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

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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。

Understanding Quantum Theory in Terms of Geometry

4 A scalar–tensor model of the idea 4.1 Making the conformal factor dynamical . . . . . . . . . . . . . 4.2 Making the quantum potential dynamical . . . . . . . . . . . . 5 Some results of the idea 5.1 Conformally ﬂat solution . . . . . . . . . . . . . . . . . . . . . 5.2 Conformally quantic solution . . . . . . . . . . . . . . . . . . .
2
5.3 Conformally highly quantic solution . . . . . . . . . . . . . . . 5.4 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Initial singularity . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Production of inhomogenity . . . . . . . . . . . . . . . . . . . 5.7 Non–locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Generalized equivalence principle 7 Formulation of the idea in terms of Weyl Geometry 7.1 Weyl–Dirac theory . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Weyl–invariant quantum gravity . . . . . . . . . . . . . . . . . 7.3 Application to cosmology . . . . . . . . . . . . . . . . . . . . . 8 Extension of the results 8.1 Many–particle systems . . . . . . . . . . . . . . . . . . . . . . 8.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusion

Effective action for scalar fields in two-dimensional gravity

a rXiv:g r-qc/1133v27May22Eﬀective action for scalar ﬁelds in two-dimensional gravity M.O.Katanaev ∗Steklov Mathematical Institute,Gubkin St.8,Moscow 117966,Russia 1October 2001Abstract We consider a general two-dimensional gravity model minimally or non-minimally coupled to a scalar ﬁeld.The canonical form of the model is eluci-dated,and a general solution of the equations of motion in the massless case is reviewed.In the presence of a scalar ﬁeld all geometric ﬁelds (zweibein and Lorentz connection)are excluded from the model by solving exactly their Hamiltonian equations of motion.In this way the eﬀective equations of mo-tion and the corresponding eﬀective action for a scalar ﬁeld are obtained.It is written in a Minkowskian space-time and does not include any geometric vari-ables.The eﬀective action arises as a boundary term and is nontrivial both for open and closed universes.The reason is that unphysical degrees of free-dom cannot be compactly supported because they must satisfy the constraint equation.As an example we consider spherically reduced gravity minimallycoupled to a massless scalar ﬁeld.The eﬀective action is used to reproduce the Fisher and Roberts solutions.1IntroductionIn recent years great attention has been paid to two-dimensional gravity models mainly for two reasons:a close relation to string theory and good laboratories to get deeper insights in classical and quantum properties of gravity models.The ﬁrst and the simplest two-dimensional constant curvature gravity was proposed in [1].It attracted much interest after the papers [2,3].Constant curvature surfaces are described by the integrable Liouville equation,and the main concern was given to physical interpretation of the solutions and inclusion of matter ﬁelds.In the present paper we consider a wide class of two-dimensional gravity models with torsion and equivalent generalized dilaton models.A two-dimensional gravity model with torsion was proposed in [4,5,6]to provide dynamics for the metric ona string world sheet already at the classical level.Two-dimensional dilaton gravity [7,8]was proposed as an eﬀective model coming from the string theory.Both models turned out to be integrable.A general solution to the equations of motion of two-dimensional gravity with torsion wasﬁrst given in the conformal gauge[9,10,11]. All solutions are divided into two classes:(i)constant curvature and zero torsion and(ii)nonconstant curvature and nonzero torsion.In this way constant curvature gravity models are included in two-dimensional gravity with torsion with a well deﬁned purely geometric action and a natural way of introduction of matterﬁelds. The equations of motions of two-dimensional gravity with torsion were integrated in the light-cone gauge[12]and without any gaugeﬁxing[13,14,15].At that time the equivalence of two-dimensional gravity with torsion and dilaton gravity was unknown,and solution of the equations of motion for a generalized dilaton gravity was independently obtained in the light-cone gauge[16].Later it was realized that in fact the two-dimensional gravity models with torsion and dilaton gravity models are equivalent[17,18,19].From a geometric point of view the dilatonﬁeld is the momentum canonically conjugate to the space component of the Lorentz connection.The equivalence of these models is nontrivial because they contain a diﬀerent set ofﬁelds.Throughout this paper we consider a general two-dimensional gravity model quadratic in torsion.The integrability of the model is connected to the existence of a Killing vectorﬁeld in the absence of matter.It is worth noting that without matterﬁelds the two-dimensional gravity models do not describe any propagating degree of freedom.This suggests a generalization of the models by addition of matter ﬁelds to attack the problems arising in black hole formation and quantum gravity. Unfortunately,addition of matter destroys integrability.Some exact solutions of two-dimensional gravity with torsion coupled to scalarﬁelds were found[20,21,22]. We mention also nontrivial solutions for dilaton gravity coupled to scalarﬁelds found in[16,23,24].In Section7.2we consider a Lagrangian with arbitrary dependence on scalar curvature and torsion.There we clarify the statement that this general model yields integrable equations of motion made in the literature[14,25,26].Let us note that the two-dimensional gravity model nonminimally coupled to a scalarﬁeld is important for general relativists working in four dimensions.It is well known that spherically reduced general relativity minimally coupled to a scalarﬁeld is equivalent to the dilaton gravity model nonminimally coupled to a scalarﬁeld in two-dimensional space-time[27].This model may be the simplest one to describe the dynamics of a black hole formation by spherical scalar waves.Recently it attracted much interest due to the discovery of critical phenomena in the black hole formation [28](for review see[29])found in numerical simulations.A satisfactory analytical calculation of the critical exponent and scaling period of discrete self-similarity has been missing up to now,and hence our understanding of critical phenomena should be considered incomplete.In the present paper we consider the two-dimensional gravity model with torsion or the equivalent dilaton gravity coupled to a scalarﬁeld.We admit nonminimal coupling to be suﬃciently general to include spherically reduced gravity.After de-scribing the Lagrangian we elucidate the Hamiltonian formulation of the model in full detail.The canonical form is essential for our approach and I am not aware how to write down the eﬀective action in the Lagrangian formulation.Afterwards a gen-2eral solution of the equations of motion in the matterless case and the equivalence with the dilaton model are brieﬂy reviewed.In the presence of a scalarﬁeld the equations of motion are not integrable.Nevertheless they may be partly integrated. We solve the geometric part of the equations of motion with respect to the zweibein and Lorentz connection assuming the scalarﬁeld to be arbitrary.Then the solution is substituted into the equations of motion for a scalarﬁeld.In this way we obtain the eﬀective equations of motion only for a scalarﬁeld and its conjugate momentum. Next we show that the eﬀective action yielding these equations arises as a boundary term.It appears because unphysical variables cannot be compactly supported func-tions as the consequence of constraint equations.The eﬀective equations of motion are written in a Minkowskian space-time and provide a general solution to the whole problem because the metric can be easily reconstructed for a given solution to the eﬀective equations of motion.The eﬀective action for spherically reduced gravity in a special case wasﬁrst derived in[30].Here we generalize this action by considering the more general two-dimensional gravity part and arbitrary coupling to scalars. For spherically reduced gravity we give a more general eﬀective action depending on one arbitrary function on time to beﬁxed by boundary conditions on the metric.The reduction of the whole system of the equations of motion to the equations for a scalarﬁeld and its conjugate momentum does not provide a general solution of the model because the eﬀective equations are integro-diﬀerential and complicated. As an example I considered spherically reduced gravity.The eﬀective action has clear limiting cases and for a small scalarﬁeld reduces to an ordinary action for spherical waves on a Minkowskian or Schwarzschild background.In the static case the equations may be integrated in elementary functions and yield the same solution as found by Fisher[31].As an example we consider also the Roberts solution[32] which provides a self-similar solution to the eﬀective equations of motion.2The actionLet us consider a Lorentzian surface with coordinates xα={τ,σ},α=0,1.We assume that it is equipped with a Riemann–Cartan geometry deﬁned by a zweibein eαa(x)and a Lorentz connectionωα(x).A general type action for a scalarﬁeld coupled to two-dimensional gravity has the formS= dx2(L G+L X),(1) where the Lagrangian for a scalarﬁeld X(x)of mass m=const is1L X=−geometric part of the Lagrangian is given by a two-dimensional gravity with torsion written in theﬁrst order formL G=−12p a p a U+V ,(3)where the densities of two-dimensional scalar curvature and the pseudotrace of tor-sion are given byR=eR=2ˆεαβ∂αωβ,(4)T∗a=eT∗a=2ˆεαβ(∂αeβa−ωαεa b eβb).(5)Hereˆεαβ=eεαβ=eεab eαa eβb is the totally antisymmetric tensor density,ˆε01=−ˆε10=−1,πand p a are considered as independent variables,and U=U(π) and V=V(π)are arbitrary functions ofπ.In the second order form,when it exists,this Lagrangian is quadratic in torsion and arbitrarily depends on the scalar curvature.A hat over a symbol means that it is a tensor density of weight−1.For the nonminimally coupled scalarﬁeld we haveρ=ρ(π).The case U=0and V=0describes surfaces of zero torsion and curvature and is not interesting from the geometric point of view.Therefore we assume that at least one of the functions diﬀers from zero.The Lagrangian(3)is quite general. If U=0then one has the gravity model with zero torsion,and the Lagrangian in the second order form is an arbitrary function of the scalar curvature deﬁned by V. For U=1and V=π2+const one immediately recovers two-dimensional gravity with torsion quadratic in curvature and torsion[4,5,6].It is essentially a unique purely geometric invariant model yielding second order equations of motion for the zweibein and Lorentz connection.In this caseπand p a are proportional to the scalar curvature and the pseudotrace of torsion provided equations of motion are fulﬁlled.For arbitrary functions U(π)and V(π)when matterﬁelds do not interact with the Lorentz connectionωαthe latter can be excluded from the model by the use of its algebraic equations of motion(Section6).This leads to a generalized two-dimensional dilaton gravity,the functionπ(x)being the dilatonﬁeld.Then the matter Lagrangian(2)describes scalars minimally,ρ=const,or nonminimally,ρ=ρ(π),coupled to dilaton gravity.Among these models of particular interest is the spherically reduced gravity(see Section11)for whichU=1κ,ρ=π3Equations of motionEquations of motion following from the action(1)can be written in the form1δπ:−12p a p a U′+V′ −1e δS2T∗a−p a U=0,(8)ˆεβαδSδeβa:∇αp a+εαa 1eδS2gαβ(∂X2−m2X2)(12)is the energy-momentum tensor of the scalarﬁeld.Primes on functions U,V,andρalways mean derivatives with respect to the argumentπ.Transformation of Greek indices into Latin ones and vice versa is everywhere performed using the zweibein ﬁeld and its inverse,and ∇denotes the covariant derivative with Christoﬀel’s sym-bols.The action(1)is invariant under local Lorentz rotations and general coordinate transformations which produce linear relations between the equations.Local Lorentz rotations by the angleω(x)δeαa=−eαbεb aω,δωα=∂αω,δp a=εa b p bω,δπ=0,(13)δX=0lead to the following dependence between equations of motion∇αδSδeαa eαbεb a−δStogether with the identity(14)produce two linear relations between equations of motione αa ∇βδS δe βa T αβa +1δωβεαβR −δSδp a ∇αp a −δS∂(∂0ω0)=0,(17)π1=∂(L G +L X )∂(∂0e 0a )=0,(19)p 1a =∂(L G +L X )∂(∂0X )=ρg 11e ∂1X.(21)The last momentum has dimension [P ]=1.(For deﬁnition of dimensions of the ﬁelds see Appendix.)We see that the gravity Lagrangian (3)is already written in the canonical form.The Hamiltonian for the whole system is given byH = dσ(ω0G +e 0a G a ),(22)whereG =−∂1π+p a εa b e 1b ,(23)G a =−∂1p a −ω1p b εb a +e 1b εab12m 2X 2(24)+e 1b εab2ρP 2+ρg 11P ∂1X.The functions G and G a are a Lorentz scalar and vector,respectively.Note that Hamiltonian (22)for polynomial U (π)and V (π)is polynomial in the ﬁelds in the absence of scalars.Addition of a scalar ﬁeld makes these functions nonpolynomial because of the denominator g 11and possible nonminimal coupling ρ(π).6The equal time Poisson brackets are deﬁned as usual{e1a,p′b}=δa bδ(σ−σ′),{ω1,π′}=δ(σ−σ′),{X,P′}=δ(σ−σ′),(25)where a prime over a function means that it is taken at a pointσ′.Computing the evolution of primary constraints(17)and(19)one gets the secondary constraints∂0π0={p0,H}=−G=0,(26)∂0p0a={p0a,H}=−G a=0.(27) Thus the Hamiltonian(22)is given by a linear combination of secondary constraints. The secondary constraints form a closed algebra{G a,G′b}=εab Up c G c+ 1eρL X G δ,(28){G a,G′}=εa b G bδ,(29){G,G′}=0,(30) where U′,V′,andρ′denote derivatives with respect to the argumentπ,δ=δ(σ−σ′), and L X is the Lagrangian for scalars(2)expressed through canonical variablesL X=−1ρg11P2+ρHere for brevity we introduced a light-like vectork a=e1a+e1bεbag11.Straightforward calculations yield{ G a, G′b}=εab(Up c+ω1k c) G cδ,(33){ G a,G′}=εa b G bδ−k a Gδ′.