Does a relativistic metric generalization of Newtonian gravity exist in 2+1 dimensions

格式：pdf
大小：128.87 KB
文档页数：6

下载文档原格式

单位内部认证船舶英语考试(试卷编号211)

单位内部认证船舶英语考试(试卷编号211)1.[单选题]The AUTOCHIEF-IV main engine remote control system includes ______.A)AC-5B)AC-5C)digitalD)hydraulic答案:C解析:2.[单选题]The star formation is most commonly used and requires _____ on the alternator.A)oneB)twoC)threeD)four答案:D解析:3.[单选题]The difference between the fire detectors of the traditional and bus control type fire alarm systems is ______.A)theyB)theC)theD)the答案:D解析:4.[单选题]VLCC stands for______.A)veryB)veryC)veryD)very答案:C解析:5.[单选题]Once the power is recovered after blackout, the sequential start of automatic power plant would enable the motors in operation before the breakdown to start ______ automatically.C)respectivelyD)immediately答案:A解析:6.[单选题]Voltage will always lead current in a/an _____.A)capacitiveB)inductiveC)magneticD)resistive答案:B解析:【注】在电感性电路中，电压总是超前电流。

inductive circuit：电感性电路；capacitive circuit：电容电路7.[单选题]Which one is the function of steering gear?A)ToB)ToC)ToD)To答案:C解析:8.[单选题]When the voltage remains constant and the resistance is increased in a series circuit, the flow of current _____.A)increasesB)increasesC)remainsD)decreases答案:D解析:9.[单选题]The emergency generator or emergency battery is connected to _____ on most large ships.A)distributionB)sectionC)emergencyD)main答案:C解析:答案:B解析:11.[单选题]Switchboards may be of the dead-front type in which all live parts are installed behind _____ and only the operation handles and instruments are on the front.A)theB)theC)theD)the答案:C解析:12.[单选题]The Maritime Labour Convention, 2006, was issued by the _____.A)UNB)IMOC)ILOD)ITU答案:C解析:13.[单选题]The difference between magnetic heading and compass heading is called______.A)variationB)deviationC)compassD)drift答案:B解析:14.[单选题]Internet Explorer, Firefox, Google Chrome, Safari, and Opera are the major ______.A)webB)uniformC)fileD)Java答案:A解析:D)It's答案:D解析:16.[单选题]The number of cycles per second occurring in AC voltage is known as the_____.A)phaseB)frequencyC)waveD)half答案:B解析:17.[单选题]Copper is often used as an electrical conductor because it _____.A)hasB)hasC)isD)holds答案:C解析:【注】electrical conductor：导电体；opposition：阻挠，反对18.[单选题]A ground can be defined as an electrical connection between the wiring of a motor and its _____.A)shuntB)circuitC)metalD)inter-pole答案:C解析:19.[单选题]In more recent years, ______ has been used by civilians in many new ways to determine positions, such as in automobile and boat navigation, hiking, emergency rescue, and precision agriculture and mining.A)GPSB)GMDSSC)AISD)Navtex20.[单选题]The podded propulsor is widely adopted in the electric propulsion system. In this system, ______.A)theB)theC)theD)the答案:A解析:21.[单选题]_____ is used to produce electric power.A)AnB)AC)AD)A答案:A解析:22.[单选题]Prior to closing the breaker when paralleling two AC generators, the recommended practice is to have the frequency of the incoming machine _____.A)slightlyB)theC)slightlyD)have答案:C解析:23.[单选题]All echo-sounders can measure the ______.A)actualB)actualC)averageD)average答案:B解析:24.[单选题]The field coils _____ and the armature is _____. This is in fact the arrangement adopted for large, heavy duty alternators.A)stationaryB)stationaryC)rotate25.[单选题]What feature(s) may be found on certain satellite EPIRB units?A)StrobeB)EmergencyC)Float-freeD)All答案:D解析:【注】卫星EPIRB有闸门照明，406MHz紧急发射和自浮释放支架。

爱因斯坦全英文

of his?
“It doesn't matter，” answered Albert Einstein. “No one knows ห้องสมุดไป่ตู้e here in New York.”
2. What did the friend want him to buy? 3. How about Einstein‘sovercoat’?
I.
General Relativity
I.
广义相对论
II. Special relativity III. Brownian motion IV. Photoelectric effect V. E=mc² VI. Einstein field equation
II. 狭义相对论 III. 布朗运动 IV. 光电效应 V. 质能等价理论
VII. Bose - Einstein statisticsVI. 爱因斯坦场方程
VII. 玻色-爱因斯坦统计
爱因斯坦是1879出生在一个
德国的城市。作为一个男孩，他
Einstein was born in 1879 in Ulm, a city in Germany. As a boy, he was slow to learn to talk, but later in his childhood he showed great curiosity about nature and ability to solve difficult mathematical problems. After he left school, he went to Switzerland, where he graduated from the university with a degree in mathematics.

Introduction to General Relativity

Introduction to General Relativity – HandoutLin “Jimmie” Haipeng, Wang “Richie” Yunchong2013.11.13What is General Relativity?…the geometric theory of gravitation published by Albert Einstein in 1916 and the current description of gravitation in modern physics.Geometric means that the presence of mass “curves” spacetime like a trampoline and results in gravity.Why do we need it?War of Theories:1905: Albert Einstein published his theory of special relativity reconciling Newton's laws of motion with electrodynamics.Special relativity changed physics’ basic frameworks like “Space” and “Time”.Quick Review of Special Relativity-Speed of Light does not change, anywhere, any way.o Time and Space are not absolute.-There is no absolute “fast” or “slow” or “at the same time”.o All rules of physics are the same in any inertial reference frame.o All inertial reference frames are equal. (You can’t distinguish between any one)Problems: VS Classical Physics- Time and Space are no longer absolute^^ This resulted in a new framework for Physics. Existing theories like Newton’s Gravity Theory no longer worked.(Since mass changes, time and space are no longer absolute, etc.)Several physicists, including Einstein, searched for a theory that would reconcile Newton's law of gravity and special relativity.Newton’s Gravitational Model is failing•Time and space are no longer absolute, mass isn’t either•Half of what Newton’s Gravitational Model is using is failing•Astrophysics says it doesn’t work out•Light is deflecting? Time is passing differently due to Gravity?•Newton isn’t saying it allWith that, let’s follow the steps of Einstein for a basic understanding of GR1 Equivalence PrincipleSpecial Relativity: You can’t distinguish between inertial reference framesGeneral Relativity: You can't distinguish between ANY reference frames.“Roughly speaking, the principle states that a person in a free-falling elevator cannot tell that they are in free fall. Every experiment in such a free-falling environment has the same results as it would for an observer at rest or moving uniformly in deep space, far from all sources of gravity.”2 Accelerating Reference FramesYou experi ence acceleration (“Gravity”) in accelerating reference framesSuch an additional force due to non-uniform relative motion of two reference frames is called a pseudo-force.3 Gravity “acceleration” also causes time to go slower.Imagine a disk spinning. On the outer part, v is larger (v=wR) so time is slower there. Acceleration is larger, and according to equivalence principles – its gravity, so gravity causes slower time too.Imagine rays of light.4 Curvature of Space results in GravityApplications & EffectsNamely Astrophysics.(Richie you go, one two)ReferencesGiancoli, Douglas C. Physics for Scientists and Engineers. Addison-Wesley, 2008. Book.Iro, Harald. A Modern Approach to Classical Mechanics. World Scientific, 2002. Book. Xihua, Zhong and Chen Ximou. Modern Physics. Beijing: Peking University Press, 2011. Book.。

中考英语经典科学实验与科学理论深度剖析阅读理解20题

中考英语经典科学实验与科学理论深度剖析阅读理解20题1<背景文章>Isaac Newton is one of the most famous scientists in history. He is known for his discovery of the law of universal gravitation. Newton was sitting under an apple tree when an apple fell on his head. This event led him to think about why objects fall to the ground. He began to wonder if there was a force that acted on all objects.Newton spent many years studying and thinking about this problem. He realized that the force that causes apples to fall to the ground is the same force that keeps the moon in orbit around the earth. He called this force gravity.The discovery of the law of universal gravitation had a huge impact on science. It helped explain many phenomena that had previously been mysteries. For example, it explained why planets orbit the sun and why objects fall to the ground.1. Newton was sitting under a(n) ___ tree when he had the idea of gravity.A. orangeB. appleC. pearD. banana答案：B。

Cylindrically Symmetric Solution in Teleparallel Theory of Gravitation

Møller [10] in connection also with attempts to deﬁne the energy of the gravitational ﬁeld. Teleparallel theories of gravity are deﬁned on the Weitzenb¨ ock spacetime [16], which is endowed with the aﬃne connection. The curvature tensor constructed out of this connection vanishes identically. This connection deﬁnes a space-time with an absolute parallelism or teleparallelism of vector ﬁelds [3]. In this geometrical framework the gravitational eﬀects are due to the torsion tensor corresponding to the above mentioned connection. As stated above that the calculations of energy within the framework of general relativity theory have some problems [10]. It is the aim of the present work to study a tetrad having cylindrical symmetry and apply it to the ﬁeld equation of gravitation and electromagnetic. Solving the resulting non linear diﬀerential equation an exact solution is obtained. We then, calculate the energy using the superpotential of Mikhail et al. [18]. In section 2, we give a brief review of the gravitational and electromagnetic theory. The tetrad having cylindrical symmetry is applied to the ﬁeld equations of the gravitational and electromagnetic theory in section 3. The solution of the resulting ﬁeld equations is also given in section 3. The singularities of this solution are given in section 4. In section 5, the calculations of energy using the energy-momentum complex derived by Mikhail et al. are given. Section 6 is devoted for the discussion and a summary of the results. A comparison between the energy calculated and that of general relativity is also given in section 6.

