Spacetime singularity resolution by M-theory fivebranes calibrated geometry, Anti-de Sitter

关于太阳黑洞的英文作文Title: Exploring the Mysteries of Solar Black Holes。

Introduction:Solar black holes, also known as stellar black holes, are enigmatic celestial objects that continue to fascinate astronomers and physicists alike. These cosmic phenomenaare born from the collapse of massive stars, leading to the formation of incredibly dense regions in space with gravitational fields so strong that not even light can escape from them. In this essay, we delve into theintricacies of solar black holes, exploring their formation, characteristics, and the profound impact they have on the universe.Formation of Solar Black Holes:Solar black holes originate from the explosive deathsof massive stars, known as supernovae. When a star exhaustsits nuclear fuel, it can no longer withstand its own gravitational force, causing it to collapse under its own weight. This collapse results in a supernova explosion, where the outer layers of the star are expelled into space, leaving behind the core.If the core of the star is sufficiently massive –typically several times the mass of the Sun – it undergoes further collapse, forming a black hole. This process compresses the mass of the core into an infinitesimally small point known as a singularity, surrounded by an event horizon beyond which no information or matter can escape.Characteristics of Solar Black Holes:Solar black holes possess several uniquecharacteristics that distinguish them from other celestial objects:1. Gravitational Singularity: At the center of a black hole lies a gravitational singularity, where matter is infinitely dense and spacetime curvature is infinitelysteep. The laws of physics, as we understand them, break down at this point, leading to a theoretical realm where our current understanding fails.2. Event Horizon: Surrounding the singularity is the event horizon, the boundary beyond which escape is impossible. Once an object crosses the event horizon, it is inexorably drawn towards the singularity, never to return.3. No Hair Theorem: According to the no-hair theorem, black holes are characterized by only three properties: mass, electric charge, and angular momentum. All other information about the material that formed the black hole is lost beyond the event horizon.Impact on the Universe:Solar black holes play a significant role in shaping the cosmos, influencing the dynamics of galaxies, stars, and even spacetime itself:1. Galactic Evolution: Black holes, including solarblack holes, are thought to reside at the centers of most galaxies. Their immense gravitational pull can influence the motion of stars and gas within galaxies, affectingtheir evolution over cosmic timescales.2. Stellar Dynamics: In binary star systems, a solar black hole can gravitationally interact with its companion star, leading to phenomena such as accretion disk formation and the emission of X-rays. These interactions can provide valuable insights into the properties of black holes and their surrounding environments.3. Time Dilation: The extreme gravitational fields near black holes cause time to dilate significantly. This phenomenon, predicted by Einstein's theory of general relativity, has practical implications for space exploration and our understanding of the nature of spacetime.Conclusion:Solar black holes stand as some of the most enigmaticand intriguing objects in the universe. From their mysterious formation to their profound impact on the cosmos, these cosmic behemoths continue to captivate theimagination of scientists and laypeople alike. As we probe deeper into the nature of black holes, we unlock newinsights into the fundamental laws that govern the universe and our place within it.。

a r X i v :g r -q c /0209001v 1 31 A u g 2002CIRI/02-swrskg01What is the spacetime of physically realizable spherical collapse?Sanjay M.Wagh 1,Ravindra V.Saraykar 1,2,and Keshlan inder 31Central India Research Institute,Post Box 606,Laxminagar,Nagpur 440022,India E-mail:cirinag@.in 2Also at:Department of Mathematics,Nagpur University Campus,Nagpur 440010,India.E-mail:sarayaka@.in3School of Mathematical &Statistical Sciences,University of Natal,Durban 4041,South Africa.E-mail:govinder@nu.ac.za (Dated:August 31,2002)We argue that a particular spacetime,a spherically symmetric spacetime with hyper-surface orthogonal,radial,homothetic Killing vector,is a physically meaningful spacetime that describes the problem of spherical gravitational collapse in its full “physical”generality.PACS numbers:04.20.-q,04.20.CvKeywords:Gravitational collapse -spherical symmetry -radial homothety -physically realizableI.INTRODUCTIONAny sufficiently sparsely distributed ordinary “neutral”matter is dusty,that is,collision-less and pressureless.Further,emission of radiation and,hence,radiation is not expected in such dust.In Newtonian gravity,as well as in general rel-ativity,we can then study gravitational collapse from this dusty “initial”state of matter.Under the action of its self-gravity,dust mat-ter collapses.Then,self-gravity leads to mass or energy-flux in some preferential direction,the ra-dial direction for a spherical spacetime.But,this is not the flux of radiation.Therefore,there is no mass-flux in the rest frame of collapsing dusty matter,but it is present for other observers in the spacetime.Next stage of collapse is reached when particles of dust begin to collide with each other.Negligi-ble amount of radiation,but existing nonetheless,is expected from whatever atomic excitations or from whatever free electrons get created in atomic collisions in such matter.Therefore,dusty matter evolves into matter with pressure and radiation,both simultaneously non-vanishing.The energy-flux can no longer be removed by going to the rest frame of matter.As far as Newtonian gravity and general relativ-ity,both,are concerned,we may study the grav-itational collapse beginning even as matter with non-vanishing pressure and radiation.At some stage,exothermic,thermonuclear reac-tions begin in matter with non-negligible pressure.With it,a star is born in the spacetime.This is the manner of gravitational collapse of dusty matter leading to the birth of a star.Till theexothermic thermonuclear reactions in the stellar core support the overlying stellar layers,such a stellar object is gravitationally stable.But,the spacetime continues to be dynamic since radiation is present in it.The stellar object may also accrete matter from its surrounding while emitting radiation.Once again,in Newtonian gravity and in general relativity,both,we may study the collapse of this “initial”stellar configuration of matter.Now,as and when “heating”of the overlying stellar layers decreases due to changes in exother-mic thermonuclear processes in the core of the star,the self-gravity of the stellar object leads to its gravitational contraction.These are,in general,very slow and involved processes.Gravitational contraction leads to generation of pressure by compression and by the occurrence of exothermic thermonuclear reactions of heavier nu-clei.The star may stabilize once more.This chain,of gravitational contraction of star,followed by pressure increase,followed by subse-quent stellar stabilization,continues as long as thermonuclear processes produce enough heat to support the overlying stellar layers.The theory of the atomic nucleus shows that exothermic nuclear processes do not occur when Iron nucleus forms.With time,the rate of heat generation in iron-dominated-core becomes insuf-ficient to support the overlying stellar layers which may then bounce offthe iron-core resulting into a stellar explosion,a supernova.Then,many,different such,stages of evolution are the results of physical processes that are un-related to the phenomenon of gravitation.These are,for example,collisions of particles of matter,electromagnetic and other forces between atomic or sub-atomic constituents of matter etc.As an example,let some non-gravitational pro-cess,opposing collapse,result into pressure that does not appreciably rise in response to small con-traction of the stellar matter.That is,pressure does not appreciably rise when gravitationalfield is increased by a small amount Then,the collapse of a sufficiently massive object would not be halted by that particular non-gravitational process.There-fore,a mass limit is obtained in this situation.For example,electron degeneracy pressure leads to the Chandrasekhar limit[1].Clearly,some of the non-gravitational processes determine the gravitational stability of physical objects.This is true in Newtonian gravity as well as in general relativity,both.In general relativity,non-gravitational processes are included via the energy-momentum tensor for matter.Non-gravitational processes determine the relation of density and pressure of matter.The temporal evolution of matter is to be determined from such a relation,and from other physical rela-tions,if any.It is therefore that,by physically realizable grav-itational collapse,we mean collapse that leads matter,step by step,through the above different “physical”stages of evolution.Hence,the spacetime of“physically realizable”collapse of matter must be able to begin with any stage in the chain of evolution of matter under the action of its self-gravity.The temporal evolution from any“initial”data,any“physical”stage in question,is to be obtained from applicable non-gravitational properties of matter.But,a supernova remnant,or a star that failed to explode,may be quite massive for its self-gravity to dominate over all conceivable competing reasons opposing it at various stages of further evolution. This may also happen as a result of mass-accretion taking the object in question over some mass-limit in operation.The collapse is,now,unstoppable.A spacetime singularity is expected to form in such unstoppable collapse.Associated with studies of unstoppable collapse is the issue of whether the physically realizable gravitational collapse leads to a black hole or to a naked spacetime singularity.This is the issue of the Cosmic Censorship Hypothesis(CCH)[2]. Clearly,the answer to this very important ques-tion in general relativity can then be obtained only on the basis of the spacetime of the physically re-alizable gravitational collapse.Now,we show below that,for spherical symme-try,a spacetime with hyper-surface orthogonal,ra-dial,homothetic Killing vector provides“all”the above steps of evolution of matter.II.SPACETIME OF PHYSICALLYREALIZABLE COLLAPSEOne radially homothetic spacetime has the fol-lowing metric[3]in co-moving coordinates: ds2=−y2dt2+γ2(y′)2B2dr2+y2Y2dΩ2(1) with y=y(r),an overhead prime indicating a derivative with respect to r,B≡B(t),Y≡Y(t) andγbeing a constant.As can be easily verified,the metric(1)admits a spacelike Homothetic Killing Vector(HKV)of the formX a=(0,yy2Y B+2¨By2γ2B2+2y2Y2+4¨Ysome t =t s .Thus,the singularity of first type is asingular hyper-surface for (1).The singularity of the second type is a singular sphere of coordinate radius r .The singular sphere reduces to a singular point for r =0that is the center of symmetry.Singularities of the second type constitute a part of the initial data,singular data,for the evolution.The metric (1)has evident degeneracies when y (r )=0,y (r )→∞either on a degenerate sphere of coordinate radius r ,for some “thick”shell or globally.Another degeneracy occurs for y (r )=constant for some “thick”shell or globally.In what follows,we shall assume that there is no singular initial-data and that there are no evi-dently degenerate situations for the metric (1).Temporal evolution in (1)The Einstein tensor for (1)is:G tt =1γ2B 2+˙Y2BY (5)G rr =γ2B 2y ′2Y −˙Y2γ2B 2−1B−Y˙Y ˙B γ2B 2(7)G φφ=sin 2θG θθ(8)G tr =2˙By′2σ33=1Y−˙B6σ.Therefore,the spacetime of (1)is,in general,shearing and radiating,both.With our assumptions of no singular and degen-erate initial data,we then have a “cosmological”situation -continued spherical collapse of matter from the assumed “initial”state.Now,for the co-moving observer with four-velocity U =1∂t ,the radial velocity of thefluid is V r =˙Ywhere an overhead dot denotes a time derivative.The co-moving observer is ac-celerating for (1)since ˙Ua =U a ;b U b is,in gen-eral,non-vanishing for y ′=0.The expansion isΘ=1B+2˙Yy 2˙Y2Y 2−1y 2˙Y2Y B+1γ2B 2(14)We also obtain2¨YB =22(ρ+3p )(15)Then,from (15),the relation of pressure anddensity of matter is the required additional “phys-ical”information.Also required is other relevant “physical”information to determine the radiation generation in the spacetime of (1).To provide for the required information of “phys-ical”nature is a non-trivial task in general relativ-ity just as it is for Newtonian gravity.The details of these considerations are,of course,beyond the scope of this letter.However,it is clear that thefield equations de-termine only the temporal functions from the prop-erties of matter in the spacetime of(1). Moreover,it is also clear that matter will con-tinue to pile up on such a star in a“cosmolog-ical setting”and,hence,such a star will always be taken over any mass-limit in operation at any stage of its evolution that will,ultimately,lead to the singular hyper-surface of the spacetime of(1). The radial dependence of matter properties is “specified”as1/y2but thefield equations of gen-eral relativity do not determine the metric function y(r)in(1).Therefore,the radial distribution of matter is ar-bitrary in terms of the co-moving radial coordinate r.This is the“maximal”physical freedom com-patible with the assumption of spherical symme-try,we may note.Note,however,that the phys-ical generality here is not be taken to mean the “geometrical”generality.III.ISSUE OF REGULARITY OF CENTER A spherical spacetime admits an SO(3)group of rotational symmetry.The orbits of the symmetry group are closed ones.The center of the spheri-cally symmetric spacetime geometry is defined to be the“invariant point”of the SO(3)group of ro-tations,as it must be.Galilean invariance of the Newtonian equations implies that every observer observes the shrinkage of orbits of the rotation group to zero radius at the center of a spherically symmetric object. Consequently,we may demand that the orbits of the rotation group also shrink to zero radius for a spherically symmetric spacetime.Such a space-time is said to possess a regular center.Now,for(1),y(r)is the“area radius”.When y|r=0=0,the orbits of the rotation group SO(3) do not shrink to zero radius at the center of the spacetime although the curvature invariants re-mainfinite at the center.Also,when y|r=0=0,the orbits of the rotation group shrink to zero radius at the center but the curvature invariants blow up at the center,then.It is well-known[5]that the center and the initial data for matter,both,are not simultaneously reg-ular for a spherical spacetime with hyper-surface orthogonal HKV.Therefore,the spacetime of(1)does not possess a regular center and regular matter data,simulta-neously,and we may consider it to be“unphysical”even when its matter follows the expected“physi-cal”evolution.However,this issue requires careful analysis.It is crucial to ask:who,which observer,is observ-ing the orbits of the rotation group shrink to zero radius?This is important in general relativity though not in Newtonian gravity.Consider the Schwarzschild spacetime.The asymptotic observer does not see any sphere,cen-tered on the mass-point,shrink to any radius below r=2M,the“infinite red-shift surface”.Then,for the asymptotic observer,r=2M is the center of the spacetime!Thus,the“area ra-dius”has this minimum value for the asymptotic observer.It is no coincidence that this minimum value of the“area radius”as observed by the asymptotic observer is related to the gravitational mass or the Schwarzschild mass M.Recall the well-know Newtonian theorem:“The gravitational force on a body that lies outside a closed spherical shell of matter is the same as it would be if all the shell’s matter were concentrated into a point at its center.”The general relativistic manifestation of this Newtonian result is that the spacetime of a spheri-cal body must possess non-vanishing central value for mass in that spacetime.For suitable observer, the orbits of the rotation group are not expected to shrink to zero radius at the center of a spherical spacetime with matter.The co-moving observer of(1)is the“equiv-alent”of the asymptotic observer of the Schwarzschild spacetime in that it is also the “cosmological”observer for(1).Then,for this observer,orbits of the rotation group SO(3)are not expected to shrink to zero“area radius”at the center of the spacetime of(1).Therefore,the conflict of the“non-regularity”of the center and the“physical”evolution of matter in(1)can be resolved with this observation. Now,let us call a star for which the orbits of the rotation group do not shrink to zero radius at its center a strange star,in contrast to the“standard”spherical star for which the center is regular.Then, it is clear from the above that all the spherical stars embedded in a cosmological surrounding are expected to be strange stars in the sense described above.IV.CONCLUDING REMARKS Many spacetimes of spherically symmetric na-ture are known[6].For example,the“original”Schwarzschild[7],the“standard”Schwarzschild (or,Hilbert-Droste,[7]),the Vaidya,the Tolman-Bondi class,the Friedmann-Lemaitre-Robertson-Walker spacetimes.Some of these known exam-ples of spherically symmetric spacetime geometries contain no matter,only radiation dust,only mat-ter dust,etc.To construct the spacetime of a“physically re-alizable”spherical collapse of matter,we therefore need to“match”different,appropriate,such space-time geometries.Of course,this is a non-trivial and,mostly,very difficult task.But,these different spacetime geometries are, clearly,different choices of y(r)in(1).For ex-ample,consider collapse from initial dusty matter with vacuum“exterior”.Then,y(r)is infinite in the exterior.The spacetime of(1)then provides the“final”spacetime that may be obtained after matching many different spacetimes with appropriate physi-cal conditions of matter.It is,therefore,the space-time of physically realizable collapse of spherically symmetric matter.We have,therefore,studied the shear-free col-lapse in[8]and the collapse with shear and energy flux in[9].We have also obtained source free elec-tromagneticfields using the Hertz-Debye formal-ism[10]in[9].In these studies,some general con-clusions as well as the explicit forms for temporal metric functions have been presented for a simple equation of state of the barotropic form p=αρwhereαis a constant.We have also studied,in [11],the phenomenon of Hawking radiation in a radially homothetic spacetime of the metric(1).[1]Chandrasekhar S(1931)Ap J7481Chandrasekhar S(1958)An introduction to the study of stellar structure(New York:Dover) [2]Penrose R(1998)in Black Holes and Singlarities:S.Chandrasekhar Symposium(Ed.R.M.Wald, Yale:Yale University Press)[3]Wagh S M,Govender M,Govinder K S,MaharajS D,Muktibodh P S and Moodley M(2001)Class.Quantum Grav.182147-2162[4]Synge J L(1964)in Relativity,Groups and Topol-ogy,Les Houches Lectures(New York:Gordon and Breach)[5]Macintosh,C.B.(1975)Gen.Rel.Grav.7199[6]Kramer, D.,Stephani,H.,MacCallum,M. A.H.and Herlt,E.(1980)Exact Solutions of Ein-stein’s Field Equations(Cambridge UniversityPress,Cambridge)[7]Abrams L S(1979)Phys.Rev.D202474.Alsoavailable as Database:gr-qc/0201044[8]Wagh S M,Saraykar R V,Muktibodh P S andGovinder K S(2002)Title:Radially homoth-etic,Shear-free,Spherical gravitational collapse Preprint:CIRI/07swpmkg[9]Wagh S M(2002)Title:Spherical gravitationalcollapse and electromagneticfields in radially ho-mothetic spacetimes Preprint:CIRI/02-sw01 [10]Cohen J M and Kegeles L S(1974)Phys.Rev.D101070-1084[11]Wagh S M(2002)Title:Hawking radiation inspatially homothetic,spherical collapse Preprint: CIRI/smw06。