(34) The algebra of G a is related to the conformal algebra generated by two scalar(with respect to Lorentz rotations)constraintsH0=−e1aεa b G b=−e1aεa b G b+ω1G,(35)H1=e1a G a=e1a G a+ω1G.(36) The constraints H1and H0are Lorentz invariant projections of the vector con-straint G a on the directions parallel and perpendicular to the vector e1a,respec-tively.Straightforward calculations show that the new set of constraints satisfy the following algebra[35]{H0,H′0}=−(H1+H′1)δ′,(37){H0,H′1}=−(H0+H′0)δ′,(38){H1,H′1}=−(H1+H′1)δ′,(39){H0,G′}={H1,G′}=−Gδ′,(40){G,G′}=0,(41) whereδ′=∂g11 −e11(H0−ω1G)+e10(H1−ω1G) ,(42) G1=12p a p a U+V−ρ2ρP2−ρ5The canonical transformationAt the classical level two models related by a canonical transformation are equiva-lent.At the quantum level this property is not valid in general:There are canonical transformations resulting in diﬀerent quantum models.Special care must be given to nonlinear canonical transformations.This means that in the canonical quanti-zation the correct choice of canonical variables is of primary importance.Nobody knows the correct choice of variables because gravity is not yet quantized.Therefore one is free to choose any set of canonical variables if it leads to a simpler quantum model.Although we do not consider quantization of the model in the present paper the canonical variables introduced in this section are essential for the solution of the constraints in Section8already at the classical level.In this section we make the canonical transformation e1a,p a→q,q⊥,p,p⊥which explicitly separates the Lorentz angle and simpliﬁes many formulas[35].Consider a generating functional depending on old coordinates and new momentaF=1e10−e11.(46)Varying it with respect to the old coordinates one obtains the relation between old and new momentap a=p e1ag11.(47)It shows thatp=e1a p a,p⊥=p aεa b e1b,that is,p and p⊥are projections of the momentum p a on the vector e1a and the perpendicular direction.Variation of the generating functional(46)with respect to the momenta yields the relation between the coordinatese10=e q sh q⊥,e11=e q ch q⊥.(48) To drop the modulus signs in Eq.(46)we assume for deﬁniteness thate11>0and e11>e10.(49) The space component of the metric equals tog11=−e2qand is always negative.We see that the coordinate q⊥coincides with the Lorentz angle while q parame-terizes the length of the vector e1a.The square of the momentum isp a p a=(p2⊥−p2)e−2q.(50)9The constraints in new variables have the formH0=−∂1p⊥+p∂1q⊥+p⊥∂1q−ω1p+ω1(−∂1π+p⊥)+12m2X2 −12∂1X2,(51)H1=−∂1p−ω1∂1π+p∂1q+p⊥∂1q⊥+P∂1X,(52)G=−∂1π+p⊥.(53) For U=const the quadratic part in the momenta12eπ R+12p a T∗a−e 12T∗a−εβγ∂βeγa .(56)Therefore instead of the Lorentz connection one can choose the torsion components T∗a as independent variables.Solving their algebraic equations of motion one arrives at the generalized dilaton modelL D=−12e∂π2U−eV,(57) 10depending on two arbitrary functions U(π)and V(π).Here we used the obvious abbreviation∂π2=gαβ∂απ∂βπ.Thus,if the matter Lagrangian does not contain the Lorentz connection then the dilaton model is equivalent to the two-dimensional gravity with torsion.In fact,we have proved that if the original variables eαa,ωα, p a,π,and X satisfy equations of motion(7)–(11),then the variables gαβ,π,and X satisfy equations of motion following from the Lagrangian L D+L X.The inverse statement is as follows.Let gαβ,π,and X satisfy equations of motion for L D+L X. Solve the algebraic equation eαa eβbηab=gαβfor zweibein(a solution is unique up to a local Lorentz rotations),construct p a using Equation(9),and solve equations (8)and(56)for T∗a andωα,respectively.Then the original equations of motion (7)–(11)will be satisﬁed.The equivalence is a global one because the transformation of variables is non-degenerate(56)as far as nondegenerate is the zweibein.This equivalence yields the geometric meaning for the dilatonﬁeld:It is the momentum conjugate to the space component of the Lorentz connectionω1.Variation of the action for the Lagrangian(57)with respect to the metric and the dilatonﬁeld yields the equations of motion1δgαβ:−12 ∇απ ∇βπU+12∂π2U−V=0,(58)1δπ:−12∂π2U′−V′=0,(59)where =gαβ ∇α ∇βis the Laplace–Beltrami operator.This system of equations was solved in the conformal gauge in[16],but we are not aware how to solve directly this system of nonlinear equations of motion for arbitrary functions U and V without gaugeﬁxing.In the next section we write down a general solution to the equivalent two-dimensional gravity with torsion we started with in an arbitrary coordinate system.The latter model turns out to be simpler,and a general solution will be written even without gaugeﬁxing.7A general solution without matterThe important feature of two-dimensional gravity with torsion described by the Lagrangian(3)alone is its integrability.It has a long history.First the quadratic model was solved in the conformal[9,10]and light-cone[12]gauge.In[13,14, 15]a solution for the quadratic model was in fact obtained without gaugeﬁxing. Afterwards this solution was clariﬁed and generalized in the papers[37,19].In this section we summarize all approaches and write a general local solution of the equations of motion for arbitrary functions U and V in the absence of matterﬁelds. It is naturally written in the canonical formulation.This solution has one Killing vectorﬁeld,and using the conformal blocks technique one is able to construct easily all global(maximally extended along extremals)solutions of two-dimensional gravity with torsion or,equivalently,dilaton gravity for arbitrary given functions U and V. In this section we set X=0.7.1Local solutionThe integration of the equations of motion is most easily performed for the light cone components of the vectors in the tangent spacep ±=12(p 0±p 1),e 1±=12(e 10±e 11).(60)The Lorentz metric and antisymmetric tensor for the tangent space indices a ={+,−}become η±±= 0110 ,ε±±= 0−110.(61)The raising and lowering of the light cone indices is performed according to the rules p +=p −and p −=p +.Then equations of motion(7)–(10)taketheform −12T ∗+−p +U =0,(63)−12(p 2⊥−p 2)e −2q −Q −W =const ,(71)where Q and W are primitivesQ(π)= πdsU(s),W(π)= πdsV(s)e−Q(s).(72)They have dimensions[Q]=1,[W]=[A]=l−2,as the consequence of(285)and(284).The constants of integration in Equations (72)do not matter because A by itself is a constant on the equations of motion∂αA=0.This equality is proved by direct calculations using equations of motion(65)–(67). Note that this integral of motion depends only on the momenta and its conservation is the consequence of eqs.(65)–(67)only.Its existence will be clariﬁed in the next section.This integral of motion was independently found in dilaton gravity[38,39].The second important observation is that the formdxαfα=√p−e Q(73)is closed on the equations of motion.Here we introduced a dimensionfull gravita-tional constant[κ]=l−2in order for fαto be dimensionless[fα]=1.To prove the closedness of the form(73)one has to verify that the expressionˆεαβ∂αfβ=0vanishes identically when equations of motion(63),(65),and(67)are satisﬁed.It means that at least locally the one form(73)may be written as the gradient of some scalar functionfα=∂αf.(74) Afterwards a general solution to the equations of motion(62)–(67)is written imme-diatelyeα+=1κp−e−Q∂αf,(75)eα−=1√p−−1κ (A+W)U+e−Q V ∂αf,(77)p+=1is the consequence of the linear dependence of the equations of motion given by (14)and(16).Note that the solution(75)–(78)was obtained without any gauge ﬁxing and contains three arbitrary functions f,πand p−.Theﬁrst two functions correspond to the invariance of the model under general coordinate transformations. They must satisfy the restrictiondπ∧d f=0orεαβ∂απ∂βf=0.(79) The third function p−corresponds to a Lorentz rotation and must be strictly positive p−>0or negative p−<0.The second class of solutions describes surfaces with nonzero torsion and noncon-stant scalar curvature which are most easily calculated using Equations(62)–(64). The corresponding metric isds2=2eα+eβ−dxαdxβ=2e−Q 1√√∂f.(81) Its square isK2=2√κ∂1f[√√κdxαeα−is closed on the equations of motion (64),(65),and (66).A similar procedure results in a general solution to the equations of motione α−=1κp +e −Q ∂αf,(87)e α+=1√p +−1κ (A +W )U +e −Q V ∂αf,(89)p −=1κ(A +W )d f 2+1κd fdπ .(91)This diﬀerence may be eliminated by the redeﬁnition f →−f .At the end of this section we make a short comment on the conserved quantity A (71).For the Schwarzschild solution it equals a mass of the black hole up to a constant factor (see Section 11).The equationA =const (92)is a ﬁrst class constraint of the model.One may check that its space derivative is expressed in terms of the secondary constraints∂1A =e −Q −p a G a +12p c p c U +V k a e −Q Gδ,{A, G ′0}={A, G ′1}= (p ⊥−p )e −2q ω1+1g 11H 0−e 1a εa b p b 2p a p a U +V G e −Q δ,= pe −2q H 0−p ⊥e −2q H 1+ (p ⊥−p )e −2q ω1+1g 11H 0+e 1a p a 2p a p a U +V G e −Q δ= p⊥e−2q H0−pe−2q H1+ −(p⊥−p)e−2qω1+1(π R+p a T∗a)−e U,(94)2where U(℘,π)is an arbitrary function of two scalar functions℘=p a p a andπ.For the linear function U=℘U/2+V in℘we recover the original gravity model(3). In addition we clarify the statement that equations of motion are integrable in this more general case[14,25,26].Equations of motion for the Lagrangian(94)in light-cone coordinates take the formδS:−ˆεαβ(∂αeβ+−ωαeβ++2p−U,℘eα+eβ−)=0,(96)δp+δS:∂απ−p+eα++p−eα−=0,(98)δωβδSˆεβα:∂αp−−ωαp−−eα+U=0,(100)δeβ−whereU,℘=∂U∂π.We have nine equations for nine independent variablesπ,p±,ωα,eα±.There are three linear relations between equations of motion(14),(16)due to the symmetry under general coordinate transformations and local Lorentz rotations.It means that toﬁnd a general solution to the equations of motion one has to solve only six independent equations.Equations(95)–(97)may be written in the form−1T⋆±−2p±U,℘=0.(102)2where the scalar curvature and torsion are deﬁned by equations(4),(5).If these equations have a unique solution with respect toπand p±then the Lagrangian(94) is nothing else then the Legendre transform of some function=eF(R,T2)(103)L(2)Gdepending on the scalar curvature R and the torsion squared term T2=T∗a T∗a with respect to three variables−R/2and−T∗a/2.The Lagrangian written in theﬁrst order form(94)is more general because we do not assume that a function U(℘,π) admits the Legendre transformation.Let us try toﬁnd the six unknownsωα,eα±in terms of the conjugate momenta considered as arbitrary functions.Forﬁxed indexαEquations(98)–(100)constitute a set of three linear algebraic inhomogeneous equations forωα,eα±.Its determinant vanishes identically,and hence for any nontrivial solution there must exist a rela-tion between the momenta.Toﬁnd it one may take the linear combination of the equations(99)p−+(100)p+and using equation(98)ﬁnd∂α℘−2∂απU=0.It means that momenta must satisfy the following ordinary diﬀerential equationd℘(−∂απ+p+eα+),(105)p−1ωα=(p+eβ+) =0.(107)℘Without loss of generality we setp+eα+=℘e−Q fα,where fαis a one form and Q(℘,π)is some unknown scalar function of two variables to be speciﬁed later.Then Equation(107)takes the form℘e−Qˆεαβ[∂αfβ−∂απfβ(2U Q,℘+Q,π−2U,℘)]=0.(108) Solution of this equation always exists for reasonable functionsπ,U,and Q,but a solution cannot be written explicitly in a general case.It can be exactly integrated in a particular case.If we choose the function Q satisfying the partial diﬀerential equation2U Q,℘+Q,π−2U,℘=0,(109) then a one form fαmust be closed.It means that locally this one form is exact(74). Hence we get a general solutioneα+=2p−e−Q∂αf,(110) where f is an arbitrary function,and Q is a solution of the partial diﬀerential Equation(109).The rest of the equations of motion are satisﬁed as the consequence of their linear dependence.Equations for characteristics℘(π),Q(℘(π),π)for Equation(109)ared℘dπ=2∂Uone has to extend the solution along extremals (geodesics).This can be done using the constructive conformal block method developed for the Lorentzian [42,43,44]and Euclidean [45]signaturemetric.An equivalent set of rules in the Eddington–Finkelstein coordinates may be found in [46,47].To apply the conformal blocks technique one has to rewrite local solutions obtained in the previous section in the conformal form.This should be done in every domain where the solution is deﬁned because a global solution is obtained by gluing all patches together.Before doing this we write the metric (80)in the diagonal gauge for comparison with the metric in the matterfull case.In a domain with A +W >0we leave the coordinate πas it is and transform the coordinate f onlyf =1κ2e −Q 1A +W .(114)For A +W >0coordinates τand πare timelike and spacelike,respectively.This metric can be easily rewritten in the conformal gauge suitable for the global analysis.Introducing the space coordinate σdeﬁned by the equationdπ√2κe −Q (A +W )(dτ2−dσ2),A +W >0.(115)Introducing the invariant variable ˆq related to πby ordinary diﬀerential equationd ˆq √dσ=N (ˆq ),(118)and the conformal factor isN(ˆq)=12σ+g(π),where g(π)is deﬁned by the same Equation(113).Then solution(80)takes adiagonal formds2=−1A+W−1dτ=±A+Wκ,the metric takes a conformallyﬂat formds2=−1dτ=−N(ˆq).Solutions in domains A+W>0and A+W<0may be united in the wayds2=|N(ˆq)|(dτ2−dσ2),dˆqfunctions.This will be done in two steps.First,we solve three constraints(51)–(53),which are equivalent to the variational derivatives of the action with respect to the time components of the zweibein and Lorentz connectionδS/δe0a andδS/δω0. Afterwards we solve the rest of the geometric part of the equations of motion.The constraints G and G a in the light cone coordinates areG=−∂1π+e1+p+−e1−p−,(121) G±=−∂1p±∓ω1p±∓e1∓ p+p−U+V−ρ4e1±P2∓,(122) where we introduced a shorthand notation1P±=。

The Lorentz Force Law (2)