Relativity- The Special and General Theory

Relativity:The Special and General TheorybyAlbert EinsteinRelativity: The Special and General TheoryPreface (4)Part I: The Special Theory of Relativity (5)The System of Co-ordinates (7)Space and Time in Classical Mechanics (9)The Galileian System of Co-ordinates (10)The Principle of Relativity (in the restricted sense) (11)The Theorem of the Addition of Velocities Employed in Classical Mechanics (13)The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity (14)On the Idea of Time in Physics (16)The Relativity of Simulatneity (18)On the Relativity of the Conception of Distance (20)The Lorentz Transformation (21)The Behaviour of Measuring-Rods and Clocks in Motion (24)The Heuristic Value of the Theory of Relativity (29)General Results of the Theory (30)Experience and the Special Theory of Relativity (33)Minkowski's Four-Dimensional Space (36)Special and General Principle of Relativity (38)The Gravitational Field (40)The Equality of Inertial and Gravitational Mass as an argument for the General Postule of Relativity (42)In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory? (43)A Few Inferences from the General Principle of Relativity (44)Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference (46)Euclidean and Non-Euclidean Continuum (48)Gaussian Co-ordinates (50)The Space-Time Continuum of the Speical Theory of Relativity Considered as a Euclidean Continuum (52)The Space-Time Continuum of the General Theory of Realtivity is Not a Euclidean Continuum (53)Exact Formulation of the General Principle of Relativity (55)The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity (57)Cosmological Difficulties of Newton's Theory (59)The Possibility of a "Finite" and yet "Unbounded" Universe (60)The Structure of Space According to the General Theory of Relativity (62)Appendix I: Simple Derivation of the Lorentz Transformation (Supplementary to Section 11) (63)Appendix II: Minkowski's Four-Dimensional Space ("World")(supplementary to section 17) (67)Appendix III: The Experimental Confirmation of the General Theory of Relativity (68)Appendix IV: The Structure of Space According to the General Theory of Relativity (Supplementary to Section 32) (73)PrefaceThe present book is intended, as far as possible, to give an exact insight into the theory of Relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. The work presumes a standard of education corresponding to that of a university matriculation examination, and, despite the shortness of the book, a fair amount of patience and force of will on the part of the reader. The author has spared himself no pains in his endeavour to present the main ideas in the simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated. In the interest of clearness, it appeared to me inevitable that I should repeat myself frequently, without paying the slightest attention to the elegance of the presentation. I adhered scrupulously to the precept of that brilliant theoretical physicist L. Boltzmann, according to whom matters of elegance ought to be left to the tailor and to the cobbler. I make no pretence of having withheld from the reader difficulties which are inherent to the subject. On the other hand, I have purposely treated the empirical physical foundations of the theory in a "step-motherly" fashion, so that readers unfamiliar with physics may not feel like the wanderer who was unable to see the forest for the trees. May the book bring some one a few happy hours of suggestive thought!December, 1916A. EINSTEINPart I: The Special Theory of RelativityIn your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What, then, do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little consideration.Geometry sets out form certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines," to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry "true." Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a "distance" two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation.If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of ageometrical proposition in this sense we understand its validity for a construction with rule and compasses.Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.Notes1) It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when the points A and C being given, B is chosen such that the sum of the distances AB and BC is as short as possible. This incomplete suggestion will suffice for the present purpose.The System of Co-ordinatesOn the basis of the physical interpretation of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a " distance " (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry ; then, starting from A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length. 1)Every description of the scene of an event or of the position of an object in space is based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place specification " Times Square, New York," [A] I arrive at the following result. The earth is the rigid body to which the specification of place refers; " Times Square, New York," is a well-defined point, to which a name has been assigned, and with which the event coincides in space.2)This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other. But we can free ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering over Times Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed.•(a) We imagine the rigid body, to which the place specification is referred, supplemented in such a manner that the object whose position we require is reached by. the completed rigid body.•(b) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.•(c) We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud.From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates.This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the threeperpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry.In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available ; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be sought in accordance with the above considerations. 3)We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for "distances;" the "distance" being represented physically by means of the convention of two marks on a rigid body.Notes1) Here we have assumed that there is nothing left over i.e. that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method.[A] Einstein used "Potsdamer Platz, Berlin" in the original text. In the authorised translation this was supplemented with "Tranfalgar Square, London". We have changed this to "Times Square, New York", as this is the most well known/identifiable location to English speakers in the present day. [Note by the janitor.]2) It is not necessary here to investigate further the significance of the expression "coincidence in space." This conception is sufficiently obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice.3) A refinement and modification of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book.Space and Time in Classical MechanicsThe purpose of mechanics is to describe how bodies change their position in space with "time." I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins.It is not clear what is to be understood here by "position" and "space." I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. A pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola? Moreover, what is meant here by motion "in space" ? From the considerations of the previous section the answer is self-evident. In the first place we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference." The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of " body of reference " we insert " system of co-ordinates," which is a useful idea for mathematical description, we are in a position to say : The stone traverses a straight line relative to a system of co-ordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (embankment) it describes a parabola. With the aid of this example it is clearly seen that there is no such thing as an independently existing trajectory (lit. "path-curve" 1)), but only a trajectory relative to a particular body of reference.In order to have a complete description of the motion, we must specify how the body alters its position with time ; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction ; the man at the railway-carriage window is holding one of them, and the man on the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.Notes1) That is, a curve along which the body moves.The Galileian System of Co-ordinatesAs is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia, can be stated thus: A body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line. This law not only says something about the motion of the bodies, but it also indicates the reference-bodies or systems of coordinates, permissible in mechanics, which can be used in mechanical description. The visible fixed stars are bodies for which the law of inertia certainly holds to a high degree of approximation. Now if we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed star describes a circle of immense radius in the course of an astronomical day, a result which is opposed to the statement of the law of inertia. So that if we adhere to this law we must refer these motions only to systems of coordinates relative to which the fixed stars do not move in a circle. A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a " Galileian system of co-ordinates." The laws of the mechanics of Galflei-Newton can be regarded as valid only for a Galileian system of co-ordinates.The Principle of Relativity (in the restricted sense)In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. We call its motion a uniform translation ("uniform" because it is of constant velocity and direction, " translation " because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the embankment, is uniform and in a straight line. If we were to observe the flying raven from the moving railway carriage. we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line. Expressed in an abstract manner we may say : If a mass m is moving uniformly in a straight line with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system K1 provided that the latter is executing a uniform translatory motion with respect to K. In accordance with the discussion contained in the preceding section, it follows that: If K is a Galileian co-ordinate system. then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K1 the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K.We advance a step farther in our generalisation when we express the tenet thus: If, relative to K, K1 is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K1 according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. At this juncture the question of the validity of the principle of relativity became ripe for discussion, and it did not appear impossible that the answer to this question might be in the negative.Nevertheless, there are two general facts which at the outset speak very much in favour of the validity of the principle of relativity. Even though classical mechanics does not supply us with a sufficiently broad basis for the theoretical presentation of all physical phenomena, still we must grant it a considerable measure of " truth," since it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful. The principle of relativity must therefore apply with great accuracy in the domain of mechanics. But that a principle of such broad generality should hold with such exactness in one domain of phenomena, and yet should be invalid for another, is a priori not very probable.We now proceed to the second argument, to which, moreover, we shall return later. If the principle of relativity (in the restricted sense) does not hold, then the Galileian co-ordinate systems K, K1, K2, etc., which are moving uniformly relative to each other, will not be equivalent for the description of natural phenomena. In this case we should be constrained to believe that natural laws are capable of being formulated in a particularlysimple manner, and of course only on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one (K 0) of a particular state of motion as our body of reference. We should then be justified (because of its merits for the description of natural phenomena) in calling this system " absolutely at rest," and all other Galileian systems K " in motion." If, for instance, our embankment were the system K 0 then our railway carriage would be a system K , relative to which less simple laws would hold than with respect to K 0. This diminished simplicity would be due to the fact that the carriage K would be in motion (i.e. "really") with respect to K 0. In the general laws of nature which have been formulated with reference to K, the magnitude and direction of the velocity of the carriage would necessarily play a part. We should expect, for instance, that the note emitted by an organpipe placed with its axis parallel to the direction of travel would be different from that emitted if the axis of the pipe were placed perpendicular to this direction.Now in virtue of its motion in an orbit round the sun, our earth is comparable with a railway carriage travelling with a velocity of about 30 kilometres per second. If the principle of relativity were not valid we should therefore expect that the direction of motion of the earth at any moment would enter into the laws of nature, and also that physical systems in their behaviour would be dependent on the orientation in space with respect to the earth. For owing to the alteration in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the hypothetical system K 0 throughout the whole year. However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions. This is very powerful argument in favour of the principle of relativity.The Theorem of the Addition of Velocities Employed inClassical MechanicsLet us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v, and that a man traverses the length of the carriage in the direction of travel with a velocity w. How quickly or, in other words, with what velocity W does the man advance relative to the embankment during the process ? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage, and hence also relative to the embankment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total be covers the distance W=v+w relative to the embankment in the second considered. We shall see later that this result, which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained ; in other words, the law that we have just written down does not hold in reality. For the time being, however, we shall assume its correctness.The Apparent Incompatibility of the Law of Propagation of Light with the Principle of Relativity There is hardly a simpler law in physics than that according to which light is propagated in empty space. Every child at school knows, or believes he knows, that this propagation takes place in straight lines with a velocity c= 300,000 km./sec. At all events we know with great exactness that this velocity is the same for all colours, because if this were not the case, the minimum of emission would not be observed simultaneously for different colours during the eclipse of a fixed star by its dark neighbour. By means of similar considerations based on observations of double stars, the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of motion of the body emitting the light. The assumption that this velocity of propagation is dependent on the direction "in space" is in itself improbable.In short, let us assume that the simple law of the constancy of the velocity of light c (in vacuum) is justifiably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties? Let us consider how these difficulties arise.Of course we must refer the process of the propagation of light (and indeed every other process) to a rigid reference-body (co-ordinate system). As such a system let us again choose our embankment. We shall imagine the air above it to have been removed. If a ray of light be sent along the embankment, we see from the above that the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v, and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity w of the man relative to the embankment is here replaced by the velocity of light relative to the embankment. w is the required velocity of light with respect to the carriage, and we havew = c-v.The velocity of propagation ot a ray of light relative to the carriage thus comes cut smaller than c.But this result comes into conflict with the principle of relativity set forth in Section V. For, like every other general law of nature, the law of the transmission of light in vacuo [in vacuum] must, according to the principle of relativity, be the same for the railway carriage as reference-body as when the rails are the body of reference. But, from our above consideration, this would appear to be impossible. If every ray of light is propagated relative to the embankment with the velocity c, then for this reason it would appear that another law of propagation of light must necessarily hold with respect to the carriage — a result contradictory to the principle of relativity.。

