Brane-Black Hole Correspondence and Asymptotics of Quantum Spectrum

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动态广义球对称含荷黑洞对Dirac粒子的热辐射

文章编号：００６（００００５０１６— ４４２１）６— ５４— ８０
动态广义球对称含荷黑洞对Ｄｒｃ子的热辐射ｉ粒ａ
刘定兴
（重庆三峡学院物理与电子工程学院，重庆万州摘４４０）０００
要：分析讨论动态广义球对称含荷黑洞视界附近的狄拉克方程，准确地定出了动态广义球对称含荷黑洞的
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动态广义球对称含荷黑洞的时空线元为
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将式（）出的线元用号差（＋，一，重新表示为１给一，一）
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为Ｈｗｉ辐射温度。此后人们在黑洞热力学的研究上做了大量工作，ａｋｇｎ特别是在计算黑洞的Ｈｗｉ温度ａｋｇｎ与其辐射谱方面，取得了许多有价值的成果。继Ｈａｋｎｗｉｇ之后，人们又采用各种各样的方法来研究Ｈｗｉｇａｋｎ温度和黑洞辐射谱。主要的方法有路径积分法ｊ温度格林函数法ＪＤｍｕ－Ｒｆｎ方法』Ｕｒｈ法Ｊ、、ａｏｒｕｉｉ－、ｎｕ和Ｚａｈｏ等人在１９９１年提出的决定动态黑洞热效应的新方法ｌ等等。值得说明的是，ｈｏ等人提出的这６一Ｚａ
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一ｅｒｓｎ０４ｉ

圈量子引力和空间分立性

子几何已发展成为具有稳固数学基础、好定义、景无关、物质有常规耦合的理论ｌ，中包括给出了很背和４其］
Ｐａｃｌｋ标度的量子几何绘景、算技术和研究黑洞热力学经典问题的一种方法．然还存在一些问题有待解ｎ计当决，如与经典广义相对论的一致性、子自旋动力学等．例量
的量子化表式作出本体诠释．
关键词：圈量子引力；面积量子化表式；积量子化表式；体本体诠释
中图分类号：４２２Ｏ１．
文献标识码：Ａ
当代基础物理学中的根本性挑战，过于量子力学和广义相对论相互结合的问题Ｌ．于广义相对论实莫１由］质上是经典的相对论性引力场理论，们便把量子力学和广义相对论相谐和的单一理论，为量子引力理人称论，称量子引力．（ｏｐ量子引力是近年来量子引力研究中相当活跃的一种，和超弦／理论构成了简圈Ｌｏ）它Ｍ
１二十世纪８）０年代
１８９２年，度物理学家Ａ．ｅ印Ｓｎ在Ｐｙ．ｅ．Ｐｙ．ｅｔ上相继发表两篇文章，广义相对论引力场ｈｓＲｖ和ｈｓＬｔ．把方程表述成简单而精致的形式，在理论物理学家中曾引起不小的轰动．们把这样简单得多的表述，用这他应到量子引力研究中，在当时并没有取得有价值的成果．但

圈量子引力的回顾

圈量子引力的回顾
龙芸
（北第二师范学院物理与电子工程系，武汉湖４００３２５）
摘要：圈量子引力在物理学中占据非常重要的地位。０纪９２世０年代后，圈量子引力得到了很大的发展，建立了数学上定义严格的运动学框架，引入自结网圈作为量子引力态，出了很多有意义的物理结论，几何算符的量子化，旋给如并且它还在黑洞物理和宇宙学等方向上展现了强有力的发展势头它是一个非常活跃的量子引力理论，越来越广泛地应用到普朗克尺度物理被中去，这也加强了理论学家们的信心。本文试着对圈量子引力的这些主要结果做个回顾。
图２三阶顶角
三阶展开表示的一个例子
ｆ６＋叶ｃ＝ｅｅｎｖ
正则量子广义相对论希望能从数学上广义地定义出一个非
微扰的、背景度规无关的量子引力理论。与大约２０年前．ｓｔａＡｈｅｒｋ率先提出，重建的广义相对论相空间由广义坐标与广义动量形他成，杨密Ｓ２规范理论相同，义坐标变成一个三维规范矩与Ｕ（）广阵向量场，应的广义动量和三维度规的关系为：对
维普资讯
２ｏｏ８年２月
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亨德森-哈塞尔巴尔赫方程

亨德森-哈塞尔巴尔赫方程
哈及伦-赫伯特方程(Hilbert-Einstein equation),又称为亨德森-哈塞尔巴尔赫方程，是一个描述宇宙结构形成的物理方程。

它是20世纪物理学诺贝尔奖获得者阿尔伯特·亨德森(Albert Einstein)与荷瑟里安.哈及伦(Herbert Hilbert)的结合结果。

关于宇宙的探索,20世纪初,宇宙的结构仍然未知,但通过观察宇宙的变化,物理学家发现,当黑洞的重力足够大时,它不仅仅影响了物质的周围,还会影响周围的空间与时间。

1915年,阿尔伯特·亨德森提出了广义相对论,它预测宇宙结构包括时间。

1920年,荷瑟里安·哈塞尔巴尔赫把宇宙假设为一个充满黑洞结构的大量体系,他发现一个有趣的方程,它预测宇宙结构变化与其初始状态之间的关系。

亨德森-哈塞尔巴尔赫方程是由这个预测演变而来,它描述了宇宙结构形成的物理过程。

今天,使用该方程可以探索宇宙的结构与变化,由此可以研究宇宙的演化,以及宇宙的构造。

许多天文学家和物理学家正在尝试研究宇宙的性质。

它还能帮助人们了解高维空间,尤其是有关宇宙的运动的高维力学。

总的来说,亨德森-哈塞尔巴尔赫方程是一个重要的物理方程,它有助于人们理解宇宙的演变和结构形成,探索宇宙的性质和构造,以及了解宇宙运动的高维力学。

它已经成为物理学领域的一个重要研究方面,同时也给人类带来无限可能性。

物理学名词

1/4波片quarter-wave plateCG矢量耦合系数Clebsch-Gordan vector coupling coefficient; 简称“CG[矢耦]系数”。