Derivation of the Lorentz Force Law and the Magnetic Field Concept using an Invariant Formulation of the LorentzTransformationJ.H.FieldD´e partement de Physique Nucl´e aire et Corpusculaire Universit´e de Gen`e ve.24,quaiErnest-Ansermet CH-1211Gen`e ve4.e-mail;john.ﬁeld@cern.chAbstractIt is demonstrated how the right hand sides of the Lorentz Transformation equa-tions may be written,in a Lorentz invariant manner,as4–vector scalar products.The formalism is shown to provide a short derivation,in which the4–vector elec-tromagnetic potential plays a crucial role,of the Lorentz force law of classical elec-trodynamics,and the conventional deﬁnition of the magneticﬁeld in terms spatialderivatives of the4–vector potential.The time component of the relativistic gen-eralisation of the Lorentz force law is discussed.An important physical distinctionbetween the space-time and energy-momentum4–vectors is also pointed out.Keywords;Special Relativity,Classical Electrodynamics.PACS03.30+p03.50.De1IntroductionNumerous examples exist in the literature of the derivation of electrodynamical equa-tions from simpler physical hypotheses.In Einstein’s original paper on Special Relativ-ity[1],the Lorentz force law was derived by performing a Lorentz transformation of the electromagneticﬁelds and the space-time coordinates from the rest frame of an electron (where only electrostatic forces act)to the laboratory system where the electron is in motion and so also subjected to magnetic forces.A similar demonstration was given by Schwartz[2]who also showed how the electrodynamical Maxwell equations can be derived from the Gauss laws of electrostatics and magnetostatics by exploiting the4-vector char-acter of the electromagnetic current and the symmetry properties of the electromagnetic ﬁeld tensor.The same type of derivation of electrodynamic Maxwell equations from the electrostatic and magnetostatic ones has recently been performed by the present author on the basis of‘space-time exchange symmetry’[3].Frisch and Wilets[4]discussed the derivation of Maxwell’s equations and the Lorentz force law by application of relativistic transforms to the electrostatic Gauss law.Dyson[5]published a proof,due originally to Feynman,of the Faraday-Lenz law of induction,based on Newton’s Second Law and the quantum commutation relations of position and momentum,that excited considerable interest and aﬂurry of comments and publications[6,7,8,9,10,11]about a decade ndau and Lifshitz[12]presented a derivation of Amp`e re’s Law from the electro-dynamic Lagrangian,using the Principle of Least Action.By relativistic transformation of the Coulomb force from the rest frame of a charge to another inertial system in rela-tive motion,Lorrain,Corson and Lorrain[13]derived both the Biot-Savart law,for the magneticﬁeld generated by a moving charge,and the Lorentz force law.In many text books on classical electrodynamics the question of what are the funda-mental physical hypotheses underlying the subject,as distinct from purely mathematical developments of these hypotheses,used to derive predictions,is not discussed in any de-tail.Indeed,it may even be stated that it is futile to address the question at all.For example,Jackson[14]states:At present it is popular in undergraduate texts and elsewhere to attempt to derive magneticﬁelds and even Maxwell equations from Coulomb’s law of electrostatics and the theory of Special Relativity.It should immediately obvious that,without additional assumptions,this is impossible.’This is,perhaps,a true statement.However,if the additional assumptions are weak ones,the derivation may still be a worthwhile exercise.In fact,in the case of Maxwell’s equations,as shown in References[2,3],the‘additional assumptions’are merely the formal deﬁnitions of the electric and magneticﬁelds in terms of the space–time derivatives of the 4–vector potential[15].In the case of the derivation of the Lorentz force equation given below,not even the latter assumption is required,as the magneticﬁeld deﬁnition appears naturally in the course of the derivation.In the chapter on‘The Electromagnetic Field’in Misner Thorne and Wheeler’s book ‘Gravitation’[16]can be found the following statement:Here and elsewhere in science,as stressed not least by Henri Poincar´e,that view isout of date which used to say,“Deﬁne your terms before you proceed”.All the laws and theories of physics,including the Lorentz force law,have this deep and subtle chracter, that they both deﬁne the concepts they use(here B and E)and make statements about these concepts.Contrariwise,the absence of some body of theory,law and principle deprives one of the means properly to deﬁne or even use concepts.Any forward step in human knowlege is truly creative in this sense:that theory concept,law,and measurement —forever inseperable—are born into the world in union.I do not agree that the electric and magneticﬁelds are the fundamental concepts of electromagnetism,or that the Lorentz force law cannot be derived from simpler and more fundamental concepts,but must be‘swallowed whole’,as this passage suggests. As demonstrated in References[2,3]where the electrodynamic and magnetodynamic Maxwell equations are derived from those of electrostatics and magnetostatics,a more economical description of classical electromagentism is provided by the4–vector potential. Another example of this is provided by the derivation of the Lorentz force law presented in the present paper.The discussion of electrodynamics in Reference[16]is couched entirely in terms of the electromagneticﬁeld tensor,Fµν,and the electric and magnetic ﬁelds which,like the Lorentz force law and Maxwell’s equations,are‘parachuted’into the exposition without any proof or any discussion of their interrelatedness.The4–vector potential is introduced only in the next-but-last exercise at the end of the chapter.After the derivation of the Lorentz force law in Section3below,a comparison will be made with the treatment of the law in References[2,14,16].The present paper introduces,in the following Section,the idea of an‘invariant for-mulation’of the Lorentz Transformation(LT)[17].It will be shown that the RHS of the LT equations of space and time can be written as4-vector scalar products,so that the transformed4-vector components are themselves Lorentz invariant quantities.Consid-eration of particular length and time interval measurements demonstrates that this is a physically meaningful concept.It is pointed out that,whereas space and time intervals are,in general,physically independent physical quantities,this is not the case for the space and time components of the energy-momentum4-vector.In Section3,a derivation of the Lorentz force law,and the associated magneticﬁeld concept,is given,based on the invariant formulation of the LT.The derivation is very short,the only initial hypothesis being the usual deﬁnition of the electricﬁeld in terms of the4-vector potential,which,in fact,is also uniquely speciﬁed by requiring the deﬁnition to be a covariant one.In Section 4the time component of Newton’s Second Law in electrodynamics,obtained by applying space-time exchange symmetry[3]to the Lorentz force law,is discussed.Throughout this paper it is assumed that the electromagneticﬁeld constitutes,to-gether with the moving charge,a conservative system;i.e.eﬀects of radiation,due to the acceleration of the charge,are neglected2Invariant Formulation of the Lorentz Transforma-tionThe space-time LT equations between two inertial frames S and S’,written in a space-time symmetric manner,are:x =γ(x−βx0)(2.1)y =y(2.2)z =z(2.3)x 0=γ(x0−βx)(2.4) The frame S’moves with velocity,v,relative to S,along the common x-axis of S and S’.βandγare the usual relativistic parameters:vβ≡√−(∆x0)2+∆x2=∆x(2.12) since,for the measurement procedure just described,∆x0=0.Notice that∆x is not necessarily deﬁned in terms of such a measurement.If,following Einstein[1],the interval ∆x is associated with the length, ,of a measuring rod at rest in S and lying parallel to the x-axis,measurements of the ends of the rod can be made at arbitarily diﬀerent times in S.The same result =∆x will be found for the length of the rod,but the corresponding invariant interval,S x,as deﬁned by Eqn(2.12)will be diﬀerent in each case.Similarly,∆x0may be identiﬁed with the time-like invariant interval corresponding to successive observations of a clock at aﬁxed position(i.e.∆x=0)in S:S0≡the same value,∆x0,for the time diﬀerence between two events in S,but with diﬀerent values of the invariant interval deﬁned by Eqn(2.13).In virtue of Eqns(2.12)and(2.13)the LT equations(2.8)and(2.11)may be written the following invariant form:S x=−¯U(β)·S(2.14)S 0=U(β)·S(2.15) where the following4–vectors have been introduced:S≡(S0;S x,0,0)=(∆x0;∆x,0,0)(2.16)U(β)≡(γ;γβ,0,0)(2.17)¯U(β)≡(γβ;γ,0,0)(2.18)The time-like4-vector,U,is equal to V/c,where V is the usual4–vector velocity,whereas the space-like4–vector,¯U,is‘orthogonal to U in four dimensions’:U(β)·¯U(β)=0(2.19) Since the RHS of(2.14)and(2.15)are4–vector scalar products,S x and S 0are manifestly Lorentz invariant quantites.These4–vector components may be deﬁned,in terms of speciﬁc space-time measurements,by equations similar to(2.12)and(2.13)in the frame S’.Note that the4–vectors S and S are‘doubly covariant’in the sense that S·S and S ·S are‘doubly invariant’quantities whose spatial and temporal terms are,individually, Lorentz invariant:S·S=S20−S2x=S ·S =(S 0)2−(S x)2(2.20) Every term in Eqn(2.20)remains invariant if the spatial and temporal intervals described above are observed from a third inertial frame S”moving along the x-axis relative to both S and S’.This follows from the manifest Lorentz invariance of the RHS of Eqn(2.14)and (2.15)and their inverses:S x=−¯U(−β)·S (2.21)S0=U(−β)·S (2.22) Since the LT Eqns(2.1)and(2.4)are valid for any4–vector,W,it follows that:W x=−¯U(β)·W(2.23)W 0=U(β)·W(2.24) Again,W x and W 0are manifestly Lorentz invariant.An interesting special case is the energy-momentum4–vector,P,of a physical object of mass,m.Here the‘doubly in-variant’quantity analagous to S·S in Eqn(2.20)is equal to m2c2.Choosing the x-axis parallel to p andβto correspond to the object’s velocity,so that S’is the object’s proper frame,and since P≡mcU(β),Eqns(2.23)and(2.24)yield,for this special case:P x=−mc¯U(β)·U(β)=0(2.25)P 0=mcU(β)·U(β)=mc(2.26)Since the Lorentz transformation is determined by the single parameter,β,then it follows from Eqns(2.25)and(2.26)that,unlike in the case of the space and time intervals in Eqns(2.8)and(2.11),the spatial and temporal components of the energy momentum 4–vector,in an arbitary inertial frame,are not independent.In fact,P0is determined in terms of P x and m by the relation,that follows from the inverse of Eqns(2.25)and(2.26):P0=Thus,from rotational invariance,the general covariant deﬁnition of the electricﬁeld is:E i=∂i A0−∂0A i(3.4) This is the‘additional assumption’,mentioned by Jackson in the passage quoted above, that is necessary,in the present case,to derive the Lorentz force law.However,as written, it concerns only the physical properties of the electricﬁeld:the magneticﬁeld concept has not yet been introduced.A further a posteriori justiﬁcation of Eqn(3.4)will be given after derivation of the Lorentz force law.Here it is simply noted that,if the spatial part of the4–vector potential is time-independent,Eqn(3.4)reduces to the usual electrostatic deﬁnition of the electricﬁeld.The force F on an electric charge q at rest in the frame S’is given by the deﬁnition of the electricﬁeld,and Eqn(3.4)as:F i=q(∂ i A 0−∂ 0A i)(3.5) Equations analagous to(2.24)may be written relating A and∂ to the corresponding quantities in the frame S moving along the x’axis with velocity−v relative to S’:∂ 0=U(β)·∂(3.6)A 0=U(β)·A(3.7) Substituting(3.6)and(3.7)in(3.5)gives:F i=q∂ i(U(β)·A)−(U(β)·∂)A i(3.8)This equation expresses a linear relationship between F i,∂ i and A i.Since the coeﬃcients of the relation are Lorentz invariant,the same formula is valid in any inertial frame,in particular,in the frame S.Hence:F i=q∂i(U(β)·A)−(U(β)·∂)A i(3.9)This equation gives,in4–vector notation,a spatial component of the Lorentz force on the charge q in the frame S,and so completes the derivation.To express the Lorentz force formula in the more familiar3-vector notation,it is convenient to introduce the relativistic generalisation of Newton’s Second Law[19]:dPdτ=γdP iIntroducing now the magneticﬁeld according to the deﬁnition[20]:B k≡− ijk(∂i A j−∂j A i)=( ∇× A)k(3.12) enables Eqn(3.11)to be written in the compact form:dP idγ βt=mc(3.15)∂twhere Eqn(3.12)has been used.Eqn(3.15)is just the Faraday-Lenz induction law,i.e.the magnetodynamic Maxwell equation.This is only apparent,however,once the‘magnetic ﬁeld’concept of Eqn(3.12)has been introduced.Thus the initial hypothesis,Eqn(3.4),is actually a Maxwell equation.This is the a posteriori justiﬁcation,mentioned above,for this covariant deﬁnition of the electricﬁeld.It is common in discussions of electromagnetism to introduce the second rank electro-magneticﬁeld tensor,Fµνaccording to the deﬁnition:Fµν≡∂µAν−∂νAµ(3.16) in terms of which,the electric and magneticﬁelds are deﬁned as:E i≡F i0(3.17)B k≡− ijk F ij(3.18) From the point of view adopted in the present paper both the electromagneticﬁeld tensor and the electric and magneticﬁelds themselves are auxiliary quantities introduced only for mathematical convenience,in order to write the equations of electromagnetism in a compact way.Since all these quantities are completly deﬁned by the4–vector potential, it is the latter quantity that encodes all the relevant physical information on any electro-dynamic problem[21].This position is contrary to that commonly taken in the literature and texbooks where it is often claimed that only the electric and magneticﬁelds have physical signiﬁcance,while the4–vector potential is only a convenient mathematical tool. For example R¨o hrlich[22]makes the statement:These functions(φand A)known as potentialsmanner!In other cases(e.g.Maxwell’s equations)simpler expessions may be written interms of the4–vector potential.The quantum theory,quantum electrodynamics,thatunderlies classical electromagnetism,requires the introduction the4–vector photonﬁeld Aµin order to specify the minimal interaction that provides the dynamical basis of the theory.Similarly,the introduction of Aµis necessary for the Lagrangian formulation ofclassical electromagnetism.It makes no sense,therefore,to argue that a physical conceptof such fundamental importance has‘no physical meaning’.The initial postulate used here to derive the Lorentz force law is Eqn(3.4),whichcontains,explicitly,the electrostatic force law and,implicitly,the Faraday-Lenz inductionlaw.The actual form of the electrostatic force law(Coulomb’s inverse square law)is notinvoked,suggesting that the Lorentz force law may be of greater generality.On theassumption of Eqn(3.4)(which has been demonstrated to be the only possible covariantdeﬁnition of the electricﬁeld),the existence of the‘magneticﬁeld’,the‘electromagneticﬁeld tensor’,andﬁnally the Lorentz force law itself have all been derived,without furtherassumptions,by use of the invariant formulation of the Lorentz transformation.It is instructive to compare the derivation of the Lorentz force law given in the presentpaper with that of Reference[13]based on the relativistic transformation properties of theCoulomb force3–vector.Coulomb’s law is not used in the present paper.On the otherhand,Reference[13]makes no use of the4–vector potential concept,which is essential forthe derivation presented here.This demonstrates an interesting redundancy among thefundamental physical concepts of classical electromagnetism.In Reference[2],Eqns(3.4),(3.12)and(3.16)were all introduced as a priori initialpostulates without further justiﬁcation.In fact,Schwartz gave the following explanationfor his introduction of Eqn(3.16)[23]:So far everything we have done has been entirely deductive,making use only ofCoulomb’s law,conservation of charge under Lorentz transformation and Lorentz in-variance for our physical laws.We have now come to the end of this deductive path.Atthis point when the laws were being written,God had to make a decision.In generalthere are16components of a second-rank tensor in four dimensions.However,in anal-ogy to three dimensions we can make a major simpliﬁcation by choosing the completelyantisymmetric tensor to represent ourﬁeld quantities.Then we would have only6inde-pendent components instead of the possible16.Under Lorentz transformation the tensorwould remain antisymmetric and we would never have need for more than six independentcomponents.Appreciating this,and having a deep aversion to useless complication,Godnaturally chose the antsymmetric tensor as His medium of expression.Actually it is possible that God may have previously invented the4–vector potentialand special relativity,which lead,as shown above,to Eqn(3.4)as the only possible co-variant deﬁnition of the electricﬁeld.As also shown in the present paper,the existence ofthe remaining elements of the antisymmetricﬁeld tensor,containing the magneticﬁeld,then follow from special relativity alone.Schwartz derived the Lorentz force law,as inEinstein’s original Special Relativity paper[1],by Lorentz transformation of the electricﬁeld,from the rest frame of the test charge,to one in which it is in motion.This requiresthat the magneticﬁeld concept has previously been introduced as well as knowledge ofthe Lorentz transformation laws of the electric and magneticﬁelds.In the chapter devoted to special relativity in Jackson’s book[24]the Lorentz forcelaw is simply stated,without any derivation,as are also the deﬁning equations of theelectric and magneticﬁelds and the electromagneticﬁeld tensor just mentioned.Noemphasis is therefore placed on the fundamental importance of the4–vector potential inthe relativistic description of electromagnetism.In order to treat,in a similar manner,the electromagnetic and gravitationalﬁelds,thediscussion in Misner Thorne and Wheeler[16]is largely centered on the properties of thetensor Fµν.Again the Lorentz force equation is introduced,in the spirit of the passagequoted above,without any derivation or discussion of its meaning.The deﬁning equationsof the electric and magneticﬁelds and Fµν,in terms of Aµ,appear only in the eighteenthexercise of the relevant chapter.The main contents of the chapter on the electromagneticﬁeld are an extended discussion of purely mathematical tensor manipulations that obscurethe essential simplicity of electromagnetism when formulated in terms of the4–vectorpotential.In contrast to References[2,24,16],in the derivation of the Lorentz force law andthe magneticﬁeld presented here,the only initial assumption,apart from the validityof special relativity,is the chosen deﬁnition,Eqn(3.4),of the electricﬁeld in terms ofthe4–vector potential Aµ,which is the only covariant one.Thus,a more fundamentaldescription of electromagnetism than that provided by the electric and magneticﬁeldconcepts is indeed possible,contrary to the opinion expressed in the passage from MisnerThorne and Wheeler quoted above.4The time component of Newton’s Second Law in ElectrodynamicsSpace-time exchange symmetry[3]states that physical laws inﬂat space are invariantwith respect to the exchange of the space and time components of4-vectors.For example,the LT of time,Eqn(2.4),is obtained from that for space,Eqn(2.1),by applying the space-time exchange(STE)operations:x0↔x,x 0↔x .In the present case,application of the STE operation to the spatial component of the Lorentz force equation in the secondline of Eqn(3.11)leads to the relation:dP00(4.1)where Eqns(2.5)and(3.4)and the following properties of the STE operation[3]have been used:∂0↔−∂i(4.2)A0↔−A i(4.3)C·D↔−C·D(4.4)Eqn(4.1)yields an expression for the time derivative of the relativistic energy,E=P0:d E(4.5)Integration of Eqn(4.5)gives the equation of energy conservation for a particle moving from an initial position, x I,to aﬁnal position, x F,under the inﬂuence of electromagnetic forces:E F E I d E=qxFx IE·d x(4.6)Thus work is done on the moving charge only by the electricﬁeld.This is also evident from the Lorentz force equation,(3.14),since the magnetic force β× B is perpendicular to the velocity vector,so that no work is performed by the magneticﬁeld.A corollary is that the relativistic energy(and hence the magnitude of the velocity)of a charged particle moving in a constant magneticﬁeld is a constant of the motion.Of course,Eqn(4.5) may also be derived directly from the Lorentz force law,so that the time component of the relativistic generalisation of Newton’s Second Law,Eqn(4.1),contains no physical information not already contained in the spatial components.This is related to the fact that,as demonstrated in Eqns(2.25)and(2.26),the spatial and temporal components of the energy-momentum4–vector are not independent physical quantities.AcknowledgementsI should like to thank O.L.de Lange for asking the question whose answer,presented in Section4,was the original motivation for the writing of this paper,and an anonymous referee of an earlier version of this paper for informing me of related material,in the books of Jackson and Misner,Thorne and Wheeler,which is discussed in some detail in this version.References[1]A.Einstein,17891(1905).[2]M.Schwartz,‘Principles of Electrodynamics’,(McGraw-Hill,New York,1972)Ch3.[3]J.H.Field,Am.J.Phys.69569(2001).[4]D.H.Frisch and L.Wilets,Am.J.Phys.24574(1956).[5]F.J.Dyson,Am.J.Phys.58209(1990).[6]N.Dombey,Am.J.Phys.5985(1991).[7]R.W.Breheme,Am.J.Phys.5985(1991).[8]J.L.Anderson,Am.J.Phys.5986(1991).[9]I.E.Farquhar,Am.J.Phys.5987(1991).[10]S.Tanimura,Ann.Phys.(N.Y.)220229(1992).[11]A.Vaidya and C.Farina,Phys.Lett.153A265(1991).[12]ndau and E.M.Lifshitz,‘The Classical Theory of Fields’,(Pergamon Press,Oxford,1975)Section30,P93.[13]P.Lorrain, D.R.Corson and F.Lorrain,‘Electromagnetic Fields and Waves’,(W.H.Freeman,New York,Third Edition,1988)Section16.5,P291.[14]J.D.Jackson,‘Classical Electrodynamics’,(John Wiley and Sons,New York,1975)Section12.2,P578.[15]Actually,a careful examination of the derivation of Amp`e re’s from the Gauss lawof electrostatics in Reference[3]shows that,although Eqn(3.4)of the present paper is a necessary initial assumption,the deﬁnition of the magneticﬁeld in terms of the spatial derivatives of the4–vector potential occurs naturally in the course of the derivation(see Eqns(5.16)and(5.17)of Reference[3])so it is not necessary to assume, at the outset,the expression for the spatial components of the electromagneticﬁeld tensor as given by Eqn(5.1)of Reference[3].[16]C.W.Misner,K.S.Thorne and J.A.Wheeler,‘Gravitation’,(W.H.Freeman,San Fran-cisco,1973)Ch3,P71.[17]This should not be confused with a manifestly covariant expression for the LT,whereit is written as a linear4-vector relation with Lorentz-invariant coeﬃcients,as in:D.E.Fahnline,Am.J.Phys.50818(1982).[18]A time-like metric is used for4-vector products with the components of a4–vector,W,deﬁned as:W t=W0=W0,W x,y,z=W1,2,3=−W1,2,3and an implied summation over repeated contravariant(upper)and covariant(lower) indices.Repeated Greek indices are summed from0to3,repeated Roman ones from1to3.Also∂µ≡(∂∂x1,−∂∂x3)=(∂0;− ∇)[19]H.Goldstein,‘Classical Mechanics’,(Addison-Wesley,Reading Massachusetts,1959)P200,Eqn(6-30).[20]The alternating tensor, ijk,equals1(−1)for even(odd)permutations of ijk.[21]The explicit form of Aµ,as derived from Coulomb’s law,is given in standard text-books on classical electrodynamics.For example,in Reference[13],it is to be found in Eqns(17-51)and(17-52).Aµis actually proportional to the4-vector velocity,V, of the charged particle that is the source of the electromagneticﬁeld.[22]F.R¨o hrlich,‘Classical Charged Particles’,(Addison-Wesley,Reading,MA,1990)P65.[23]Reference[2]above,Ch3,P127.[24]Reference[14]above,Section11.9,P547.。