SkyrmionIn particle theory,the skyrmion (/ˈskɜrmi.ɒn/)is a hy-pothetical particle related originally [1]to baryons .It was described by Tony Skyrme and consists of a quantum su-perposition of baryons and resonance states.[2]Skyrmions as topological objects are also important in solid state physics ,especially in the emerging technology of spintronics .A two-dimensional magnetic skyrmion ,as a topological object,is formed,e.g.,from a 3D effective-spin “hedgehog”(in the field of micromagnetics :out of a so-called "Bloch point "singularity of homotopy degree +1)by a stereographic projection ,whereby the positive north-pole spin is mapped onto a far-offedge circle of a 2D-disk,while the negative south-pole spin is mapped onto the center of the disk.1Mathematical definitionIn field theory,skyrmions are homotopically non-trivial classical solutions of a nonlinear sigma model with a non-trivial target manifold topology –hence,they are topological solitons .An example occurs in chiral models [3]of mesons,where the target manifold is a homogeneous space of the structure group (SU (N )L ×SU (N )RSU (N )diag)where SU (N )L and SU (N )R are the left and right parts of the SU (N )matrix,and SU (N ) ₐ is the diagonal subgroup .If spacetime has the topology S 3×R ,then classical con-figurations can be classified by an integral winding num-ber [4]because the third homotopy groupπ3(SU (N )L ×SU (N )R SU (N )diag ∼=SU (N ))is equivalent to the ring of integers,with the congruence sign referring to homeomorphism .A topological term can be added to the chiral Lagrangian,whose integral depends only upon the homotopy class ;this results in superselection sectors in the quantised model .A skyrmion can be approximated by a soliton of the Sine-Gordon equation ;after quantisation by the Bethe ansatz or otherwise,it turns into a fermion inter-acting according to the massive Thirring model .Skyrmions have been reported,but not conclu-sively proven,to be in Bose-Einstein condensates ,[5]superconductors ,[6]thin magnetic films [7]and also chiral nematic liquid crystals .[8]2Skyrmions in an emerging tech-nologyOne particular form of the skyrmions is found in mag-netic materials that break the inversion symmetry and where the Dzyaloshinskii-Moriya interaction plays an im-portant role.They form “domains”as small as a 1nm (e.g.in Fe on Ir(111)[9]).The small size of mag-netic skyrmions makes them a good candidate for fu-ture data storage solutions.Physicists at the University of Hamburg have managed to read and write skyrmions using scanning tunneling microscopy.[10]The topological charge,representing the existence and non-existence of skyrmions,can represent the bit states “1”and “0”.3References[1]At later stages the model was also related to mesons .[2]Wong,Stephen (2002).“What exactly is a Skyrmion?".arXiv :hep-ph/0202250[hep/ph ].[3]Chiral models stress the difference between “left-handedness”and “right-handedness”.[4]The same classification applies to the mentioned effective-spin “hedgehog”singularity":spin upwards at the north-pole,but downward at the southpole.See also Döring,W.(1968).“Point Singularities in Mi-cromagnetism”.Journal of Applied Physics 39(2):1006.Bibcode :1968JAP....39.1006D .doi :10.1063/1.1656144.[5]Al Khawaja,Usama;Stoof,Henk (2001).“Skyrmionsin a ferromagnetic Bose–Einstein condensate”.Nature 411(6840):918–20.Bibcode :2001Natur.411..918A .doi :10.1038/35082010.PMID 11418849.[6]Baskaran,G.(2011).“Possibility of Skyrmion Superconductivity in Doped Antiferromagnet K$_2$Fe$_4$Se$_5$".arXiv :1108.3562[cond-mat.supr-con ].[7]Kiselev,N.S.;Bogdanov,A.N.;Schäfer,R.;Rößler,U.K.(2011).“Chiral skyrmions in thin magnetic films:New objects for magnetic storage technologies?".Journal of Physics D:Applied Physics 44(39):392001.arXiv :1102.2726.Bibcode :2011JPhD...44M2001K .doi :10.1088/0022-3727/44/39/392001.[8]Fukuda,J.-I.;Žumer,S.(2011).“Quasi-two-dimensional Skyrmion lattices in a chiralnematic liquid crystal”.Nature Communica-tions 2:246.Bibcode :2011NatCo...2E.246F .doi :10.1038/ncomms1250.PMID 21427717.123REFERENCES [9]Heinze,Stefan;Von Bergmann,Kirsten;Menzel,Matthias;Brede,Jens;Kubetzka,André;Wiesen-danger,Roland;Bihlmayer,Gustav;Blügel,Ste-fan(2011).“Spontaneous atomic-scale magneticskyrmion lattice in two dimensions”.Nature Physics7(9):713–718.Bibcode:2011NatPh...7..713H.doi:10.1038/y summary(Jul31,2011).[10]Romming,N.;Hanneken, C.;Menzel,M.;Bickel,J. E.;Wolter, B.;Von Bergmann,K.;Kubet-zka, A.;Wiesendanger,R.(2013).“Writing andDeleting Single Magnetic Skyrmions”.Science341(6146):636–9.Bibcode:2013Sci...341..636R.doi:10.1126/ysummary–(Aug8,2013).3 4Text and image sources,contributors,and licenses4.1Text•Skyrmion Source:/wiki/Skyrmion?oldid=637550141Contributors:Michael Hardy,Charles Matthews,Phys, Icairns,Lumidek,Pjacobi,Jag123,Fwb22,Rjwilmsi,Conscious,Wikid77,Headbomb,Lincoln F.Stern,Tarotcards,KylieTastic,Pix-elBot,Doprendek,Addbot,Luckas-bot,Yobot,Citation bot,Obersachsebot,Omnipaedista,Citation bot1,Merongb10,Meier99,Korepin, EmausBot,JSquish,ZéroBot,StringTheory11,AManWithNoPlan,Isocliff,Parcly Taxel,Bibcode Bot,BattyBot,ChrisGualtieri,Andy-howlett,1andreasse,Nicohoho,NorskMaelstrom,Noah Van Horne and Anonymous:74.2Images•File:Portal-puzzle.svg Source:/wikipedia/en/f/fd/Portal-puzzle.svg License:Public domain Contributors:?Original artist:?4.3Content license•Creative Commons Attribution-Share Alike3.0。

关于空间时间的英语作文Title: Exploring the Mysteries of Spacetime。

Space and time, two fundamental concepts that have intrigued humanity for centuries, are intricately intertwined in the fabric of the universe. From the elegant equations of Einstein's theory of relativity to the mind-bending phenomena of black holes and wormholes, the exploration of spacetime has opened doors to a deeper understanding of our cosmos. In this essay, we delve into the mysteries of spacetime and its profound implicationsfor our understanding of the universe.The Unity of Space and Time:In classical physics, space and time were treated as separate entities. However, with the advent of Einstein's theory of relativity in the early 20th century, it became clear that space and time are intimately connected. According to relativity, space and time form a four-dimensional continuum known as spacetime. In this framework, events are described not just by their spatial coordinates but also by their temporal coordinates, and the fabric of spacetime itself can be warped by the presence of matterand energy.Einstein's Theory of General Relativity:At the heart of our understanding of spacetime lies Einstein's theory of general relativity, whichrevolutionized our conception of gravity. According to general relativity, massive objects like stars and planets warp the fabric of spacetime around them, causing other objects to move along curved paths in response to this curvature. This explains the phenomenon of gravity as the bending of spacetime itself, rather than a force acting ata distance.Black Holes: Portals to the Unknown:One of the most intriguing consequences of general relativity is the existence of black holes. Black holes areregions of spacetime where the gravitational pull is so intense that nothing, not even light, can escape fromwithin a certain boundary called the event horizon. Beyond this point, the laws of physics as we know them break down, leading to a singularity—a point of infinite density where spacetime itself becomes infinitely curved. Black holes challenge our understanding of the universe and raise profound questions about the nature of reality.Wormholes: Gateways to Other Realms:In the realm of theoretical physics, wormholes are hypothetical tunnels through spacetime that could potentially connect distant regions of the universe or even different universes altogether. Wormholes are predicted by the equations of general relativity, but whether they actually exist and whether they could be traversable remains an open question. If traversable wormholes were to exist, they could provide a means of interstellar travel or even allow for journeys through time, leading tofascinating possibilities for exploration and discovery.Spacetime and the Fate of the Universe:The study of spacetime also has profound implicationsfor cosmology—the study of the origin, evolution, and eventual fate of the universe. By modeling the dynamics of spacetime on cosmic scales, scientists have developed various theories about the ultimate destiny of our universe. Will it continue expanding indefinitely, eventually becoming a cold, dark void? Or will it reach a point of maximum expansion and begin to contract, culminating in a fiery cataclysm known as the Big Crunch? The answers to these questions depend on the properties of spacetime andthe distribution of matter and energy within it.Conclusion:In conclusion, the exploration of spacetime has opened up new vistas of understanding and discovery, challenging our preconceived notions of space, time, and reality. From the elegant equations of general relativity to the mysterious realms of black holes and wormholes, the studyof spacetime continues to captivate the imagination ofscientists and laypeople alike. As we peer ever deeper into the fabric of the cosmos, we may yet uncover even more profound truths about the nature of existence itself.。