X射线摄谱仪X-ray spectrographX射线衍射X-ray diffractionX射线衍射仪X-ray diffractometer[玻耳兹曼]H定理[Boltzmann] H-theorem[玻耳兹曼]H函数[Boltzmann] H-function[彻]体力body force[冲]击波shock wave[冲]击波前shock front[狄拉克]δ函数[Dirac] δ-function[第二类]拉格朗日方程Lagrange equation[电]极化强度[electric] polarization[反射]镜mirror[光]谱线spectral line[光]谱仪spectrometer[光]照度illuminance[光学]测角计[optical] goniometer[核]同质异能素[nuclear] isomer[化学]平衡常量[chemical] equilibrium constant[基]元电荷elementary charge[激光]散斑speckle[吉布斯]相律[Gibbs] phase rule[可]变形体deformable body[克劳修斯-]克拉珀龙方程[Clausius-] Clapeyron equation[量子]态[quantum] state[麦克斯韦-]玻耳兹曼分布[Maxwell-]Boltzmann distribution[麦克斯韦-]玻耳兹曼统计法[Maxwell-]Boltzmann statistics[普适]气体常量[universal] gas constant[气]泡室bubble chamber[热]对流[heat] convection[热力学]过程[thermodynamic] process[热力学]力[thermodynamic] force[热力学]流[thermodynamic] flux[热力学]循环[thermodynamic] cycle[事件]间隔interval of events[微观粒子]全同性原理identity principle [of microparticles][物]态参量state parameter, state property[相]互作用interaction[相]互作用绘景interaction picture[相]互作用能interaction energy[旋光]糖量计saccharimeter[指]北极north pole, N pole[指]南极south pole, S pole[主]光轴[principal] optical axis[转动]瞬心instantaneous centre [of rotation][转动]瞬轴instantaneous axis [of rotation]t 分布student's t distributiont 检验student's t testＫ俘获K-captureＳ矩阵S-matrixＷＫＢ近似WKB approximationＸ射线X-rayΓ空间Γ-spaceα粒子α-particleα射线α-rayα衰变α-decayβ射线β-rayβ衰变β-decayγ矩阵γ-matrixγ射线γ-rayγ衰变γ-decayλ相变λ-transitionμ空间μ-spaceχ 分布chi square distributionχ 检验chi square test阿贝不变量Abbe invariant阿贝成象原理Abbe principle of image formation阿贝折射计Abbe refractometer阿贝正弦条件Abbe sine condition阿伏伽德罗常量Avogadro constant阿伏伽德罗定律Avogadro law阿基米德原理Archimedes principle阿特伍德机Atwood machine艾里斑Airy disk爱因斯坦-斯莫卢霍夫斯基理论Einstein-Smoluchowski theory 爱因斯坦场方程Einstein field equation爱因斯坦等效原理Einstein equivalence principle爱因斯坦关系Einstein relation爱因斯坦求和约定Einstein summation convention爱因斯坦同步Einstein synchronization爱因斯坦系数Einstein coefficient安[培]匝数ampere-turns安培[分子电流]假说Ampere hypothesis安培定律Ampere law安培环路定理Ampere circuital theorem安培计ammeter安培力Ampere force安培天平Ampere balance昂萨格倒易关系Onsager reciprocal relation凹面光栅concave grating凹面镜concave mirror凹透镜concave lens奥温电桥Owen bridge巴比涅补偿器Babinet compensator巴耳末系Balmer series白光white light摆pendulum板极plate伴线satellite line半波片halfwave plate半波损失half-wave loss半波天线half-wave antenna半导体semiconductor半导体激光器semiconductor laser半衰期half life period半透[明]膜semi-transparent film半影penumbra半周期带half-period zone傍轴近似paraxial approximation傍轴区paraxial region傍轴条件paraxial condition薄膜干涉film interference薄膜光学film optics薄透镜thin lens保守力conservative force保守系conservative system饱和saturation饱和磁化强度saturation magnetization本底background本体瞬心迹polhode本影umbra本征函数eigenfunction本征频率eigenfrequency本征矢[量] eigenvector本征振荡eigen oscillation本征振动eigenvibration本征值eigenvalue本征值方程eigenvalue equation比长仪comparator比荷specific charge; 又称“荷质比(charge-mass ratio)”。

从爱因斯坦到霍金的宇宙课后习题答案

从爱因斯坦到霍金的宇宙2019尔雅满分答案物理学的起源1Physics这个词最先是谁想出的?(B)A、柏拉图B、亚里士多德C、欧几里得D、阿基米德2颐和园宝云阁的“物含妙理总堪寻”是由康熙题词。

(X)“物理”一词在中国1谁认为“格物致知”中的“格”意思是“变革”?(D)A、朱熹B、王阳明C、王艮D、毛泽东2王阳明强调人心,良知,冉伟革去外物,良知自存。

(对)物理学的诞生1谁首先指出物理学是一门“实验的科学”、“测量的科学”?(B)A、阿基米德B、伽利略C、牛顿2阿基米德的重要发现是(BC)。

A、自由落体定律B、浮力定律C、杠杆原理D、相对性原理3下列哪些定律是伽利略首先确认的?(ACD)A、相对性原理B、杠杆原理C、自由落体定律D、惯性定律“1642年”在物理学上的意义1牛顿的主要成就是(AB)。

A、力学三定律B、万有引力定律C、光的波动说D、能量守恒定律2库伦从介质的弹性振动导出了电磁学的基本方程组。

(对)3麦克斯韦从介质的弹性振动导出了电磁学的基本方程组。

(对)热学的发展1热力学的哪一条定律说"不能从单一热源吸热做功,而对外界不产生影响"?(B) A、第一定律C、第三定律D、第零定律2开尔文提出不能从单一热源吸热做工而不产生其他影响。

对明朗天空的两朵乌云1爱因斯坦提出下列理论中的哪一个,用以解释光电效应?(D)A、量子论B、光子说C、波动说D、光量子论2瑞利—金斯曲线在短波波段与实验曲线完全符合,在长波波段变得无穷大。

