真实之路 宇宙法则的完全指南 (彭罗斯)19麦克斯韦和爱.pdf

19 The classical fields of Maxwell and Einstein 19 1Evolution away from Newtonian dynamics In the period between the introduction of Newton s superb dynamical scheme which we can best date as the publication of his Principia in 1687 and the appearance of special relativity theory which could reasonably be dated at Einstein s Wrst publication on the subject in 1905 many import ant developments in our pictures of fundamental physics took place The biggest shift that occurred in this period was the realization mainly through the 19th century work of Faraday and Maxwell that some notion of physical Weld permeating space must coexist with the previously held Newtonian reality of individual particles interacting via instantaneous forces 1Later this Weld notion also became a crucial ingrediant of Einstein s 1915 curved spacetime theory of gravity What are now called the classical Welds are indeed the electromagnetic Weld of Maxwell and the gravitational Weld of Einstein But we now know that there is much more to the nature of the physical world than just classical physics Already in 1900 Max Planck had revealed the Wrst hints of the need for a quantum theory although more than another quarter century was required before a well formulated and comprehensive theory could be provided It should also be made clear that in addition to all these profound changes to the Newtonian foun dations of physics that have taken place there had been other important developments both prior to these changes and coexistent with some of them in the form of powerful mathematical advances within Newtonian theory itself These mathematical advances will be the subject of Chapter 20 They have important interrelations with the theory of classical Welds and even more signiWcantly they form an essential prerequisite to the proper understanding of quantum mechanics as will be described in subsequent chapters As a further important area of advance the subject of thermodynamics and its reWnement referred to as statistical mechanics should certainly be considered This concerns the behaviour of systems of large numbers of bodies where the details of the motions are not regarded as important the behaviour of the system being described in terms of 440 averages of appropriate quantities This was an achievement initiated in the mid 19th to early 20th centuries and the names of Carnot Clausius Maxwell Boltzmann Gibbs and Einstein feature most strongly I shall address some of the most fundamental and puzzling issues raised by thermodynamics later in Chapter 27 In this chapter I shall describe the physical Weld theories of Maxwell and Einstein the classical physics of electromagnetism and gravitation The theory of electromagnetism also plays an important part in quantum theory providing the archetypical Weld for the further development of quantum Weld theory which we shall encounter in Chapter 26 On the other hand the appropriate quantum approach to the gravita tional Weld remains enigmatic and controversial Addressing these quan tum gravitational issues will be an important part of the later chapters in this book Chapter 28 onwards For the physics that we shall be examin ing next however we shall conWne our investigation to physical Welds in their classical guise I referred at the beginning of this chapter to the fact that a profound shift in Newtonian foundations had already begun in the 19th century before the revolutions of relativity and quantum theory in the 20th The Wrst hint that such a change might be needed came from the wonderful experimental Wndings of Michael Faraday in about 1833 and from the pictures of reality that he found himself needing in order to accommodate these Basically the fundamental change was to consider that the New tonian particles and the forces that act between them are not the only inhabitants of our universe Instead the idea of a Weld with a disembod ied existence of its own was now having to be taken seriously It was the great Scottish physicist James Clark Maxwell who in 1864 formulated the equations that this disembodied Weld must satisfy and he showed that these Welds can carry energy from one place to another These equations uniWed the behaviour of electric Welds magnetic Welds and even light and they are now known simply as Maxwell s equations the Wrst of the relativ istic Weld equations From the vantage point of the 20th century when profound advances in mathematical technique have been made and here