a rXiv:q uant-ph/981189v13N ov1998Cosmic quantum measurementbyI.C.PercivalDepartment of Physics Queen Mary and Westfield College,University of London Mile End Road,London E14NS,England Abstract Hardy’s theorem states that the hidden variables of any realistic theory of quan-tum measurement,whose predictions agree with ordinary quantum theory,must have a preferred Lorentz frame.This presents the conflict between special rel-ativity and any realistic dynamics of quantum measurement in a severe form.The conflict is resolved using a ‘measurement field’,which provides a timelike function of spacetime points and a definition of simultaneity in the context of a curved spacetime.Locally this theory is consistent with special relativity,but globally,special relativity is not enough;the time dilation of general relativity and the standard cosmic time of the Robertson-Walker cosmologies are both essential.A simple but crude example is a relativistic quantum measurement dynamics based on the nonrelativistic measurement dynamics of L¨u ders.98Nov 30,QMW-PH-98-??Submitted to ??11Introduction2Hardy’s theorem3Simultaneity4Not proper time5Cosmology6Measurementfield7A simple relativistic measurement dynamics8Discussion and conclusions1IntroductionThe theoretical ideas of Hardy[24],Bell[7],Wheeler[41]and Hawking and Seifert[21,36],combined with the long-range entanglement experiments of Tit-tel,Gisin and collaborators[37,38],lead to an elementary cosmological dynam-ics of quantum measurement that satisfies the principles of special and general relativity.According to Bohr[11,42],the result of a quantum measurement is influenced by the condition of the measuring apparatus.Dynamical theories of quantum measurement ascribe this influence to a dynamical process.Here the influence is due to interaction with a cosmic backgroundfield,the measurementfield which plays the role of hidden variables in some other dynamical theories.By an ex-tension of the Einstein,Podolsky and Rosen thought experiment[16],John Bell showed[8],[9]that quantum theories which represent quantum measurement as a dynamical process are nonlocal.Hardy[24]showed the need for a preferred Lorentz frame.The measurementfield provides the frame.Generalized quantum measurement is a physical process by which the state of a quantum system influences the value of a classical variable.It includes any such process,for example laboratory measurements,but also other,very different, processes[28].These include the cosmic rays that produced small but detectable dislocations in mineral crystals during the Jurassic era,and the quantumfluc-tuations in the early universe that may have caused today’s anisotropies in the universe.It includes those quantumfluctuations that are amplified by chaotic dynamics to produce significant changes in classical dynamical variables.For such measurements,these are the dynamical variables of the measurer,although there is no measuring apparatus in the usual sense.Assume,therefore,that quantum measurement is universal,that the measure-ment process is taking place throughout spacetime,with the possible exception of the neighbourhood of some spacetime singularities.Here I also assume that2quantum measurement dynamics includes a specific representation for the evo-lution of an individual quantum system:the evolution of an ensemble is not enough.And assume that there are no causal loops in spacetime.This picture and the results that follow are based on the followingfive principles PR1-5and two theorems PR6,7:PR1.Kepler-Galileo-Humans are not at the centre of the universe,in any sense.PR2.Newton-Laplace-Every physical process has a dynamical expla-nation.PR3.Einstein-Special relativity.PR4.Einstein-General relativity.PR5.Cosmological-On sufficiently large scales the universe is spatially isotropic and therefore uniform.PR6.Hardy[24]-Measurement dynamics inflat spacetime and consis-tent with ordinary quantum theory needs a special Lorentz frame.PR7.Hardy simultaneity-Measurement dynamics in curved spacetime needs a definition of simultaneity for events with spacelike separation.I also assume that The principles PR1and PR2together are incompatible with ordinary quantum mechanics,in which quantum measurement is the preserve of humans,and does not require a dynamical explanation.Despite this,it is our most successful theory.A good reason for the recent revival of alternative realis-tic theories which are compatible with these principles is the greater control we have over quantum systems,which raises the possibility of distinguishing differ-ent quantum theories of measurement experimentally.Notice that the(weak) version of the Newton-Laplace principle PR2does not require the dynamics to be deterministic.Aharanov and Albert[1,2,3]pointed to the particular difficulties of reconciling special relativity and realistic quantum theories.According to Shimony[33], special relativity and quantum mechanics might live in‘peaceful coexistence’, but Hardy’s theorem suggests a fundamental conflict between the expected re-sults of quantum measurement and invariance under Lorentz transformations. The principal purpose of this paper is to resolve this apparent conflict between special relativity and the dynamics of quantum measurement.There are now many alternative quantum theories that provide a nonrelativis-tic dynamics of quantum measurement.To my knowledge,there has been no alternative theory that resolves the major problem of reconciling special relativ-ity and the dynamics of quantum measurement in general,or Hardy’s theorem PR6in particular.Nor have I been able to formulate a consistent relativistic dynamical theory of quantum measurement that applies to our universe without including both general relativity and cosmology.3In order to make the paper more accessible,sections2to4include short reviews of the relevant quantum theory for general relativists and cosmologists,and some relevant general relativity and cosmology for quantum theorists.Section2describes Bell’s theorem and Hardy’s theorem.Section3presents an alternative proof of Hardy’s theorem based on a combination of two classically connected Bell experiments.Their spacetime configuration is similar to that of the no-simultaneity thought experiment of Einstein’s original work on spe-cial relativity.It goes on to sketch a proof the Hardy simultaneity theorem, which is an extension of his original theorem,but for curved spacetime,and to state the Hawking-Seifert theorem on timelike functions in some general curved spacetimes.Section4uses general relativistic time dilation to demonstrate that for two clocks at different heights on the Earth or at different locations in the universe, local proper times do not provide a global definition of simultaneity.Two reasons for a cosmological theory of quantum measurement are given in section5,and section6introduces the measurementfield,which provides the simultaneity needed for such a theory,and for the resolution of the conflict between quantum measurement and relativity.Section7sketches a simple example of relativistic measurement dynamics based on the measurementfield,which leaves much room for improvement,and the final section8includes a brief discussion of the relation between this dynamics and some other alternative quantum theories.2Hardy’s theoremHardy’s theorem PR6[24]goes further than Bell’s theorem on nonlocality of quantum measurement.From this theorem it follows that any dynamical theory of measurement,in which the results of the measurements agree with those of ordinary quantum theory,must have a preferred Lorentz frame.The theorem does not determine this frame.Bell’s theorem shows that measurement dynamics is nonlocal if the results of measurements follow the rules of ordinary quantum theory[6,7].Bell demon-strated his theorem by a thought experiment illustrated infigure1,in which two entangled particles,each with spin nonzero,and with total spin zero,are produced from a source S.A component of spin perpendicular to the direction of motion of each particle is measured,one at A,and the other at B,where typically ASB is a straight line,with S at its centre.The alignment of the spin measuring apparatus at A or B is the preparation event,or input,A1or B1, and the measurement and recording of the spin component is the measurement event,or output,A2or B2.Both the events A1,A2at A have spacelike separa-tion from both the events B1,B2at B.In the illustrated example the particles are photons,and a line at45deg represents a photon at the velocity of light.4Figure1:Spacetime diagram of Bell’s experiment.Thin diagonal lines at45o represent the velocity of light.At both A1and B1,the angle is the setting of the angle of spin or polarization measurement,which is an input event,and at A2and B2,+/−represents the recording of the spin or polarization,an output event.Bell’s theorem then states that for any realistic dynamics of quantum measure-ment,if the results agree with ordinary quantum theory,either the input at A1 affects the output at B2or the input at B1affects the output at A2,or both. There must be nonlocal space-like causality.Such an experiment was carried out by Aspect and his collaborators[4,5,30], although there remains at least one loophole to be closed because,despite the considerable care that was taken,it is not clear that the events fully satisfied the spacelike separation condition.[18,39].Hardy demonstrates his theorem through a thought experiment involving two matter interferometers,one for electrons and one for positrons,with an intersec-tion between them that allows annihilation of the particles to produce gamma rays.In its original form,this experiment is likely to remain a thought exper-iment.Improved versions which depend on similar principles and are experi-mentally feasible are given in[13,25].A different derivation based on classical5links between two Bell experiments is given in[29]and section3.3SimultaneityIn classical special relativity withflat spacetimes there is no unique simultaneity for events with spacelike separation.This was demonstrated by Einstein in the famous classical thought experiment,which we describe for later convenience. In Einstein’s experiment,illustrated infigure2,Figure2:Spacetime diagram of Einstein’s simultaneity experiment.The straight lines ASB and A′S′B′represent a source and two receivers at rest in their respective frames.one part consists of a source S which emits aflash of light,and two receivers A and B,equidistant from S in the same straight line,where A,S and B are at6rest in a frame L.It is received simultaneously at A and B with respect to this frame.The other part of the experiment consists of an identical trio A′S′B′,in the same straight line as ASB,which are at rest in a different frame L′,moving with respect to L in the direction AOB,where S and S′are nearly coincident at the time t0when both of themflash.The light from S′is received simultaneously at A′and B′with respect to L′,and this is clearly not simultaneous with respect to L.Hence relativity.There is an alternative proof[29]of Hardy’s theorem,which depends on two Bell experiments in a similar spacetime configuration to Einstein’s simultaneity experiment,and labelled similarly infigure3.This is the double Bell exper-iment.The two Bell experiments are independent at the quantum level,butFigure3:Spacetime diagram of the double Bell experiment,with photons in opticalfibres.The meaning of the symbols corresponds to their meaning in the first twofigures.On the scale of thisfigure,the photons remain at rest while in the delay coils.CL are classical links,by which an output from each experiment determines an input to the other.they are linked classically so that the output A2controls the input A1′,which is in its future lightcone,and the output B2′controls the input B1.According to Bell’s theorem,there are nonlocal interactions NI,such as the setting of the7angle at A1affecting the measurement at B2,or the setting of the angle at B1’affecting the measurement at A2’.If both of these are present,then there is a causal loop A2′−→CL A1−→NI B2−→CL B1′−→NI A2′(1)The assumption that there are no causal loops in spacetime or the equivalent assumption that there is no backward causality,then leads to the exclusion of one of the nonlocal interactions,which makes the dynamics dependent on the Lorentz frame,and so leads to Hardy’s theorem.Details are in the letter [28].As it stands,Hardy’s theorem makes no statement about simultaneity,but this also can be derived using the double Bell experiment.Thus a configuration similary to that used by Einstein to show that classical special relativity has no universal simultaneity,can be used to show that quantum measurement requires universal simultaneity,which is the Hardy simultaneity theorem PR7of the introduction.In flat spacetimes,a preferred Lorentz frame or a standard of rest defines a universal time and simultaneity between distant events.For curved spacetimes,only a local Lorentz frame or standard of rest has meaning,and even if there is a standard of rest for every point,this does not necessarily provide a definition of simultaneity.The examples of the next section show that in the presence of gravitational fields,the local times defined by local preferred Lorentz frames are not necessarily consistent with a universal simultaneity defined throughout a region.For curved spacetimes,simultaneity is a stronger condition.For flat spacetimes they are equivalent.It is therefore important to extend Hardy’s theorem by showing that universal quantum measurement dynamics requires universal simultaneity.It is not feasible to set up the double Bell experiment so that the curvature of spacetime has a significant and relevant effect,but the thought experiment is needed to study the effect of the curvature on the dynamics of quantum measurement.Just as for flat spacetime,the condition that there is no backward causality or equivalently that there is no causal loop [29]requires that there is a time ordering for an event at A with respect to an event at B,which is spatially separated from the event at A.Unfortunately this time ordering depends on the hidden variables or background field,which are not accessible to current experiments.Universal quantum measurement therefore requires a universal time ordering for events with spacelike separation,which is equivalent to a universal simultaneity.This is the Hardy simultaneity theorem,which applies to curved spacetime.It is an extension of the original Hardy theorem and is based on the assumption that the events occur at spacetime points.Some latitude is allowed when they take a finite time or occupy a finite region of space.Hawking [21]and Seifert[32]proved that in all universes in the neighbourhood8of which there is no backward causality,there are timelike functions,spacelike foliations of spacetime and corresponding definitions of simultaneity[36].It is interesting to note that‘no backward causality’is the same condition as that used in[29]to prove that measurement dynamics needs a local Lorentz frame,and is used here to show that it needs simultaneity.Hardy showed that special frames are needed.However,the example of section4illustrates that the existence of a special frame at every point of a curved spacetime does not imply that there is a consistent definition of simultaneity.4Not proper timeThe most obvious choice of a time to define simultaneity is the proper time of local matter,but this is inconsistent,because,as in Einstein’s experiment,the local matter can have different frames.In the neighbourhood of the Earth,we could use the Earth as a standard of rest,but this also is inconsistent,because in gravitationalfields it leads to‘simultaneity’between events that have timelike separation.As an example take two small clocks at rest with respect to the surface of the Earth,one vertically at a height h above the other,where h is small compared with the radius of the Earth.They could be at the top and bottom of a tall building,or one on a table in a laboratory,and one on thefloor beneath it. Suppose they are synchronized at time t=0in the rest frame at this time. At later times,the general relativistic time dilation due to their gravitational potential difference is much greater that the special relativistic time dilation due to their different velocities around the Earth’s centre.The clocks will show a time difference∆t after time t,where∆tc2(2) and g is the acceleration due to the Earth’sfield near its surface.The separation between‘simultaneous’events as given by the two clocks be-comes timelike after a time t for which a signal from one to the other takes a time∆t,where∆t=hg≈1year,(3)which is independent of h.The greater height leads to greater time shifts because of the greater gravitational potential difference,but the time taken for light to travel between the bodies increases in proportion.The fact that t=c/g is so close to the time taken for the Earth to orbit the Sun is a well-known coincidence. One could try to define simultaneity by using some kind of average over local times,but it is not at all clear over what scale the average should be taken:the9scale of the apparatus?of the Earth?of the Solar System?the galaxy?or the universe?An answer to this question is suggested in section6.There is the same problem if we try to use the rest frame of afield to define simultaneity.Suppose that we follow Hardy’s example(see also[34,14])of the universal background radiation.This provides a local standard of rest.We could use any kind of clock in this standard of rest to define the local‘time’. But then there is a gravitational time shift between the clocks in a gravitational potential well,and outside it.So again,using these measures of local time,‘simultaneous’events can have a timelike separation,which is not allowed.The same applies applies to the universal background neutrinofields,or any other backgroundfield,of zero or any other rest mass.Also,as pointed out by Hardy,there is no clear dynamical process whereby the quantum measurement of individual systems can be made to depend on the background radiation.Section6shows that the Hawking-Seifert theorem leads to a possible resolution of this problem of simultaneity.5CosmologyWheeler[41]considered the possibility that entanglement and localization might occur on cosmological scales.We have no observational evidence that entan-glement survives over such distances,but the Hardy simultaneity theorem and the experiments of Tittel,Gisin et al[37]provide two strong arguments for the importance of cosmology to quantum measurement.According to the Hardy simultaneity theorem,quantum measurement needs simultaneity between distant events.There is already a cosmological definition of simultaneity for the standard models like the Robertson-Walker metric for isotropic spacetime.Cosmic standard time provides a time function,dividing(or foliating)spacetime into three-dimensional spacelike surfaces of constant cosmic standard time,which are maximally symmetric subspaces of the whole of the spacetime[40].Any timelike function provides a definition of simultaneity,in which events with the same functional value are simultaneous.However,on scales smaller than the cosmological,the spacetime of the universe universe in our epoch is not isotropic,which leads to the problems of defining simultaneity given in the previous section.The second follows from the experiments that demonstrate entanglement over a given distance,which is currently greatest for the Tittel et.al.experiment.A system AB consisting of two parts A and B is in an entangled pure state when AB is in a pure state,but neither A nor B separately is in a pure state. Suppose that the systems A and B are distant from one another.Then spacelike separated measurements of A and of B lead to the nonlocality and simultane-10ity problems of dynamical theories of measurement.The experiment of Tittel and his collaborators[37]shows directly that there can be entanglement over 10km,so simultaneity must be defined over regions of this size.Assuming that generalized measurement is universal,and that the Earth is typical regarding measurement,following the Kepler-Galilean principle PR1,entanglement and its destruction by measurement over distances of10km is present always and everywhere.Nowfill spacetime with overlapping regions of linear dimension 10km in space and10km/c in time.For every overlap region the definitions of simultaneity must be consistent,so by iteration they must be consistent throughout spacetime,or at least where and when generalized measurement takes place.We have no means of checking whether this includes the neighbourhood of singularities in spacetime or very strong gravitationalfields,including the very early universe,the late stages of a closed universe,or in the neighbourhood of a black hole,but the rest of spacetime needs a definition of simultaneity,with a corresponding timelike function and spacelike foliation.6MeasurementfieldExperimenters in anyfield of physics who work in a nearlyflat spacetime,and find that the results of their experiments depend on the Lorentz frame of the apparatus,do not immediately conclude that special relativity is wrong.They look for some previously unsuspected background influence that depends on the environment and which determines a special frame.This influence comes from a background physical system which interacts with the system being studied.No one has found a convincing example for which this procedure fails.Environment is used in a broad sense,and may includefields that penetrate the system. Similarly,the need for a consistent definition of simultaneity for measurement does not contradict special relativity.But it requires a physical system that defines simultaneity and interacts with the measured system.Suppose that this physical system is a measurementfieldµ(x),where x=(x0,x1,x2,x3)are the time and space coordinates of a spacetime point.Make the following further assumptions:11MU1.For the purposes of the present paper,µ(x)is a real classical scalarfiter it will have to be quantized,it may have imaginary components and may not be scalar.MU2.There was an epoch in the early universe with a a cosmic time t. MU3.In this early epochµwas a monotonically increasing function of t.MU4.In local inertial frames and in epochs like ours,µ(x)satisfies the zero-mass Klein-Gordon equation,or wave equation(c=1):∂2The required cosmological principle says that on sufficiently large scales,av-erages over spherical shells are uniform.This is a stronger than the usual cosmological principle,that,usually by implication,refers to averages over solid spheres or similar regions.7A simple relativistic measurement dynamicsA nonrelativistic stochastic measurement dynamics,which is consistent with the Kepler-Galileo and Newton-Laplace principles,was proposed long ago by L¨u ders[26].The mathematics comes from the Copenhagen school,particularly von Neumann and Heisenberg,but the physics is consistent with measurement as a nonrela-tivistic dynamical process,not with the Copenhagen interpretation.Heisenberg gave a descriptive account in[22].Gisin used L¨u ders dynamics as a starting point for the quantum state diffusion approach to measurement[19]. Because the measurements in this nonrelativistic theory are localized in space-time,it can be assumed that they have a definite time order.Consider just one measurement of a dynamical variable of the quantum system with corre-sponding nondegenerate Hermitean operator G with eigenstates|g .Before the measurement,the quantum system is in the initial state|i and afterwards it is in afinal state|f ,which is one of the eigenstates|g .The measurement dynamics is stochastic,and the probability that the system willfinish in state |g isPr |f =|g = i|g 2.(5) In general the classical system also changes its state,from some initial configu-ration to afinal configuration corresponding to the measurement of the value g whose probability(5)depends on|i .This is the influence of the initial quan-tum state of the quantum system on thefinal state of the classical system,that characterizes generalized quantum measurement.The stochastic evolution of classical and quantum systems consists of continuous evolution of each according to their own deterministic dynamics,with sudden stochastic jumps which correspond to the measurements that take place at times determined by the classical system,in which the quantum and classical systems influence each other.In this theory,the classical system can influence the quan-tum system through time-dependent Hamiltonians whose current value depends on the state of a classical system.But the quantum system can only influence a classical system through a measurement.In the corresponding relativistic theory,the jumps do not occur at constant time,but at constant values ofµ.In this way the measurementfield affects the dynamics of quantum measurement,there are no causal loops,and relativistic principles are preserved,at least formally.13This picture of measurement is unsatisfactory in several ways.In particular,the timing of the jumps is not normally determined by the classical system alone. In the modern theory of continuous laboratory measurements,originating with Davies[15],developed by many authors[31,35,12]and now used widely in quantum optics,the timing of the jumps is determined also by the state of the quantum system.Further,the L¨u ders picture assumes that there are distinctly classical and distinctly quantal degrees of freedom,with a‘shifty split’between them[7].This split is convenient for conventional quantum theory,but there is no evidence that the world is divided in this way into purely classical and purely quantum domains.Modern nonrelativistic dynamical theories of measurement have neither of these problems,but the theory of the measurementfield has not yet been extended to relativistic versions of these theories.8Discussion and conclusionsIt may seem surprising that tachyons have played no role[27].There is a rea-son for this.The usual theory of tachyons has no preferred frame[17].This is normally considered an advantage,but without the preferred frame,interaction with normal matter leads to causal loops.Hence the difficulty of giving a phys-ical interpretation to such interaction.Here the preferred frame is a necessity, through a a nonlocal interaction with the measurementfield,and without using tachyons.There are many versions of nonrelativistic measurement dynamics without the faults of the L¨u ders theory discussed in section7.All of them are nonlocal, following Bell’s theorem.Thefirst was the pilot wave theory of de Broglie and Bohm,in which there are both waves and particles,and measurement dynamics is the result of the effect of the quantum waves on the classical particles[23,10]. Since then there have been dynamical theories based on waves alone,in which the Schr¨o dinger equation is modified by a weak stochastic process of localization, leading to the‘collapse’of the quantum wave.Particles are just very localized waves[28].The nonlocal processes depend on simultaneous changes that take place at spacelike separated points,where the simultaneity is determined by the universal time variable t.The theory presented here assumed that events like the orientation of a polarizer or the recording of a spin take place at spacetime points.In fact preparation and recording occupyfinite regions of space and take afinite time.This complicates the theory,but quantum state diffusion models of nonrelativistic measurement show that this complication leads to no fundamentally new problems.[28].To each of the nonrelativistic dynamical theories there corresponds a relativistic theory in which the definition of simultaneity is provided by the measurement field variableµ,just as in the case of L¨u der’s dynamics.But none of these14。