X 并非神童的爱因斯坦1爱因斯坦在苏黎世工业大学上学期间,其物理教授是(A)。

A、韦伯B、卢瑟福C、玻尔D、狄拉克求职不顺的爱因斯坦1爱因斯坦从苏黎世工业大学毕业后曾向著名的物理学家奥斯特瓦尔德求职。

对爱因斯坦的丰收年127岁那年,是爱因斯坦的丰收年,他做出了如下的创新工作(ABD)。

A、光量子说B、狭义相对论C、广义相对论D、E = mc2相对论的建立（Ⅰ）1迈克耳孙实验结果与当时天文学的光行差现象相矛盾。

引力波观测原文PhysRevLett.116.061102

Observation of Gravitational Waves from a Binary Black Hole MergerB.P.Abbott et al.*(LIGO Scientific Collaboration and Virgo Collaboration)(Received21January2016;published11February2016)On September14,2015at09:50:45UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal.The signal sweeps upwards in frequency from35to250Hz with a peak gravitational-wave strain of1.0×10−21.It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole.The signal was observed with a matched-filter signal-to-noise ratio of24and a false alarm rate estimated to be less than1event per203000years,equivalent to a significance greaterthan5.1σ.The source lies at a luminosity distance of410þ160−180Mpc corresponding to a redshift z¼0.09þ0.03−0.04.In the source frame,the initial black hole masses are36þ5−4M⊙and29þ4−4M⊙,and the final black hole mass is62þ4−4M⊙,with3.0þ0.5−0.5M⊙c2radiated in gravitational waves.All uncertainties define90%credible intervals.These observations demonstrate the existence of binary stellar-mass black hole systems.This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.DOI:10.1103/PhysRevLett.116.061102I.INTRODUCTIONIn1916,the year after the final formulation of the field equations of general relativity,Albert Einstein predicted the existence of gravitational waves.He found that the linearized weak-field equations had wave solutions: transverse waves of spatial strain that travel at the speed of light,generated by time variations of the mass quadrupole moment of the source[1,2].Einstein understood that gravitational-wave amplitudes would be remarkably small;moreover,until the Chapel Hill conference in 1957there was significant debate about the physical reality of gravitational waves[3].Also in1916,Schwarzschild published a solution for the field equations[4]that was later understood to describe a black hole[5,6],and in1963Kerr generalized the solution to rotating black holes[7].Starting in the1970s theoretical work led to the understanding of black hole quasinormal modes[8–10],and in the1990s higher-order post-Newtonian calculations[11]preceded extensive analytical studies of relativistic two-body dynamics[12,13].These advances,together with numerical relativity breakthroughs in the past decade[14–16],have enabled modeling of binary black hole mergers and accurate predictions of their gravitational waveforms.While numerous black hole candidates have now been identified through electromag-netic observations[17–19],black hole mergers have not previously been observed.The discovery of the binary pulsar system PSR B1913þ16 by Hulse and Taylor[20]and subsequent observations of its energy loss by Taylor and Weisberg[21]demonstrated the existence of gravitational waves.This discovery, along with emerging astrophysical understanding[22], led to the recognition that direct observations of the amplitude and phase of gravitational waves would enable studies of additional relativistic systems and provide new tests of general relativity,especially in the dynamic strong-field regime.Experiments to detect gravitational waves began with Weber and his resonant mass detectors in the1960s[23], followed by an international network of cryogenic reso-nant detectors[24].Interferometric detectors were first suggested in the early1960s[25]and the1970s[26].A study of the noise and performance of such detectors[27], and further concepts to improve them[28],led to proposals for long-baseline broadband laser interferome-ters with the potential for significantly increased sensi-tivity[29–32].By the early2000s,a set of initial detectors was completed,including TAMA300in Japan,GEO600 in Germany,the Laser Interferometer Gravitational-Wave Observatory(LIGO)in the United States,and Virgo in binations of these detectors made joint obser-vations from2002through2011,setting upper limits on a variety of gravitational-wave sources while evolving into a global network.In2015,Advanced LIGO became the first of a significantly more sensitive network of advanced detectors to begin observations[33–36].A century after the fundamental predictions of Einstein and Schwarzschild,we report the first direct detection of gravitational waves and the first direct observation of a binary black hole system merging to form a single black hole.Our observations provide unique access to the*Full author list given at the end of the article.Published by the American Physical Society under the terms of the Creative Commons Attribution3.0License.Further distri-bution of this work must maintain attribution to the author(s)and the published article’s title,journal citation,and DOI.properties of space-time in the strong-field,high-velocity regime and confirm predictions of general relativity for the nonlinear dynamics of highly disturbed black holes.II.OBSERVATIONOn September14,2015at09:50:45UTC,the LIGO Hanford,W A,and Livingston,LA,observatories detected the coincident signal GW150914shown in Fig.1.The initial detection was made by low-latency searches for generic gravitational-wave transients[41]and was reported within three minutes of data acquisition[43].Subsequently, matched-filter analyses that use relativistic models of com-pact binary waveforms[44]recovered GW150914as the most significant event from each detector for the observa-tions reported here.Occurring within the10-msintersite FIG.1.The gravitational-wave event GW150914observed by the LIGO Hanford(H1,left column panels)and Livingston(L1,rightcolumn panels)detectors.Times are shown relative to September14,2015at09:50:45UTC.For visualization,all time series are filtered with a35–350Hz bandpass filter to suppress large fluctuations outside the detectors’most sensitive frequency band,and band-reject filters to remove the strong instrumental spectral lines seen in the Fig.3spectra.Top row,left:H1strain.Top row,right:L1strain.GW150914arrived first at L1and6.9þ0.5−0.4ms later at H1;for a visual comparison,the H1data are also shown,shifted in time by this amount and inverted(to account for the detectors’relative orientations).Second row:Gravitational-wave strain projected onto each detector in the35–350Hz band.Solid lines show a numerical relativity waveform for a system with parameters consistent with those recovered from GW150914[37,38]confirmed to99.9%by an independent calculation based on[15].Shaded areas show90%credible regions for two independent waveform reconstructions.One(dark gray)models the signal using binary black hole template waveforms [39].The other(light gray)does not use an astrophysical model,but instead calculates the strain signal as a linear combination of sine-Gaussian wavelets[40,41].These reconstructions have a94%overlap,as shown in[39].Third row:Residuals after subtracting the filtered numerical relativity waveform from the filtered detector time series.Bottom row:A time-frequency representation[42]of the strain data,showing the signal frequency increasing over time.propagation time,the events have a combined signal-to-noise ratio(SNR)of24[45].Only the LIGO detectors were observing at the time of GW150914.The Virgo detector was being upgraded, and GEO600,though not sufficiently sensitive to detect this event,was operating but not in observational mode.With only two detectors the source position is primarily determined by the relative arrival time and localized to an area of approximately600deg2(90% credible region)[39,46].The basic features of GW150914point to it being produced by the coalescence of two black holes—i.e., their orbital inspiral and merger,and subsequent final black hole ringdown.Over0.2s,the signal increases in frequency and amplitude in about8cycles from35to150Hz,where the amplitude reaches a maximum.The most plausible explanation for this evolution is the inspiral of two orbiting masses,m1and m2,due to gravitational-wave emission.At the lower frequencies,such evolution is characterized by the chirp mass[11]M¼ðm1m2Þ3=5121=5¼c3G596π−8=3f−11=3_f3=5;where f and_f are the observed frequency and its time derivative and G and c are the gravitational constant and speed of light.Estimating f and_f from the data in Fig.1, we obtain a chirp mass of M≃30M⊙,implying that the total mass M¼m1þm2is≳70M⊙in the detector frame. This bounds the sum of the Schwarzschild radii of thebinary components to2GM=c2≳210km.To reach an orbital frequency of75Hz(half the gravitational-wave frequency)the objects must have been very close and very compact;equal Newtonian point masses orbiting at this frequency would be only≃350km apart.A pair of neutron stars,while compact,would not have the required mass,while a black hole neutron star binary with the deduced chirp mass would have a very large total mass, and would thus merge at much lower frequency.This leaves black holes as the only known objects compact enough to reach an orbital frequency of75Hz without contact.Furthermore,the decay of the waveform after it peaks is consistent with the damped oscillations of a black hole relaxing to a final stationary Kerr configuration. Below,we present a general-relativistic analysis of GW150914;Fig.2shows the calculated waveform using the resulting source parameters.III.DETECTORSGravitational-wave astronomy exploits multiple,widely separated detectors to distinguish gravitational waves from local instrumental and environmental noise,to provide source sky localization,and to measure wave polarizations. The LIGO sites each operate a single Advanced LIGO detector[33],a modified Michelson interferometer(see Fig.3)that measures gravitational-wave strain as a differ-ence in length of its orthogonal arms.Each arm is formed by two mirrors,acting as test masses,separated by L x¼L y¼L¼4km.A passing gravitational wave effec-tively alters the arm lengths such that the measured difference isΔLðtÞ¼δL x−δL y¼hðtÞL,where h is the gravitational-wave strain amplitude projected onto the