I refer particularly to the calculus on manifolds that we have seen in Chapters 12 15 Maxwell s equations seem to have a compelling naturalness and simplicity that almost make us wonder how the electric magnetic Welds could ever have been considered to obey any other laws But such a perspective on things ignores the fact that it was the Maxwell equations themselves that led to a very great many of these mathematical developments It was the form of these equations that led Lorentz Poincare and Einstein to the spacetime transformations of special relativity which in turn led to Minkowski s conception of spacetime In the spacetime The classical fields of Maxwell and Einstein 19 1 441 framework these equations found a form that developed naturally into Cartan s theory of diVerential forms 12 6 and the charge and magnetic Xux conservation laws of Maxwell s theory led to the body of integral expressions that are now encapsulated so beautifully by that marvellous formula referred to in 12 5 6 as the fundamental theorem of exterior calculus Perhaps in seeming to attribute all these advances to the inXuence ofMaxwell sequations Ihavetakenasomewhattooextreme position with these comments Indeed while Maxwell s equations un doubtedly had a key signiWcance in this regard many of the precursors of these equation such as those of Laplace D Alembert Gauss Green Ostrogradski Coulomb Ampe re and others have also had im portant inXuences Yet it was still the need to understand electric and magnetic Welds that largely supplied the driving force behind these devel opments these and the gravitational Weld also The remainder of this chapter is devoted to understanding the electromagnetic and the gravitational Welds and how they Wt in with the modern mathematical framework 19 2Maxwell s electromagnetic theory What then are the Maxwell equations They are partial diVerential equations see 10 2 which describe the time evolutions of the three components E1 E2 E3of the electric Weld and of the three components B1 B2 B3of the magnetic Weld where the electric charge density r and the three components of the electric current density j1 j2 j3are considered as given quantities Certain other Weld quantities having to do with an ambient material within which the Welds may be considered to be propa gating can also be incorporated In discussions of fundamental physics as is our concern here it is usual to ignore those aspects of Maxwell s equations that relate to such an ambient medium since the medium itself would in reality consist of many tiny constituents each of which could in principle be treated at the more fundamental level It will be convenient also to choose what are called Gaussian units and use standard Minkowski coordinates of 18 1 namely x0 t x1 x x2 y x3 z signature with spacetime units so that the velocity of light c is taken to be unity c 1 The electromagnetic Weld and the charge current density are respect ively collected together according to a prescription originally due in eVect to Minkowski into a spacetime 2 form F called the Maxwell Weld tensor and a spacetime vector J called the charge current vector with components displayed in matrix form as 19 2CHAPTER 19 442 F00F01F02F03 F10F11F12F13 F20F21F22F23 F30F31F32F33 0 B B B 1 C C C A 0E1E2E3 E10 B3B2 E2B30 B1 E3 B2B10 0 B B B 1 C C C A J0 J1 J2 J3 0 B B B 1 C C C A r j1 j2 j3 0 B B B 1 C C C A Note that the antisymmetry Fba Fabholds as is required for a 2 form I shall also make use of what are referred to as the Hodge duals of F and J these being respectively the 2 form F and the 3 form J deWned by F00 F01 F02 F03 F10 F11 F12 F13 F20 F21 F22 F23 F30 F31 F32 F33 0 B B B 1 C C C A 0 B1 B2 B3 B10 E3E2 B2E30 E1 B3 E2E10 0 B B B 1 C C C A J123 J023 J013 J012 0 B B B 1 C C C A r j1 j2 j3 0 B B B 1 C C C A Where the required antisymmetry properties Fab F ab and Jabc J abc hold In terms of the Levi Civita tensor e 12 7 with totally antisym metric components eabcd e abcd and normalized so that e0123 1 the duals can be written as Fab 1 2eabcdF cd and Jabc eabcdJd where the raised version Fabof Fabis simply gacgbdFcd in accordance with 14 7 Note that the raised version eabcd gapgbqgcrgdsepqrssatisWes e0123 1 whence the e of 12 7 is given by 19 1 Eabcd eabcd See Fig 19 1 for the diagrammatic form of these dualizing operations and also of the Maxwell equations themselves We shall Wnd that the notion of a dual in this sense and other related senses will have importance for us later in various diVerent