a rXiv:g r-qc/93228v 122Feb1993KFKI-RMKI-16-FEB-1993K´a rolyh´a zy’s quantum space-time generates neutron star density in vacuum Lajos Di´o si 1and B´e la Luk´a cs 2KFKI Research Institute for Particle and Nuclear Physics H-1525Budapest 114,POB 49,HungaryAbstractBy simple arguments,we have shown that K´a rolyh´a zy’s model overestimates the quantum uncertainty of the space-time geometry and leads to absurd physi-cal consequences.The given model can thus not account for gradual violation of quantum coherence and can not predict tiny experimental effects either.In a pioneering paper[1],it was suggested that the quantum mechanics of macroscopic objects ought to be modified due to a certain eventual unsharpness of space-time ter on,the possibility of experimental verification of the model,too,has been developed[2,3].The idea went as follows.By combining Heisenberg’s uncertainty principle with gravitation,the following relation has been obtained for the minimum uncertainty∆s of a single(timelike) geodesic:∆s2=α4/3s2/3,(1) where s is the length of the geodesic andαis the Planck length[c.f.Eq.(3.1)of Ref.1].Then this uncertainty is believed to be a universal lower bound,and so must appear in the space time in an objective way.This was done via random ”gravitational waves”.The present authors[4]reanalysed the concept leading to Eq.(1).A result is that in Refs.1and2the value M of mass realizing the least uncertainty along the given geodesic takes irrealistically high values∼¯hThe random coefficients c k are uncorrelated.Their average is zero while the spreads are given byL3¯c2k=α4/3k−5/3(4) where L is the normalization volume[c.f.Eqs.(3.4)and(3.5)of Ref.1].The above equation is the only one which is conform to the uncertainty relation(1).According to the intentions implicit in Refs.1and2,the space-time geometries defined by Eqs.(2)and(3)must be approximate solutions of the Einstein equations. However,it turns out that they will not.Though they satisfy,by construction,the linearized vacuum Einstein equations(2),the conditions for the linear approxima-tion will seriously fail.We are going to test two rather trivial conditions.Thefirst will hold but the second will not.Let us calculate the mean squared deviation of the metric tensor from its Minkowski value.Squaring both sides of the Eq.(3)and taking stochastic aver-ages of the coefficients c k,one obtains:¯γ2∼k ¯c2k∼α4/3L−3 k k−5/3∼(αk max)4/3.(5)One needs afinite cutoffon k otherwise the amplitude of the random waves would diverge.K´a rolyh´a zy suggests k max=1013cm−1and this assures thatγis much smaller than the unity.This was thefirst condition for applying the linear form(2) of the Einstein equations.As for the second condition,let usfirst invoke the expansion of the scalar curvature R up to the second order inγ[c.f.Ref.5]:R=12γij¯⊔γij+14(γij,k−γik,j)(γij,k−γik,j)+ (6)Now,by substituting the waves(3)into this equation,thefirst order term indeed vanishes.The magnitude of the average of the remaining terms can be estimated by invoking Eq.(4);one obtains:¯R∼α4/3k10/3max.(7)This curvature is extremely ing the previous cutoffwe are led to¯R∼1cm−2.So the correspondingfluctuating metric is not at all the”extremely small smearing”[1]of theflat space-time,thought before.According to the exact Einstein equation R=8πα2∼1026g/cm3,(9)c2i.e.11orders of magnitude above neutron star density.In Ref.1the details of the cutoffwere thought of no importance.We have, however,pointed out that the original cutoffwould imply absurd results for cosmo-logical mass density.Since the cutoffk max is the only free parameter in the model one may hope to save the theory by choosing a lower value for it.Unfortunately, the choice k max=105cm−1,familiar from e.g.the model of Ghirardi et al.[6], yields still water density.Further decrease of k max is needed.Then,however,there would be only macroscopic wavelengths1/k and the gravitationalfluctuations(3) would not play a rˆo le in the quantum-classical transition anymore.The trace(9) in itself could be removed by means of an incredibly high cosmologic constantΛ, but in the Robertson-Walker Universe geometries two nontrivial components of the Einstein equations survive,and one cannot remove the problem from both.Obviously,the K´a rolyh´a zy model[1]has shown to overestimate something in the assumed quantum smearing of the space time.The spectrum(4)of gravitational fluctuations is certainly wrong whatever cutoffis chosen.The proposals outlined in Refs.[2,3]derive extremelyfine effects to observe experimentally.In the light of the cosmological absurdity of the model we wonder if such tiny effects would have to be taken serious.The necessity and timeliness to perform the present research were recognized in a discussion with Prof.P.Gn¨a dig of the E¨o tv¨o s University.This work was supported by the Hungarian Scientific Research Fund under Grant No1822/1991.References[1]F.K´a rolyh´a zy,NuovoCim.XLIIA,1506(1966)[2]F.K´a rolyh´a zy,A.Frenkel and B.Luk´a cs,in:Physics as natural philosophy,eds.A.Shimony and H.Feschbach(MIT Press,Cambridge,MA,1982)[3]F.K´a rolyh´a zy,A.Frenkel and B.Luk´a cs,in:Quantum concepts in space and time, eds.R.Penrose and C.J.Isham(Clarendon,Oxford,1986)[4]L.Di´o si and B.Luk´a cs,Phys.Lett.142A,331(1989)[5]C.W.Misner,K.S.Thorne,J.A.Wheeler:Gravitation(Freeman,San Francisco, 1973)[6]G.C.Ghirardi,A.Rimini and T.Weber,Phys.Rev.D34,470(1986)。

黑洞介绍英语作文带翻译Title: Exploring the Enigma of Black Holes。

Introduction。

Black holes have long captured the imagination of scientists and the public alike. These enigmatic cosmic entities, formed from the collapse of massive stars, possess gravitational forces so intense that not even light can escape their grasp. In this essay, we will delve into the fascinating world of black holes, exploring their properties, formation, and the profound implications they hold for our understanding of the universe.Properties of Black Holes。

At the heart of every black hole lies a singularity, a point of infinite density where the laws of physics, as we currently understand them, break down. Surrounding this singularity is the event horizon, the boundary beyond whichnothing can escape the black hole's gravitational pull. It is this event horizon that gives black holes their name, as it appears "black" to outside observers.Formation of Black Holes。