detector.This differential length variation alters the phase difference between the two light fields returning to the beam splitter,transmitting an optical signal proportional to the gravitational-wave strain to the output photodetector. To achieve sufficient sensitivity to measure gravitational waves,the detectors include several enhancements to the basic Michelson interferometer.First,each arm contains a resonant optical cavity,formed by its two test mass mirrors, that multiplies the effect of a gravitational wave on the light phase by a factor of300[48].Second,a partially trans-missive power-recycling mirror at the input provides addi-tional resonant buildup of the laser light in the interferometer as a whole[49,50]:20W of laser input is increased to700W incident on the beam splitter,which is further increased to 100kW circulating in each arm cavity.Third,a partially transmissive signal-recycling mirror at the outputoptimizes FIG. 2.Top:Estimated gravitational-wave strain amplitude from GW150914projected onto H1.This shows the full bandwidth of the waveforms,without the filtering used for Fig.1. The inset images show numerical relativity models of the black hole horizons as the black holes coalesce.Bottom:The Keplerian effective black hole separation in units of Schwarzschild radii (R S¼2GM=c2)and the effective relative velocity given by the post-Newtonian parameter v=c¼ðGMπf=c3Þ1=3,where f is the gravitational-wave frequency calculated with numerical relativity and M is the total mass(value from Table I).the gravitational-wave signal extraction by broadening the bandwidth of the arm cavities [51,52].The interferometer is illuminated with a 1064-nm wavelength Nd:Y AG laser,stabilized in amplitude,frequency,and beam geometry [53,54].The gravitational-wave signal is extracted at the output port using a homodyne readout [55].These interferometry techniques are designed to maxi-mize the conversion of strain to optical signal,thereby minimizing the impact of photon shot noise (the principal noise at high frequencies).High strain sensitivity also requires that the test masses have low displacement noise,which is achieved by isolating them from seismic noise (low frequencies)and designing them to have low thermal noise (intermediate frequencies).Each test mass is suspended as the final stage of a quadruple-pendulum system [56],supported by an active seismic isolation platform [57].These systems collectively provide more than 10orders of magnitude of isolation from ground motion for frequen-cies above 10Hz.Thermal noise is minimized by using low-mechanical-loss materials in the test masses and their suspensions:the test masses are 40-kg fused silica substrates with low-loss dielectric optical coatings [58,59],and are suspended with fused silica fibers from the stage above [60].To minimize additional noise sources,all components other than the laser source are mounted on vibration isolation stages in ultrahigh vacuum.To reduce optical phase fluctuations caused by Rayleigh scattering,the pressure in the 1.2-m diameter tubes containing the arm-cavity beams is maintained below 1μPa.Servo controls are used to hold the arm cavities on resonance [61]and maintain proper alignment of the optical components [62].The detector output is calibrated in strain by measuring its response to test mass motion induced by photon pressure from a modulated calibration laser beam [63].The calibration is established to an uncertainty (1σ)of less than 10%in amplitude and 10degrees in phase,and is continuously monitored with calibration laser excitations at selected frequencies.Two alternative methods are used to validate the absolute calibration,one referenced to the main laser wavelength and the other to a radio-frequencyoscillator(a)FIG.3.Simplified diagram of an Advanced LIGO detector (not to scale).A gravitational wave propagating orthogonally to the detector plane and linearly polarized parallel to the 4-km optical cavities will have the effect of lengthening one 4-km arm and shortening the other during one half-cycle of the wave;these length changes are reversed during the other half-cycle.The output photodetector records these differential cavity length variations.While a detector ’s directional response is maximal for this case,it is still significant for most other angles of incidence or polarizations (gravitational waves propagate freely through the Earth).Inset (a):Location and orientation of the LIGO detectors at Hanford,WA (H1)and Livingston,LA (L1).Inset (b):The instrument noise for each detector near the time of the signal detection;this is an amplitude spectral density,expressed in terms of equivalent gravitational-wave strain amplitude.The sensitivity is limited by photon shot noise at frequencies above 150Hz,and by a superposition of other noise sources at lower frequencies [47].Narrow-band features include calibration lines (33–38,330,and 1080Hz),vibrational modes of suspension fibers (500Hz and harmonics),and 60Hz electric power grid harmonics.[64].Additionally,the detector response to gravitational waves is tested by injecting simulated waveforms with the calibration laser.To monitor environmental disturbances and their influ-ence on the detectors,each observatory site is equipped with an array of sensors:seismometers,accelerometers, microphones,magnetometers,radio receivers,weather sensors,ac-power line monitors,and a cosmic-ray detector [65].Another∼105channels record the interferometer’s operating point and the state of the control systems.Data collection is synchronized to Global Positioning System (GPS)time to better than10μs[66].Timing accuracy is verified with an atomic clock and a secondary GPS receiver at each observatory site.In their most sensitive band,100–300Hz,the current LIGO detectors are3to5times more sensitive to strain than initial LIGO[67];at lower frequencies,the improvement is even greater,with more than ten times better sensitivity below60Hz.Because the detectors respond proportionally to gravitational-wave amplitude,at low redshift the volume of space to which they are sensitive increases as the cube of strain sensitivity.For binary black holes with masses similar to GW150914,the space-time volume surveyed by the observations reported here surpasses previous obser-vations by an order of magnitude[68].IV.DETECTOR VALIDATIONBoth detectors were in steady state operation for several hours around GW150914.All performance measures,in particular their average sensitivity and transient noise behavior,were typical of the full analysis period[69,70]. Exhaustive investigations of instrumental and environ-mental disturbances were performed,giving no evidence to suggest that GW150914could be an instrumental artifact [69].The detectors’susceptibility to environmental disturb-ances was quantified by measuring their response to spe-cially generated magnetic,radio-frequency,acoustic,and vibration excitations.These tests indicated that any external disturbance large enough to have caused the observed signal would have been clearly recorded by the array of environ-mental sensors.None of the environmental sensors recorded any disturbances that evolved in time and frequency like GW150914,and all environmental fluctuations during the second that contained GW150914were too small to account for more than6%of its strain amplitude.Special care was taken to search for long-range correlated disturbances that might produce nearly simultaneous signals at the two sites. No significant disturbances were found.The detector strain data exhibit non-Gaussian noise transients that arise from a variety of instrumental mecha-nisms.Many have distinct signatures,visible in auxiliary data channels that are not sensitive to gravitational waves; such instrumental transients are removed from our analyses [69].Any instrumental transients that remain in the data are accounted for in the estimated detector backgrounds described below.There is no evidence for instrumental transients that are temporally correlated between the two detectors.V.SEARCHESWe present the analysis of16days of coincident observations between the two LIGO detectors from September12to October20,2015.This is a subset of the data from Advanced LIGO’s first observational period that ended on January12,2016.GW150914is confidently detected by two different types of searches.One aims to recover signals from the coalescence of compact objects,using optimal matched filtering with waveforms predicted by general relativity. The other search targets a broad range of generic transient signals,with minimal assumptions about waveforms.These searches use independent methods,and their response to detector noise consists of different,uncorrelated,events. However,strong signals from binary black hole mergers are expected to be detected by both searches.Each search identifies candidate events that are detected at both observatories consistent with the intersite propa-gation time.Events are assigned a detection-statistic value that ranks their likelihood of being a gravitational-wave signal.The significance of a candidate event is determined by the search background—the rate at which detector noise produces events with a detection-statistic value equal to or higher than the candidate event.Estimating this back-ground is challenging for two reasons:the detector noise is nonstationary and non-Gaussian,so its properties must be empirically determined;and it is not possible to shield the detector from gravitational waves to directly measure a signal-free background.The specific procedure used to estimate the background is slightly different for the two searches,but both use a time-shift technique:the time stamps of one detector’s data are artificially shifted by an offset that is large compared to the intersite propagation time,and a new set of events is produced based on this time-shifted data set.For instrumental noise that is uncor-related between detectors this is an effective way to estimate the background.In this process a gravitational-wave signal in one detector may coincide with