国家电网公司专业技术人员电力英语水平考试题库（英语短文判断）

国家电网公司专业技术人员电力英语水平考试题库（英语短文判断）第一篇：国家电网公司专业技术人员电力英语水平考试题库(英语短文判断)1.Feature ofpower generationThe simultaneousness of the electric power generation means that ……P2822.Types of circuit breakerThe high voltage circuit breaker is mainly composed of contactors ,……P283 3.Optical fiber communicationOptical fiber communication is a 10-pound note kind of information communication by optical fiber.……P2844.Power plantAccording to the mode of energy conversion ,power plants can be classified into fossil-fired……P2835.Selection of metal material for the boiler in units 1000mwgradeTaking a 10-pound note general view of the 1000mw grade high-efficiency supercritical unit designed ……P2866.The role of the condenserThe condenser is a 10-pound note surface heat exchanger in which cooling water passing through the tubes ……P2877.Hydraulic structureThe selected type of dam of hydraulic power plant depends principally on topographic,……P2888.Heat treatmentThe purpose of post-weld heat treatment is :to diminish the residual stress in the welded……P2899.Business and riskscapitalism ……P29010.ElectricityElectricity may be dangerous.it always takes the shortest way to the ground ……P29011.Undersea lifeThe undersea world is very mysterious.……P29112.Advice on friendshipWe all need friends.without friends we may feel empty and sad ……P292 13.AustralisAustralis is a vast continent,the sixth largest in wor ld.……P29214.BiomassBiomass is a cost-effective source of energy.……P29315.Nuclear radiationNuclear power’s danger to health ,safety ,and even to lifeitselfcan be summed up in one world ……P29416.Livestock’s long shadowWhen you think about the growth of human population over the last century or so ,it is all ……P29517.Pain managementYears ago ,doctors often said that pain was a normal part of life.……P29518.The obama administration’sbank rescue proposalAmong the criticisms of the boama administration’s bank ……P296第二篇：国家电网专业技术人员电力英语水平考试(英语短文判断)1.Advice on friendshipWe all need friends.Without friends we may feel empty and sad.……P3142.Business and riskscapitalism, ……P3113.Closed loop operation of power gridThe closed loop operation of power grid refers to the mode of connecting the substations or transformers ……P3034.ElectricityElectricity may be dangerous.It always takes the shortest way to the ground.……P3125.Feature of power generationThe simultaneousness of the electric power generation means that the electric power generation,……P3026.Grounding of electric equipmentConnecting electric equipment with a grounded conductor in the earth is called grounding.……P3057.Heat treatmentThe purpose of post-weld heat treatment is: to diminish the residual stress in the welded joints;……P310structureThe selected type of dam of hydraulic power plant depends principally on 8.Hydraulic topographic……P3109.Optical fiber communicationOptical fiber communication is a kind information by optical fiber.……P30610.PaperDo you know the key to the following question?……P31311.Power plantAccording to the mode of energy conversion, power plants can be classified into fossil-fired,……P30712.Selection of metal material for the boiler in units of 1 000MW gradeTaking a general view of the 1 000MW high-efficiency supercritical unit designed and made in China,……P30813.Types of circuit breakerThe high voltage circuit breaker is mainly composed of contactors,……P30414.The role of the condenserThe condenser is a surface heat exchange in which cooling water passing through the tubers ……P30815.UnderseaThe undersea world is very mysterious.In the daytime, there is enough light.……P313第三篇：国家电网公司专业技术人员电力英语水平考试题库(英语短文判断)1.Feature of power generation The simultaneousness of the electric power generation means that ……P2822.Types of circuit breaker The high voltage circuit breaker is mainly composed of contactors ,……P2833.Optical fiber communication Optical fiber communication is a 10-pound note kind of information communication by optical fiber.……P2844.Power plant According to the mode of energy conversion ,power plants can be classified into fossil-fired……P2835.Selection of metal material for the boiler in units 1000mw gradeTaking a 10-pound note general view of the 1000mw grade high-efficiency supercritical unit designed ……P286 6.The role of the condenser The condenser is a 10-pound note surface heat exchanger in which cooling water passing through the tubes ……P287 7.H ydraulic structure The selected type of dam of hydraulic power plant depends principally on topographic,……P288 8.Heat treatment The purpose of post-weldheat treatment is :to diminish the residual stress in the welded……P289 9.Business and risks Marx once q uoted a famous saying in his work capitalism ……P290 10.ElectricityElectricity may be dangerous.it always takes the shortest way to the ground ……P29011.Undersea life The undersea world is very mysterious.……P291 12.Advice on friendship We all need friends.without friends we may feel empty and sad ……P292 13.Australis Australis is a vast continent,the sixth largest in world.……P292 14.BiomassBiomass is a cost-effective source of energy.……P293 15.Nuclear radiationNuclear power’s danger to health ,safety ,and even to lifeitself can be summed up in one world ……P294 16.Livestock’s long shadow When you think about the growth of human population over the last century or so ,it is all ……P295 17.Pain management Years ago ,doctors often said that pain was a normal par t of life.……P295 18.The obama administration’s bank rescue proposal Among the criticisms of the boama administration’s bank ……P296第四篇：国家电网公司专业技术人员电力英语水平考试题库-4英语阅读理解阅读理解Passage 1Have you ever seen a moon that looks unbelievably big?1.To what do—harvest moon(All of these)2.The main purpose—is to(inform)3.The author knew—the moon(mysterious)4.The moon looks bigger if(it is--horizon)5.The autumm moon(help farmers--crops)Passage 2Strange thing happens to time when you travel.1.The best title—is(how time--world)2.The difference in—is(one hour)3.From this –ocean(is divided--zones)4.The international—name for(the point--begins)5.If you cross—clock(ahead one--zone)Passage 3Holidays in the United States usually occur at least once a month1.The government—have a(3-day)2.Workers in the—from(Tuesday to Friday)3.Which statement—passage?(All the--vacation)4.The reason—that(no one--place)5.Which of the—passage?(Something—U.S)Passage 4Sarah Winchester was a very rich woman.1.What did—house(Making it bigger)2.The story—had(7 floors)3.Who did—house(Carpenters--workers)4.How long—continue(For 38 years)5.Sarah’s—finished(when she died)Passage 5The diner is only a humble restaurant,1.What’s the—2(The attraction--people)2.The purpose—to(gove a--passage)3.Why do—diner?(It’s--loneliness)4.Diners attract(many--people)5.Diners are(fascinating)Passage 6In the past two years,millions-1.The word—to(make use of)2.It can—fitness,(bicyle--rise)3.The bicycle is(enjoying--revival)4.The reader—are(concerned--lives)5.in the—means(a rapid--sale)Passage 7Doctors have known for a long time that— Or loss1.Doctors have—that(one many--noise)2.This passage—hearing(will be--second)3.According to—aspirin(makes hearing--worse)lions of—they(take--aspirin)5.The purpose—find(whether aspirin--noises)Passage 8Just two month ago,Ana,a teenager,was—1.Ana realizes that(she must--exam)2.Ana has—for(seven years)3.Ana experiences—with(the--lectures)4.Ana tells—about(her family)5.The best—is(Ana comes--colors)Passage 9Any mistake made in the printing of a –Collectors.1.A postage—if(a mistake--printing)2.In 1847—were(not--stamps)3.In 1847—in the(wording)4.＄16800—of(the--blue)5.The valuable—by(British printers)Passage 10In the English educational system,1.The purpose—to(describe--on)2.The exam—age of(fifteen)3.We may—that(the exam--exams)4.The passage—that(schooling--England)5.As used—means(to take--of)Passage 11For centuries,in the countries of south and—the1.What can—passage?(It is hard--them)2.Thailand was—because(white—1920s)3.Why is—author?(Because--owners)4.Which of—times?(Today--5150)5.The passage—from(a research report)Passage 12The communications explosion is on the scale of the rail,1.By saying—to(display--life)2.The author—is(amazing)3.Which of—true?(The--functionally)4.According—us to(talk and--are)5.The phrase—by(each car)Passage 13Many private institutions of higher education around the—danger.1.According to—of(their characteristics)2.The author—mean(get into difficulties)3.We can—support(private schools)4.Which of-NOT-schools?(Private--schools)5.Which of--schools?(National--support)Passage 14Japan is getting tough about recycling—and------kind of way.1.According to—of(the consumers)2.Which of—plastics?(It retains--reprocessing)3.According to—to(a kind--layer)4.In the—that(21-inch—so far)5.The author—to(inform)Passage 15A friend of mine,in response to aconversation—of life,1.The author—because(like--unfair)2.Surrendering—will(make--things)3.The second—discusses(it’s—of life)4.In the—fact that(life--fair)5.From the—life is(positive)Passage 16People appear to be born to compute.1.What does—discuss?(The--children)2.From the—children(begin—and talk)3.In his—is(objective)4.According to—children(didn’t think)5.Which of—of?(Children--easily)Passage 17The small coastal town of Broome,in northwest Australia,1.The first—that(Broome--vast)2.Sun Pictures—in that(the—the grass)3.Gregory Peck—(a movie star)4.The non—refers to(an insect--incident)5.It can—by(the Sun Pictures)Passage 18A new technology is going to ripe,one that could transform—lives,1.As is—superconductivity(is--development)2.The new—that(it is being--world)3.What does—wold?(Dramatic)4.From the—that(Asian--technology)5.Which of—passage?(Superconductivity:)Passage 19More surprising,perhaps,than the current difficulties—and thriving.1.By calling—that(more--Europeans)2.From the—that(traditional--difficulty)3.Which of—families?(Many--acceptable)4.Part-time children(are--spouses)5.Even though—families,(the--marriage)Passage 20People become quite illogical when they try to decide—cannot.1.The wold—means(disgusting)2.We can—author(was angry--plants)3.The author—snails(are the--food)4.The best—be(One--Poison05.As indicated—because(they learn--families)Passage 21All the use ful energy at the surface of the earth comes from the activity of the sun.1.The sun is the source---EXCEPT(atomic power)2.Radiant energy is stored---by(plants)3.The sun’s energy provides---all EXCEPT(water)4.The largest part lf the---earth is(absorbed by the earth’s atmosphere)5.Of the sun’s total---receives(a very samall portion)Passage 22The market is a concept---for the passage?1.Which of the following---passage?(What Is the Market?)2.All of the following---EXCEPT(attending a night school)3.You are buying---when you(dine at a restaurant)4.The word---probably mean(concrete)5.In what way is the market---something?(It tells you what to produce)Passage 23X-rays wer first discovered by a German scientist---togethe.1.What puzzled Rontgen---was(some radiation---tube)2.The screen didn’t---when(it was moved to the next room)3.Rontgen put his hand---to(find out more about the rays)4.The rays proved to---through(bone)5.From the passage---are(invisible)Passage 24“Body clocks”are biological methods of controlling---were doing.1.According to the passage(one can help---“body clocks”)2.Irregular signs shown---warning of(possible illnesses)3.We tend to do physical---because(our body is most active then)4.The author suggests---study is(at night)5.According to the---day-dream(every hour in the day time)Passage 25Plastics are materials which are softened---cheaply1.The word“sympathetic”in---means(agreeable)2.It can be concluded from this passage that(plastics are cheap as antiques)3.Which of the following---plastics?(Carbon)4.Plastics that harden----called(thermosetting)5.Which lf the following---passage?(The Development---Material)Passage 26When we analyze the salt salinity of ocean---of the world1.This passage mainly tells us about(the causes of the--salinity)2.It can be inferred from—by(evaporation)3.Which of the following---salinity(Formation of sea ice)4.Which of the following---passage(The temperature---salinity)5.The purpose--Weddell Sea is(to give an example of---salinity)Passage 27The science of meteorology is concerned with---meteorology1.Which of the following is –-passage(Approaches to--Meteorology)2.The predictions of synoptic---based on the(preparation r----maps)3.Which of the following is not---weather forecasting(Sports)4.The author implies—will lead to(greater protection--property)5.In the last sentenceof-refers to(mathematics and physics)Passage 28As we have seen,the focus of medical care in our—of daily life1.Today medical care is placing-on(removing peoples bad living habits)2.In the first paragraph—that(good health is more than not being ill)3.Traditionally,a person—if he(is free from any kind of disease)4.According to the author---(to strive to maintain—possible health)5.According to what—healthy?(People who try to--limitations)Passage 29IF you want to teach your---is not1.If a mother adds—(the childs may feel that--apolgy)2.According to the author—(I’m aware--blame)3.It is not advisable—(it is vague and ineffective)4.We lean from—(their ages should--account)5.It can be inferred—(not as-seems)Passage 30Scratchy throats---catching one1.According to the—(shorten the--illness)2.We learn from the passage that(over-the-counter---)3.According to the passage—(one should take_disease)4.Which of the—cold(A high temperature)5.If children have –(are advised not to--aspirin)Passage 31Sign has become a scientific—stuff1.The study of sign—(a challengeto---)2.The present growing—(an English--deaf)3.According to Stokoe—(a genuine language)4.Most educators objected—(a language could--sounds)5.Stokoe’s argument is—(language is a product of the brain)Passage 32It is hard to track the blue—miles1.The passage is chiefly about(the civilian--system)2.The underwater---(to trace and locate---)3.The deep-sea---(the unique property--)4.It can be inferred—(military---)5.Which of the following—(it is now partly--)Passage 33You never see them—recovered1.What does the author—(It is an indispensable--)2.What information –(Data for analyzing--)3.Why was the black—(The early models often---)4.Why did the Federal—(To make them--)5.What do we know—(there is still---)Passage 34New technology links------to the firm1.What is the author’s attitude—(positive)2.With the increased---(are attaching more-----)3.In this passage—(missing opportunities for---)4.According to the—(Ability to speak---)5.The advantage of—(better control the---)第五篇：国家电网公司专业技术人员电力英语水平考试宝典(补全短文)补全短文Passage 1 Functions of power transmissionCEADBThe function of(1)is to send power from power plants to load center or to exchange。

