真实之路 宇宙法则的完全指南 (彭罗斯)10表面.pdf

10 Surfaces 10 1Complex dimensions and real dimensions One of the most impressive achievements in the mathematics of the past two centuries is the development of various remarkable techniques that can handle non Xat spaces of various dimensions It will be important for our purposes that I convey something of these ideas to the reader for modern physics depends vitally upon them Up to this point we have been considering spaces of only one dimen sion The reader might well be puzzled by this remark since the complex plane the Riemann sphere and various other Riemann surfaces have featured strongly in several of the previous chapters However in the context of holomorphic functions these surfaces are really to be thought of as being in essence of only one dimension this dimension being a complex dimension as was indeed remarked upon in 8 2 The points of such a space are distinguished from one another locally by a single parameter albeit a parameter that happens to be a complex number Thus these surfaces are really to be thought of as curves namely complex curves Of course one could split a complex number z into its real and imaginary parts x y where z x iy and think of x and y as being two independent real parameters But the process of dividing a complex number up in this way is not something that belongs within the realm of holomorphic operations So long as we are concerned only with holo morphic structures as we have been up until now when considering our complex spaces we must regard a single complex parameter as providing just a single dimension This at least is the attitude of mind that I recommend should be adopted On the other hand one may take an opposing position namely that holomorphic operations constitute merely particular examples of more general operations whereby x and y can if desired be split apart to be considered as separate independent parameters The appropriate way of achieving this is via the notion of complex conjugation which is a non holomorphic operation The complex conjugate of the complex number 179 z x iy z x iy Real axis z x iy where x and y are real numbers is the complex number z z given by z z x iy In the complex z plane the operation of forming the complex conjugate of a complex number corresponds to a reXection of the plane in the real line see Fig 10 1 Recall from the discussion of 8 2 that holomorphic oper ations always preserve the orientation of the complex plane If we wish to consider a conformal mapping of a part of the complex plane which reverses the orientation such as turning the complex plane over on itself then we need to include the operation of complex conjugation But when included with the other standard operations adding multiplying taking a limit complex conjugation also allows us to generalize our maps so that they need not be conformal at all In fact any map of a portion of the complex plane to another portion of the complex plane let us say by a continuous transformation can be achieved by bringing the operation of complex conjugation in with the other operations Let me elaborate on this comment We may consider that holomorphic functions are those built up from the operations of addition and multipli cation as applied to complex numbers together with the procedure of taking a limit because these operations are suYcient for building up power series an inWnite sum being a limit of successive partial sums 10 1 If we also incorporate the operation of complex conjugation then we can generate general say continuous functions of x and y because we can express x and y individually by x z z z 2 y z z z 2i Any continuous function of x and y can be built up from real numbers by sums products and limits I shall tend to use the notation F z z z with z z mentioned explicitly when a non holomorphic function of z is being considered This serves to emphasize the fact that as soon as we move 10 1 Explain why subtraction and division can be constructed from these Fig 10 1The complex conjugate of z x iy x y real is z z x iy obtained as a reXection of the z plane in the real axis 180 10 1CHAPTER 10 outside the holomorphic realm we must think of our functions as being deWned on a 2 real dimensional space rather than on a space of a single complex dimension Our function F z z z can be considered equally well to be expressed in terms of the real and imaginary parts x and y of z and we can write this function as f x y say Then we have f x y F z z z although of course f s explicit mathematical expression will in general be quite diVerent from that of F For example if F z z z z2 z z2 then f x y 2x2 2y2 As another example we might consider F z z z z z z then f x y x2 y2 which is the square of the modulus jzj of z that is 10 2 z z z jzj2 10 2Smoothness partial derivatives Since by considering functions of more than one variable we are now beginning to venture into higher dimensional spaces some remarks are needed here concerning calculus on such spaces As we shall be seeing explicitly in the chapter following the next one spaces referred to as manifolds can be of any dimension n where n is a positive integer An n dimensional manifold is often referred to simply as an n manifold Einstein s general relativity uses a 4 manifold to describe spacetime and many modern theories employ manifolds of higher dimension still We shall explore general n manifolds in Chapter 12 but for simplicity in the present chapter we just consider the situation of a real 2 manifold or surface S S Then local real coordinates x and y can be used to label the diVerent points of S S in some local region of S S In fact the discussion is very representative of the general n dimensional case A 2 dimensional surface could for example be an ordinary plane or an ordinary sphere But the surface is not to be thought of as a complex plane or a Riemann sphere because we shall not be concerned with assigning a structure to it as a complex space i e with the attendant notion of holomorphic