英语关于黑洞的作文The Mysterious and Fascinating Black Holes.In the vast and enigmatic universe, black holes standas one of the most intriguing and perplexing phenomena. These regions of space, characterized by their intense gravity and complete absence of light, have captivated the imagination of scientists and laypeople alike for centuries. Despite their otherworldly nature, black holes play acrucial role in understanding the evolution and structureof our universe.The concept of black holes emerged in the late 18th century, with the pioneering work of scientists like John Michell and Pierre-Simon Laplace. They theorized the existence of objects so massive that not even light could escape their intense gravitational pull. However, it wasnot until the 20th century that astronomers began to gather evidence that supported the existence of these mysterious objects.One of the most significant milestones in the study of black holes was the work of Albert Einstein. His theory of general relativity provided a mathematical framework to describe the behavior of gravity and its interaction with matter. This theory laid the foundation for understanding the properties of black holes, including their formation, evolution, and interaction with their environment.Black holes are formed when a massive star collapses under its own weight at the end of its life cycle. This collapse compresses the star's matter into a tiny, ultra-dense region known as a singularity. The gravity around this singularity is so intense that nothing, including light, can escape its pull. The boundary of this region, known as the event horizon, marks the point where the escape velocity exceeds the speed of light.There are two main types of black holes: stellar-mass black holes and supermassive black holes. Stellar-mass black holes are formed when a star of about 10 to 30 times the mass of the Sun collapses. These black holes have adiameter of only a few kilometers but possess a mass comparable to that of a small star. On the other hand, supermassive black holes have masses ranging from millionsto billions of times the mass of the Sun. They are believed to reside at the centers of most galaxies, including ourown Milky Way.The study of black holes has revealed much about the structure and dynamics of the universe. For instance, black holes play a crucial role in the evolution of galaxies. By accreting matter and emitting radiation, they can significantly impact the star formation and gas dynamics of their host galaxies. Additionally, the merging of black holes, a common occurrence in the universe, can emit gravitational waves, ripples in the fabric of spacetimethat can be detected by advanced telescopes like the Laser Interferometer Gravitational-Wave Observatory (LIGO).Despite their otherworldly nature, black holes are not entirely devoid of life. In fact, there are theories that suggest the existence of accretion disks around black holes. These disks are formed when matter from a nearby star orgas cloud is attracted to the black hole and begins toorbit it. As the matter spirals inward, it heats up and emits radiation, creating a bright and energetic environment.The study of black holes also holds the key to understanding some of the most fundamental questions about our universe. For instance, black holes provide a unique laboratory to test the limits of Einstein's theory of general relativity. By studying the behavior of matter and light near the event horizon, scientists can gain insights into the nature of gravity and its interaction with quantum mechanics.In conclusion, black holes are one of the most mysterious and fascinating phenomena in the universe. They challenge our understanding of gravity, matter, and the structure of the cosmos. As we continue to explore and study these enigmatic objects, we may unlock the secrets of the universe and gain a deeper understanding of our placein the cosmos.。

中考英语太空探索成就单选题40题1. The first artificial satellite was launched by the Soviet Union in 1957. What was the name of this satellite?A. Sputnik 1B. Explorer 1C. Apollo 1D. Shenzhou 1答案：A。

解析：1957年苏联发射的第一颗人造卫星叫斯普特尼克1号。

选项B“Explorer 1”是美国发射的第一颗人造卫星；选项C“Apollo 1”是美国阿波罗计划中的一艘飞船，但不是第一颗人造卫星；选项D“Shenzhou 1”是中国发射的神舟一号飞船，与苏联发射的第一颗人造卫星无关，所以正确答案是A。

2. In which year did the United States send the first man to the moon?A. 1961B. 1969C. 1971D. 1979答案：B。

解析：1969年美国实现了首次载人登月。

1961年是苏联宇航员加加林首次进入太空的年份；1971年不是首次载人登月的时间；1979年与首次载人登月事件无关，所以正确答案是B。

3. Who was the first man in space?A. Neil ArmstrongB. Yuri GagarinC. Buzz AldrinD. Alan Shepard答案：B。

解析：尤里·加加林是第一个进入太空的人。

尼尔·阿姆斯特朗是第一个登上月球的人；巴兹·奥尔德林是第二个登上月球的人；艾伦·谢泼德是美国第一位进入太空的宇航员，但不是世界上第一个进入太空的人，所以正确答案是B。

4. The Apollo program was carried out by the United States. How many manned lunar landings were there in this program?A. 5B. 6C. 7D. 8答案：B。

0
where f has dimension 2 − ∆ in the unperturbed sigma model (1.2). We will henceforth
July 2005
1. Introduction Closed string tachyon condensation affects the dynamics of spacetime in interesting and tractable ways in many systems [1-5]. In this paper, we study circumstances in which closed string tachyon condensation plays a crucial role in the dynamics of a system which has a spacelike singularity at the level of general relativity. The singular region of the spacetime is replaced by a phase of tachyon condensate which lifts the closed string degrees of freedom, effectively ending ordinary spacetime. We will focus primarily on a simple set of examples with shrinking circles, in which we can make explicit calculations exhibiting this effect. Before specializing to this, let us start by explaining the relevant structure of the stringy corrections to spacelike singularities appearing in a more general context. Much of this general discussion appeared earlier in [6].1 Consider a general relativistic solution approaching a curvature singularity in the past or future. The metric is of the form

four-volume bounded by Σ1 and Σ2, and, if necessary, time-like boundaries on which f1 = f2 = 0. Then we may write
(f1, f2)Σ2 − (f1, f2)Σ1 = i
∂V
(f2∗
↔
∂µ
f1)dΣµ
This is not a crucial problem in the first two steps listed above. The formulation
of a classical field theory and its formal quantization may be carried through in an
• Lecture 2. The Hawking effect - particle creation by black holes. • Lecture 3. Ultraviolet and infrared divergences, renormalization of the expecta-
tion value of the stress tensor; global symmetry breaking in curved spacetime. • Lecture 4. Negative energy in quantum field theory, its gravitational effects,
2ϕ + m2ϕ + ξRϕ = 0.
(1.2)
Here R is the scalar curvature, and ξ is a new coupling constant. There are two

0
1
Introduction
Exact solutions representing the head on collision of two gravitational plane waves are some of the simplest exact dynamical spacetimes. They provide clear examples of highly non linear behavior in general relativity: when two plane waves collide the focusing effects of each exact plane wave lead to mutual fucusing. This is revealed by the formation of either a spacetime singularity or of a non-singular Killing-Cauchy horizon at the focusing points of the two waves [1, 2, 3]. They may also be useful to provide local models for processes that may be taking place in our universe as a result of gravitational waves produced in black hole collisions [4, 5], the decay of cosmological inhomogeneous singularities [6], or by travelling waves in strongly gravitating cosmic strings [7, 8]. Quantum effects in such dynamical spacetimes must surely be important and one expects particle production and vacuum polarization when a quantum field is coupled to such background. Yurtsever [6] was the first to study field quantization on a colliding wave background, he considered the Kahn and Penrose [1] solution which may be interpreted as the collision of two impulsive plane waves. The solution which has curvature singularities at the focusing points of the plane waves allows for the definition of two physically meanigful vacuum states: an “in” vacuum state associated to the flat space before the collision of the two plane waves and an “out” vacuum state related to the flat spacetime regions behind the shock fronts also before the collision. The Bogoliubov coefficients relating the “in” and “out” creation and annihilation operators could be found only approximately in the longwavelenght limit. In this approximation the spectrum of created particles is consistent with a thermal distribution. In a recent paper [9] we have considered the quantization of a massless scalar field in the background of the collision of two plane waves which form a non singular Killing-Cauchy horizon. This solution describes the collision of two gravitational shock waves followed by trailing gravitational radiation [10]. The interaction region of the two waves is locally isometric to a region inside the event horizon of a Schwarzschild black hole with the KillingCauchy horizon corresponding to the event horizon [2, 11, 12]. Two unambiguous and physically meaningful quantum vacuum states may be defined: an “in” vacuum associated to the positive frequency mode solutions in the flat region before the collision of the waves and an “out” vacuum related to the positive frequency modes defined through the two null vector fields in the Killing-Cauchy horizon. Such state, which is invariant under the symmetries associated to the horizon corresponds to the unique preferred vacuum state defined by Kay and Wald [13] in spacetimes with bifurcate Killing horizons. It was found that the “in” vacuum contains a number of “out” particles which is proportional to the inverse of the frequency. In the long-wavelength limit the spectrum is consistent with a thermal spectrum at a temperature which is inversely proportional to the focusing time of the gravitational plane waves, in agreement with Yurtsever’s result [6]. Our result however is exact: although the Klein-Gordon equation in the interaction region cannot be solved exactly the “in” modes become blueshifted towards the trailing points of the waves and can be propagated through this region by using the geometrical optics approximation. This is somewhat similar to the situation in a Schwarzschild black hole [14]. In this paper we want to reconsider this problem in the light of wave packet mode quantization instead of the monocromatic modes we have used in ref. [9]. There are several reasons 1

时空扩展MUSIC的空速估计算法张晓光【摘要】针对基于声矢量传感器的空气流动速度(简称空速)测量问题,提出基于时空扩展MUSIC空速估计算法.首先,经过理论分析,得出在超音速气流形成的马赫锥内,任何一点有且仅有两个波阵面相叠加,这两个波阵面可以被认为是由两个相干“等效声源”产生的,在此项基础上,建立了声波在超音速稳定气流中的传播模型;其次,结合声矢量传感器的测量模型,基于时空扩展MUSIC算法实现超音速空速估计,并对测量模型进行了失效分析,推导了估计的CRB界;仿真实验结果验证了算法的有效性.【期刊名称】《电光与控制》【年(卷),期】2018(025)007【总页数】5页(P58-62)【关键词】超音速气流;空速估计;声矢量传感器;MUSIC;失效分析【作者】张晓光【作者单位】辽东学院,辽宁丹东110000【正文语种】中文【中图分类】V211.3;TP391.10 引言现代航空工业的发展，使得飞行器进入超音速时代，基于空速管的传统测量系统，在超音速下受到各种因素的影响而无法正常工作。

美国早在20世纪就开始嵌入式大气数据传感(FADS)系统的研究[1]，此系统在测量精度、系统可靠性和适用范围上都有很大优势，目前，国内也相继开展其算法研究[2-3]。

但FADS系统的动静压测量模型是非线性的，且某些系数需风洞试验标定，限制了系统的实用性[3]。

声传感器或超声传感器广泛应用于各种运动物体的声速测量[4-5]，但其仅能感受声场的声压信息，信息量较少。

声矢量传感器是由1个声压传感器和3个相互垂直放置的质点振速传感器组成的新型传感器，它可以同步测量声场同一点处声压和质点振速矢量[6]。

由于在连续流体介质声场中，任何一点附近的运动状态可用声压、密度以及介质运动速度唯一表示，那么声场中某一点的振速矢量就包含了该声场中的流体介质运动速度信息。

基于此，文献[7-8]将声矢量传感器应用于FADS系统，用声矢量传感器测得某点的质点振速，进而借助一定的算法得到飞机的空速；文献[9]借助有效声速概念，建立声矢量传感器阵列在稳定气流作用下的包含待估空速的近场输出模型，提出一种基于多重信号分类(Multiple Signal Classification,MUSIC)的近场空速估计算法；文献[10]将声波在连续、均匀稳定气流中的传播原理引入到声矢量传感器输出模型中，构建了稳定气流作用下近场质点振速测量模型，提出一种近场空速估计的MUSIC算法，但是，这些模型和算法仅适用于亚音速范围[10]，而对于超音速条件下形成的马赫锥，它们不再有效。