On the particle acceleration near the light surface of radio pulsars

a r X i v :a s t r o -p h /0002525v 1 29 F eb 2000Mon.Not.R.Astron.Soc.000,1–??(1999)Printed 1February 2008(MN L A T E X style ﬁle v1.4)On the particle acceleration near the light surface of radiopulsarsV.S.Beskin 1,2and R.R.Raﬁkov 31National Astronomical Observatory,Osawa 2–21–1,Mitaka,Tokyo 181–8588,Japan 2P.N.Lebedev Physical Institute,Leninsky prosp.,53,Moscow,117924,Russia 3Princeton University Observatory,Princeton,NJ,08544,USAAccepted 1999.Received 1999;in original form 1999ABSTRACTThe two–ﬂuid eﬀects on the radial outﬂow of relativistic electron–positron plasma are considered.It is shown that for large enough Michel (1969)magnetization param-eter σ≫1and multiplication parameter λ=n/n GJ ≫1one–ﬂuid MHD approxima-tion remains correct in the whole region |E |<|B |.In the case when the longitudinal electric current is smaller than the Goldreich–Julian one,the acceleration of particles near the light surface |E |=|B |is determined.It is shown that,as in the previously considered (Beskin Gurevich &Istomin 1983)cylindrical geometry,almost all electro-magnetic energy is transformed into the energy of particles in the narrow boundary layer ∆̟/̟∼λ−1.Key words:two–ﬂuid relativistic MHD:radio pulsars—particle acceleration1INTRODUCTIONDespite the fact that the structure of the magnetosphere of radio pulsars remains one of the fundamental astrophysical problems,the common view on the key theoretical question –what is the physical nature of the neutron star braking –is absent (Michel 1991,Beskin Gurevich &Istomin 1993,Mestel 1999).Nevertheless,very extensive theoretical studies in the seventies and the eighties allowed to obtain some model-independent results.One of them is the absence of magnetodipole energy loss.This result was ﬁrst obtained theoretically (Henriksen &Norton 1975,Beskin et al 1983).It was shown that the electric charges ﬁlling the magnetosphere screen fully the magnetodipole radiation of a neutron star for an arbitrary inclination angle χbetween the rotational and magnetic axes if there are no longitudinal currents ﬂowing in the ter this result was also conﬁrmed by observations.The direct detections of the interaction of the pulsar wind with a companion star in close binaries (see e.g.Djorgovsky &Evans 1988,Kulkarni &Hester 1988)have shown that it is impossible to explain the heating of the companion by a low–frequency magnetodipole wave.On the other hand,the detailed mechanism of particle acceleration remains unclear.Indeed,a very high magnetization parameter σ(Michel 1969)in the pulsar magnetosphere demonstrates that within the light cylinder r <R L =c/Ωthe main part of the energy is transported by the Poynting ﬂux.It means that the additional mechanism of particle acceleration must work in the vicinity of the light cylinder.It is necessary to stress that an eﬀective particle acceleration can only take place for small enough longitudinal electric currents I <I GJ when the plasma has no possibility to pass smoothly through the fast magnetosonic surface and when the light surface |E |=|B |is located at a ﬁnite distance.As to the case of the large longitudinal currents I >I GJ ,both analytical (Tomimatsu 1994,Begelman &Li 1994,Beskin et al 1998)and numerical (Bogovalov 1997)considerations demonstrate that the acceleration becomes ineﬀective outside the fast magnetosonic surface,and the particle-to-Poynting ﬂux ratio remains small:∼σ−2/3(Michel 1969,Okamoto 1978).The acceleration of an electron–positron plasma near the light surface was considered by Beskin Gurevich and Istomin (1983)in the simple 1D cylindrical geometry for I ≪I GJ .It was shown that in a narrow boundary layer ∆̟/̟∼1/λalmost all electromagnetic energy is actually converted to the particles energy.Nevertheless,cylindrical geometry does not provide the complete picture of particle acceleration.In particular,it was impossible to include self–consistently the disturbance of a poloidal magnetic ﬁeld and an electric potential,the later playing the main role in the problem of the plasma acceleration (for more details see e.g.Mestel &Shibata 1994).Hence,a more careful 2D consideration is necessary.c1999RAS2V.S.Beskin and R.R.RaﬁkovIn Sect.2we formulate a complete system of2D two–ﬂuid MHD equations describing the electron–positron outﬂow from a magnetized body with a monopole magneticﬁeld.The presence of an exact analytical force–free solution(Michel 1973)allows us to linearize this system which results in the existence of invariants(energy and angular momentum)along unperturbed monopoleﬁeld lines similar to the ideal one–ﬂuid MHDﬂow.In Sect.3it is shown that forσ≫1andλ≫1 (λ=n/n GJ is the multiplication factor)the one–ﬂuid MHD approximation remains true in the entire region within the light surface.Finally,in Sect.4the acceleration of particles near the light surface|E|=|B|is considered.It is shown that,as in the case of cylindrical geometry,in a narrow boundary layer∆̟/̟∼λ−1almost all the electromagnetic energy is converted into the energy of particles.2BASIC EQUATIONSLet us consider a stationary axisymmetric outﬂow of a two–component plasma in the vicinity of an active object with a monopole magneticﬁeld.It is necessary to stress that,of course,the monopole magneticﬁeld is a rather crude approximation for a pulsar magnetosphere.Nevertheless,even for a dipole magneticﬁeld near the origin,at large distances r≫R L in the wind zone the magneticﬁeld can have a monopole–like structure.For this reason the disturbance of a monopole magnetic ﬁeld can give us an important information concerning particle acceleration far from the neutron star.The structure of theﬂow is described by the set of Maxwell‘s equations and the equations of motion∇E=4πρe,∇×E=0,∇B=0,∇×B=4πc×B.(2)Here E and B are the electric and magneticﬁelds,ρe and j are the charge and current densities,and v±and p±are the speed and momentum of particles.In the limit of inﬁnite particle energyγ=∞,v0r=c,v0ϕ=0,v0θ=0,(3) and for charge and current densityρ0e=ρs R2sr2cosθ,(4)the monopole poloidal magneticﬁeldB0r=B sR2scR sccosθ,(7) and theﬂux functionΨ(r,θ),so thatB0p=∇Ψ×eϕ2πce R2s2cosθ+η+(r,θ) ,(9)n−=ΩB sr2 λ+1c[−cosθ+δ(r,θ)],(11)Ψ(r,θ)=2πB s R2s[1−cosθ+εf(r,θ)],(12)c 1999RAS,MNRAS000,1–??On the particle acceleration near the light surface of radio pulsars3v ±r=c1−ξ±r (r,θ),v ±θ=cξ±θ(r,θ),v ±ϕ=cξ±ϕ(r,θ),(13)B r =B sR 2ssin θ∂fr sin θ∂f cR s c ∂δcr−sin θ−∂δsin θ∂2cos θξ+r−λ+1∂rr2∂δsin θ∂∂θ=0,(20)∂ζrλ−12cos θξ−θ,(21)−ε∂r 2−ε∂θ 1∂θ=2Ω2cos θξ+ϕ−λ+1∂rξ+θγ++ξ+θγ+r ∂δr −sin θΩr2ξ+ϕ,(23)∂r=−4λσ −1∂θ+ζrξ−r +c∂rγ+=4λσ−∂δr ξ+θ,(25)∂∂r−sin θ∂rξ+ϕγ++ξ+ϕγ+Ωr sin θ∂fΩr 2ξ+θ,(27)∂r=−4λσ −εc∂r−c4λmc 3(29)is the Michel‘s (1969)magnetization parameter,m is the electron mass,and all deﬂecting functions are supposed to be ≪1.It is necessary to stress that for applications the magnetic ﬁeld B s is to be taken near the light cylinder R s ≈R L because in the internal region of the pulsar magnetosphere B ∝r −3.As it has already been mentioned,only outside the light cylinder the poloidal magnetic ﬁeld may have quasi monopole structure.As a result,σ=Ω2eB 0R 34V.S.Beskin and R.R.Raﬁkov1975,Arons Scharlemann 1979),γin ≤102for secondary plasma.For this reason in what follows we consider in more details the caseγ3in ≪σ,(34)when the additional acceleration of particles inside the fast magnetosonic surface takes place (see e.g.Beskin Kuznetsova Raﬁkov 1998).It is this case that can be realized for fast pulsars.Moreover,it has more general interest because the relation(34)may be true also for AGNs.As to the case γ3in ≫σcorresponding to ordinary pulsars,the particle energy remains constant (γ=γin )at any way up to the fast magnetosonic surface (see Bogovalov 1997for details).Further,one can putδ(R s ,θ)=0,(35)εf (R s ,θ)=0,(36)η+(R s ,θ)−η−(R s ,θ)=0.(37)These conditions result from the relation c E s +ΩR s e ϕ×B s =0corresponding rigid rotation and perfect conductivity of the surface of a star.Finally,as will be shown in Sect.3.2,the derivative ∂δ/∂r actually determines the phase of plasma oscillations only and plays no role in the global structure.Finally,the determination of the electric current and,say,the derivative ∂f/∂r depend on the problem under consideration.Indeed,as is well–known,the cold one–ﬂuid MHD outﬂow contains two singular surfaces,Alfv´e nic and fast magnetosonic ones.It means that for the transonic ﬂow two latter functions are to be determined from critical conditions (Heyvaerts 1996).In particular,the longitudinal electric current within this approach is not a free parameter.On the other hand,if the electric current is restricted by some physical reason,the ﬂow cannot pass smoothly through the fast magnetosonic surface.In this case,which can be realized in the magnetosphere of radio pulsars (Beskin et al 1983,Beskin &Malyshkin 1998),near the light surface |E |=|B |an eﬀective particle acceleration may take place.Such an acceleration will be considered in Sect.4.3THE ELECTRON–POSITRON OUTFLOW 3.1The MHD LimitIn the general case Eqns.(19)–(28)have several integrals.Firstly,Eqns.(21),(25),and (26)result in ζ−22σλsin θ=1sin θ,(38)where l (θ)describe the disturbance of the electric current at the star surface by the equation I (R,θ)=I A sin 2θ+l (θ).Expression (38)corresponds to conservation of the total energy ﬂux along a magnetic ﬁeld line.Furthermore,Eqns.(25)–(28)together with the boundary conditions (35),(36)result in δ=εf −1c ξ+ϕ+14λσγ−1−Ωr sin θ4λσγin .(40)They correspond to conservation of the z –component of the angular momentum for both types of particles.It is necessary to stress that the complete nonlinearized system of equations contains no such simple invariants.As σλ≫1,we can neglect in Eqns.(23)–(28)their left-hand sides.In this approximation we have ξ+=ξ−i.e.γ−=γ+=γ,so that −1∂θ+ζr ξr +c Ωr sin θ∂fΩr 2ξθ=0,(42)and γ1−Ωr sin θtan θεf +l (θ)σsin θ(γ−γin ).(45)c1999RAS,MNRAS 000,1–??On the particle acceleration near the light surface of radio pulsars5 Substituting these expressions into(41)and using Eqns.(19)–(22),we obtain the following equation describing the disturbance of the magnetic surfacesε(1−x2sin2θ)∂2fx2∂sinθ∂f∂x−2εsinθcosθ∂fsinθdσ(γ−γin)−sinθ∂θ−2λsin2θ(ξ+r−ξ−r)+2λxsinθ(ξ+ϕ−ξ−ϕ)≈0,(47)so actually there is perfect agreement with the MHD approximationε(1−x2sin2θ)∂2fx2∂sinθ∂f∂x−2εsinθcosθ∂fsinθdσ(γ−γin)−sinθ∂θ=0.As was shown earlier(Beskin et al1998),to pass through the fast magnetosonic surface it’s necessary to have|l|<σ−4/3.(49) Hence,within the fast magnetosonic surface r≪r F one can neglect terms containingδ=εf andζ.Then,relations(41)and (42)result inγ(1−x sinθξϕ)=γin,(50)ξr=ξϕ2ξr−ξ2ϕ,(53) we obtain forσ≫γ3in for r≪r Fγ2=γ2in+x2sin2θ,(54)ξϕ= x sinθ x sinθ,(55)ξr= x2sin2θ x2sin2θ,(56)in full agreement with the MHD results.Next,to determine the position of the fast magnetosonic surface r F,one can analyze the algebraic equations(38)and (41)which give−∂δtanθδ−1xξϕ=0.(57) Using now expressions(43)and(53),one canﬁnd2γ3−2σ K+1∂θ.(59) Equation(58)allows us to determine the position of the fast magnetosonic surface and the energy of particles.Indeed, determining the derivative r∂γ/∂r,one can obtainr ∂γ3γ−σ(2K+x−2).