinberg(1972)Gravitation and Cosmology(John Wiley&Sons,New York)pp.189-90,194-96.[6]K.Lee,V.P.Nair and E.J.Weinberg,(1992)Gen.Rel.and Grav.24,1203.[7]J.J.Bisognano and E.H.Wichmann(1975)J.Math.Phys.16,985.12。