a r X i v :0707.2303v 2 [h e p -t h ] 29 O c t 2007Preprint typeset in JHEP style -HYPER VERSIONTimothy J.Hollowood and Graham M.Shore Department of Physics,University of Wales Swansea,Swansea,SA28PP,UK.E-mail:t.hollowood@,g.m.shore@ Abstract:It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime.Assuming the validity of the Kramers-Kronig dispersion relation,this would imply a superluminal wavefront velocity and the violation of causality.Here,we calculate for the first time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic.We find that the high-frequency limit of the phase velocity (i.e.the wavefront velocity)is always equal to c andcausality is assured.However,the Kramers-Kronig dispersion relation is violated due to a non-analyticity of the refractive index in the upper-half complex plane,whose origin may be traced to the generic focusing property of null geodesic congruences and the existence of conjugate points.This puts into question the issue of micro-causality,i.e.the vanishing of commutators of field operators at spacelike separated points,in local quantum field theory in curved spacetime.1.IntroductionQuantumfield theory in curved spacetime is by now a well-understood subject.How-ever,there remain a number of intriguing puzzles which hint at deeper conceptual implications for quantum gravity itself.The best known is of course Hawking radia-tion and the issue of entropy and holography in quantum black hole physics.A less well-known effect is the discovery by Drummond and Hathrell[1]that vacuum po-larization in QED can induce a superluminal phase velocity for photons propagating in a non-dynamical,curved spacetime.The essential idea is illustrated in Figure1. Due to vacuum polarization,the photon may be pictured as an electron-positron pair, characterized by a length scaleλc=m−1,the Compton wavelength of the electron. When the curvature scale becomes comparable toλc,the photon dispersion relation is modified.The remarkable feature,however,is that this modification can induce a superluminal1low-frequency phase velocity,i.e.the photon momentum becomes spacelike.Figure1:Photons propagating in curved spacetime feel the curvature in the neighbourhood of their geodesic because they can become virtual e+e−pairs.Atfirst,it appears that this must be incompatible with causality.However, as discussed in refs.[2–4],the relation of causality with the“speed of light”is far more subtle.For our purposes,we may provisionally consider causality to be the requirement that no signal may travel faster than the fundamental constant c defining local Lorentz invariance.More precisely,we require that the wavefront velocity v wf, defined as the speed of propagation of a sharp-fronted wave pulse,should be less than,or equal to,c.Importantly,it may be shown[2,4,5]that v wf=v ph(∞),the high-frequency limit of the phase velocity.In other words,causality is safe even if the low-frequency2phase velocity v ph(0)is superluminal provided the high-frequency limit does not exceed c.This appears to remove the potential paradox associated with a superluminal v ph(0).However,a crucial constraint is imposed by the Kramers-Kronig dispersion relation3(see,e.g.ref.[6],chpt.10.8)for the refractive index,viz.Re n(∞)−Re n(0)=−2ωIm n(ω).(1.1)where Re n(ω)=1/v ph(ω).The positivity of Im n(ω),which is true for an absorptive medium and is more generally a consequence of unitarity in QFT,then implies that Re n(∞)<Re n(0),i.e.v ph(∞)>v ph(0).So,given the validity of the KK dispersion relation,a superluminal v ph(0)would imply a superluminal wavefront velocity v wf= v ph(∞)with the consequent violation of causality.We are therefore left with three main options[4],each of which would have dramatic consequences for our established ideas about quantumfield theory: Option(1)The wavefront speed of light v wf>1and the physical lightcones lie outside the geometric null cones of the curved spacetime,inapparent violation of causality.It should be noted,however,that while this would certainly violate causality for theories in Minkowski spacetime,it could still be possible for causality to be preserved in curved spacetime if the effective metric characterizing the physical light cones defined by v wf nevertheless allow the existence of a global timelike Killing vectorfield. This possible loophole exploits the general relativity notion of“stable causality”[8,9] and is discussed further in ref.[2].Option(2)Curved spacetime may behave as an optical medium ex-hibiting gain,i.e.Im n(ω)<0.This possibility was explored in the context ofΛ-systems in atomic physics in ref.[4], where laser-atom interactions can induce gain,giving rise to a negative Im n(ω)and superluminal low-frequency phase velocities while preserving v wf=1and the KKdispersion relation.However,the problem in extending this idea to QFT is that the optical theorem,itself a consequence of unitarity,identifies the imaginary part of forward scattering amplitudes with the total cross section.Here,Im n(ω)should be proportional to the cross section for e+e−pair creation and therefore positive.A negative Im n(ω)would appear to violate unitarity.Option(3)The Kramers-Kronig dispersion relation(1.1)is itself vio-lated.Note,however,that this relation only relies on the analyticity ofn(ω)in the upper-half plane,which is usually considered to be a directconsequence of an apparently fundamental axiom of local quantumfieldtheory,viz.micro-causality.Micro-causality in QFT is the requirement that the expectation value of the com-mutator offield operators 0|[A(x),A(y)]|0 vanishes when x and y are spacelike separated.While this appears to be a clear statement of what we would understand by causality at the quantum level,in fact its primary rˆo le in conventional QFT is as a necessary condition for Lorentz invariance of the S-matrix(see e.g.ref.[6], chpts.5.1,3.5).Since QFT in curved spacetime is only locally,and not globally, Lorentz invariant,it is just possible there is a loophole here allowing violation of micro-causality in curved spacetime QFT.Despite these various caveats,unitarity,micro-causality,the identification of light cones with geometric null cones and causality itself are all such fundamental elements of local relativistic QFT that any one of these options would represent a major surprise and pose a severe challenge to established wisdom.Nonetheless,it appears that at least one has to be true.To understand how QED in curved spacetime is reconciled with causality,it is therefore necessary to perform an explicit calculation to determine the full frequency dependence of the refractive index n(ω)in curved spacetime.This is the technical problem which we solve in this paper.The remarkable result is that QED chooses option(3),viz.analyticity is violated in curved spacetime.Wefind that in the high-frequency limit,the phase velocity always approaches c,so we determine v wf= 1.Moreover,we are able to confirm that where the background gravitationalfield induces pair creation,γ→e+e−,Im n(ω)is indeed positive as required by unitarity. However,the refractive index n(ω)is not analytic in the upper half-plane,and the KK dispersion relation is modified accordingly.One might think that this implies a violation of microcausality,however,there is a caveat in this line of argument which requires a more ambitious off-shell calculation to settle definitively[7].–3–In order to establish this result,we have had to apply radically new techniques to the analysis of the vacuum polarization for QED in curved spacetime.The original Drummond-Hathrell analysis was based on the low-energy,O(R/m2)effective action for QED in a curved background,L=−1m2 aRFµνFµν+bRµνFµλFνλ+cRµνλρFµνFλρ +···.(1.2) derived using conventional heat-kernel or proper-time techniques(see,for example, [10–14].A geometric optics,or eikonal,analysis applied to this action determines the low-frequency limit of the phase velocity.Depending on the spacetime,the photon trajectory and its polarization,v ph(0)may be superluminal[1,15,16].In subsequent work,the expansion of the effective action to all orders in derivatives,but still at O(R/m2),was evaluated and applied to the photon dispersion relation[11,12,17, 18].However,as emphasized already in refs.[2,3,18],the derivative expansion is inadequate tofind the high-frequency behaviour of the phase velocity.The reason is that the frequencyωappears in the on-shell vacuum polarization tensor only in the dimensionless ratioω2R/m4.The high-frequency limit depends non-perturbatively on this parameter4and so is not accessible to an expansion truncated atfirst order in R/m2.In this paper,we instead use the world-line formalism which can be traced back to Feynman and Schwinger[19,20],and which has been extensively developed in recent years into a powerful tool for computing Green functions in QFT via path integrals for an appropriate1-dim world-line sigma model.(For a review,see e.g.ref.[21].) The power of this technique in the present context is that it enables us to calculate the QED vacuum polarization non-perturbatively in the frequency parameterω2R/m4 using saddle-point techniques.Moreover,the world-line sigma model provides an extremely geometric interpretation of the calculation of the quantum corrections to the vacuum polarization.In particular,we are able to give a very direct interpretation of the origin of the Kramers-Kronig violating poles in n(ω)in terms of the general relativistic theory of null congruences and the relation of geodesic focusing to the Weyl and Ricci curvatures via the Raychoudhuri equations.A further key insight is that to leading order in R/m2,but still exact inω2R/m4, the relevant tidal effects of the curvature on photon propagation are encoded in thef(ωm2Penrose plane-wave limit[22,23]of the spacetime expanded about the original null geodesic traced by the photon.This is a huge simplification,since it reduces the problem of studying photon propagation in an arbitrary background to the much more tractable case of a plane wave.In fact,the Penrose limit is ideally suited to this physical problem.As shown in ref.[24],where the relation with null Fermi normal coordinates is explained,it can be extended into a systematic expansion in a scaling parameter which for our problem is identified as R/m2.The Penrose expansion therefore provides us with a systematic way to go beyond leading order in curvature.The paper is organized as follows.In Section2,we introduce the world-line formalism and set up the geometric sigma model and eikonal approximation.The relation of the Penrose limit to the R/m2expansion is then explained in detail, complemented by a power-counting analysis in the appendix.The geometry of null congruences is introduced in Section3,together with the simplified symmetric plane wave background in which we perform our detailed calculation of the refractive index. This calculation,which is the heart of the paper,is presented in Section4.The interpretation of the result for the refractive index is given in Section5,where we plot the frequency dependence of n(ω)and prove that asymptotically v ph(ω)→1. We also explain exactly how the existence of conjugate points in a null congruence leads to zero modes in the sigma model partition function,which in turn produces the KK-violating poles in n(ω)in the upper half-plane.The implications for micro-causality are described in Section6.Finally,in Section7we make some further remarks on the generality of our results for arbitrary background spacetimes before summarizing our conclusions in Section8.2.The World-Line FormalismFigure2:The loop xµ(τ)with insertions of photon vertex operators atτ1andτ2.–5–In the world-line formalism for scalar QED5the1-loop vacuum polarization is given byΠ1-loop=αT3 T0dτ1dτ2Z V∗ω,ε1[x(τ1)]Vω,ε2[x(τ2)] .(2.1)The loop with the photon insertions is illustrated in Figure(2).The expectation value is calculated in the one-dimensional world-line sigma model involving periodic fields xµ(τ)=xµ(τ+T)with an actionS= T0dτ 15Since all the conceptual issues we address are the same for scalars and spinors,for simplicity we perform explicit calculations for scalar QED in this paper.The generalization of the world-line formalism to spinor QED is straightforward and involves the addition of a further,Grassmann,field in the path integral.For ease of language,we still use the terms electron and positron to describe the scalar particles.6This will require some appropriate iǫprescription.In particular,the T integration contour should lie just below the real axis to ensure that the integral converges at infinity.7In general,one has to introduce ghostfields to take account of the non-trivial measure for the fields, [dxµ(τ)of geometric optics where Aµ(x)is approximated by a rapidly varying exponential times a much more slowly varying polarization.Systematically,we haveAµ(x)= εµ(x)+ω−1Bµ(x)+··· e iωΘ(x).(2.4) We will need the expressions for the leading order piecesΘandε.This will necessitate solving the on-shell conditions to thefirst two non-trivial orders in the expansion in R1/2/ω.To leading order,the wave-vector kµ=ωℓµ,whereℓµ=∂µΘis a null vector (or more properly a null1-form)satisfying the eikonal equation,ℓ·ℓ≡gµν∂µΘ∂νΘ=0.