Lectures on Loop Quantum Gravity

5
6 6 9 15 16 20 22 22 25
I.2
I.3
II Mathematical and Physical Foundations of Quantum General Relativity 28
II.1 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.1.1 Polarization and Preferred Poisson Algebra B . . . . . . . . . . . . . . . . . . II.1.2 Representation Theory of B and Suitable Kinematical Representations . . . . II.1.2.1 Curves, Paths, Graphs and Groupoids . . . . . . . . . . . . . . . . . II.1.2.2 Topology on A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.1.2.3 Measures on A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.1.2.4 Representation Property . . . . . . . . . . . . . . . . . . . . . . . . II.2 Quantum Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II.2.1 The Space of Solutions to the Gauss and Spatial Diﬀeomorphism Constraint II.2.2 Kinematical Geometrical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 29 33 33 35 36 39 40 41 42

Continuum theory of memory effect in crack patterns of drying pastes

NTS FOR THE MODEL
∗ E-mail:
ooshida@damp.tottori-u.ac.jp
The strategy in this paper includes the construction of a set of model equations acceptable as a continuum description of CaCO3 paste. A useful precedent for this model construction can be found in the continuum mechanics of gases and simple liquids [7, 8], in which the Navier-Stokes equation is deduced from several macroscopic requirements, such as homogeneity, isotropy, and the postulation that the deviatoric stress tensor is a linear function of the rate-of-strain tensor (without time lag). Following this precedent, let us list the analogous
arXiv:0804.0966v1 [cond-mat.soft] 7 Apr 2008
As the plastic behavior of soft glassy materials has been attracting increasing interest [1], it was reported by Nakahara and Matsuo [2, 3, 4] that a drying paste exhibits a memory eﬀect. They observed a drying process for a paste containing calcium carbonate (CaCO3 ) and water in a shallow container in order to study the resulting crack pattern. The crack pattern was typically found to be isotropic, but they discovered a way to introduce anisotropy into the paste before the drying process commences: by applying a horizontal oscillation to the container immediately after the paste is poured into it, a memory of the oscillation is imprinted into the paste, which determines how it should break in the future. Through systematic experiments, Nakahara and Matsuo also found that plasticity is essential to the memory eﬀect in CaCO3 pastes. No memory eﬀect is observed if the strength of the applied oscillation is below the threshold value corresponding to the plastic yield stress of the paste. Just above the threshold value, the paste remembers the oscillation that caused the plastic ﬂow, developing cracks perpendicular to the direction of the oscillation. If the oscillation is too strong or the paste contains too much water, waves and global ﬂows are induced, eliminating the memory eﬀect. A diﬀerent kind of paste (mixture of magnesium carbonate hydroxide with water) [5] exhibits not only a memory eﬀect similar to that of CaCO3 that occurs just above the threshold of plastic ﬂows and causes cracks perpendicular to the external oscillation, but also a diﬀerent type of memory eﬀect in its water-rich condition where the cracks are parallel to the direction of the global laminar ﬂow caused by the oscillation. Too strong an oscillation and too much water also destroy the memory in this paste, with the emergence of chaotic, turbulence-like ﬂows [49] characterized by ﬂuid motion in every direction. Here we focus our attention on the former type of memory eﬀect that causes cracks perpendicular to the external forcing, which we refer to as the Type-I Nakahara eﬀect. The latter type, which could be called the Type-II Naka-