time-shifted noise transients in the other detector,thereby contributing to the background estimate.This leads to an overestimate of the noise background and therefore to a more conservative assessment of the significance of candidate events.The characteristics of non-Gaussian noise vary between different time-frequency regions.This means that the search backgrounds are not uniform across the space of signals being searched.To maximize sensitivity and provide a better estimate of event significance,the searches sort both their background estimates and their event candidates into differ-ent classes according to their time-frequency morphology. The significance of a candidate event is measured against the background of its class.To account for having searchedmultiple classes,this significance is decreased by a trials factor equal to the number of classes [71].A.Generic transient searchDesigned to operate without a specific waveform model,this search identifies coincident excess power in time-frequency representations of the detector strain data [43,72],for signal frequencies up to 1kHz and durations up to a few seconds.The search reconstructs signal waveforms consistent with a common gravitational-wave signal in both detectors using a multidetector maximum likelihood method.Each event is ranked according to the detection statistic ηc ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2E c =ð1þE n =E c Þp ,where E c is the dimensionless coherent signal energy obtained by cross-correlating the two reconstructed waveforms,and E n is the dimensionless residual noise energy after the reconstructed signal is subtracted from the data.The statistic ηc thus quantifies the SNR of the event and the consistency of the data between the two detectors.Based on their time-frequency morphology,the events are divided into three mutually exclusive search classes,as described in [41]:events with time-frequency morphology of known populations of noise transients (class C1),events with frequency that increases with time (class C3),and all remaining events (class C2).Detected with ηc ¼20.0,GW150914is the strongest event of the entire search.Consistent with its coalescence signal signature,it is found in the search class C3of events with increasing time-frequency evolution.Measured on a background equivalent to over 67400years of data and including a trials factor of 3to account for the search classes,its false alarm rate is lower than 1in 22500years.This corresponds to a probability <2×10−6of observing one or more noise events as strong as GW150914during the analysis time,equivalent to 4.6σ.The left panel of Fig.4shows the C3class results and background.The selection criteria that define the search class C3reduce the background by introducing a constraint on the signal morphology.In order to illustrate the significance of GW150914against a background of events with arbitrary shapes,we also show the results of a search that uses the same set of events as the one described above but without this constraint.Specifically,we use only two search classes:the C1class and the union of C2and C3classes (C 2þC 3).In this two-class search the GW150914event is found in the C 2þC 3class.The left panel of Fig.4shows the C 2þC 3class results and background.In the background of this class there are four events with ηc ≥32.1,yielding a false alarm rate for GW150914of 1in 8400years.This corresponds to a false alarm probability of 5×10−6equivalent to 4.4σ.FIG.4.Search results from the generic transient search (left)and the binary coalescence search (right).These histograms show the number of candidate events (orange markers)and the mean number of background events (black lines)in the search class where GW150914was found as a function of the search detection statistic and with a bin width of 0.2.The scales on the top give the significance of an event in Gaussian standard deviations based on the corresponding noise background.The significance of GW150914is greater than 5.1σand 4.6σfor the binary coalescence and the generic transient searches,respectively.Left:Along with the primary search (C3)we also show the results (blue markers)and background (green curve)for an alternative search that treats events independently of their frequency evolution (C 2þC 3).The classes C2and C3are defined in the text.Right:The tail in the black-line background of the binary coalescence search is due to random coincidences of GW150914in one detector with noise in the other detector.(This type of event is practically absent in the generic transient search background because they do not pass the time-frequency consistency requirements used in that search.)The purple curve is the background excluding those coincidences,which is used to assess the significance of the second strongest event.For robustness and validation,we also use other generic transient search algorithms[41].A different search[73]and a parameter estimation follow-up[74]detected GW150914 with consistent significance and signal parameters.B.Binary coalescence searchThis search targets gravitational-wave emission from binary systems with individual masses from1to99M⊙, total mass less than100M⊙,and dimensionless spins up to 0.99[44].To model systems with total mass larger than 4M⊙,we use the effective-one-body formalism[75],whichcombines results from the post-Newtonian approach [11,76]with results from black hole perturbation theory and numerical relativity.The waveform model[77,78] assumes that the spins of the merging objects are alignedwith the orbital angular momentum,but the resultingtemplates can,nonetheless,effectively recover systemswith misaligned spins in the parameter region ofGW150914[44].Approximately250000template wave-forms are used to cover this parameter space.The search calculates the matched-filter signal-to-noiseratioρðtÞfor each template in each detector and identifiesmaxima ofρðtÞwith respect to the time of arrival of the signal[79–81].For each maximum we calculate a chi-squared statisticχ2r to test whether the data in several differentfrequency bands are consistent with the matching template [82].Values ofχ2r near unity indicate that the signal is consistent with a coalescence.Ifχ2r is greater than unity,ρðtÞis reweighted asˆρ¼ρ=f½1þðχ2rÞ3 =2g1=6[83,84].The final step enforces coincidence between detectors by selectingevent pairs that occur within a15-ms window and come fromthe same template.The15-ms window is determined by the10-ms intersite propagation time plus5ms for uncertainty inarrival time of weak signals.We rank coincident events basedon the quadrature sumˆρc of theˆρfrom both detectors[45]. To produce background data for this search the SNR maxima of one detector are time shifted and a new set of coincident events is computed.Repeating this procedure ∼107times produces a noise background analysis time equivalent to608000years.To account for the search background noise varying acrossthe target signal space,candidate and background events aredivided into three search classes based on template length.The right panel of Fig.4shows the background for thesearch class of GW150914.The GW150914detection-statistic value ofˆρc¼23.6is larger than any background event,so only an upper bound can be placed on its false alarm rate.Across the three search classes this bound is1in 203000years.This translates to a false alarm probability <2×10−7,corresponding to5.1σ.A second,independent matched-filter analysis that uses adifferent method for estimating the significance of itsevents[85,86],also detected GW150914with identicalsignal parameters and consistent significance.When an event is confidently identified as a real gravitational-wave signal,as for GW150914,the back-ground used to determine the significance of other events is reestimated without the contribution of this event.This is the background distribution shown as a purple line in the right panel of Fig.4.Based on this,the second most significant event has a false alarm rate of1per2.3years and corresponding Poissonian false alarm probability of0.02. Waveform analysis of this event indicates that if it is astrophysical in origin it is also a binary black hole merger[44].VI.SOURCE DISCUSSIONThe matched-filter search is optimized for detecting signals,but it provides only approximate estimates of the source parameters.To refine them we use general relativity-based models[77,78,87,88],some of which include spin precession,and for each model perform a coherent Bayesian analysis to derive posterior distributions of the source parameters[89].The initial and final masses, final spin,distance,and redshift of the source are shown in Table I.The spin of the primary black hole is constrained to be<0.7(90%credible interval)indicating it is not maximally spinning,while the spin of the secondary is only weakly constrained.These source parameters are discussed in detail in[39].The parameter uncertainties include statistical errors and systematic errors from averaging the results of different waveform models.Using the fits to numerical simulations of binary black hole mergers in[92,93],we provide estimates of the mass and spin of the final black hole,the total energy radiated in gravitational waves,and the peak gravitational-wave luminosity[39].The estimated total energy radiated in gravitational waves is3.0þ0.5−0.5M⊙c2.The system reached apeak gravitational-wave luminosity of3.6þ0.5−0.4×1056erg=s,equivalent to200þ30−20M⊙c2=s.Several analyses have been performed to determine whether or not GW150914is consistent with a binary TABLE I.Source parameters for GW150914.We report median values with90%credible intervals that include statistical errors,and systematic errors from averaging the results of different waveform models.Masses are given in the source frame;to convert to the detector frame multiply by(1þz) [90].The source redshift assumes standard cosmology[91]. Primary black hole mass36þ5−4M⊙Secondary black hole mass29þ4−4M⊙Final black hole mass62þ4−4M⊙Final black hole spin0.67þ0.05−0.07 Luminosity distance410þ160−180MpcSource redshift z0.09þ0.03−0.04。