(2.5) A solution of the eikonal equation determines a family or congruence of null geodesics in the following way.9The contravariant vectorfieldℓµ(x)=∂µΘ(x),(2.6) is the tangent vector to the null geodesic in the congruence passing through the point xµ.In the particle interpretation,kµ=ωℓµis the momentum of a photon travelling along the geodesic through that particular point.It will turn out that the behaviour of the congruence will have a crucial rˆo le to play in the resulting behaviour of the refractive index.The general relativistic theory of null congruences is considered in detail in Section3.Now we turn to the polarization vector.To leading order in the WKB approxima-tion,this is simply orthogonal toℓ,i.e.ε·ℓ=0.Notice that this does not determine the overall normalization ofε,the scalar amplitude,which will be a space-dependent function in general.It is useful to splitεµ=Aˆεµ,whereˆεµis unit normalized.At the next order,the WKB approximation requires thatˆεµis parallel transported along the geodesics:ℓ·Dˆεµ=0.(2.7) The remaining part,the scalar amplitude A,satisfies1ℓ·D log A=−εµD·ℓ.(2.9)2Since the polarization vector is defined up to an additive amount of k,there are two linearly independent polarizationsεi(x),i=1,2.Since there are two polarization states,the one-loop vacuum polarization is ac-tually a2×2matrixΠ1-loop ij =αT3 T0dτ1dτ2Z× εi[x(τ1)]·˙x(τ1)e−iωΘ[x(τ1)]εj[x(τ2)]·˙x(τ2)e iωΘ[x(τ2)] .(2.10)In order for this to be properly defined we must specify how to deal with the zero mode of xµ(τ)in the world-line sigma model.Two distinct–but ultimately equiv-alent–methods for dealing with the zero mode have been proposed in the litera-ture[25–29].In thefirst,the position of one particular point on the loop is defined as the zero mode,while in the other,the“string inspired”definition,the zero mode is defined as the average position of the loop:xµ0=1Now notice that the exponential pieces of the vertex operators in(2.1)act as source terms and so the complete action including these ism2S=−T+can always be brought into the formds2=2du dΘ−C(u,Θ,Y a)dΘ2−2C a(u,Θ,Y b)dY a dΘ−C ab(u,Θ,Y c)dY a dY b.(2.14) It is manifest that dΘis a null1-form.The null congruence has a simple description as the curves(u,Θ0,Y a0)forfixed values of the transverse coordinates(Θ0,Y a0).The geodesicγis the particular member(u,0,0,0).It should not be surprising that the Rosen coordinates are singular at the caustics of the congruence.These are points where members of the congruence intersect and will be described in detail in the next section.With the form(2.14)of the metric,onefinds that the classical equations of motion of the sigma model action(2.13)have a solution with Y a=Θ=0whereu(τ)satisfies¨u=−2ωTm2δ(τ).(2.15)More general solutions with constant but non-vanishing(Θ,Y a)are ruled out by the constraint(2.12).The solution of(2.15)is˜u(τ)=−u0+ 2ωT(1−ξ)τ/m20≤τ≤ξ2ωTξ(1−τ)/m2ξ≤τ≤1.(2.16)where the constantu0=ωTξ(1−ξ)/m2(2.17) ensures that the constraint(2.12)is satisfied.The solution describes a loop which is squashed down onto the geodesicγas illustrated in Figure(3).The electron and positron have to move with different world-line velocities in order to accommodate the fact that in generalξis not equal to1Now that we have defined the Rosen coordinates and found the classical saddle-point solution,we are in a position to set up the perturbative expansion.The idea is to scale the transverse coordinatesΘand Y i in order to remove the factor of m2/T in front of the action.The affine coordinate u,on the other hand,will be left alone since the classical solution˜u(τ)is by definition of zeroth order in perturbation theory. The appropriate scalings are precisely those needed to define the Penrose limit[22]–in particular we closely follow the discussion in[23].The Penrose limit involvesfirst a boost(u,Θ,Y a)−→(λ−1u,λΘ,Y a),(2.18) whereλ=T1/2/m,and then a uniform re-scaling of the coordinates(u,Θ,Y a)−→(λu,λΘ,λY a).(2.19) As argued above,it is important that the null coordinate along the geodesic u is not affected by the combination of the boost and re-scaling;indeed,overall(u,Θ,Y a)−→(u,λ2Θ,λY a).(2.20) After these re-scalings,the sigma model action(2.13)becomesS=−T+1m2Θ(ξ)+ωT4 10dτ 2˙u˙Θ−C ab(u,0,0)˙Y a˙Y b −ωT m2Θ(0)+···.(2.22) The leading order piece is precisely the Penrose limit of the original metric in Rosen coordinates.Notice that we must keep the source terms because the combination ωT/m2,or more precisely the dimensionless ratioωR1/2/m2,can be large.However, there is a further simplifying feature:once we have shifted the“field”about the clas-sical solution u(τ)→˜u(τ)+u(τ),it is clear that there are no Feynman graphs with-out externalΘlines that involve the vertices∂n u C ab(˜u,0,0)u n˙Y a˙Y b,n≥1;hence, we can simply replace C ab(˜u+u,0,0)consistently with the background expression C ab(˜u,0,0).This means that the resulting sigma model is Gaussian to leading order in R/m2:S(2)=14 10dτC ab(˜u,0,0)˙Y a˙Y b,(2.23)wherefinally we have dropped the˙u˙Θpiece since it is just the same as inflat space and the functional integral is normalized relative toflat space.This means that all the non-trivial curvature dependence lies in the Y a subspace transverse to the geodesic.10It turns out that the Rosen coordinates are actually not the most convenient co-ordinates with which to perform explicit calculations.For this,we prefer Brinkmann coordinates(u,v,y i).To define these,wefirst introduce a“zweibein”in the subspace of the Y a:C ab(u)=δij E i a(u)E j b(u),(2.24) with inverse E a i.This quantity is subject to the condition thatΩij≡dE iadE ia2E a j.(2.29)du2We have introduced these coordinates at the level of the Penrose limit.However, they have a more general definition for an arbitrary metric and geodesic.They are in fact Fermi normal coordinates.These are“normal”in the same sense as the more common Riemann normal coordinates,but in this case they are associated to the geodesic curveγrather than to a single point.This description of Brinkmann coordinates as Fermi normal coordinates and their relation to Rosen coordinates and the Penrose limit is described in detail in ref.[24].In particular,this reference givestheλexpansion of the metric in null Fermi normal coordinates to O(λ2).To O(λ) this isds2=2du dv−R iuju y i y j du2−dy i2+λ −2R uiuv y i v du2−43R uiuj;k y i y j y k du2 +O(λ2),(2.30)which is consistent with(2.28)since R iuju=−h ij for a plane wave.It is worth pointing out that Brinkmann coordinates,unlike Rosen coordinates,are not singular at the caustics of the null congruence.One can say that Fermi normal coordinates (Brinkmann coordinates)are naturally associated to a single geodesicγwhereas Rosen coordinates are naturally associated to a congruence containingγ.In Brinkmann coordinates,the Gaussian action(2.23)for the transverse coordi-nates becomesS(2)=−12m2Ωij y i y j τ=ξ−ωTexplicitly.In doing so,we discover many surprising features of the dispersion relation that will hold in general.The symmetric plane wave metric is given in Brinkmann coordinates by(2.28), with the restriction that h ij is independent of u.This metric is locally symmetric in the sense that the Riemann tensor is covariantly constant,DλRµνρσ=0,and can be realized as a homogeneous space G/H with isometry group G.12With no loss of generality,we can choose a basis for the transverse coordinates in which h ij is diagonal:h ij y i y j=σ21(y1)2+σ22(y2)2.(3.1) The sign of these coefficients plays a crucial role,so we allow theσi themselves to be purely real or purely imaginary.For a general plane-wave metric,the only non-vanishing components of the Rie-mann tensor(up to symmetries)areR uiuj=−h ij(u).(3.2) So for the symmetric plane wave,we have simplyR uu=σ21+σ22,(3.3)R uiui=−σ2iand for the Weyl tensor,1C uiui=−σ2i+12Notice that,contrary to the implication in ref.[4,18],the condition that the Riemann tensor is covariantly constant only implies that the spacetime is locally symmetric,and not necessarily maximally symmetric[13,23].A maximally symmetric space has Rµνρσ=1plane wave background,then explain how the key features are described in the gen-eral theory of null congruences.The geodesic equations for the symmetric plane wave(2.28),(3.1)are:¨u=0,¨v+2˙u2i=1σ2i y i˙y i=0,¨y i+˙u2σ2i y i=0.(3.5)We can therefore take u itself to be the affine parameter and,with the appropriate choice of boundary conditions,define the null congruence in the neighbourhood of, and including,γas:v=Θ−122i=1σi tan(σi u+a i)y i2.(3.9)The tangent vector to the congruence,defined asℓµ=gµν∂νΘ,is therefore ℓ=∂u+1The polarization vectors are orthogonal to this tangent vector,ℓ·εi=0,and are further constrained by(2.9).Solving(2.7)for the normalized polarization(one-form) yields13ˆεi=dy i+σi tan(σi u+a i)y i du.(3.11) The scalar amplitude A is determined by the parallel transport equation(2.8),from which we readilyfind(normalizing so that A(0)=1)A=2i=1 cos(σi u+a i)(3.12)The null congruence in the symmetric plane wave background displays a number of features which play a crucial role in the analysis of the refractive index.They are best exhibited by considering the Raychoudhuri equation,which expresses the behaviour of the congruence in terms of the optical scalars,viz.the expansionˆθ, shearˆσand twistˆω.These are defined in terms of the covariant derivative of the tangent vector as[30]:ˆθ=1112Rµνℓµℓν,Ψ0=Cµρνσℓµℓνmρmσ.14As demonstrated in refs.[31],the effectof vacuum polarization on low-frequency photon propagation is also governed by the two curvature scalarsΦ00andΨ0.Indeed,many interesting results such as the polarization sum rule and horizon theorem[31,32]are due directly to special properties ofΦ00andΨ0.As we now show,they also play a key rˆo le in the world-line formalism in determining the nature of the full dispersion relation.√2R uu=1 2(C u1u1−C u2u2)=1By its definition as a gradientfield,it is clear that D[µℓν]=0so the null con-gruence is twist-freeˆω=0.The remaining Raychoudhuri equations can then be rewritten as∂u(ˆθ+ˆσ)=−(ˆθ+ˆσ)2−Φ00−|Ψ0|,∂u(ˆθ−ˆσ)=−(ˆθ−ˆσ)2−Φ00+|Ψ0|.(3.15) The effect of expansion and shear is easily visualized by the effect on a circular cross-section of the null congruence as the affine parameter u is varied:the expansionˆθgives a uniform expansion whereas the shearˆσproduces a squashing with expansion along one transverse axis and compression along the other.The combinationsˆθ±ˆσtherefore describe the focusing or defocusing of the null rays in the two orthogonal transverse axes.We can therefore divide the symmetric plane wave spacetimes into two classes, depending on the signs ofΦ00±|Ψ0|.A Type I spacetime,whereΦ00±|Ψ0|are both positive,has focusing in both directions,whereas Type II,whereΦ00±Ψ0 have opposite signs,has one focusing and one defocusing direction.Note,however, that there is no“Type III”with both directions defocusing,since the null-energy condition requiresΦ00≥0.For the symmetric plane wave,the focusing or defocusing of the geodesics is controlled byeq.(3.6),y i=Y i cos(σi u+a i).Type I therefore corresponds toσ1and σ2both real,whereas in Type II,σ1is real andσ2is pure imaginary.The behaviour of the congruence in these two cases is illustrated in Figure(4).1y21y2Figure4:(a)Type I null congruence with the special choiceσ1=σ2and a1=a2so that the caustics in both directions coincide as focal points.(b)Type II null congruence showing one focusing and one defocusing direction.To see this explicitly in terms of the Raychoudhuri equations,notefirst that the curvature scalarsΦ00−|Ψ0|=σ21,Φ00+|Ψ0|=σ22are simply the eigenvalues of h ij.The optical scalars areˆθ=−12 σ1tan(σ1u+a1)−σ2tan(σ2u+a2) (3.16)and we easily verify∂uˆθ=ˆθ2−ˆσ2−12(σ21−σ22).(3.17)It is clear that provided the geodesics are complete,those in a focusing direction will eventually cross.In the symmetric plane wave example,with y i=Y i cos(σi u+ a i),these“caustics”occur when the affine parameterσi u=π(n+115This does not necessarily mean that the conjugate points are joined by more than one actual geodesic,only that an infinitesimal deformation ofγter we shall see that the existence of conjugate points relies on the existence of zero modes of a linear problem.Conversely,the existence of a geodesic other thanγjoining p and q does not necessarily mean that p and q are conjugate[8,33].16Whether these deformed geodesics become actual geodesics is the question as to whether they lift from the Penrose limit to the full metric.4.World-line Calculation of the Refractive IndexIn this section,we calculate the vacuum polarization and refractive index explicitly for a symmetric plane wave.As we mentioned at the end of Section 2,the explicit calculations are best performed in Brinkmann coordinates.We will need the expres-sions for Θand εi for the symmetric plane wave background:these are in eqs.(3.9),(3.11)and (3.12).From these,we have the following explicit expression for the vertex operator 17V ω,εi [x µ(τ)]= ˙y i +σi tan(σi ˜u +a i )˙˜u y i 2 j =1 cos(σj ˜u +a j )×exp iω v +1410dτ ˙y i 2−˙˜u 2σ2i yi 2 −ωT σi −det g [x (τ)]which can be exponenti-ated by introducing appropriate ghosts [25–29].However,in Brinkmann coordinatesafter the re-scaling (2.27),det g =−1+O (λ)and so to leading order in R/m 2the determinant factor is simply 1and so plays no rˆo le.The same conclusion would not be true in Rosen coordinates.The y i fluctuations satisfy the eigenvalue equation¨y i+˙˜u 2σ2i y i −2ωT σi 17Notice that at leading order in R/m 2we are at liberty to replace u (τ)by its classical value ˜u (τ).The argument is identical to the one given in Section 2.。