function deWned on the surface Its only structure needs to be that of a smooth manifold Geometrically this means that we do not need to keep track of anything like a local conformal structure as we did for our Riemann surfaces in 8 2 but we do need to be able to tell when a function deWned on the space i e a function whose domain is the space is to be considered as smooth For an intuitive notion of what a smooth manifold is think of a sphere as opposed to a cube where of course in each case I am referring to the surface and not the interior For an example of a smooth function 10 2 Derive both of these 181 Surfaces 10 2 h h hh h2 h a b c Fig 10 2Functionsonasphere S S picturedassittinginEuclidean3 space whereh measuresthedistanceabovetheequatorialplane a Thefunctionhitselfissmooth on S S negativevaluesindicatedbybrokenlines b Themodulusjhj seeFig 6 2b is not smooth along the equator c The square h2is smooth all over S S on the sphere we might think of a height function say the distance above the equatorial plane the sphere being pictured as sitting in ordinary Euclidean 3 space in the normal way distances beneath the plane being counted negatively See Fig 10 2a On the other hand if our function is the modulus of this height function see 6 1 and Fig 10 2b so that distances beneath the equator also count positively then this function is not smooth along the equator Yet if we consider the square of the height function then this function is smooth on the sphere Fig 10 2c It is instructive to note that in all these cases the function is smooth at the north and south poles despite the singular appearance at the poles of the contour lines of constant height The only instance of non smoothness occurs in our second example at the equator In order to understand what this means a little more precisely let us introduce a system of coordinates on our surface S S These coordinates need apply only locally and we can imagine gluing S S together out of local pieces coordinate patches in a similar manner to our procedure for Riemann surfaces in 8 1 For the sphere for example we do need more than one patch Within one patch smooth coordinates label the diVerent points see Fig 10 3 Our coordinates are to take real number values and let us call them x and y without any suggestion intended that they ought to be combined together in the form of a complex number Suppose now S y x Fig 10 3Within one local patch smooth real number coordinates x y label the points 182 10 2CHAPTER 10 that we have some smooth function F deWned onS S In the modern mathematical terminology F is a smooth map from S S to the space of real numbers R or complex numbers C in case F is to be a complex valued function on S S because F assigns to each point of S S a real or complex number i e F maps S S to the real or complex numbers Such a function is sometimes called a scalar Weld on S S On a particular coordin ate patch the quantity F can be represented as a function of the two coordinates let us say F f x y where the smoothness of the quantity F is expressed as the diVerentiability of the function f x y I have not yet explained what diVerentiability is to mean for a function of more than one variable Although intuitively clear the precise deWnition is a little too technical for me to go into thoroughly here 1Some clarifying comments are nevertheless appropriate First of all for f be diVerentiable as a function of the pair of variables x y it is certainly necessary that if we consider f x y in its capacity as a function of only the one variable x where y is held to some constant value then this function must be smooth at least C1 as a function of x in the sense of functions of a single variable see 6 3 moreover if we consider f x y as a function of just the one variable y where it is x that is now to be held constant then it must be smooth C1 as a function of y However this is far from suYcient There are many functions f x y which are separately smooth in x and in y but for which would be quite unreason able to call smooth in the pair x y 10 3 A suYcient additional require ment for smoothness is that the derivatives with respect to x and y separately are each continuous functions of the pair x y Similar state ments of particular relevance to 4 3 would hold if we consider functions of more than two variables We use the partial derivative symbol to denote diVerentiation with respect to one variable holding the other s Wxed The partial derivatives of f x y with respect to x and with respect to y respectively are written 10 3 Consider the real function f x y xy x2 y2 N in the respective cases N 2 1 and 1 2 Show that in each case the function is diVerentiable C o with respect to x for any Wxed y value and that the same holds with the roles of x and y reversed Nevertheless f is not smooth as a function of the pair x y Show this in the case N 2 by demonstrating that the function is not even bounded in the neighbourhood of the origin 0 0 i e it takes arbitrarily large values there in the case N 1 by demonstrating that the function though bounded is not actually continuous as a function of x y and in the case N 1 2 by showing that though the function is now continuous it is not smooth along the line x y Hint Examine the values of each function along straight lines through the origin in the x y plane Some readers may Wnd it illuminating to use a suitable 3 dimensional graph plotting computer facility if this is available but this is by no means necessary 183 Surfaces 10 2 f x and f y Asanexample wenotethatiff x y x2 xy2 y3 then f x 2x y2and f y 2xy 3y2 If these quantities exist and are continuous then we say that F is a C1 smooth function on the surface We can also consider higher orders of derivative denoting the second partial derivative of f with respect to x and y respectively by 2f x2 and 2f y2 Now we need C2 smoothness of course There is also a mixed second derivative 2f x y which means f y x namely the partial deriva tive with respect to x of the partial