(60)c 1999RAS,MNRAS000,1–??6V.S.Beskin and R.R.RaﬁkovAs the fast magnetosonic surface is the X–point,both the nominator and denominator are to be equal to zero here.As a result,evaluating r∂K/∂r as K,we obtainδ∼σ−2/3;(61) r F∼σ1/3R L;(62)γ(r F)=σ1/3sin2/3θ,(63) where the last expression is exact.These equations coincide with those obtained within the MHD consideration.It is the self–consistent analysis whenδ=εf,and hence K depends on the radius r that results in theﬁnite value for the fast magnetosonic radius r F.On the other hand,in a given monopole magneticﬁeld,whenεf does not depend on the radius,the critical conditions result in r F→∞for a cold outﬂow(Michel,1969,Li et al1992).Near the fast magnetosonic surface r∼σ1/3R L the MHD solution givesγ∼σ1/3,(64)εf∼σ−2/3.(65) Hence,Eqns.(53),(55),and(56)result inξr∼σ−2/3,(66)ξθ∼σ−2/3,(67)ξϕ∼σ−1/3.(68) As we see,theθ–component of the velocity plays no role in the determination of theγ.However,analyzing the left-hand sides of the Eqns.(23)–(28)one can evaluate the additional(nonhydrodynamic)varia-tions of the velocity components∆ξ±r∼λ−1σ−4/3,(69)∆ξ±θ∼λ−1σ−2/3,(70)∆ξ±ϕ∼λ−1σ−1.(71) Hence,for nonhydrodynamic velocities∆ξ±r≪ξr and∆ξ±ϕ≪ξϕto be small,it is necessary to have a large magnetizationparameterσ≫1only.On the other hand,∆ξ±θ/ξθ∼λ−1.In other words,for a highly magnetized plasmaσ≫1even outside the fast magnetosonic surface the velocity components(and,hence,the particle energy)can be considered hydrodynamically. The diﬀerence∼λ−1appears in theθcomponent only,but it does not aﬀect the particle energy.Finally,one can obtain from (39),(40)thatδ−εftanθδ−σ−11∂θ+sinθξr.(75) Together with(21)one can obtain for r≫r Fγ=σ 2cosθεf−εsinθ∂f∂r r2∂f∂θξr+1∂θ(ξϕsinθ)=0.(77) Together with(76)this equation in the limit r≫r F coincides with the asymptotic version of the trans–ﬁeld equation (Tomimatsu1994,Beskin et al1998)ε∂2f∂r−sinθD+1∂θ=0,(78)where g(θ)=K(θ)/sin2θ,andc 1999RAS,MNRAS000,1–??On the particle acceleration near the light surface of radio pulsars7D +1=1∂rr 2∂δ∂θ+1∂θ(ξϕsin θ)+2λ(ξ+r −ξ−r )=0.(80)Indeed,one can see from equations (19)and (20)that near the origin x =R s in the case γ+in =γ−in (and for the small variationof the current ζ∼σ−4/3which is necessary,as was already stressed,to pass through a fast magnetosonic surface)the densityvariation on the surface is large enough:(η+−η−)∼γ−2in ≫ζ.Hence,the derivative ∂2δ/∂r 2here is of the order of γ−2in .Onthe other hand,according to (22),the derivative ε∂2f/∂r 2is x 2times smaller.This means that in the two–component system the longitudinal electric ﬁeld is to appear resulting in a redistribution of the particle energy.Clearly,such a redistribution is impossible for the charge–separated outﬂow.In other words,for a ﬁnite particle energy a one–component plasma cannot maintain simultaneously both the Goldreich charge and Goldreich current density (4).In a two–component system with λ≫1it can be realized by a small redistribution of particle energy (Ruderman &Sutherland 1975,Arons &Scharlemann 1989).For simplicity,let us consider only small distances x ≪1.In this case one can neglect the changes of the magnetic ing now (25)and (26),we haveγ+=γin −4λσδ;(81)γ−=γin +4λσδ.(82)Finally,taking into account that ξθand ξϕare small here,one can obtain from (20)r2∂2δ∂r +1∂θsin θ∂δγ2in,(83)where A =16λ2σ16λ2σ,(86)and µ≈√∂rr2∂δ∂θξr +1∂θ(ξϕsin θ)=0.(89)c1999RAS,MNRAS 000,1–??8V.S.Beskin and R.R.RaﬁkovAsδ∼εf≪σ−2/3for r≪r F,andξr∼γ−20≫δ,theﬁrst term in(89)can be omitted.As a result,the solution of Eqn.(89)coincides exactly with the MHD expression,i.e.γ2=γ2in+x2sin2θ(54).Finally,using(87),(88),and(55)–(56), one can easily check that the nonhydrodynamical terms(47)in the trans–ﬁeld equation(48)do actually vanish.4THE BOUNDARY LAYERLet us now consider the case when the longitudinal electric current I(R,θ)in the magnetosphere of radio pulsars is too small (i.e.the disturbance l(θ)is too large)for theﬂow to pass smoothly through the fast magnetosonic surface.First of all,it can be realized when the electric current is much smaller than the Goldreich one.This possibility was already discussed within the Ruderman–Sutherland model of the internal gap(Beskin et al1983,Beskin&Malyshkin1998).But it may take place in the Arons model(Arons&Scharlemann1979)as well.Indeed,within this model the electric current is determined by the gap structure.Hence,in general case this current does not correspond to the critical condition at the fast magnetosonic surface.In particular,it may be smaller than the critical current(of course,the separate consideration is necessary to check this statement).For simplicity let us consider the case l(θ)=h sin2θ.Neglecting now the last terms∝σ−1in the trans–ﬁeld equation (48),we obtainε(1−x2sin2θ)∂2fx2∂sinθ∂f∂x−2εsinθcosθ∂f(2|h|)1/4.(92) As we see,for l(θ)=h sin2θthis surface has the form of a cylinder.It is important that the disturbance of magnetic surfaces εf∼(|h|)1/2remains small here.Comparing now the position of the light surface(92)with that of the fast magnetosonic surface(62),one canﬁnd that the light surface is located inside the fast magnetosonic one ifσ−4/3≪|h|≪1,(93) which is opposite to(49).One can check that the condition(93)just allows to neglect the non force–free term in Eqn.(48).Using now the solution(91)and the MHD conditionδ=εf,one canﬁnd from(58)2γ3−2σ hx2sin4θ+1√x0−x sinθ,(95) wherex0=14(2|h|)1/21On the particle acceleration near the light surface of radio pulsars 9 Figure1.The behavior of the Lorentz factor in the caseσ−4/3≪|h|≪1.One can see that the one–ﬂuid MHD solution(95)existsforγ<σ1/3only.But in the two–ﬂuid approximation in the narrow layer∆̟=̟c/λthe particle energy increases up to the value∼σcorresponding to the full conversion of the electromagnetic energy to the energy of particles.invariants(39)and(40)can be used to deﬁneξ±ϕ:ξ+ϕ=1γ+ ;(99)ξ−ϕ=1γ− .(100)Furthermore,one can deﬁne2ξ+r=1(γ−)2+(ξ−ϕ)2+(ξ−θ)2.(102) As to the energy integral(38),it determines the variation of the currentζ.Now it can be rewritten asζ=22σsinθ.(103)Finally,Eqns.(19)–(28)look like̟2c d2δ2cosθ ξ+θ− λ+12cosθ ξ+r− λ+1d̟2=−2sin2θ λ−12cosθ ξ−ϕ ,(105)̟c d2σsinθ−̟ccosθd̟−sinθξ+r+sinθd̟ ξ−θγ− =−4λσ −γ++γ−sinθdδx0ξ−ϕ ,(107)̟cdd̟−sinθξ+θ,(108)c 1999RAS,MNRAS000,1–??10V.S.Beskin and R.R.Raﬁkov̟c dd̟−sinθξ−θ,(109)where we neglected the terms∝δ/r in(106)and(107).Comparing the leading terms,we have inside the layer∆̟/R L∼λ−1γ±∼h1/2cσ,(110)ξ±θ∼h1/4c,(111)ξ±r∼h1/2c,(112)∆δ∼h3/4c/λ,(113) where h c=|h|.Then the leading terms in(99)–(103)for∆̟>λ−1R L areξ+ϕ=1x0,(114)ξ−ϕ=1x0,(115)2ξ+r=(ξ+ϕ)2+(ξ+θ)2,(116) 2ξ−r=(ξ−ϕ)2+(ξ−θ)2,(117)ζ=−(γ++γ−)d̟2=2λsinθcosθ(ξ+θ−ξ−θ),(119)̟c d2σsinθ−̟ccosθd̟−sinθξ+r,(120)̟c d2σsinθ−̟ccosθd̟−sinθξ−r,(121)̟cdd̟ γ− =4λσsinθξ−θ,(123)with all the terms in the right–hand sides of(120)and(121)being of the same order of magnitude.As a result,the nonlinear equations(119)–(123)and(105)give the following simple asymptotic solutionγ±=4sin2θσ(λl)2,(124)ξ±θ=∓2sinθλl,(125)∆δ=−4On the particle acceleration near the light surface of radio pulsars11F(rad) x =−2m2c4γ2 (E y−B z)2+(E z−B y)2 ,(129)which can be important for large enough particle paring(129)with appropriate terms in(120)–(123)one can conclude that the radiation force can be neglected forσ<σcr,whereσcr= cc <3×10−3B−3/712λ2/74(131)which givesP>0.06B3/712λ−2/74s.(132)Hence,for most radio pulsars the radiation force indeed can be neglected.As to the pulsars withσ>σcr,it is clear that for γ>σcr the radiation force becomes larger than the electromagnetic one and strongly inhibits any further acceleration.As a result,we can evaluate the maximum gamma–factor which can be reached during the acceleration asγmax≈σcr≈106.(133)5DISCUSSIONThus,on a simple example it was demonstrated that for real physical parameters of the magnetosphere of radio pulsars (σ≫1andλ≫1)the one–ﬂuid MHD approximation remains true in the whole region within the light surface|E|=|B|. On the other hand,it was shown that in a more realistic2D case the main properties of the boundary layer near the light surface existing for small enough longitudinal currents I<I GJ(eﬀective energy transformation from electromagneticﬁeld to particles,current closure in this region,smallness of the disturbance of electric potential and poloidal magneticﬁeld)remain the same as in the1D case considered previously(Beskin et al1983).It is necessary to stress the main astrophysical consequences of our results.First of all,the presence of such a boundary layer explains the eﬀective energy transformation of electromagnetic energy into the energy of particles.As was already stressed,now the existence of such an acceleration is conﬁrmed by observations of close binaries containing radio pulsars(as to the particle acceleration far from a neutron star,see e.g.Kennel&Coroniti1984,Hoshino et al1992,Gallant&Arons 1994).Simultaneously,it allows us to understand the current closure in the pulsar magnetosphere.Finally,particle acceleration results in the additional mechanism of high–energy radiation from the boundary of the magnetosphere(for more details see Beskin et al1993).Nevertheless,it is clear that the results obtained do not solve the whole pulsar wind problem.Indeed,as in the cylindrical case,it is impossible to describe the particle motion outside the light surface.The point is that,as one can see directly from Eqn.(126),for a complete conversion of electromagnetic energy into the energy of particles it is enough for them to pass onlyλ−1of the total potential drop between pulsar magnetosphere and inﬁnity.It means that the electron–positron wind propagating to inﬁnity has to pass the potential drop which is much larger than their energy.It is possible only in the presence of electromagnetic waves even in an axisymmetric magnetosphere which is stationary near the origin.Clearly,such aﬂow cannot be considered even within the two–ﬂuid approximation.In our opinion,it is only a numerical consideration that can solve the problem completely and determine,in particular,the energy spectrum of particles and the structure of the pulsar wind.Unfortunately,up to now such numerical calculations are absent.ACKNOWLEDGMENTSThe authors are grateful to I.Okamoto and H.Sol for fruitful discussions.VSB thanks National Astronomical Observatory, Japan for hospitality.This work was supported by INTAS Grant96–154and by Russian Foundation for Basic Research(Grant 96–02–18203).REFERENCESArons J.,Scharlemann E.T.,1979,ApJ,231,854Begelman M.C.,Li Z.-Y.,1994,ApJ,426,269Beskin V.S.,Gurevich A.V.,Istomin Ya.N.,1983,Soviet Phys.JETP,58,235Beskin V.S.,Gurevich A.V.,Istomin Ya.N.,1993,Physics of the Pulsar Magnetosphere,Cambridge Univ.Press,CambridgeBeskin V.S.,Kuznetsova I.V.,Raﬁkov R.R.,1998,MNRAS299,341c 1999RAS,MNRAS000,1–??12V.S.Beskin and R.R.RaﬁkovBeskin V.S.,Malyshkin L.M.,1998,MNRAS298,847Bogovalov S.V.,1997,A&A,327,662Djorgovsky S.,Evans C.R.,1988,ApJ,335,L61Gallant Y.A.,Arons J.,1994,ApJ,435,230Goldreich P.,Julian,W.H.,1969,ApJ,157,869Henriksen R.N.,Norton J.A.,1975,ApJ,201,719Heyvaerts J.,1996,in Chiuderi C.,Einaudi G.,ed,Plasma Astrophysics,Springer,Berlin,p.31Hoshino M.,Arons J.,Gallant Y.A.,Langdon A.B.,1992,ApJ,390,454Kennel C.F.,Coroniti,F.V.,1984,ApJ,283,694Kulkarni S.R.,Hester J.,1988,Nature,335,801Li Zh.–Yu.,Chiueh T.,Begelman M.C.,1992,ApJ,394,459Mestel L.,1999,Cosmical Magnetism,Clarendon Press,OxfordMestel L.,Shibata S.,1994,MNRAS,271,621Michel F.C.,1969,ApJ,158,727Michel F.C.,1973,ApJ,180,L133Michel F.C.,1991,Theory of Neutron Star Magnetosphere,The Univ.of Chicago Press,ChicagoOkamoto I.,1978,MNRAS,185,69Ruderman M.A.,Sutherland P.G.,1975,ApJ,196,51Shibata S.,1997,MNRAS,287,262Tomimatsu A.,1994,Proc.Astron.Soc.Japan,46,123c 1999RAS,MNRAS000,1–??。

Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory

† ∗
Contents
1 Introduction and summary 2 Quantum ﬁeld theory on noncommutative spaces 2.1 2.2 2.3 2.4 Sc . . . . . . . . . . . . . . . . . . . . . . . . . . Noncommutative Yang-Mills theory . . . . . . . . . . . . . . . . . . . . . . . Star-gauge invariant observables . . . . . . . . . . . . . . . . . . . . . . . . . The noncommutative torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 9 10 13 14 15 17 23 26 27 31 33 34 38 40 41 44
J. Ambjørn1) ∗ , Y.M. Makeenko1) 2) † , J. Nishimura1) ‡ and R.J. Szabo1) §
1)
The Niels Bohr Institute Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
2)
Institute of Theoretical and Experimental Physics B. Cheremushkinskaya 25, 117218 Moscow, Russia
2
geometry provides a natural framework to describe nonperturbative aspects of string theory [2, 5]. This belief is further supported by the fact that Matrix Theory [6] and the IIB matrix model [7], which are conjectured to provide nonperturbative deﬁnitions of string theories, give rise to noncommutative Yang-Mills theory on toroidal compactiﬁcations [8]. The particular noncommutative toroidal compactiﬁcation is interpreted as being the result of the presence of a background Neveu-Schwarz two-form ﬁeld, and it can also be understood in the context of open string quantization in D-brane backgrounds [9, 10]. Furthermore, in Ref. [11] it has been shown that the IIB matrix model with D-brane backgrounds is described by noncommutative Yang-Mills theory. The early motivation [12] for studying quantum ﬁeld theory on noncommutative spacetimes was that, because of the spacetime uncertainty relation, the introduction of noncommutativity would provide a natural ultraviolet regularization. However, more recent perturbative calculations [13]–[16] have shown that planar noncommutative Feynman diagrams contain exactly the same ultraviolet divergences that their commutative counterparts do, which implies that the noncommutativity does not serve as an ultraviolet regulator. One therefore needs to introduce some other form of regularization to study the dynamics of noncommutative ﬁeld theories. On the other hand, it has been found that the ultraviolet divergences in non-planar Feynman diagrams [16, 17] exhibit an intriguing mixing of ultraviolet and infrared scales, which can also be described using string-theoretical approaches [18, 19]. Heuristically, this UV/IR mixing can be understood in terms of the induced uncertainty relations among the spacetime coordinates. If one measures a given spacetime coordinate with some high precision, then the remaining spacetime directions will generally extend because of the smearing. Furthermore, noncommutative solitons which do not have counterparts in ordinary ﬁeld theory have been discovered [20] for suﬃciently large values of the noncommutativity parameters, and it has also been shown [19] that noncommutative Yang-Mills theory in four dimensions naturally includes gravity. In order to investigate further the non-trivial dynamics of noncommutative ﬁeld theories, it is important therefore to develop a nonperturbative regularization of these theories. Such a program has been put forward in Refs. [11, 15, 19],[21]–[24] and it is similar to earlier works [25] based on the mapping between large N matrices and spacetime ﬁelds. In particular, in Ref. [22] a uniﬁed framework was presented which naturally interpolates between the two ways that noncommutative Yang-Mills theory has appeared in the context of matrix model formulations of string theory, namely the compactiﬁcation of Matrix theory and the twisted large N reduced model. The model proposed was a ﬁnite N matrix model deﬁned by the twisted Eguchi-Kawai model [26, 27] with a quotient condition analogous to the ones considered in Refs. [8, 28]. It was interpreted as a lattice formulation of noncommutative

船舶英语考试(试卷编号111)

船舶英语考试(试卷编号111)1.[单选题]Electro-technical officers are working under the leadship of the ______.A)masterB)shipC)engineerD)chief答案:D解析:2.[单选题]The primary function of the GMDSS is ______.A)dailyB)bridgeC)MSID)distress答案:D解析:【注】GMDSS的主要功能是遇险船舶的遇险通信。

3.[单选题]_____ is NOT a part of the main switchboard.A)Bus-barB)LoadC)ParallelingD)Shore答案:D解析:【注】shore connection box：岸电接线箱4.[单选题]The echo sounder is widely used to measure the depth of waterway. The incorrect description about this system is ______.A)theB)itC)theD)in答案:D解析:5.[单选题]The letter F in MF and HF stands for ______.A)faradB)FrenchC)frequency6.[单选题]The preference trip of automatic power plant can make some load switches______ when the generator overloads.A)closeB)tripC)toD)to答案:B解析:7.[单选题]The difference between magnetic heading and compass heading is called ______.A)variationB)deviationC)compassD)drift答案:B解析:8.[单选题]The term "oil", as used in the Pollution Prevention Regulations, means _____.A)fuelB)crudeC)liquefiedD)petroleum答案:D解析:9.[单选题]All echo-sounders can measure the ______.A)actualB)actualC)averageD)average答案:B解析:10.[单选题]Which one of the following electric power systems is not used in the marine high voltage power plant?A)TheB)TheC)The11.[单选题]In_____ the control action is independent of the output.A)aB)anC)aD)a答案:B解析:12.[单选题]The steering gear provides a movement to the rudder in response to a signal from the______.A)bridgeB)MCRC)transmissionD)engine答案:A解析:13.[单选题]If both the "high level" and "low level" alarms come on for the same address of a centralized control console, the most likely problem is a/an _____.A)sensorB)failedC)lowD)extremely答案:A解析:14.[单选题]The LAN can be extended by certain mechanism and devices, such as ______.A)fiberB)modemsC)repeatersD)all答案:D解析:15.[单选题]The Integrated Bridge System is used to______.A)maintainB)monitorC)plan16.[单选题]The resistance of electric wire will decrease as its _____.A)lengthB)cross-sectionalC)temperatureD)percent答案:B解析:【注】cross-sectional area：横截面积；metallic purities：金属纯度17.[单选题]The International Sewage Pollution Prevention Certificate shall be issuedfor a period specified by the Administration, which shall not exceed from the date of issue.A)sixB)oneC)threeD)five答案:D解析:18.[单选题]The ______ can change an AC power source into a DC one.A)rectifierB)ransformerC)convertorD)invertor答案:A解析:19.[单选题]Which component is NOT equipped in the engine control room AutoChief Control Panel?A)TheB)TheC)AD)A答案:B解析:20.[单选题]Modem is ______.A)aB)a21.[单选题]The main objective of the SOLAS Convention is ______.A)toB)toC)toD)to答案:C解析:22.[单选题]A printer would be considered a(n) ______.A)controllerB)peripheralC)inputD)US答案:B解析:23.[单选题]What is international NAVTEX based on?A)TerrestrialB)NBDPC)Satellite.D)Digital答案:B解析:【注】国际NAVTEX业务是基于NBDP技术的。