contexts A remark should be made about the geometrical signiWcance of the Hodge dual We recall from 12 7 that the operation of passing from a bivector H as described by the antisymmetric quantity Hab to its dual 2 form H as given by 1 2eabcdH cd does not make much diVerence to 19 1 Check both these statements The classical fields of Maxwell and Einstein 19 2 443 FabFab Fab eabcd abcd abcd eabcd Fab 24 4 0 0 1 2 1 2 1 6 1 6 1 6 4 3 Ja Jabc Fig 19 1Diagrams for Hodge duals and Maxwell equations The quantities eabcd e abcd and Eabcd E abcd normalized so that E 0123 E0123 1 in a stand ard Minkowski frame are related to their raised lowered versions via gaband gab by eabcd Eabcdand Eabcd eabcd In the diagrams left middle lower two lines this sign change is absorbed by an eVective index reversal Boxed oV at the top right are the Maxwell equations Wrst using the Weld tensor F with its raised form Fab gacgbdFcd cf Fig 14 21 so the equations are raFab 4pJb r aFbc 0 and beneath that correspondingly using the dual F where Fab 1 2eabcdF cd Jabc eabcdJd so the equations are r aFbc 4p 3 Jabc r aF ab 0 its geometrical interpretation If H were a simple bivector for example so that the 2 form H would also be simple see the end of 12 7 then the 2 plane element determined by H would be precisely the same as the 2 plane element determined by H the only diVerence being that strictly H has the quality of a density as pointed out in 12 7 On the other hand the index raising that takes us from a 2 form Habto a bivector Hab Hcdgcagdb has a more signiWcant geometrical eVect In the case of a simple bivector the 2 plane element determined by Habis the orthogonal complement of the 2 plane element determined by Hab see 18 3 The Hodge dual as applied to the 2 form Hab taking us to 1 2eabcdH cd i e to H employs the index raising Hab7 Haband therefore involvespassingtotheorthogonalcomplement SeeFig 19 2 Accordingly the Hodge dual taking us from F to F also involves an orthogonal complement 19 2CHAPTER 19 444 Hab or abcd Hcd Hab or abcd Hcd Fig 19 2In 4 space a simple bivector HHab represents the same 2 plane element as its dual 2 form H 1 2eabcdH cd But the index lowered version of H the simple 2 form Hab which is equivalent to its dual bivector 1 2E abcdHcd repre sents the orthogonal complement 2 plane element see Fig 18 4 Hence it is the index raising lowering in the Hodge dual that leads to the passage to the orthog onal complement Having set up this notation we can now write Maxwell s equations very simply as 19 2 dF 0 d F 4p J We can also write the Maxwell equations entirely in index form as 19 3 r aFbc 0 raFab 4pJb Note that if we apply the exterior derivative operator d to both sides of the second Maxwell equation d F 4p J and use the fact that d2 0 12 6 we deduce that the charge current vector J satisWes the vanishing divergence equation 19 4 d J 0or equivalentlyraJa 0 At this point as a slight digression which will have considerable import ance for us later 32 2 and 33 6 8 11 see 18 3 it is worth while to point out the self dual and anti self dual parts of the Maxwell tensor given respectively by 19 2 Write these out fully in terms of the electric and magnetic Weld components showing how these equations provide a time evolution of the electric and magnetic Welds in terms of the operator q qt 19 3 Show the equivalence to the previous pair of equations 19 4 Show that the two versions of this vanishing divergence are equivalent The classical fields of Maxwell and Einstein 19 2 445 F 1 2 F i F and F 1 2 F i F which are complex conjugates of one another It turns out that in the quantum theory these complex quantities describe respectively the right spinning and left spinning photons quanta of the electromagnetic Weld see 22 7 12 Fig 22 7 The self dual anti self dual properties are ex pressed in 19 5 F i F Bearing in mind that J is real we can combine the two Maxwell equations as imaginary and real parts respectively as d F 2pi J Photons provide the particle description of light and we shall be seeing in Chapter 21 how quantum theory allows a particle and wave descrip tion of light to coexist It was one of Maxwell s supreme achievements to show by means of his equations that there are electromagnetic waves which travel with the speed of light and have all the known polarization properties that light has and which we shall be examin ing in 22 7 In accordance with these remarkable facts Maxwell pro posed that light is indeed an electromagnetic phenomenon In 