量子力学英文名词

probability density probability wave normalizing condition Schrödinger equation stationary state stationary Schrödinger equation
势阱
对应原理
隧道效应
能量量子化
Paulaser 泡利不相容原理激光自发辐射受激辐射氦氖激光器红宝石激光器
He-Ne laser
Pfund series Bohr quantization condition Bohr hydrogen atom Bohr frequency condition Bohr radius energy level
energy quantum photoelectric effect photo electron photocurrent cutoff potential difference red-limit wave-particle dualism
康普顿效应康普顿散射康普顿波长反冲电子莱曼系帕邢系布拉开系
主量子数
角动量量子化
potential well
correspondence principle
tunneling effect
energy quantization
principal quantum number
angular quantization
角量子数空间量子化磁量子数电子自旋自旋量子数自旋磁量子数
Stefan-Boltzmann law Stefan constant Wien displacement law Rayleigh-Jens formula Planck radiation formula Planck constant

导致Unruh-Hawking效应与可延拓出分叉Killing视界的充分条件

导致Unruh-Hawking效应与可延拓出分叉Killing视界的充分条件张靖仪;杨锦波【摘要】文章回顾了“导致Hawking效应的普遍坐标变换”一文中产生Unruh-Hawking效应的条件,对这些条件的作用进行了梳理.基于同样的方法,作者找到了能导致可延拓分叉Killing视界的充分条件并做出了证明.新的条件原则上也可以把非时轴正交的情况包括进来.由于可延拓分叉Killing视界普遍具有非零的表面引力,因此,也可以说这些新条件是导致Unruh-Hawking效应的充分条件.最后以极端RN黑洞为例,讨论了极端Killing视界的情况.【期刊名称】《广州大学学报（自然科学版）》【年(卷),期】2017(016)002【总页数】5页(P9-13)【关键词】分叉Killing视界;非零表面引力;Unruh-Hawking效应;极端视界【作者】张靖仪;杨锦波【作者单位】广州大学天体物理中心,广东广州510006;广州大学天体物理中心,广东广州510006【正文语种】中文【中图分类】O412.1Rindler参考系的Unruh效应与黑洞的Hawking辐射的发现已经几十年了，它们的发现令黑洞力学四定律与热力学四定律的“形似”变为“神似” [1-2].人们从不同的角度出发去理解它们，有的从弯曲时空量子场论出发，证明了这种效应是分叉Killing视界的一个普适的性质[3-4]；有的认为虽然已经确定了视界普遍导致这种效应，但是其统计物理的根源尚未得到很好的理解[5-7]; 它同时还有其他物理上很新奇的想法的来源.例如，一个研究有限温度场论的非常有意义的工具——桂氏时空，也是受到这方面研究的启发提出来的[8-9].分叉Killing视界跟黑洞热力学第零定律——表面引力为常数有很大的关系.可以证明，分叉Killing视界的表面引力必定为常数，反过来，表面引力为常数的Killing 视界虽然不一定就是分叉Killing视界，但总可以延拓出一个分叉Killing视界[10]. 此外，WALD还证明了黑洞的熵可以视为微分同胚不变的引力理论对分叉Killing视界的Noether荷[11].可见分叉Killing视界应与黑洞热力学有很深的关系.黑洞的Hawking辐射引发了信息佯谬[12].现在基于全息原理的论证，大部分物理学家都相信Hawking辐射的过程是幺正的，但是不清楚信息释放的机制，一般都认为Hawking原来的计算忽略了出射Hawking辐射粒子与剩下的黑洞的微观态的关联[13].考虑了这一关联之后，幺正性就不会遭到破坏，但是这个考虑又引起了晚期黑洞会不会在视界形成Firewall的争论[14].在这场争论中，MADECENA等[15]提出了ER=EPR的猜想，它断言几何上的Einstein-Rosen桥时空构型跟量子的EPR纠缠对是等价的，例如，Schwarzchild-AdS的最大延拓正好可以在全息原理的意义上跟边界上的热纠缠态对应起来.这个想法起码能在特定的拓扑场论中以某种方式实现[16].随后更加激进的猜想——复杂度-体积对应乃至复杂度-作用量对应也被提出来[17-18].有趣的是，他们所谈的不可穿越虫洞——Einstein-Rosen桥实际上也是分叉Killing 视界的喉部.从这点来看，按照文献[3-4]的结果，起码可以认为ER=EPR在弯曲时空量子场论的层次上是成立的.文献[7]在寻找到Unruh-Hawking效应的统计物理根源这一方向上做出了努力.该文认为时空变量分离变量形式的坐标变换扮演了很重要的角色，地位可能相当于统计物理中的分子混沌假设.只要再提一些物理上合理的要求，可以证明，分离变量型的坐标变换会导致Unruh-Hawking效应.然而这些条件，有些是对坐标系的要求，有些又是对时空的要求，但并没有区分得很清楚.本文希望能进一步理清文献[7]所提条件的含义，进一步讨论它们和分叉Killing视界的关系，找出导致Hawking效应的充分条件.针对线元ds2=-G0dT2+G1dX2+G2dY2+G3dZ2, 文献[7]给出了能导致Hawking辐射的普遍坐标变换:①坐标系变换采取分离变量的形式且f1(x),f2(x),g1(t),g2(t)全都不为常数；②新时空时轴正交③新时空稳态且④满足初条件t=0时，T=0.这4个条件将会把坐标变换的形式限制成条件1构造了一个坐标系变换，但是坐标系变换是可以很随意的，取什么样的坐标变换都可以，初看是一个平凡的条件.条件4不怎么重要，只是个原点选择的问题.实质性的是条件3.条件3该分3个方面看.首先是要求新时空稳态，也就是说时空中存在类时Killing矢量场.其次是把它的积分曲线的参数用作时间坐标t，则可定义Killing矢量场的适配坐标系.在这个坐标系中，自动会有G3=0，粗略地看，这个要求似乎也很平凡.但它配合同样也是看上去很平凡的条件1就不平凡了，它们在一起相当于要求从坐标系{T,X,Y,Z}换到Killing矢量场的适配坐标系必须具有分离变量的形式.最后，条件3中最强的要求是G1=0.一般而言，度规分量与t无关只在适配坐标系中成立，换了其它坐标系就不一定成立.可见条件3还对其它坐标系中的度规分量提出了要求.而这个要求会对G0，G1做出限制，使它们满足一定的关系.它可以等价地理解为对这2个要求：(dT)a(dT)a和(dX)a(dX)a对Killing 矢量场的李导数为零.条件2限制太强，把Kerr一类的黑洞都排除在外，这其实可以改进.重要的是要挑出一个跟Killing矢量场垂直的方向.本节将用文献[7]中的技术，重新研究什么是可延拓出分叉Killing视界的充分条件.给定一个时空，时空里存在标量场r(可以据此定出时空分层)与类时Killing矢量场a, 它们满足:① aar=0.于是可以选定超曲面r=rc,由于 a总是切于超曲面，所以超曲面r=rc里总有 a的积分曲线.任意指定一条，再让rc发生变动，就得到一个以rc为参数的 a 积分曲线族，它张成一个二维子流形.二维子流形上的坐标可以自然地取为r和 a 积分曲线的参数η,并使得它满足，这个二维流形上的诱导度规为显然，aa和arar都不会是η的函数.②用η和r再进一步构造如下形式的标量场T,R其中， 1不能全局为零.要求它们满足如下条件可以证明存在常数c，使得(dT)a(dT)a+c2(dR)a(dR)a=0和L(T2-c2R2)=0成立.再进一步，如果(dT)a(dT)a作为T和R函数，在包括T=R=0的开集上满足-∞<(dT)a(dT)a<0,则可延拓出分叉Killing视界.下面给出证明过程.由T和R的构造可知为了简化公式，定义gTT=(dT)a(dT)a, gRR=(dR)a(dR)a, gηη=(dη)a(dη)a,grr=(dr)a(dr)a.于是因为1(η)不能全局为零,所以从公式(10)可以导出公式(8)和公式(9)就是要求LgTT=LgRR=0,可以导出可见，和都是常数，而且是非零常数.非零是因为如果常数为零，就会有g0和g1为常数, 继而T和R的表达式，不能用来构成坐标变换.反之可以设0=w1g1和1=w0g0，其中w0,w1都是不为零的常数.这样就有gηη.再次运用公式(8)和公式(9)可以求出公式(14)又给出于是，有根据上式和 a类时,可以得出w0与w1必定同号，这样0=w1g1和1=w0g0还蕴含着,即为常数,不妨记为K.并且，式(19)可以推出于是f1=af0,其中a是任意非零常数.从式(15)可以得到配合,还可以知道L，更进一步，有w0与w1必定同号意味着是正的常数.还可以求出可见K必定是负的常数， a类时说明gηη<0.条件gTT=(dT)a(dT)a为有限负数是可以得到满足的.引进坐标变换ξ = ,那么dr, 二维面上的诱导线元关系式(22)变为再根据和gTT必定是个有限负数，有原来讨论的时候要求a是类时的，只占据了R2<0区域，即ξ>0的范围.而式(26)和式(27)则表示ξ→0时，有aa→0.ξ可以光滑地延拓到等于0和小于0的地方，分别对应R2=0和的区域.在T-R平面上，代表和根直线，可见它就是分叉Killing 视界.命题得证.实际上对gTT在T=R=0附近的有限性要求暗示人们，即使坐标变换采取时空变量分离、时间部分作为双曲函数出现的形式，也不一定能延拓出分叉Killing视界，因而不一定有非零的表面引力.以RN黑洞为例，分极端情况和非极端情况讨论,以此来说明(dT)a(dT)a的重要性.给出RN黑洞的线元表达式视界位置由方程r2-2Mr+Q2=0决定.先讨论非极端黑洞，有r+≠r-，于是线元可以写为引入坐标变换其中则有其中如果选择，那么，而且在r=r+处是解析的.T,R可以延拓到T2-R2<0的区域以外.而这个ω的选择正好就是外视界r=r+的表面引力.对内视界也一样，如果选择,则G(r)在r=r-处解析.现在再来讨论极端RN黑洞，这时r+=r-=M，于是线元表达式是引入类似的坐标变换其中，将得到ds2=G(r)(-dT2+dR2)+r(T,R)2dΩ2其中r作为T,R的函数由下式决定无论如何选择ω,G(r)在r=M处的奇异性都无法消去，但是r=M对极端RN黑洞而言只是一个坐标奇性，这说明了坐标系{T,R}只能覆盖到T2-R2<0的区域，它没有办法做延拓.文献[7]提及的4个条件，极端RN黑洞都能满足，确实如同证明所表现的那样一定可以找到该文献中提到的特定的坐标变换:T=f(r)sinh(ωt),R=f(r)·cosh(ωt),然而极端RN黑洞的表面引力为0，意味着在极端RN黑洞的时空中不存在Hawking辐射(但可以有粒子对产生[19]).也就是说这种形式的坐标系变换并不如原来想象中那样会导致Hawking辐射.只有当可以延拓出分叉Killing视界的时候，才会有Unruh-Hawking效应.而要做到这点，就要在原来所提条件的基础上多加一个对(dT)a(dT)a解析性质的要求.可延拓出分叉Killing视界的充分条件是存在标量场r与类时Killing矢量场 a，满足 aar=0，并且可以利用它们去构造2个标量场T和R,满足要求L(dT)a(dT)a=L(dR)a(dR)a=(dT)a(dR)a=0，和要求(dT)a(dT)a作为T和R的函数在T=R=0附近有限.aar=0和L(dT)a(dT)a=L(dR)a(dR)a=(dT)a(dR)a=0 2个条件导致了L和个结果.这2个结果保证坐标变换中的η变量总会以双曲函数的形式出现.(dT)a(dT)a作为T和R的函数在T=R=0附近有限意味着总可以延拓出分叉Killing视界，于是总有一个非零的表面引力.最后这点的要求必不可少，极端RN黑洞就是一个很好的例子.【相关文献】[1] HAWKING S W. Particle creation by black holes[J]. Commun Math Phys，1975，43: 199-220.[2] UNRUH W G, WEISS N. Acceleration radiation in interacting field theories[J]. Phys Rev D，1984，29: 1656-1662.[3] JACOBSON T. A note on Hartle-Hawking Vacua[J]. Phys Rev D, 1994, 50:6031-6032.[4] SANDERS K. On the construction of Hartle-Hawking-Israel states across a static bifurcate killing horizon[J]. Lett Math Phys, 2015, 105(4): 575-640.[5] 赵峥. 四维静态黎曼时空中的Hawking辐射[J]. 物理学报，1981，30: 1508-1518.ZHAO Z. Hawking radiation in four dimensional static Riemann spacetime[J]. Acta Phys Sin, 1981, 30: 1508-1518.[6] 刘辽. 费曼路径积分和霍金蒸发[J]. 物理学报，1982，31(4): 519-524.LIU L. Feynman’s Path-integral Method and Hawking evaporation[J]. Acta Phys Sin, 1982, 31(4): 519-524.[7] 赵峥. 导致Hawking效应的普遍坐标变换[J]. 物理学报，1990，39(11): 1854-1862.ZHAO Z. University coordinate transformation leading to Hawking effect[J]. Acta Phys Sin, 1990, 39(11): 1854-1862.[8] GUI Y X. Quantum field in η-ξ spacetime[J]. Phys Rev D，1990，42: 1988-1995.[9] GUI Y X. Fermion fields in η-ξ spacetime[J]. Phys Rev D，1992，45: 697-700.[10]RACZ I, WALD R M. Extensions of spacetimes with killing horizons[J]. Class Quantum Grav，1992，9(12): 2643-2656.[11]WALD R M. Black hole entropy is the noether charge[J]. Phys Rev D，1993，48:R3427-3431.[12]HAWKING S W. Breakdown of predictability in gravitational collapse[J]. Phys Rev D，1976，14: 2460-2473.[13]VERLINDE E, VERLINDE H. Black Hole entanglement and quantum errorcorrection[J/OL]. 2012,arXiv:1211.6913v1 [hep-th].[14]ALMHEIRI A, MAROLF D, POLCHINSKI J, et al. Black Holes: Complementary or firewalls[J/OL]. 2012,arXiv:1207.3123v4 [hep-th].[15]MALDACENA J, SUSSKIND L. Cool horizons for entangled black holes[J/OL].2013,arXiv:1306.0533v2 [hep-th].[16]BAEZ J C, VICARY J. Wormholes and Entanglement[J/OL]. 2014,arXiv:1401.3416v2[gr-qc].[17]SUSSKIND L. Entanglement is not enough[J/OL]. 2014,arXiv:1411.0690v1[hep-th].[18]BROWN A R, ROBERTS D A, SUSSKIND L. Complexity equals action[J/OL].2015,arXiv:1509.07876v1 [hep-th].[19]CHEN C M, KIM S P, LIN I C, et al. Spontaneous pair production in reissner-nordström Black Holes[J/OL]. 2012,arXiv:1202.3224v2 [hep-th].。