Classical General Relativity

a rXiv:g r-qc/5665v 218Aug25To appear in:Handlbook of the Philosophy of Physics ,eds.J.Butterﬁeld and J.Earman,Elsevier∗I am grateful to Jeremy Butterﬁeld,Erik Curiel,and John Earman for comments on earlier drafts.1Contents1Introduction32The Structure of Relativity Theory42.1Relativistic Spacetimes (4)2.2Proper Time (9)2.3Space/Time Decomposition at a Point and Particle Dynamics..132.4Matter Fields (16)2.5Einstein’s Equation (21)2.6Congruences of Timelike Curves and“Public Space” (28)2.7Killing Fields and Conserved Quantities (32)3Special Topics363.1Relative Simultaneity in Minkowski Spacetime (36)3.2Geometrized Newtonian Gravitation Theory (44)3.3Recovering Global Geometric Structure from“Causal Structure”5221IntroductionThe essay that follows is divided into two parts.In theﬁrst,I give a brief account of the structure of classical relativity theory.1In the second,I discuss three special topics.My account in theﬁrst part(section2)is limited in several respects.I do not discuss the historical development of classical relativity theory,nor the evidence we have for it.I do not treat“special relativity”as a theory in its own right that is superseded by“general relativity”.And I do not describe known exact solutions to Einstein’s equation.(This list could be continued at great length.2) Instead,I limit myself to a few fundamental ideas,and present them as clearly and precisely as I can.The account presupposes a good understanding of basic diﬀerential geometry,and at least passing acquaintance with relativity theory itself.3In section3,Iﬁrst consider the status of the relative simultaneity relation in the context of Minkowski spacetime.At issue is whether the standard relation, the one picked out by Einstein’s“deﬁnition”of simultaneity,is conventional in character,or is rather in some signiﬁcant sense forced on us.Then I describe the “geometrized”version of Newtonian gravitation theory(also known as Newton-Cartan theory).It is included here because it helps to clarify what is and is not distinctive about classical relativity theory.Finally,I consider to what extent the global geometric structure of spacetime can be recovered from its“causal structure”.42The Structure of Relativity Theory2.1Relativistic SpacetimesRelativity theory determines a class of geometric models for the spacetime struc-ture of our universe(and subregions thereof such as,for example,our solar sys-tem).Each represents a possible world(or world-region)compatible with the constraints of the theory.It is convenient to describe these models in stages. We start by characterizing a broad class of“relativistic spacetimes”,and dis-cussing their ter we introduce further restrictions involving global spacetime structure and Einstein’s equation.We take a relativistic spacetime to be a pair(M,g ab),where M is a smooth, connected,four-dimensional manifold,and g ab is a smooth,semi-Riemannian metric on M of Lorentz signature(1,3).5We interpret M as the manifold of point“events”in the world.6The in-terpretation of g ab is given by a network of interconnected physical principles. We list three in this section that are relatively simple in character because they make reference only to point particles and light rays.(These objects alone suf-ﬁce to determine the metric,at least up to a constant.)In the next section,we list a fourth that concerns the behavior of(ideal)clocks.Still other principles involving generic matterﬁelds will come up later.We begin by reviewing a few deﬁnitions.In what follows,let(M,g ab)be a ﬁxed relativistic spacetime,and let∇a be the derivative operator on M deter-mined by g ab,i.e.,the unique(torsion-free)derivative operator on M satisfyingthe compatibility condition∇a g bc=0.Given a point p in M,and a vectorηa in the tangent space M p at p,we say ηa is:timelike ifηaηa>0null(or lightlike)ifηaηa=0causal ifηaηa≥0spacelike ifηaηa<0.In this way,g ab determines a“null-cone structure”in the tangent space at every point of M.Null vectors form the boundary of the cone.Timelike vectors form its interior.Spacelike vectors fall outside the cone.Causal vectors are those that are either timelike or null.This classiﬁcation extends naturally to curves.We take these to be smooth maps of the formγ:I→M where I⊆R is a(possibly inﬁnite,not necessarily open)interval.7γqualiﬁes as timelike(respectively null, causal,spacelike)if its tangent vectorﬁeld γis of this character at every point.A curveγ2:I2→M is called an(orientation preserving)reparametrization of the curveγ1:I1→M if there is a smooth mapτ:I2→I1of I2onto I1,with positive derivative everywhere,such thatγ2=(γ1◦τ).The property of being timelike,null,etc.is preserved under reparametrization.8So there is a clear sense in which our classiﬁcation also extends to images of curves.9A curveγ:I→M is said to be a geodesic(with respect to g ab)if its tangent ﬁeldξa satisﬁes the condition:ξn∇nξa=0.The property of being a geodesic is not,in general,preserved under reparametrization.So it does not transfer to curve images.But,of course,the related property of being a geodesic up to reparametrization does carry over.(The latter holds of a curve if it can be reparametrized so as to be a geodesic.)Now we can state theﬁrst three interpretive principles.For all curvesγ: I→M,C1γis timelike iﬀits imageγ[I]could be the worldline of a massive point particle(i.e.,a particle with positive mass);10I⊆R,with I⊂γ:γ(s)=γ(s)for all s∈I.8This follows from the fact that,in the case just described, γ2=dτ>0.ds 9The diﬀerence between curves and curve images,i.e.,between mapsγ:I→M and sets γ[I],matters.We take worldlines to be instances of the latter,i.e.,construe them as point sets rather than parametrized point sets.10We will later discuss the concept of mass in relativity theory.For the moment,we take it to be just a primitive attribute of particles.5C2γcan be reparametrized so as to be a null geodesic iﬀγ[I]could be the trajectory of a light ray;11P1γcan be reparametrized so as to be a timelike geodesic iﬀγ[I]could be the worldline of a free12massive point particle.In each case,a statement about geometric structure(on the left)is correlated with a statement about the behavior of particles or light rays(on the right).Several comments and qualiﬁcations are called for.First,we are here work-ing within the framework of relativity as traditionally understood,and ignoring speculations about the possibility of particles(“tachyons”)that travel faster than light.(Their worldlines would come out as images of spacelike curves.) Second,we have built in the requirement that“curves”be smooth.So,depend-ing on how one models collisions of point particles,one might want to restrict attention here to particles that do not experience collisions.Third,the assertions require qualiﬁcation because the status of“point par-ticles”in relativity theory is a delicate matter.At issue is whether one treats a particle’s own mass-energy as a source for the surrounding metricﬁeld g ab–in addition to other sources that may happen to be present.(Here we anticipate our discussion of Einstein’s equation.)If one does,then the curvature associated with g ab may blow up as one approaches the particle’s worldline.And in this case one cannot represent the worldline as the image of a curve in M,at least not without giving up the requirement that g ab be a smoothﬁeld on M.For this reason,a more careful formulation of the principles would restrict attention to“test particles”,i.e.,ones whose own mass-energy is negligible and may be ignored for the purposes at hand.Fourth,the modal character of the assertions(i.e.,the reference to possibility) is essential.It is simply not true,to take the case of C1,that all timelike curve images are,in fact,the worldlines of massive particles.The claim is that,asleast so far as the laws of relativity theory are concerned,they could be.Of course,judgments concerning what could be the case depend on what conditions are heldﬁxed in the background.The claim that a particular curve image could be the worldline of a massive point particle must be understood to mean that it could so long as there are,for example,no barriers in the way.Similarly, in C2there is an implicit qualiﬁcation.We are considering what trajectories are available to light rays when no intervening material media are present,i.e., when we are dealing with light rays in vacua.Though these four concerns are important and raise interesting questions about the role of idealization and modality in the formulation of physical theory, they have little to do with relativity theory as such.Similar diﬃculties arise when one attempts to formulate corresponding principles within the framework of Newtonian gravitation theory.It follows from the cited interpretive principles that the metric g ab is deter-mined(up to a constant)by the behavior of point particles and light rays.We make this claim precise in a pair of propositions about“conformal structure”and“projective structure”.Let¯g ab be a second smooth metric of Lorentz signature on M.We say that ¯g ab is conformally equivalent to g ab if there is a smooth mapΩ:M→R on M such that¯g ab=Ω2g ab.(Ωis called a conformal factor.It certainly need not be constant.)Clearly,if¯g ab and g ab are conformally equivalent,then they agree in their classiﬁcation of vectors and curves as timelike,null,etc..The converse is true as well.13Conformally equivalent metrics on M do not agree,in general, as to which curves on M qualify as geodesics or even just as geodesics up to reparametrization.But,it turns out,they do necessarily agree as to which null curves are geodesics up to reparametrization.14And the converse is true,onceagain.15Putting the pieces together,we have the following proposition.Clauses(1) and(2)correspond to C1and C2respectively.Proposition2.1.1.Let¯g ab be a second smooth metric of Lorentz signature on M.Then the following conditions are equivalent.(1)¯g ab and g ab agree as to which curves on M are timelike.(2)¯g ab and g ab agree as to which curves on M can be reparameterized so asto be null geodesics.(3)¯g ab and g ab are conformally equivalent.In this sense,the spacetime metric g ab is determined up to a conformal factor, independently,by the set of possible worldlines of massive point particles,and by the set of possible trajectories of light rays.Next we turn to projective structure.Let∇a and∇a are projectively equivalent if they agree as to which curves are geodesics up to reparametrization(i.e.,if,for all curvesγ,γcan be reparametrized so as to be a geodesic with respect to∇a is projectively equivalent to∇a.It is a basic result,due to Hermann Weyl[1921],that if¯g ab and g ab are conformally and projectively equivalent,then the conformal factor that relates them must be constant.It is convenient for our purposes,with interpretive principle P1in mind,to cast it in a slightly altered form that makes reference only to timelike geodesics(rather than arbitrary geodesics).Proposition2.1.2.Let¯g ab be a second smooth metric on M with¯g ab=Ω2g ab. If¯g ab and g ab agree as to which timelike curves can be reparametrized so as to be geodesics,thenΩis constant.The spacetime metric g ab,we saw,is determined up to a conformal factor, independently,by the set of possible worldlines of massive point particles,and by the set of possible trajectories of light rays.The proposition now makes clear the sense in which it is fully determined(up to a constant)by those sets together with the set of possible worldlines of free massive particles.16Our characterization of relativistic spacetimes is extremely loose.Many fur-ther conditions might be imposed.For the moment,we consider just one.(M,g ab)is said to be temporally orientable if there exists a continuous time-like vectorﬁeldτa on M.Suppose the condition is satisﬁed.Then two such ﬁeldsτa andˆτa on M are said to be co-oriented ifτaˆτa>0everywhere,i.e., ifτa andˆτa fall in the same lobe of the null-cone at every point of M.Co-orientation is an equivalence relation(on the set of continuous timelike vector ﬁelds on M)with two equivalence classes.A temporal orientation of(M,g ab) is a choice of one of those two equivalence classes to count as the“future”one. Thus,a non-zero causal vectorξa at a point of M is said to be future directed or past directed with respect to the temporal orientation T depending on whether τaξa>0orτaξa<0at the point,whereτa is any continuous timelike vector ﬁeld in T.Derivatively,a causal curveγ:I→M is said to be future directed (resp.past directed)with respect to T if its tangent vectors at every point are.In what follows,we assume that our background spacetime(M,g ab)is tem-porally orientable,and that a particular temporal orientation has been speciﬁed. Also,given events p and q in M,we write p≪q(resp.p<q)if there is a future-directed timelike(resp.causal)curve that starts at p and ends at q.172.2Proper TimeSo far we have discussed relativistic spacetime structure without reference to either“time”or“space”.We come to them in this section and the next.Letγ:[s1,s2]→M be a future-directed timelike curve in M with tangent ﬁeldξa.We associate with it an elapsed proper time(relative to g ab)given by|γ|= s2s1(g abξaξb)1...it can be shown that the metrical structure of the world is already fullydetermined by its inertial and causal structure,that therefore mensuration neednot depend on clocks and rigid bodies but that light signals and mass pointsmoving under the inﬂuence of inertia alone will suﬃce.(For more on Weyl’s“causal-inertial”method of determining the spacetime metric,see Cole-man and Kort´e[2001,section4.9].)17It follows immediately that if p≪q,then p<q.The converse does not hold,in general. But the only way the second condition can be true,without theﬁrst being true as well,is if the only future-directed causal curves from p to q are null geodesics(or reparametrizations of null geodesics).See Hawking and Ellis[1972,p.112].9This elapsed proper time is invariant under reparametrization ofγ,and is just what we would otherwise describe as the length of(the image of)γ.The fol-lowing is another basic principle of relativity theory.P2Clocks record the passage of elapsed proper time along their worldlines.Again,a number of qualiﬁcations and comments are called for.Our formu-lation of C1,C2,and P1was rough.The present formulation is that much more so.We have taken for granted that we know what“clocks”are.We have assumed that they have worldlines(rather than worldtubes).And we have over-looked the fact that ordinary clocks(e.g.,the alarm clock on the nightstand)do not do well at all when subjected to extreme acceleration,tidal forces,and so forth.