The Galactic Center Magnetosphere

a r X i v :a s t r o -p h /0701050v 1 2 J a n 2007The Galactic Center MagnetosphereMark MorrisDepartment of Physics &Astronomy,University of California,Los Angeles,CA 90095-1547,USA E-mail:morris@ Abstract.The magnetic ﬁeld within a few hundred parsecs of the center of the Galaxy is an essential component of any description of that region.The ﬁeld has several pronounced observational manifestations:1)morphological structures such as nonthermal radio ﬁlaments (NTFs)–magnetic ﬂux tubes illuminated by synchrotron emission from relativistic electrons –and a remarkable,large-scale,helically wound structure,2)relatively strong polarization of thermal dust emission from molecular clouds,presumably resulting from magnetic alignment of the rotating dust grains,and 3)synchrotron emission from cosmic rays.Because most of the NTFs are roughly perpendicular to the Galactic plane,the implied large-scale geometry of the magnetic ﬁeld is dipolar.Estimates of the mean ﬁeld strength vary from tens of microgauss to ∼a milligauss.The merits and weaknesses of the various estimations are discussed here.If the ﬁeld strength is comparable to a milligauss,then the magnetic ﬁeld is able to exert a strong inﬂuence on the dynamics of molecular clouds,on the collimation of a Galactic wind,and on the lifetimes and bulk motions of relativistic particles.Related to the question of ﬁeld strength is the question of whether the ﬁeld is pervasive throughout the central zone of the Galaxy,or whether its manifestations are predominantly localized phenomena.Current evidence favors the pervasive model.1.Introduction The magnetic ﬁeld at the center of the Galaxy (hereafter,the ”ﬁeld”)has been studied with a wide variety of techniques for over 20years,and while there is some consensus that thepredominant,global geometry within the central 200-300parsecs is poloidal,the discussion at this workshop has emphasized that there is no universal agreement on the strength of the ﬁeld and on the extent to which the ﬁeld strength varies from one place to another.In this review,I summarize the evidence characterizing the various points of view.Earlier reviews of the Galactic center magnetic ﬁeld have described many of the central points that have been known for some time [1,2,3,4,5,6],but recent observations have added considerably to the information that can be brought to bear on this discussion.The primary probe of the large-scale ﬁeld has been radio observations of polarized,ﬁlamentary structures which,while typically <0.5pc in width,are tens of parsecs in length.The strong radio polarization,and the occasional ﬁlamentary counterpart at X-ray wavelengths [7]indicate that the emission is synchrotron radiation,and the position angle of the polarization,once corrected for Faraday rotation,conﬁrms that the magnetic ﬁeld lies along the ﬁlaments [8,9,10,11].The almost invariant curvature of the ﬁlaments,and their absence of distortion in spite of clear interactions with the highly turbulent interstellar medium,led Yusef-Zadeh &Morris (1987[12],see also [5])to note that the implied rigidity of the ﬁlaments requires a ﬁeld strength on theorder of a milligauss,which is surprisingly large,given the scale of these structures.The orientation of the most prominent NTFs is roughly perpendicular to the Galactic plane, as illustrated in Figure1,a schematic diagram depicting allﬁlaments identiﬁed in theλ20-cm VLA survey by Yusef-Zadeh et al.(2004[13]).Because the individualﬁlaments deﬁne the localﬁeld direction,the ensemble ofﬁlaments has been interpreted in terms of a predominantly dipolarﬁeld,extending at least200pc along the Galactic plane[14].The deviations from perfect verticality of many of theﬁlaments can be ascribed to a global divergence of theﬁeld above and below the Galactic plane.The short,nonconformingﬁlaments are discussed in§2.3(and[15]).Figure1.Schematic map showing the radioﬁlaments catalogued by Yusef-Zadeh et al.(2004, [13])in the course of theirλ20-cm survey of the Galactic center.Quite a diﬀerent probe of the magneticﬁeld is provided by mid-and far-IR observations of thermal dust emission from magnetically aligned dust grains.The rotation axes of dust grains align with the magneticﬁeld by dissipative torques[16],leading to a net polarization of the thermal emission such that the E-vector is perpendicular to the magneticﬁeld.This probe, however,is strongly dominated by dense,warm clouds,so it is quite diﬀerent from the NTFs, which sample theﬁeld in the intercloud medium occupying most of the volume of the Galactic center.The magneticﬁeld implied by the polarized dust emission is parallel to the Galactic plane[17,18,19,20,21],and thus perpendicular to the large-scale intercloudﬁeld revealed by the NTFs.The perhaps surprising orthogonality of these two systems can be understood in terms of the tidal shear suﬀered by molecular clouds inhabiting the central molecular zone (CMZ).Any portion of a molecular cloud located a distance R gc pc from the Galactic center, and having a density less than104cm−3[75pc/R gc]1.8is subject to such shear[22,23],so cloud envelopes tend to get stretched into tidal streams that may subtend a large angle at the Galactic center(e.g.,[24]).Any magneticﬁeld within the clouds–presumablyﬂux-frozen to the partially ionized molecular gas–will thus be deformed into an azimuthal conﬁguration,with theﬁeldlines oriented predominantly along the direction of the shear[17].There is little evidence that the cloud and inter-cloud environments are magnetically coupled to each other in any signiﬁcant way,as might have been expected if theﬁeld lines were anchored to the cloud layer,and if the rotation of the cloud layer thus imposes a global twist upon the verticalﬁeld[25,26].The most prominent NTFs show very little deformation where they pass through the Galactic plane and interact with gas in the CMZ(e.g.,[12]).Some case can be made that Faraday rotation measurements are consistent with the geometry of a twisted,large-scale ﬁeld([6],and references therein),but these data remain too sparse to draw anyﬁrm conclusions.If,as the evidence does indicate,the magneticﬁeld is not anchored in the CMZ,then it is either anchored in the essentially non-rotating Galactic halo or beyond,or it arcs back to the Galactic plane at relatively large radii and is anchored there.In either case,theﬁeld lines do not rotate with the CMZ,and the molecular clouds move through theﬁeld with a large relative velocity.This gives rise to an induced v×B electricﬁeld at cloud surfaces(10−4B(mG)V/cm) which can accelerate particles,drive currents and contribute to the cloud heating[27,28].The residence time of clouds in the Galactic center is a few hundred million years as a result of angular momentum loss resulting from both dynamical friction and magnetic drag[29,2,30], so it is not clear how clouds forming at the outside edge of the CMZ[31]will retain any magnetic contact with their surroundings as they migrate inwards through the verticalﬁeld.Any original connection between the cloud and extra-cloudﬁelds could have pinched oﬀduring the inward migration,leaving the clouds magnetically isolated.If typical cloud lifetimes are less than the inspiral times of clouds,presumably because clouds are sheared in the tidalﬁeld,then the situation is more complex,but these comments can still apply to sheared cloud streams and the new clouds that reform as the streams interact with each other.The remainder of this review focuses on several topics of current interest–both observational and theoretical–and culminates in a description of what I think are some of the most important open questions.2.Uniformity of the Galactic Center Field2.1.Pressure Conﬁnement of Magnetic StructuresRegardless of the magneticﬁeld strength,the pressure of the interstellar medium in the CMZ is very large compared to the Galactic disk[32].A hot diﬀuse gas(T∼108K,n∼0.04cm−3) that pervades much of the volume of the Galactic center[33,34,35]has a pressure of6x 10−10dynes cm−2,and is in approximate pressure equilibrium with the warm(∼150K,low-density molecular medium[36,37],if the velocity dispersion of∼20km s−1is used to calculate a turbulent pressure.This pressure is at least two orders of magnitude higher than is characteristic of the Galactic disk.The magneticﬁeld,on the other hand,has a pressure of4x10−8B(mG)2 dynes cm−2.Consequently,if the magneticﬁeld strength in observed magneticﬁeld structures is∼a milligauss,then those structures are not conﬁned,and would expand and disappear on a short time scale.This consideration led to the argument that a milligauss magneticﬁeld must be pervasive throughout the CMZ[38];the strong and extended magneticﬁeld would then provide its own support.In this view,the NTFs are then simply illuminated magneticﬂux tubes into which relativistic electrons have been injected,and along which the electrons are constrained to ﬂow[1].A ring current at the outer edge of the CMZ,or distributed over some range of radii there,is required to generate and conﬁne the overall dipoleﬁeld[5].2.2.Models of Localized Magnetic StructuresThe alternative to a strong,pervasiveﬁeld is that the NTFs represent localized peaks in the magneticﬁeld strength.A force-free magneticﬁeld conﬁguration might be considered as a way of tying a local current to a local enhancement of the magneticﬁeld strength[39,40],but unless the overall conﬁguration is pressure conﬁned,it will be transient and short-lived.A recent suggestion by Boldyrev&Yusef-Zadeh[41]is that the NTF’s are localized structures of milligaussﬁeld strength conﬁned by the eﬀective pressure of large-scale turbulence in the Galactic center.In their model,the turbulent cells expulse theﬁeld,and concentrate it in regions between the cells.However,while theﬁeld will indeed diﬀuse out of a zone of strong turbulence,the turbulence itself is generally accompanied by the generation of newﬁeld at a rate at least as fast as the rate of outward diﬀusion.Consequently,while this mechanism raises the interesting possibility that the geometry of the boundaryﬁeld might be diﬀerent from that within the turbulent zones because of the interactions of theﬁeld emanating from the diﬀerent zones,it is not obvious how this mechanism would lead to a relative enhancement of theﬁeld strength at those boundaries.Furthermore,the turbulence in this model must be organized in such a way that the resulting magneticﬁlaments are predominantly vertical.This places a strong constraint on the overall helicity distribution of plasma motions in the Galactic center. Numerical models that address these concerns are needed to assess this model further.While other models for localized structures have been proposed[42,43,44],they lack the generality needed to account for the population and the orientations of theﬁlaments.2.3.Signiﬁcance of the Short Radio Streaks?One relatively recentﬁnding that has called the notion of a pervasive,uniformﬁeld into question is a population of short radioﬁlaments,or streaks,that occupy much of the same Galactic longitude range as the prominent NTFs[14,45,15].These structures are largely included in ﬁgure1.They diﬀer in three ways from the long-known,prominent NTFs:(i)They are quite short,∼0.1pc.