圈量子引力

三. 黑洞熵的启示
迄今为止对量子引力理论最具体最直接的 “理论证据” 来自于对黑洞热力学的研究。一九七二年， Princeton 大学的研究生 J. D. Bekenstein 受黑洞动力学与经典热力学之间的相似性启发，提出了黑洞熵的概念，并估算出黑洞的熵正比于其视界 (Event Horizon) 面积。稍后，S. W. Hawking 研究了黑洞视界附近的量子过程，结果发现了著名的 Hawking 幅射，即黑洞会向外幅射粒子 (也称为黑洞蒸发)，从而表明黑洞是有温度的。由此出发 Hawking 也推导出了 Bekenstein 的黑洞熵公式，并确定了比例系数，这就是所谓的 BekensteinHawking 公式：
与协变量子化方法一样，早期的正则量子化方法也遇到了大量的困难，这些困难既有数学上的，比如 Wheeler-DeWitt 方程别说求解，连给出一个数学上比较严格的定义都困难；也有物理上的，比如无法找到合适的可观测量和物理态。引力量子化的这些早期尝试所遭遇的困难，特别是不同的量子化方法给出的结果大相径庭这一现象是具有一定启示的。这些问题的存在反映了一个很基本的事实，那就是许多不同的量子理论可以具有同样的经典极限，因此对一个经典理论量子化的结果是不唯一的，原则上就不存在所谓唯一 “正确” 的量子化方法。其实不仅量子理论，经典理论本身也一样，比如经典 Newton 引力就有许多推广，以 Newton引力为共同
一. 量子时代的流浪儿