derivative of f with respect to y We can also take this mixed derivative the other way around to get the quantity 2f y x In fact it is a consequence of the second diVerentia bility of f that these two quantities are equal 10 4 2f x y 2f y x The full deWnition of C2 smoothness for a function of two variables requires this 10 5 For higher derivatives and higher order smoothness we have corresponding quantities 3f x3 3f x2 y 3f x y x 3f y x2 etc An important reason that I have been careful here to distinguish f from F by using diVerent letters and I may be a good deal less careful about this sort of thing later is that we may want to consider a quantity F deWned on the surface but expressed with respect to various diVerent coordinate systems The mathematical expression for the function f x y maywell changefrom patch topatch eventhoughthe value ofthe quantity F at any speciWc point of the surface covered by those patches does not change Most particularly this can occur when we consider a region of overlap between diVerent coordinate patches see Fig 10 4 If a second set of coordinates is denoted by X Y then we have a new expression 10 4 Prove that the mixed second derivatives 2f y x and 2f x y are always equal if f x y is a polynomial A polynomial in x and y is an expression built up from x y and constants by use of addition and multiplication only 10 5 Show that the mixed second derivatives of the function f xy x2 y2 x2 y2 are unequal at the origin Establish directly the lack of continuity in its second partial derivatives at the origin 184 10 2CHAPTER 10 X x S Y x F F X Y for the values of F on the new coordinate patch On an overlap region between the two patches we shall therefore have F X Y f x y But as indicated above the particular expression that F represents in terms of the quantities X and Y will generally be quite diVerent from the expression that f represents in terms of x and y Indeed X might be some complicated function of x and y on the overlap region and so might Y and these functions would have to be incorporated in the passage from f to F 10 6 Such functions representing the coordinates of one system in terms of the coordinates of the other X X x y andY Y x y and their inverses x x X Y andy y X Y are called the transition functions that express the cordinate change from one patch to the other These transition functions are to be smooth let us for simplicity say C1 smooth and this has the consequence that the smoothness notion for the quantity F is independent of the choice of coordinates that are used in some patch overlap 10 3Vector fields and 1 forms There is a notion of derivative of a function that is independent of the coordinate choice A standard notation for this as applied to the function F deWned on S S is dF where 10 6 Find the form of F X Y explicitly when f x y x3 y3 where X x y Y xy Hint What is x2 xy y2in terms of X and Y what does this have to do with f Fig 10 4To cover the whole of S S we may have to glue together several coordinate patches A smooth function F on S S would have a coordinate expression F f x y on onepatch andF F X Y on another with respective local coordinates x y X Y On an overlap region f x y F X Y where X and Y are smooth functions of x and y 185 Surfaces 10 3 dF f x dx f y dy Here we begin to run into some of the confusions of the subject and these take some while to get accustomed to In the Wrst place a quantity such as dF or dx initially tends to be thought of as an inWnitesimally small quantity arising when we apply the limiting procedure that is involved in the calculus when the derivative dy dx is formulated see 6 2 In some of the expressions in 6 5 I also considered things like d logx dx x At that stage these expressions were considered as being merely formal 2this last expression being thought of as just a convenient way multiplying throughbydx ofrepresentingthe morecorrect expression d logx dx 1 x When I write dF in the displayed formula above on the other hand I mean a certain kind of geometrical entity that is called a 1 form although this is not the most general type of 1 form see 10 4 below and 12 6 and this works for things like d logx dx x too A 1 form is not an inWnitesimal it has a somewhat diVerent kind of inter pretation a type of interpretation that has grown in importance over the years and I shall be coming to this in a moment Remarkably however despite this signiWcant change of interpretation of d the formal math ematical expressions such as those of 6 5 provided that we do not try to divide by things like dx are not changed at all There is also another issue of potential confusion in the above displayed formula which arises from the fact that I have used F on the left hand side and f on the right I did this mainly because of the warnings about the distinctionbetweenFandfthatIissuedabove ThequantityFisafunction whose domain is the manifold S S whereas the domain of f is some open regioninthe x y planethatreferstoaparticularcoordinatepatch IfIam to apply the notion of partial derivative with respect to x then I need to knowwhatitmeans toholdtheremainingvariableyconstant Itisforthis reason that f is used on the right rather than F because f knows what the coordinates x and y are whereas F doesn t Even so there is a confusion in this displayed formula because the arguments of the functions are not mentioned The F on the left is applied to a particular point p of the 2 manifold S S while f is applied to the particular coordinate values x y that the coordinate system assigns to the point p Strictly speaking this would have to be made explicit in order that the expression makes sense However it is a nuisance to have to keep saying this kind of thing and it would be much more convenient to be able to write this formula as dF F x dx F y dy or in disembodied operator form 186 10 3CHAPTER 10 d dx x dy y Indeed I am going to try to make sense of these things These formulae are instances of something referred to as the chain rule As state