(ii)Their surface brightness is typically about1/4that of the prominent NTFs.(iii)They appear to be more or less randomly oriented,and thus do not conform to the global verticality of the prominent NTFs.This point has been raised as an argument against a globally ordered,dipole magneticﬁeld.Given these pronounced diﬀerences,one could argue that the radio streaks represent a diﬀerent population with a separate origin,such as localized oblique shock structures,or strong local deformations of the large-scaleﬁeld as a result of some local,energetic disturbance.It is premature to conclude that they are inconsistent with a predominantly ordered,large-scale dipoleﬁeld.Further study of these features is warranted to determine whether they diﬀer systematically from the prominent NTFs in other ways as well,such as in terms of spectral index and polarization properties,and whether they are connected to other interstellar structures in the same way that the prominent NTFs are.2.4.Dynamical ConsequencesAs mentioned above,a pervasive,dipoleﬁeld exerts a magnetic drag force on clouds moving through it,enhancing the rate at which they spiral inwards.If suﬃciently strong,theﬁeld can also collimate winds and energetic particles that emanate from the center,creating a chimney eﬀect.This is consistent with observations of extended columnar radio features in nearby,radio-bright galactic nuclei[46,47,48],although the extent to which the energetic winds in such galaxies have been collimated by the magneticﬁeld,as opposed to the back pressure of their stratiﬁed interstellar gas layers,has not been settled.Recent work by Belmont et al.[34]has shown that at least the hydrogen in the hot,diﬀuse gas at the Galactic center is unbound,so a thermal galactic wind is implied.A dipole magnetic ﬁeld can collimate this wind to an extent that depends on theﬁeld strength,so observations of the large-scale morphology of thermal X-ray emission from the hot gas will be a useful probe of both the wind and the magneticﬁeld.Cosmic rays will also be conﬁned by a pervasive,verticalﬁeld.This has two important consequences:ﬁrst,the residence time for cosmic rays in the Galactic center will be relatively short(a few×105yr)compared to that in the Galactic disk(a few×106yr),because the constraint that cosmic rays diﬀuse primarily along theﬁeld lines implies,in the Galactic center, that they diﬀuse directly away from the Galactic plane,whereas in the Galactic disk,they are largely trapped by the azimuthalﬁeld.This relatively short residence time implies a much smaller cosmic ray density than one might infer from the volume rate of supernovae alone.This is consistent with the fact that the high-energyγ-ray emission intensity across the CMZ does not have a peak comparable in its contrast to the peak in the total column density of gas[49,50]. Second,the longitudinal diﬀusion of cosmic rays,especially electrons,would be suppressed by a pervasive verticalﬁeld.Such diﬀusion–for protons–is assumed in a recent model for the extended TeV emission observed by HESS invoking a single source of high-energy cosmic rays [51,52];this model is probably inconsistent with the presence of a strong,pervasive,vertical ﬁeld.ments on Arguments for a Weak Field3.1.The Minimum Energy AssumptionA number of researchers have estimated the strength of the Galactic center magneticﬁeld using the minimum energy assumption,also referred to as”equipartition”,applied to observations of synchrotron emission from relativistic particles(e.g.,[60]).This assumption can be applied to a medium in which energy exchange takes place between particles andﬁelds on time scales much less than the energy loss times of particles or theﬁeld generation time from macroscopic particle dynamics.This can,for example,describe environments characterized by isotropic turbulence and tangledﬁelds,such as the hot spots in the lobes of double radio source galaxies.However,it is quite generally inapplicable to the Galactic center,except perhaps in very local environments in which energetic events have recently occurred.The striking large-scale order of the Galactic center magneticﬁeld implies that its energy content is not responding in any signiﬁcant way to localﬂuid motions or relativistic particle dynamics.The relativistic particles are responding to theﬁeld,but the reverse is not true.The energy content of the Galactic centerﬁeld is far greater than that of the emitting particles,and thus theﬁeld strength can be much larger than the equipartition value.3.2.Zeeman MeasuresThe most compelling measure ofﬁeld strength would be a direct measure via the Zeeman eﬀect. Zeeman measures have indeed been made in Galactic center clouds in lines of both H and OH [53,54,55,56,57],with the result that,where any signiﬁcant Zeeman signal is seen at all,it implies aﬁeld strength on the order of a milligauss or larger.However,there are only a few places where a signiﬁcant Zeeman signal has been detected.(We do not include in these comments the Zeeman measures deduced from1720-MHz OH masers around Sgr A East and the circumnuclear disk,which giveﬁeld strengths of3-5mG[58,59],because such masers presumably arise from locally compressed gas,and may therefore not be representative of the magneticﬁeld on large scales.)One strong selection eﬀect in Zeeman measures is that the extremely broad lines of Galactic center clouds make detection of the Zeeman splitting very diﬃcult unless theﬁeld strength exceeds∼1mG.Two other points must be considered when interpreting Zeeman measurements:ﬁrst,they apply largely to the magneticﬁeld within clouds or at the surfaces of clouds.As the above discussion indicates,the magneticﬁeld geometry in clouds is not necessarily related to the large-scale intercloudﬁeld.Second,the Zeeman eﬀect measures only the mean line-of-sight component of theﬁeld,so if there areﬁeld reversals along the line of sight,or if theﬁeld direction changes across the radiotelescope beam,then there is signiﬁcant averaging and dilution of the Zeemansignal.In any case,even if Zeeman measures were able to provide insight into the strength of the intercloudﬁeld,the line-of-sight restriction makes it diﬃcult to draw conclusions about a largely vertical dipoleﬁeld.Further Zeeman measurements,not only of H and OH with improved sensitivity and spatial resolution,but also of other molecules that probe denser regions,will be very important for achieving a more complete understanding of the Galactic centerﬁeld.3.3.Synchrotron LifetimesOne argument that has been raised against a pervasiveﬁeld of milligauss strength is that the synchrotron lifetime of the electrons responsible for the nonthermal radio emission is relatively short,∼105years for electrons responsible for the330-MHz radio emission arising from the central4◦×2◦diﬀuse nonthermal source[60].So the supernova rate in the CMZ(or in the Galactic and nuclear bulges above it,since not much less than half of the relativistic electrons created in a supernova will diﬀuse along theﬁeld lines and reach the Galactic plane)must be somewhat larger than1per105yrs if supernovae alone are to account for the uniformity of the synchrotron emission.The rate of only Type Ia supernovae in the Galactic bulge has been estimated at30per105yrs,[61],and in the nuclear bulge(deﬁned in[62,63])it is about20 per105yrs,so allowing also for core collapse supernovae,the particle production rate seems abundantly suﬃcient,even if no particles diﬀuse to the Galactic center from the rest of the Galaxy[64],and if there is no particle reacceleration process operating.The synchrotron lifetimes of electrons responsible for the5-GHz radio emission from the NTFs is only∼104years,so if they diﬀuse along theﬁeld lines at the Alfv´e n speed,2200km s−1 B(mG)/n(cm−3)1/2,then the net distance they can travel before losing an appreciable amount of energy is∼20pc×B(mG)/n(cm−3)1/2,somewhat shorter than the length of the longest ﬁlaments(60pc).