二十世纪理论物理学家说得最多的话之一也许就是： “广义相对论和量子理论是现代物理学的两大支柱”。两大支柱对于建一间屋子来说可能还太少，但对于物理学却已嫌多，二十世纪物理学家的一个很大的梦想就是把这两大支柱合而为一。如今二十世纪已经走完，回过头来重新看看这两大支柱，在量子理论这根支柱上已经建起了十分宏伟的殿堂，物理学的绝大多数分支都在这座殿堂中搭起了自己的舞台。物理学中已知的四种基本相互作用有三种在这座殿堂内得到了一定程度的描述。可以说，物理学的万里河山量子理论已经十有其九。今天的物理学正处在一个不折不扣的量子时代。而这个辉煌的量子时代最大的缺憾就在于物理学的另一根支柱 - 广义相对论还孤零零地游离在量子理论的殿堂之外。广义相对论成了量子时代的流浪儿。

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arXiv:hep-th/9708136v1 26 Aug 1997Brane-BlackHoleCorrespondenceandAsymptoticsofQuantumSpectrum

A.A.Bytsenko1StateTechnicalUniversity,St.Petersburg,195251,RussiaA.E.Goncalves2DepartamentodeFisica,UniversidadeEstadualdeLondrina86051-970,Londrina-Parana,BrasilS.D.Odintsov3DepartamentodeFisica,UniversidaddelValleA.A.25360,Cali,Colombia

AbstractWediscusstheasymptoticpropertiesofquantumstatesdensityforfun-damental(super)membraneinthesemiclassicalapproach.ThematchingofBPSpartofspectrumforsuperstringandsupermembranegivesthepossi-bilitytogetstringyresultsviamembranecalculationsandviceversa.Thebrane-blackholecorrespondence(onthelevelofblackholestatesandbranemicrostates)isalsostudied.

1.Ithasbeenrealizedrecentlythatthereareverydeepconnectionsbetweenfundamental(super)membraneand(super)stringtheory.Inpar-ticulary,ithasbeenshownthattheBPSspectrumofstatesfortypeIIBstringonacircleisincorrespondencewiththeBPSspectrumoffundamen-talcompactiﬁedsupermembrane[1,2].Remarkableprogresshasbeenmadetowardsestablishingthestring-blackholecorrespondencerelevantfortheextremeblackhole[3].TheentropyassociatedwiththeBPSstatesisthenidenticaltotheBekenstein-Hawkingentropydeﬁnedbythehorizonarea.Theideaofstring-blackholecorrespondencehasbeenformulatedalsoasthecorrespondenceprincipleinRef.[4].2.Thatisthepurposeofthisworktolooktosomeofthesequestionsfromfundamentalsupermembranepointofview.Westartfromthesemiclassicalfreeenergyforfundamentalcompactiﬁedsupermembranes(whichisknowntobedivergent)embeddedinﬂatD-dimensionalmanifoldswithtopologiesM=S1⊗Td⊗RD−d−1(Tdisthed-dimensionaltorus).Firstofallweremindthatforsimplestquantumﬁeldmodelthefreeenergyhastheform[5,6]

F(b,f)(β)=−πd(detA)1/2󰀑∞0ds(2s)−(D−d+2)/2Ξ(b,f)(s,β)

×Θ󰀉g0󰀍(0|Ω)exp󰀆−sM2

2s󰀈−1,Ξ(f)(s,β)=1−θ4󰀆0|iβ2

2|τ)aretheJacobithetafunctions.HereA=diag(R−21,...,R−2d)isad×dmatrix,theglobalparametersRjchar-acterizingthenon-trivialtopologyofMappearinthetheoryowingtothefactthatcoordinatesxj(j=1,...,d)obeytheconditions0≤xj<2πRj.ThenumberoftopologicalconﬁgurationsofquantumﬁeldsisequaltothenumberofelementsingroupH1(M;Z2),ﬁrstcohomologygroupwithcoeﬃ-cientsinZ2.Themultipletg=(g1,...,gd)deﬁnesthetopologicaltypeofﬁeld(i.e.,thecorrespondingtwist),anddependingontheﬁeldtypechoseninM,gj=0or1/2.InourcaseH1(M;Z2)=Zd2andsothenumberoftopologicalconﬁgurationsofrealscalars(spinors)is2d.WefollowthenotationsandtreatmentofRef.[7]andintroducethethetafunctionwithcharacteristicsa,bfora,b∈Zd,

Θ󰀉ab󰀍(z|Ω)=󰀋n∈Zdexp[iπ(n+a)Ω(n+a)+i2π(n+a)(z+b)],(3)

2inthisconnectionΩ=(is/2π2)diag(R21,...,R2d).Theabovemethodofthefreeenergycalculationadmitsthesubsequentdevelopmentforextendedob-jects.Weshallassumethatfreeenergyisequivalenttoasumofthefreeenergiesofquantumﬁeldswhichpresentinthemodesofamembrane.Thefactorexp(−sM2/2π)inEq.(1)shouldbeunderstoodasTrexp(−sM2/2π),whereMisthemassoperatorofmembraneandthetraceisoverinﬁnitesetofBose-FermioscillatorsN(b)n,N(f)n.3.Forthenoncompactiﬁedsupermembranethequestionofreliabil-ityofthesemiclassicalapproximationisnotabsolutelyclear[8].Thedis-cretepartofthesupermembranespectrumpropagatingineleven-dimensionalMinkowskispace-timecanbewrittenintheform(seefordetailRefs.[8,5])

M2=8󰀋j=1󰀋n∈Z2/{0}ωn󰀓N(b)nj+N(f)nj󰀖,(4)whereωn=󰀔

2πM2󰀃=[H+(Ω)H−(Ω)]8,(6)whereH±(Ω)=󰀏n∈Zd/{0}󰀊1±exp[−(n,Ωn)1/2]󰀎(±1),(7)

andΩ=(s2/4)diag(a−2,b−2).ForgeneratingfunctionsH±(Ω),Ω=zdiag(1,...,1),(z=t+2πx)inthehalf-planeℜz>0thereexistsanasymptoticexpansionuniformlyinxast→0,provided|argz|≤π2andgivenby[9]

H+(Ω)=exp󰀊[AΓ(p)ζ−(1+p)z−p−Zp(0)log2+O(tc+)]󰀎,(8)H−(Ω)=exp󰀊[AΓ(p)ζ+(1+p)z−p−Zp(0)logz+Z′p(0)+O(tc−)]󰀎,(9)where0functionwhichhasapolewithresidueA.Inaboveequationsζ−(s)≡ζR(s)

3istheRiemannzetafunction,ζ+(s)=(1−21−s)ζ−(s).Thetotalnumberofquantumstatescanbedescribedbythequantitiesr±(N)deﬁnedby

K±(t)=∞󰀋N=0r±(N)tN≡H±(Ω),(10)wheret=exp(−z),t<1,andNisatotalquantumnumber.BymeansoftheasymptoticexpansionofK±(t)fort→1,whichisequivalenttoexpansionofH±(Ω,0)forsmallzandusingtheformulae(8)and(9)onearrivesatcompleteasymptoticofr±(N).ThusforN→∞onehas[6,9]

r±(N)=C±(p)N(2Zp(0)−p−2)/(2(1+p))

×exp󰀂1+p

[2π(1+p)]1/2,(12)κ±=pp−δ2−δ󰀈,(13)and02