(Try smashing the alarm clock against the wall.)Again,these concerns are important and raise interesting questions about the role of idealization in the formulation of physical theory.(One might construe an“ideal clock”as a point-sized test object that perfectly records the passage of proper time along its worldline,and then take P2to assert that real clocks are,under appropriate conditions,to varying degrees of accuracy,approximately ideal.)But as with our concerns about the status of point particles,they do not have much to do with relativity theory as such.Similar ones arise when one attempts to for-mulate corresponding principles about clock behavior within the framework of Newtonian theory.Now suppose that one has determined the conformal structure of spacetime, say,by using light rays.Then one can use clocks,rather than free particles, to determine the conformal factor.One has the following simple result,which should be compared with proposition2.1.2.18Proposition2.2.1.Let¯g ab be a second smooth metric on M with¯g ab=Ω2g ab. Further suppose that the two metrics assign the same lengths to all timelikecurves,i.e.,|γ|¯gab =|γ|gabfor all timelike curvesγ:I→M.ThenΩ=1everywhere.(Here|γ|gabis the length ofγrelative to g ab.)P2gives the whole story of relativistic clock behavior(modulo the concerns noted above).In particular,it implies the path dependence of clock readings.If two clocks start at an event p,and travel along diﬀerent trajectories to an eventq ,then,in general,they will record diﬀerent elapsed times for the trip.(E.g.,one will record an elapsed time of 3,806seconds,the other 649seconds.)This is true no matter how similar the clocks are.(We may stipulate that they came oﬀthe same assembly line.)This is the case because,as P2asserts,the elapsed time recorded by each of the clocks is just the length of the timelike curve it traverses in getting from p to q and,in general,those lengths will be diﬀerent.Suppose we consider all future-directed timelike curves from p to q .It is natural to ask if there are any that minimize or maximize the recorded elapsed time between the events.The answer to the ﬁrst question is ‘no’.Indeed,one has the following proposition.Proposition 2.2.2.Let p and q be events in M such that p ≪q .Then,for all ǫ>0,there exists a future-directed timelike curve γfrom p to q with |γ|<ǫ.(But there is no such curve with length 0,since all timelike curves have non-zero length.)Though some work is required to give the proposition an honest proof (see O’Neill [1983,pp.294-5]),it should seem intuitively plausible.If there is a timelike curve connecting p and q ,there also exists a jointed,zig-zag null curve that connects them.It has length 0.But we can approximate the jointed nullpqshort timelike curvelong timelike curve Figure 2.2.1:A long timelike curve from p to q and a veryshort one that swings back-and-forth,and approximates abroken null curve.curve arbitrarily closely with smooth timelike curves that swing back and forth.So (by the continuity of the length function),we should expect that,for all ǫ>0,there is an approximating timelike curve that has length less than ǫ.(See ﬁgure 2.2.1.)The answer to the second question (Can one maximize recorded elapsed time between p and q ?)is ‘yes’if one restricts attention to local regions of spacetime.11In the case of positive deﬁnite metrics,i.e.,ones with signature of form(n,0), we know,geodesics are locally shortest curves.The corresponding result for Lorentz metrics is that timelike geodesics are locally longest curves.Proposition2.2.3.Letγ:I→M be a future-directed timelike curve.Then γcan be reparametrized so as to be a geodesic iﬀfor all s∈I,there exists an open set O containingγ(s)such that,for all s1,s2∈I with s1≤s≤s2,if the image ofγ(and its reparametrizations)areis longer than all other timelike curves in O fromγ(s1)toγ(s2).(Hereγ|[s1,s2] the restriction ofγto the interval[s1,s2].)The proof of the proposition is very much the same as in the positive deﬁnite case.(See Hawking and Ellis[1972,p.105].)Thus of all clocks passing locally from p to q,that one will record the greatest elapsed time that“falls freely”from p to q.To get a clock to read a smaller elapsed time than the maximal value one will have to accelerate the clock.Now acceleration requires fuel,and fuel is not free.So proposition2.2.3has the consequence that(locally)“saving time costs money”.And proposition2.2.2may be taken to imply that(locally)“with enough money one can save as much time as one wants”.The restriction here to local regions of spacetime is essential.The connection described between clock behavior and acceleration does not,in general,hold on a global scale.In some relativistic spacetimes,one canﬁnd future-directed timelike geodesics connecting two events that have diﬀerent lengths,and so clocks following the curves will record diﬀerent elapsed times between the events even though both are in a state of free fall.Furthermore–this follows from the preceding claim by continuity considerations alone–it can be the case that of two clocks passing between the events,the one that undergoes acceleration during the trip records a greater elapsed time than the one that remains in a state of free fall.The connection we have been considering between clock behavior and accel-eration was once thought to be paradoxical.(I am thinking of the“clock(or twin)paradox”.)Suppose two clocks,A and B,pass from one event to another in a suitably small region of spacetime.Further suppose A does so in a state of free fall,but B undergoes acceleration at some point along the way.Then,we know,A will record a greater elapsed time for the trip than B.This was thought paradoxical because it was believed that“relativity theory denies the possibility of distinguishing“absolutely”between free fall motion and accelerated motion”. (If we are equally well entitled to think that it is clock B that is in a state of free fall,and A that undergoes acceleration,then,by parity of reasoning,it should12be B that records the greater elapsed time.)The resolution of the paradox,if one can call it that,is that relativity theory makes no such denial.The situa-tions of A and B here are not symmetric.The distinction between accelerated motion and free fall makes every bit as much sense in relativity theory as it does in Newtonian physics.In what follows,unless indication is given to the contrary,a“timelike curve”should be understood to be a future-directed timelike curve,parametrized by elapsed proper time,i.e.,by arc length.In that case,the tangentﬁeldξa of the curve has unit length(ξaξa=1).And if a particle happens to have the image of the curve as its worldline,then,at any point,ξa is called the particle’s four-velocity there.2.3Space/Time Decomposition at a Point and ParticleDynamicsLetγbe a timelike curve representing the particle O with four-velocityﬁeldξa. Let p be a point on the image ofγ,and letλa be a vector at p.There is a natural decomposition ofλa into components parallel to,and orthogonal to,ξa:λa=(λbξb)ξaparallel toξa +(λa−(λbξb)ξa)orthogonal toξa.(1)These are standardly interpreted,respectively,as the“temporal”and“spatial”components ofλa(relative toξa).In particular,the three-dimensional subspace of M p consisting of vectors orthogonal toξa is interpreted as the“inﬁnitesimal”simultaneity slice of O at p.19If we introduce the tangent and orthogonal projection operatorsk ab=ξaξb(2)h ab=g ab−ξaξb(3) then the decomposition can be expressed in the formλa=k a bλb+h a bλb.(4) We can think of k ab and h ab as the relative temporal and spatial metrics deter-mined byξa.They are symmetric and satisfyk a b k b c=k a c(5)h a b h b c=h a c.(6)Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas.For ex-ample,suppose that the worldline of a second particleξa.(Sinceξa andξa)>0.)We compute the speed ofξa relative to O and divide by its temporal magnitude relative to O:v=speed of ξbξb .(7)(Given any vectorµa,we understand µa to be(µaµa)12if it is spacelike.)From(2),(3),(5),and(6),we havek a bξb k ac2=(k bcξc)1ξbξb)(8) andh a bξb h ac2=(−h bcξc)1ξbξb)2−1)1ξbξb)2−1)1(ξbξb=11−v2.(11)It is a basic fact of relativistic life that there is associated with every point particle,at every event on its worldline,a four-momentum(or energy-momentum)vector P a.In the case of a massive particle with four-velocityξa,and the(positive)proportionality factor is just what we would otherwise call the mass(or rest mass)m of the particle.So we have P a=m2.(It is strictly positive in theﬁrst case,and0in the second.)14Now suppose a massive particle O has four-velocityξa at an event,and another particle,either a massive particle or a photon,has four-momentum P a there.We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose P a in terms ofξa.By(4) and(2),we haveP a=(P bξb)ξa+h a b P b.(12)The energy relative to O is the coeﬃcient in theﬁrst term:E=P bξb.In the case of a massive particle where P a=mξbξb)=m1−v2.(13)(If we had not chosen units in which c=1,the numerator in theﬁnal expression would have been mc2and the denominator c2.)The three-momentum relative to O is the second term in the decomposition,i.e.,the component of P a orthogonal toξa:h a b P b.In the case of a massive particle,by(9)and(11),it has magnitudep= h a b mξbξb)2−1)1√2ξn∇n(ξaξa)=1things backwards.Between jump and arrival you are not accelerating.You are in a state of free fall and moving(approximately)along a spacetime geodesic. But before the jump,and after the arrival,you are accelerating.Theﬂoor of the observation desk,and then later the sidewalk,push you away from a geodesic path.The all-important idea here is that we are incorporating the“gravita-tionalﬁeld”into the geometric structure of spacetime,and particles traverse geodesics if and only if they are acted upon by no forces“except gravity”.The acceleration of any massive particle,i.e.,its deviation from a geodesic trajectory,is determined by the forces acting on it(other than“gravity”).If the particle has mass m>0,and the vectorﬁeld F a onγ[I]represents the vector sum of the various(non-gravitational)forces acting on the particle,then the particle’s four-accelerationξn∇nξa satisﬁes:F a=mξn∇nξa.(15)This is our version of Newton’s second law of motion.Consider an example.Electromagneticﬁelds are represented by smooth,anti-symmetricﬁelds F ab.(Here“anti-symmetry”is the condition that F ba=−F ab.) If a particle with mass m>0,charge q,and four-velocityﬁeldξa is present, the force exerted by theﬁeld on the particle at a point is given by q F a bξb.If we use this expression for the left side of(15),we arrive at the Lorentz law of motion for charged particles in the presence of an electromagneticﬁeld:q F a bξb=mξb∇bξa.20(16) 2.4Matter FieldsIn classical relativity theory,one generally takes for granted that all that there is,and all that happens,can be described in terms of various matterﬁelds,e.g., materialﬂuids and electromagneticﬁelds.21Each suchﬁeld is represented by one or more smooth tensor(or spinor)ﬁelds on the spacetime manifold M.Each is assumed to satisfyﬁeld equations involving theﬁelds that represent it and the spacetime metric g ab.ξaξb(F ab+F ba),and by the anti-symmetry of F ab,(F ab+F ba)=0.221This being the case,the question arises how(or whether)one can adequately recover talk about“point particles”in terms of the matterﬁelds.We will say just a bit about the question in this section.16For present purposes,the most important basic assumption about the matter ﬁelds is the following.Associated with each matterﬁeld F is a symmetric smooth tensorﬁeld T ab characterized by the property that,for all points p in M,and all future-directed,unit timelike vectorsξa at p,T a bξb is thefour-momentum density of F at p as determined relative toξa.T ab is called the energy-momentumﬁeld associated with F.The four-momentum density vector T a bξb at p can be further decomposed into its temporal and spa-tial components relative toξa,just as the four-momentum of a massive particle was decomposed in the preceding section.The coeﬃcient ofξa in theﬁrst com-ponent,T abξaξb,is the energy density of F at p as determined relative toξa. The second component,T nb(g an−ξaξn)ξb,is the three-momentum density of F at p as determined relative toξa.Other assumptions about matterﬁelds can be captured as constraints on the energy-momentum tensorﬁelds with which they are associated.Examples are the following.(Suppose T ab is associated with matterﬁeld F.)Weak Energy Condition:Given any future-directed unit timelike vectorξa at any point in M,T abξaξb≥0.Dominant Energy Condition:Given any future-directed unit timelike vec-torξa at any point in M,T abξaξb≥0and T a bξb is timelike or null. Conservation Condition:∇a T ab=0at all points in M.Theﬁrst asserts that the energy density of F,as determined by any observer at any point,is non-negative.The second adds the requirement that the four-momentum density of F,as determined by any observer at any point,is a future-directed causal(i.e.,timelike or null)vector.It captures the condition that there is an upper bound to the speed with which energy-momentum can propagate(as determined by any observer).It captures something of theﬂavor of principle C1in section2.1,but avoids reference to“point particles”.22 The conservation condition,ﬁnally,asserts that the energy-momentum car-ried by F is locally conserved.If two or more matterﬁelds are present in the。

电动力学

Electrodynamics
Dr. M. J. Perry1 Michælmas 1997
1LATEXed by Paul Metcalfe – comments to soc-archim-notes@.
Revision: 2.8 Date: 1998-07-05 19:06:22+01
6. You must cause any edited versions to carry prominent notices stating that you edited them and the date of any change.
THESE NOTES ARE PROVIDED BY THE ARCHIMEDEANS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE ARCHIMEDEANS OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THESE NOTES, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

General Relativity and Relativistic Astrophysics

2. Potential Theory
The fundamental problem in potential theory is: given a force eld gi(uk ) which is a function of n variables uk , what is the potential (uk ) (if it exists) whose gradient returns the force eld,
" #
;
(2)
(i.e., it is curl-free), one may nd a solution which is conveniently expressed using a
line-integral
(uk ) =
Z
X
If the two endpoints are xed, all contours return the same answer. In practice, one employs the simplest contour that one can imagine: a line connecting the origin to the observation point uk . Using s, 0 s 1, to parameterize the contour, the line-integral may be rewritten as (uk ) =
Fields Institute Publication Sixth Canadian Conference on General Relativity and Relativistic Astrophysics May 25-27, 1995

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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。