(The Alfv´e n speed is assumed because the diﬀusion is usually limited by scattering of the streaming particles oﬀof Alfv´e n waves propagating along theﬁeld lines).So far,observations indicate that the radio spectral index has no noticeable variation along the length of theﬁlaments(e.g.,[10]).Consequently,if the relativistic electrons are produced at a speciﬁc location along them,then the synchrotron lifetime may present a problem unless theﬁeld strength is substantially less than a milligauss.Two possible alternatives warrant consideration:ﬁrst that the diﬀusion along theﬁeld lines is much faster than the relatively slow rate assumed here because the magneticﬁeld is much more rigid and smooth than in most situations where the Alfv´e n speed is invoked.Second,a reacceleration process may take place along theﬁlaments via shocks,wave dissipation,or reconnection,in analogy with the reacceleration processes needed to account for the persistence of highly relativistic particles in extragalactic jet sources,in spite of their synchrotron and Compton losses.4.The Double Helix NebulaA potential new probe of the Galactic center magneticﬁeld was recently revealed at24µm with the Spitzer Space Telescope[65].At a distance of∼100pc toward positive Galactic latitude from the Galactic center,a nebula having the form of an intertwined double helix extends over at least50pc,with its long axis oriented approximately perpendicular to the Galactic plane (Figure2).This feature was interpreted as a torsional Alfv´e n wave propagating away from the Galactic center along the magneticﬁeld,and driven by the rotation of the circumnuclear gas disk (CND).The few-parsec scale of the CND matches the width of the nebula,and the wavelength of the torsional wave,19pc,corresponds to the∼104-year rotation period of the CND if the Alfv´e n speed is103km s−1.This speed,in turn,constrains the magneticﬁeld to have a strength of0.5n1/2mG in the context of this hypothesis,where n is the hydrogen density in the medium through which the wave propagates.The density is not known,but for values of the magnetic ﬁeld ranging from0.1to1mG,a plausible density is found:n=0.04-4cm−3.The presence of two strands has been attributed to an apparent”dumbbell”asymmetry of the driving disk(see[65]);the magneticﬁeld threading the disk is concentrated into two diametrically opposed density maxima.A potential weakness of the torsional wave hypothesis is that the wave cannot yet be followed all the way down to its hypothetical source,the CND.However,this also raises the question of why the double helix is visible in theﬁrst place;its mid-infrared emission is most likely thermal emission from dust,so the visibility of the nebula at its present location presumably requires that the wave has levitated charged dust grains.Because of variable conditions at the base of the wave over the past105years(indeed,the CND is a rather disturbed,non-equilibrated disk [5]),such dust may not have been continuously available to highlight the wave.This may also help explain why a similar nebula is not present on the opposite side of the CND.An alternative scenario for understanding the Double Helix feature is that it be connected in some way with the linear radioﬁlaments of the Galactic Center Radio Arc.If the Northern extension of the Arc[66]is followed and extrapolated to Galactic latitudes beyond0.5◦(seeﬁg 20b of[13]),then it coincides approximately with the long axis of the Double Helix.However, there is no continuous connection in the radio maps between the linearﬁlaments and the Double Helix,and the only radio emission associated with the Double Helix lies outside the mid-IR strands(w,personal communication).There is so far no explanation for how a long bundle of linear,nonthermalﬁlaments could culminate in helically wound,thermal structures. Whether or not the CND hypothesis for the Double Helix is valid,further study of this feature should provide valuable insight into the Galactic center magneticﬁeld.5.Open QuestionsThe questions that seem now to be the most compelling for guiding near-future research on the Galactic center magneticﬁeld,besides those already mentioned above,are the following:•Whether or not the central verticalﬁeld is more or less uniform,how and where does it merge with the azimuthalﬁeld of the Galactic disk?•If the Galactic center magnetosphere is deﬁned as the region in which nonthermal radio ﬁlaments are observed,then its outer edge roughly coincides with the edge of the CMZ, with the Galaxy’s inner inner Lindblad resonance,and with the transition from X1to X2 gas orbits in the bar.What is the interplay between these phenomena,at this critical juncture in the Galaxy?•Can high-resolution observations be used to obtain more detail on the points of interaction between cloud and intercloudﬁelds?This may best be done with a combination of radio and far-infrared polarization measurements.•What process produces the relativistic particles that illuminate the NTFs via their synchrotron emission?•At the moment,we lack consensus on the power source for the108K gas occupying much of the volume of the nuclear bulge.Can we appeal to the stirring that takes place as clouds move through theﬁeld,leaving magnetosonic and Alfv´e n waves in their wake?Or can the energy be supplied by magneticﬁeld line annihilation of new verticalﬁeld constantly migrating inwards from the rest of the Galaxy?•What is the origin of the poloidalﬁeld?Dynamo models have been hard-pressed to produce a dipoleﬁeld like that observed,and a promising possibility is that the centralﬁeld represents protogalacticﬁeld that has been concentrated over the history of the Galaxy by mass inﬂow[67].Now is a propitious time to take these models to the next stage of sophistication.AcknowledgmentsI gratefully acknowledge stimulating discussions with Steve Cowley.Figure2.The Double Helix Nebula,observed with the MIPS instrument on the Spitzer Space Telescope[65].This structure was also seen at shorter wavelengths,3.6-8µm,with the IRAC camera on Spitzer.References[1]Morris,M.1990,in IAU Symp.No.140:Galactic and Intergalactic Magnetic Fields,eds:R.Beck,P.Kronberg,and R.Wielebinski,Dordrecht:Kluwer,p.361[2]Morris,M.1994,in The Nuclei of Normal Galaxies:Lessons from the Galactic Center,NATO ASI Series C:Vol.445,eds:R.Genzel and A.I.Harris,Dordrecht:Kluwer,p.185[3]Morris,M.1996,in IAU Symp.No.169:Unsolved Problems in the Milky Way,eds:L.Blitz&PJ Teuben,Dordrecht:Kluwer,p.247[4]Morris,M.1998,in IAU Symp.No.184:The Central Regions of the Galaxy and Galaxies,ed:Y.Sofue,Dordrecht:Kluwer,p.331[5]Morris,M.&Serabyn,E.1996,Ann.Rev.Astron.Ap.34,645[6]Novak,G.2005,in Magnetic Fields in the Universe:From Laboratory and Stars to Primordial Structures,eds:E.M.de Gouveia Dal Pino,G.Lugones,&zarian,New York:AIP,p329[7]Yusef-Zadeh,F.,Wardle,M.,Muno,M.,Law,C.&Pound,M.2005,Adv.Spa.Res.35,1074[8]Tsuboi,M.,Inoue,M.,Handa,T.,Tabara,H.,Kato,T.,Sofue,Y.&Kaifu,N.1986,AJ92,818[9]Yusef-Zadeh,F.,Wardle,M.&Parastaran,P.1997,ApJL475,L119[10]Lang,C.C,Morris,M.&Echevarria,L.1999,ApJ526,727.[11]Lang,C.C.2007,in this volume.[12]Yusef-Zadeh,F.&Morris,M.1987,AJ94,1178[13]Yusef-Zadeh,F.,Hewitt,J.W.&Cotton,W.2004,ApJS155,421[14]Nord,M.E.,Lazio,T.,Kassim,N.,Hyman,S.,LaRosa,T.,Brogan,C.&Duric,N.2004,ApJ128,1646[15]LaRosa,T.N.2007,in this volume.[16]Hildebrand,R.H.1988,QJRAS29,327[17]Werner,M.W.,Davidson,J.A.,Morris,M.,Novak,G.,Platt,S.R.&Hildebrand,R.H.1988,ApJ333,729[18]Morris,M.,Davidson,J.A.,Werner,M.,Dotson,J.,Figer,D.F.,Hildebrand,R.Novak,G.&Platt,S.1992,ApJL399,63[19]Hildebrand,R.H.,Davidson,J.A.,Dotson,J.,Figer,D.,Novak,G.,Platt,S.&Tao,L.1993,ApJ417,565[20]Novak,G.,Chuss,D.T.,Renbarger,T.,Griﬃn,G.S.,Newcomb,M.G.,Peterson,J.B.,Loewenstein,R.F.,Pernic,D.&Dotson,J.L.2003,ApJL583,L83[21]Chuss,D.T.,Davidson,J.A.,Dotson,J.L.,Dowell,C.D.,Hildebrand,R.H.,Novak,G.&Vaillancourt,J.E.2003,ApJ599,1116[22]G¨u sten,R.1989,in The Center of the Galaxy,IAU Symp.No.136,ed:M.Morris,Dordrecht:Kluwer,p89[23]Stark,A.A.,Martin,C.L.,Walsh,W.M.,Xiao,K.&Lane,A.P.2004,ApJL614,L41.[24]Tsuboi,M.,Handa,T.&Ukita,N.1999,ApJS,120,1[25]Uchida,Y.,Sofue,Y.&Shibata,K.1985,Nature317699[26]Shibata,K.&Uchida,Y.1987,PASJ39,559[27]Benford,G.1988,ApJ333,735[28]Morris,M.&Yusef-Zadeh,F.1989,ApJ343,703[29]Stark,A.A.,Bally,J.,Gerhard,O.E.&Binney,J.1991,MNRAS248,14P[30]Belmont,R.&Tagger,M.2006,A&A452,15[31]Binney,J.,Gerhard,O.E.,Stark,A.A.,Bally,J.&Uchida,K.I.1991,MNRAS252,210[32]Spergel,D.N.&Blitz,L.1992,Nature357,665[33]Muno,M.,Baganoﬀ,F.K.,Bautz,M.W.,Feigelson,E.D.,Garmire,G.P.,Morris,M.R.,Park,S.,Ricker,G.R.&Townsley,L.K.2004,ApJ613,326[34]Belmont,R.,Tagger,M.,Muno,M.,Morris,M.&Cowley,S.2005,ApJL631,L53[35]Belmont,R.&Tagger,M.2007,this volume[36]Dahmen,G.,H¨u ttemeister,S.,Wilson,T.L.&Mauersberger,R.1998,A&A331,959[37]Rodriguez-Fernandez,N.J.,Martin-Pintado,J.,Fuente,A.&Wilson,T.L.2004,A&A427,217[38]Morris,M.&Yusef-Zadeh,F.1989,ApJ343,703[39]Yusef-Zadeh,F.,Morris,M.&Chance,D.1984,Nature310557[40]Yusef-Zadeh,F.&Morris,M.1987,ApJ322721[41]Boldyrev,S.&Yusef-Zadeh,F.2006,ApJL637,L101[42]Bicknell,G.V.&Li,J.2001,ApJ548,L72[43]Shore,S.N.&LaRosa,T.N.1999,ApJ593,587[44]Dahlburg,R.B.,Einaudi,G.,LaRosa,T.N.&Shore,S.N.2002,ApJ568,220[45]LaRosa,T.N.Nord,M.E.,Lazio,T.J.W.&Kassim,N.E.2004,ApJ607,302[46]Hummel,E.,van Gorkom,J.H.&Kotanyi,C.G.1983,ApJL267,L5[47]Duric,N.&Seaquist,E.R.1988,ApJ326,574[48]Keel,W.C.&Wehrle,A.E.1993,AJ106,236[49]Hunter,S.D.et al.1997,ApJ481,205[50]Strong,A.W.,Moskalenko,I.V.&Reimer,O.2004,ApJ613,962[51]Aharonian,F.et al.2006,Nature439,695[52]Hinton,J.A.2007,this volume[53]Schwarz,U.J.&Lasenby,J.1990,in Galactic&Intergalactic Magnetic Fields,eds:R.Beck,P.P.Kronberg,R.Wielebinski,Dordrecht:Kluwer,p383[54]Killeen,N.E.B.,Lo,K.Y.&Crutcher,R.M.1992,ApJ385,585[55]Plante,R.L.,Lo,K.Y.&Crutcher,R.M.1995,ApJL445,L113[56]Uchida,K.I.&G¨u sten,R.1995,A&A298473[57]Marshall,A.N.,Lasenby,A.N.&Yusef-Zadeh,F.1995,MNRAS274,519[58]Yusef-Zadeh,F.,Roberts,D.A.,Goss,W.M.,Frail,D.A.&Green,A.J.1999,ApJ512,230[59]Sjouwerman,L.2007,this volume[60]LaRosa,T.N.,Brogan,C.L.,Shore,S.N.,Lazio,T.J.,Kassim,N.E.&Nord,M.E.2005,ApJL626,L23[61]Schanne,S.,Cass´e,M.,Sizun,P.,Cordier,B.&Paul,J.2006,astro-ph/0609566[62]Serabyn,E.&Morris,M.1996,Nature382,602[63]Launhardt,R.,Zylka,R.&Mezger,P.G.2002,A&A384,112[64]Cowin,B.&Morris,M.2007,in preparation[65]Morris,M.,Uchida,K.I.&Do,Tuan2006,Nature440,308[66]Yusef-Zadeh,F.&Morris,M.1988,ApJ329,729[67]Chandran,B.D.G.,Cowley,S.C.&Morris,M.2000,ApJ528,723。