2
minus infinity (neglecting back reaction) as the event horizon or cosmological singularity is approached. Thus, the negative energies become unconstrained near the horizon or the initial singularity. The quantum inequalities derived in this paper can also be used to support the quantum interest conjecture [15]. According to this conjecture any negative energy flux must be preceded or followed by a larger positive energy flux (i.e. the negative flux must be repaid with interest by the positive flux). Pretorius [16] has shown that the quantum interest conjecture for scalar fields in Minkowski space follows from the scaling properties of the corresponding quantum inequalities. Since the Dirac equation satisfies the same quantum inequalities in two dimensional Minkowski space it will also satisfy the quantum interest conjecture. Pretorius also conjectured that in curved spacetime the quantum interest conjecture applies only to deviations from the ground state, not to the total energy. This conjecture is shown to be true for the spacetimes examined in this paper. I will take h ¯ = c = G = 1 throughout this paper.

a rXiv:g r-qc/64119v 127Apr26Space Noncommutativity Corrections to the Cardy-Verlinde Formula M.R.Setare ∗Physics Dept.Inst.for Studies in Theo.Physics and Mathematics(IPM)P.O.Box 19395-5531,Tehran,IRAN Abstract In this letter we compute the corrections to the Cardy-Verlinde formula of Schwarzschild black holes.These corrections stem from the space noncommuta-tivity.Because the Schwarzschild black holes are non rotating,to the first order of perturbative calculations,there is no any effect on the properties of black hole due to the noncommutativity of space.1IntroductionThe Cardy-Verlinde formula proposed by Verlinde[1],relates the entropy of a certain CFT with its total energy and its Casimir energy in arbitrary ing the AdS d/CFT d−1[2]and dS d/CFT d−1correspondences[3],this formula has been shown to hold exactly for different black holes(see for example[4]-[13]).Black hole thermodynamic quantities depend on the Hawking temperature via the usual thermodynamic relations.The Hawking temperature undergoes corrections from many sources:the quantum corrections[14],the self-gravitational corrections[15],and the cor-rections due to the generalized uncertainty principle[16].In this letter we concentrate on the corrections due to the space non commutativity.Re-cently there has been considerable interest in the possible effects of the non commutative space[17].In[18]the author have argued that every consideration on space time mea-surement that allows gravitational effects asks for non-commutative space time.By considering a black hole as quantum state instead of a classical object[19]and accord-ing to quantum mechanics principle,one can conclude its energy and its corresponding conjugate time can not be simultaneously measured exactly.The energy of states should approximately be the hole’s gravitational energies measured at the region of the horizon. In other hand the gravitational energies are quasilocal,in the Schwarzschild black hole case,this quasilocal energy is proportional to the radius of event horizon.Therefore the uncertainty relation between energy and time in the event horizon region lead that the radial coordinate is noncommutative with time at the horizon.In section2we drive the corrections to the thermodynamic quantities due to the space noncommutativity.In section3we consider the generalized Cardy-Verlinde formula of a 4−dimensional Schwarzschild black hole[20,21],then we obtain the space noncommuta-tivity corrections to this entropy formula.2Schwarzschild black hole in noncommutative space A4−dimensional Schwarzschild black hole of mass M is described by the metricds2=−(1−2Mr)−1dr2+r2(dθ2+sin2θdϕ2)(1)Considering the black hole as quantum state,the energy of quantum state is the quasi-local energy at the horizon[22].According to the definition given by Brown and York [23]the quasi-local energy is asE QL=1σ(K−K0)(2)whereΣis the two dimension spherical surface,σis the determinant of the2−metric on Σ,K is the trace of the extrinsic curvature ofΣ,and K0is a reference term that is used to normalize the energy with respect to a reference spacetime.In the Schwarzschild metric case we haveKθθ=Kϕϕ=−r−2Mr(4)We obtainE QL(r→∞)=rr)=2MG(6)The quantum state energy E and its conjugate time t can not be simultaneously measured exactly[19,22].By considering E and t as operators,we have[t,E]=i.(7) Using Eq.(6)we can write[t,r]|r=r H=il2p.(8) where l p=√√2θij˜p j,p j=˜p j,(14) where the new variables satisfy the usual Poisson brackets{x i,x j}=0,{x i,p j}=δij,{p i,p j}=0.(15) Using the new coordinates,we havef(r)=1−2M(x i−θij p j2)(16)whereθij=1/2εijkθk.The equationf(r H)=0,(17) where r H is the modified horizon,leads us tof(r H)=1−2Mr2H− L. θ16=0(18)where L k=εijk x i p j,p2= p. p andθ2= θ. θ.The Schwarzschild black hole is non-rotating, therefore L=0,in this case the new horizon is given byr H=[4M2+P2θ2−( P. θ)22πr2H =M64M2]=T0[1−P2θ2−( P. θ)264M2]=A0[1+P2θ2−( P. θ)22πr20,A0=4πr20are Hawking-Bekenstein temperature and the horizon areain the commutative space.The corrected entropy due to noncommutativity of space is asS=A4[1+P2θ2−( P. θ)264M2](22)3Space noncommutativity corrections to the Cardy-Verlinde formulaThe entropy of a(1+1)−dimensional CFT is given by the well-known Cardy formula[26]S=2π 6(L0−c24is causedby the Casimir effect.After making the appropriate identifications for L0and c,the same Cardy formula is also valid for CFT in arbitrary spacetime dimensions d−1in the form[1]S CF T=2πRE c(2E−E c),(24)the so called Cardy-Verlinde formula,where R is the radius of the system,E is the total energy and E c is the Casimir energy,defined asE c=(d−1)E−(d−2)T S.(25)So far,mostly asymptotically AdS and dS black hole solutions have been considered[2]-[13].In[20],it is shown that even the Schwarzschild and Kerr black hole solutions,which are asymptoticallyflat,satisfy the modification of the Cardy-Verlinde formulaS CF T=2πR2EE c.(26)This result holds also for various charged black hole solution with asymptoticallyflat spacetime[21]In this section we compute the effect of space noncomutativity to the entropy of a d= 4−dimensional Schwarzschild black hole described by the Cardy-Verlinde formula(26). The energy Eq.(6)and Casimir energy Eq.(25)now will be modified asE=r HG[1+P2θ2−( P. θ)2128M2](27)E c=3E−2T BH S=3E0[1+P2θ2−( P. θ)264M2][1+P2θ2−( P. θ)2128M2.(28)We substitute expressions(27)and(28)which where computed to second order inθin the Cardy-Verlinde formula in order that space noncommutativity corrections to be consideredS CF T=πr H 2E c0E0(1+P2θ2−( P. θ)22(1+3E0128M2[1+1E C0)]).(29)4ConclusionIn this paper we have examined the effects of the space non-commutativity in the gener-alized Cardy-Verlinde formula.The event horizon of the black hole undergoes corrections from the non-commutativity of space as Eq.(19).Since the non-commutativity parameter is so small in comparsion with the length scales of the system,one can consider the non-commutative effect as perturbations of the commutative counterpart[24].Then we have obtained the corrections to the temperature and entropy as Eqs.(20,22).Because the Schwarzschild black holes are non rotating,to thefirst order of perturbative calculations, there is no any effect on the properties of black hole due to the non-commutativity of space.Then we have obtained the corrections to the entropy of a dual conformalfield theory live onflat space as Eq.(29).It is necessary to mention that,our result in the present paper is valid for a specific choice of spacetime non-commutativity which is defined by Eqs.(8,10).To see a more general kind of spacetime non-commutativity refer to[27,28,29],in these papers,the principle of Lie algebra stability of the Poincare-Heisenberg algebra leads to a more general kind of spacetime non-commutativity.In those modifications,the commutators of spacetime coordinates is given by the generators of rotations and boosts.References[1]E.Verlinde,On the Holographic Principle in a Radiation Dominated Universe,hep-th/0008140.[2]J.M.Maldacena,Adv.Theor.Math.Phys.2,231,(1998);E.Witten,Adv.Theor.Math.Phys.2,253(1998);S.Gubser,I.Klebanov and A.Polyakov,Phys.Lett.B428,105(1998);O.Aharony,S.Gubser,J.Maldacena,H.Ooguri and Y.Oz, Phys.Repts.323,183,(2000).Further references are contained therein.[3]A.Strominger,JHEP0110,034,(2001);JHEP0111,049,(2001);For review,seeM.Spradlin,A.Strominger,A.Volovich,hep-th/0110007.[4]S.Nojiri,S.D.Odintsov,Phys.Lett.B519,145,(2001);JHEP0112,033,(2001);Phys.Lett.B523,165,(2001).[5]V.Balasubramanian,J.de Boer and D.Minic,Phys.Rev.D65,123508(2002).[6]T.Shiromizu,D.Ida and T.Torii,JHEP0111,010,(2001).[7]B.McInnes,Nucl.Phys.B627,311,(2002).[8]Y.S.Myung,Mod.Phys.Lett.A16,2353,(2001).[9]U.H.Danielsson,JHEP0203,020,(2002).[10]S.Ogushi,Mod.Phys.Lett.A17,51,(2002).[11]M.R.Setare,Mod.Phys.Lett.A17,2089,(2002).[12]M.R.Setare and R.Mansouri,Int.J.Mod.Phys.A18,4443,(2003).[13]M.R.Setare and M.B.Altaie,Eur.Phys.J.C30,273,(2003).[14]S.Das,P.Majumdar and R.K.Bhaduri,Class.Quant.Grav.19,2355(2002);J.E.Lidsey,S.Nojiri,S.D.Odintsov and S.Ogushi,Phys.Lett.B544,337,(2002);M.R.Setare,Phys.Lett.B573,173(2003);Eur.Phys.J.C33,555,(2004).[15]E.Keski-Vakkuri and P.Kraus,Phys.Rev.D54,7407,(1996);M.K.Parikh and F.Wilczek,Phys.Rev.Lett.85,5042,(2000);Mohammad.R.Setare,Elias C.Vagenas, Phys.Lett.B584,127,(2004).[16]M.R.Setare,Phys.Rev.D70,087501,(2004).[17]N.Seiberg and E.Witten,JHEP,09,032,(1999).[18]D.V.Ahluwalia,Phys.Lett.B339,301,(1994).[19]G.’tHooft,Nucl.Phys.B256,727,(1985);Int.J.Mod.Phys.A11,4623,(1996).[20]D.Klemm,A.C.Petkou,G.Siopsis,D.Zanon,Nucl.Phys.B620,519,(2002).[21]D.Youm,Mod.Phys.Lett.A16,1263,(2001).[22]Mu-Lin Yan and Hua Bai,gr-qc/0406017.[23]J.D.Brown and J.W.York phys.Rev.D47,1407,(1993).[24]Xin-zhou Li,hep-th/0508128.[25]M.Chaichian,M.M.Sheikh-Jabbari,and A.Tureanu,Phys.Rev.Lett.86,2716,(2001).[26]J.L.Cardy,Nucl.Phys.B270(1986)186.[27]R.V.Mendes,J.Phys.A27,8091,(1994).[28]C.Chryssomalakos and E.Okon,Int.J.Mod.Phys.D13,2003,(2004).[29]D.V.Ahluwalia,Class.Quant.Grav.22,1433,(2005).。

毕业设计说明书外文文献及翻译学生姓名：学号：学院：信息商务学院学院专业：电子科学与技术指导教师：2011年 6 月外文资料原文：The Permitted Angular Velocity Pattern And ThePre-horizon Regime*Fernando de FeliceAbstractThere exist mechanical effects which allow an observer on circular its around ametric source to recognize the likely closeness of an event horizon. When these effects manifest lhemselves the spacetime is said to be in a pre-horizon regime. Here this concept will be made more precise in the case of the Kerr metric.IntroductionA question of physical interest is whether an observer in orbit around a black hole can realize from within his rest frame, how close he is to the event horizon. If the observer is moving on spatially circular trajectories then there are at least three mechanical effects which signal the likely proximity to an event horizon. These are: (i) The thrust needed to keep the orbit circular, increases outward with the modulus of the orbital frequency of revolution.(ii) The local inertial compass (a gyroscope) precesses forwards with respect to the orbital frequency of revolution.(iii) The forward gyroscopic precession increases with the modulus of the orbital frequency of revolution.If we call f the spatial acceleration of the orbit (the specific thrust), R the angular frequency of the orbital revolution as would be measured at infinity, Q2 the angular frequency of precession of the local compass of inertia and S2* QG'/[S2l, conditions (i) to (iii) can be expressed respectively as:When all these properties are satisfied the spacetime is then said to be in a pre-horizon regime. Evidently conditions (1) are meaningful only if the observer is able to measure the Work partially supparted by the Italian Space Agency under contract AS191-RS-49.three parameters f, z2 and z2 and also able to distinguish between inwards and outwards with respect to the metric source. z2 is the most difficult quantity to determine without some interaction with a distant observer; to know C in fact one should know its modulus and its sign with respect to a local clockwise direction. It is likely that within his orbiting frame the observer is able to measure only some function of a, in which case it will depend on the specific case under investigation whether the observer is able to verify conditions (1) or not. In Schwarzschild spacetime, for example, observers moving on a circular orbit can determine, with a suitable measurement of the local stresses, the direction of the orbital revolution and the modulus of its proper frequency, namely z2 multiplied by the redshift factor (de Felice et a1 1993). Since the redshift factor is positive, conditions (1) for a pre horizon in Schwanschild spacetime can be reformulated in terms of the proper frequency, allowing the orbiting observer to identify without ambiguities his spatial position.In the Kerr metric, on the contrary, it is not yet clear how one should determine the angular frequency of the orbital revolution Q or a convenient function of it, therefore I shall assume, for the purpose of the following discussion, that all the parameters in (I) are known a priori. Conditions (1) tell the orbiting observer that if an event horizon exists it is going to be close, but not how close; moreover, they may hold without an event horizon really existing. This situation occurs in the over-charged Reissner-Nordstrom spacetime and inside a static, incompressible, spherical star, described by the internal Schwarzschild solution (de Felice 1991) and also in the Kerr naked singularity solution (de Felice et al 1991).In the Ken metric the identification of the prehorizon made in de Felice et al (1991) included a region where not all of conditions (1) were satisfied. I will deduce here the correct pre-horizon structure (section 3) and suggest an operational procedure which allows the orbiting observer to recognize that his orbit is co-rotating with themetric source, that the local outward direction (direction away from the source) is concordant with the sense of the physical thrust and what his position is with respect to the photon circular orbits. I will further bring into evidence a mode of behaviour of the orbiting gyroscope, which could be interpreted as a memory by the background geometry of the static case it deviates from. Evidently the observer is assumed to know a priori that the background geometry is given by the Kerr m e ~ acn d that he is on a spatially circular equatorial orbit. The units are such that c = 1 = G; Greek indices run from 0 to 3.Kerr metric: circular orbits and the permitted angular velocity patternIn Boyer and Lindquist coordinates (Misner et al 1973), the Ken metric is described by the line element:where a and m are the specific angular momentum and total mass of the metric source respectively, and:The family of timelike spatially circular orbits in spacetime (1) is given by:where k and m are the time and the axial Killing vectors respectively; R = u+/u' is, as stated, the angular frequency of revolution, and e* is the redshift factor which is given by:I shall limit my considerations to the equatorial plane (6 = x/2). "he requirement that U=be timelike constrains R to the range:are solutions of the equation e-u = 0.It is convenient to introduce a new variable:The plots of the functions y = y c+(r,a), limited to the region outside the outer event horizon, are shown in figure 2. The extrema of y c+ occur at the photon circular orbits whose radial coordinates will be denoted as r+ and whose location is fixed by the equation:where：Figure 1 which is the locus of the event horimnTo complete the analysis of f(y; r ) we need to find its critical points. From (10) weFigure 2. The plots limited to the region in the equatorial plane which is outsidethe angular frequency of gravitational drag. From (7) and (15) this occurs when:As stated in the introduction, in a pre-horizon R* should increase with∣y∣,hence we search for where this is true. From (17) and the definition of R*, we have:The solutions of (20) are given by:In order to recognize the analytical behaviour of (23) we need to compare (21) and (22)with the other functions a2(r) which were essential in order to draw the PAv-pattern of figure 2. It is straightforward to find that:ConclusionThe main question which motivated the present investigation is bow an observer orbiting around a black hole can identify his spatial position with respect to the metric source, by means of measurements performed only within his rest frame (a space ship, say). More precisely, one may ask what is the minimum amount of a priori information necessary in order to set up, with local experiments, a non-ambiguous orientation in spacetime without interacting with a set of different observers.Evidently it is essential to know how mechanical systems, such as a physical thrust and a gyroscope, respond to a change of the orbital frequency of revolution for the localization of the orbit in the PAV-pattern. For this reason I have discussed the structure of the PAV-pattern in the equatorial plane of the Kerr metric. The pre-horizon is the most interesting part of this structure, since it hosts, loosely speaking, a gravitational field, the strength of which foreshadows the likely closeness to an event horizon. While this criterion is unambiguous in Schwarzschild and Reissner-Nordstrom spacetimes (de Felice 1991; de Felice et al 1993), in the Kerr metric the rotation of the source contributes to an apparent weakening of the gravitational strength for co-rotating orbits, causing a shrinking of the pre-horizon regime. However, monitoring the behaviour of the physical thrust with the angular frequency of revolution S2, one can still recover the necessary information about the observer's position with respect to an event horizon. For this to be possible one has to determine, from within the orbiting frame, the direction of the orbital revolution with respect to a local clockwise direction and the modulus of its proper frequency. This and the extention to electromagnetic types of measurements is now a matter of investigation.外文资料翻译：允许的角速度模式和预地平线制度费尔南多德费利切摘要本文假设允许存在一个观察员及其周边参数回归源认识到事件的时候可能产生力学效应。