1888 almost a quarter century after Maxwell published his equations Hein rich Hertz experimentally conWrmed Maxwell s marvellous theoretical prediction Intheexplicitdescriptionsabove Ihaveassumedthatthe background spacetime is Xat Minkowski space M and the discussions to follow in 19 3 4 and the Wrst part of 19 5 can all be taken on this basis also However this is not really necessary and all the conclusions still apply if spacetime curvature is present For this the components given above must be regarded as being taken with respect to some local Minkowskian frame and the index notation will take care of the rest 19 6 19 3Conservation and flux laws in Maxwell theory The vanishing divergence of the charge current vector provides us with the equation of conservation of electric charge The reason that it is 19 5 Show this Wrst demonstrating that dualizing twice yields minus the original quantity Does this sign relate to the Lorentzian signature of spacetime Explain 19 6 Can you spell this out What happens to the components of F and F in a general curvilinear coordinate system Why are the Maxwell equations unaVected if expressed correctly 19 3CHAPTER 19 446 referred to as a conservation equation comes from the fact that by the fundamentaltheoremofexteriorcalculus see 12 6 wehave Rd J qR J so that Q J 0 integrated over any closed 3 surface Q in Minkowski space M Any closed 3 surface in M is the boundary qR of some compact 4 dimensional region R in M See Fig 19 3 The quantity J can be interpreted as the Xux of charge or Xow of charge across Q qR Thus what the above equation tells us is that the net Xux of electric charge across this R Q R Q J 0 Time Fig 19 3Conservation of electric charge in spacetime The closed 3 surface Q is the boundary Q R of a compact 4 volume R in Minkowski spacetime M so the fundamental theorem of exterior calculus tells us Q J Rd J 0 since d J 0 The quantity J describes the Xux or Xow of charge across Q so the total charge Xowing in across Q is equal to that Xowing out expressing charge conservation The classical fields of Maxwell and Einstein 19 3 447 boundary has to be zero i e the total coming into R has to be exactly equal to the total going out of R electric charge is conserved 19 7 We can also use the second Maxwell equation d F 4p J to derive what is called a Gauss law This particular law applies at one given time t t0 so we are now using the three dimensional version of the fundamental theorem of exterior calculus This tells us the value of the total charge lying within some closed 2 surface S at time t0 see Fig 19 4 by expressing this charge as an integral over S of the dual of the Maxwell tensor F which amounts to saying that we can obtain the total charge surrounded by S if we integrate the total Xux of electric Weld E across S 19 8 More generally this applies even if S does not lie in some Wxed time t t0 Suppose that S is the spacelike 2 boundary of some compact 3 spatial region A Then the total charge w in the region A surrounded by S or in spacetime terms threaded through S see Fig 19 4 is given by t to S A 19 7 Although correct this argument has been given somewhat glibly Spell out the details more fully in the case when R is a spacetime cylinder consisting of some bounded spatial region that is constant in time for a Wxed Wnite interval of the time coordinate t Explain the diVerent notions of Xux of charge involved contrasting this for the spacelike base and top of the cylinder with that for the timelike sides 19 8 Spell out why this is just the electric Xux Fig 19 4Within the 3 surface of constant time t t0 Maxwell s d F 4p J gives us the Gauss law whereby the integral of electric Xux integral of F over a closed spatial 2 surface measures the total charge surrounded by the fundamental theorem of exterior calculus In fact this is not restricted to 2 surfaces at constant time and the Gauss law is thereby generalized 19 3CHAPTER 19 448 S F 4pw where w A J We can also obtain a related kind of conservation law from the Wrst Maxwell equation dF 0 This has just the same form as the second Maxwell equation except that F replaces F and the source corresponding to J is now zero Thus for any closed 2 surface in Min kowski space 2we always have the Xux law S F 0 Note that in passing from F to F or from F to F we simply interchange the electric and magnetic Weld vectors with a change of sign for one of them The absence of a source for F is an expression of the fact that as far as is known there are no m