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

a r X i v :g r -q c /0207070v 2 16 D e c 2002Does a relativistic metric generalization of Newtonian gravity exist in 2+1dimensions?J.L.Alonso,J.L.Cort´e s,and liena ∗Departamento de F´ısica Te´o rica,Universidad de Zaragoza,C.Pedro Cerbuna 12,E-50009Zaragoza (Spain)(Dated:October 23,2002)It is shown that,contrary to previous claims,a scalar tensor theory of Brans-Dicke type provides a relativistic generalization of Newtonian gravity in 2+1dimensions.The theory is metric and test particles follow the space-time geodesics.The static isotropic solution is studied in vacuum and in regions ﬁlled with an incompressible perfect ﬂuid.It is shown that the solutions can be consistently matched at the interface matter-vacuum,and that the Newtonian behavior is recovered in the weak ﬁeld regime.I.INTRODUCTIONNewtonian gravity,a theory of gravitational phenom-ena which is invariant under Galilean transformations and,therefore,valid only in the low energy (weak ﬁelds and slow motions)regime,must be generalized to a Rel-ativistic Theory of Gravitation (RTG).Einstein General Relativity (EGR)is a good candidate for RTG in 3+1di-mensions,but other possibilities,as Brans-Dicke theory (BDT)[1],have been proposed.Indeed,it is believed by many that a quantum theory of gravitation,which seems unavoidable if we want to deal with gravitational phenomena at the Planck scale,must contain something more than ERG [2].In 2+1dimensions,EGR is not a RTG.The Riemann-Christoﬀel tensor is uniquely determined by the Ricci tensor,which vanishes outside the sources.Hence,space-time is ﬂat in regions devoid of matter,the geodesics are straight lines and test particles do not feel any gravita-tional ﬁeld [3].A proper RTG in 2+1dimensions needs some additional ingredient besides the metric tensor of EGR [4].A minimal candidate for RTG is a scalar ten-sor theory of Brans-Dicke type.It has been claimed that even theories of this sort,which are much richer than EGR,do not describe Newtonian gravity in the low en-ergy limit [5].We will show in this paper that BDT in 2+1dimensions reproduces Newtonian gravity when the low energy regimen is consistently analysed.The additional ingredient that seems necessary to construct a quantum theory of gravitation in 3+1dimensions ap-pears at an earlier stage in lower dimensions.The construction of a RTG in dimensions lower than 3+1is interesting because it may allow to study phenom-ena characteristic of gravity,which have 3+1dimensional analogues,such as gravitational instabilities and black holes,in a simpliﬁed context [6,7].The paper is organized as follow.In section II the equa-tions of scalar-tensor theories in D -dimensional space-time are derived,the particular case of BDT is identi-g [φR +W (φ)g µνφ;µφ;ν],(1)where g µνis the metric tensor,whose signature is taken (−,+,...,+),g =−det g µν,R is the scalar of curvature,and φis the scalar ﬁeld that acts as the inverse gravita-tional coupling at each space-time point.The action con-tains an unspeciﬁed function of φ,W .Diﬀerent choices of such function give diﬀerent scalar tensor theories.The coupling of the matter to gravity depends only on the metric tensor in a covariant way,and does not depend of the scalar ﬁeld.Hence,the variation of the matter action,S m ,under variations of the metric,δg µν,can be written as δS m =1/2 d D x√2eral covariance implies a continuity equation for Tµν:Tµν;ν=0,(2) that describes the exchange of energy between matter and gravity.Furthermore,this equation ensures that test particles move along the geodesics of space-time associ-ated to the metric gµνand thus the equivalence principle is preserved.In this sense,scalar-tensor theories are met-ric theories of gravitation.The equations for the metric tensor and the scalarﬁeld that follow from the action S=S g+S m areφ Rµν−12gµνgρσ φ;ρφ;σ=−1D−2gµνgρσ+gρµgσν φ;ρ;σ−W(φ)φ;µφ;ν−1D−2gµνT ,(4a)2 D−1φ+W′(φ) gρσφ;ρφ;σ−12(1+ω)T,(5a)φRµν=− φ;µ;ν+ω2[Tµν−λgµνT].(5b)whereλis a function ofωand D:λ=1+ωG eξ(thenφ;µ=φξ;µandφ;µ;ν=φ(ξ;µ;ν+ξ;µξ;ν)),whereξis dimension-less and the constant G appears for dimensional reasons, the equations take the formgρσ(ξ;ρ;σ+ξ;ρξ;σ)=G1+ωT,(7a) Rµν=−ξ;µ;ν−(1+ω)ξ;µξ;ν−G2gµνR,andthe equations readgρσ(ξ;ρ;σ+ξ;ρξ;σ)=G3of linear equationsforξ(n ),g (n )µν,and T (n )µν.To ﬁrst order in G ,Eqs.(7)yieldηρσ∂2ξ(1)2(1+ω)T (0),(10a)R (1)µν=−∂2ξ(1)2T (0)µν−ληµνT(0).(10b)For a static ﬁeld produced by non-relativistic matter (T (0)µν=0for (µ,ν)=(0,0)and T (0)00=ρ,where ρisthe density of matter),R (1)00=14ρ.(12)The Newtonian potential is identiﬁed from the geodesicequation as V N =−G2S D −2(1−λ),(14)and for D =2G N =G ω2S D −212(ω+2)C 2ξ2ωC 2ξ−2C ξg and inte-grating in drdϕ,and assuming that a static and isotropic source is conﬁned within a ﬁnite spatial region,we getC ξ=−exp(α0−β0−ξ0)GgT .In the weak ﬁeld regime G M is small.Eqs.(9a)-(9b)imply that ξ0,α0,and β0are proportional to G .Then to ﬁrst order in G we havee 2β≈1+2Gr 0.(20)A comparison with the Newtonian solution,V N =G N M ln(r/r 0),gives the Newton constant G N =G /[4π(2+ω)],the same obtained from ﬁeld equations in presence of matter in the weak ﬁeld regime.Eqs.(7)do not admit solutions with a static point mass as source [5].Indeed,they do not admit static and isotropic solutions with concentrated sources (energy-momentum tensor that contains Dirac deltas).The rea-son is that it is not possible to ﬁnd solutions α,β,and ξof the sourceless equations with singularities cancelling the source singularities in the ﬁeld equations.We are thus lead to consider extended sources.This is interesting be-cause doubts have been cast on the existence of static and isotropic solutions of Eqs.(7)even with extended sources [5].To investigate this in depth,let us consider as sourcean incompressibleﬂuid of densityρconﬁned on a disk of radius r M.The corresponding energy-momentum tensor isTµν=p gµν+(ρ+p)UµUν,(21) where p is the pressure and Uµthe four velocity,which veriﬁes gµνUµUν=−1.For a staticﬂuid we haveT tt=e2βρ,T rr=e2αp,Tϕϕ=r2p,(22)and the remaining components vanish.Covariant conser-vation of Tµνimplies the equation of hydrostatic equilib-rium[9]:β′=−p′ˆrξ′+ξ′(β′+ξ′−α′)=−κ(1−2¯p)e2α−ξ,(24a)β′′+1ˆr α′+ξ′′+β′2+(1+ω)ξ′2−α′(β′+ξ′)=−κ[1+ω−ω¯p]e2α−ξ,(24c)1ˆr+ξ′ =ωξ′2−2As a check,it can be seen that Eq.(19)is veriﬁed.The solution up to orderκ4is displayed in Table I.The ﬁeldsβandαare positive and monotonically increasing order by order,and the preasure is positive and monoton-ically decreasing order by order,vanishing at the border of the mass distribution(ˆr=1).This is in agreement with intuition,since a positive preasure is required to bal-ance the gravitational attraction.Gravitational collapse is impossible since theﬂuid is incompressible.Up to orderκ4,the limitω→∞gives(recall that κ∼1/ω)ξ=β=¯p=Cξ=0andα=14¯κ2ˆr4+18¯κ4ˆr8,(29)where¯κ=G r2Mρ/2.On the other hand,we can take the limitω→∞of Eqs.(24).This forcesξ′=0and then the solution with the boundary conditions previously de-scribed isξ=β=¯p=0andα=−1ξ1−1ξ2164(9+6ω)ˆr4128 (15+12ω)ˆr2+(1+5ω)ˆr4−1ξ412048(31+71ω+60ω2)ˆr4+124576(1710+4457ω+3508ω2+840ω3)ˆr8β11β2164(5+4ω)ˆr4128 (12ω2+27ω+15)ˆr2+(4ω2+7ω+7)ˆr4+1β412048(121+189ω+128ω2+48ω3)ˆr4+124576(802+2291ω+1932ω2+480ω3)ˆr8α11α2132(8ω2+19ω+8)ˆr464(12ω+15)ˆr2+5384(64ω3+238ω2+251ω+73)ˆr696(15ω2+38ω+26)ˆr2+1256(35ω2+84ω+39)ˆr6+14−1¯p21¯p31348ω+5256(53+74ω+24ω2)ˆr2−1384(11+26ω+12ω2)ˆr624576(8593+17719ω+12860ω2+3360ω3)−1 4096(159+218ω+208ω2+96ω3)ˆr4−1 24576(503+1707ω+1692ω2+480ω3)ˆr8C(1)ξ−1C(2)ξ−1C(3)ξ−1C(4)ξ−1。