2. Differentiable Manifolds and Tensors

Transcription

1 . Dfferentable Manfolds and Tensors.1. Defnton of a Manfold.. The Sphere as a Manfold.3. Other Examples of Manfolds.4. Global Consderatons.5. Curves.6. Functons on M.7. Vectors and Vector Felds.8. Bass Vectors and Bass Vector Felds.9. Fber Bundles.10. Examples of Fber Bundles.11. A Deeper Look at Fber Bundles.1. Vector Felds and Integral Curves.13. Exponentaton of the Operator d d.14. Le Brackets and Noncoordnate Bases.15. When s a Bass a Coordnate Bass?.16. One-Forms.17. Examples of One-Forms.18. The Drac Delta Functon.19. The Gradent and the Pctoral Representaton of a One-Form.0. Bass One-Forms and Components of One-Forms.1. Index Notaton.. Tensors and Tensor Felds.3. Examples of Tensors.4. Components of Tensors and the Outer Product.5. Contracton.6. Bass Transformatons.7. Tensor Operatons on Components.8. Functons and Scalars.9. The Metrc Tensor on a Vector Space.30. The Metrc Tensor Feld on a Manfold.31. Specal Relatvty.3. Bblography

2 .1. Defnton of a Manfold Let R n be the set of all n-tuples of real numbers x 1,, x n. A set M of ponts s a (topologcal) manfold f each pont P n t has an open neghborhood U homeomorphc to some open set V n R n. becton (1-1 onto map) In other words, there s a b-contnuous 1 n :U V by P P x P,, x P for all P n M. The n numbers x P are called the coordnates of P and n s the dmenson of M. Thus, the topology of M s the same as R n locally. The par U, s called a chart, or a local coordnate system. An atlas on M s a set, of charts so that the domans U U covers M,.e., every P s n some U. For reasons of compatblty, an atlas of class k C 1 : U U U U (a) requres the maps to be maps of class k C. Note that s a map between open sets of R n. In 1 fact, t represents a coordnate transformaton for ponts n the overlap regon U U of two coordnate systems gven by and. A manfold wth an atlas of class k C s sad to be a k C manfold. Those wth k 1 are called dfferentable manfolds. For convenence, we shall deal only wth C or C manfolds.

3 .. The Sphere as a Manfold Consder the -sphere S 1 3 x x x a const whch conssts of the ponts n R 3 that satsfy Any suffcently small neghborhood of every pont P on S has a 1-1 map onto a regon n R. Such mappngs are n general nether angle- nor length- preservng. Consder now the map f : S R by a,, x 1, x, where a,, are the sphercal coordnates of a pont P wth Cartesan coordnates x 1, x, x 3 a sn cos, a sn sn, a cos For any small enough regon U on S, ths s a dffeomorphsm so that, can serve as coordnates for U. In fact, as long as U does not contan the arc A gven by Cartesan coordnates a sn,0, a cos wth 0,, s a good coordnate system. On the other hand, for 0,, of a sn,0, a cos under f. For 0, there are mages,,0, the mage of 0,0,a and under f s the entre lne x 1, x 0, n R. Thus, f s not defned on the arc A so that, cannot be a coordnate system for the entre S. By takng out the arc A from ts doman, the mage of f s a rectangular open regon n R gven by and 0 x. 1 0 x What these all mean s that S cannot be covered by a sngle chart: at least are needed. Of nterest s the stereographc map whch proects the sphere onto a plane tangent to t. Here, only a sngle pont fals to be covered by the map.

4 .3. Other Examples of Manfolds Roughly speakng, any set that can be parametrzed contnuously s a manfold. The number of ndependent parameters requred s the dmenson of the manfold. Some examples of manfolds are: 1. The set of all rotatons of a rgd body n 3-D space. The (contnuous) parameters are the three Euler angles.. The set of all (pure boost) Lorentz transformatons. The parameters are the three components of boost velocty. 3. The 6N-D phase space of N partcles n 3-D space. The parameters are the 3N poston and 3N momentum coordnates. 4. Gven m dependent varables y y and n ndependent ones x x 1,, m,, 1 n, the set of all ponts y,, y ; x,, x s a (m+n)-d manfold. Solutons to 1 m 1 some set of (algebrac of dfferental) equatons nvolvng these varables are surfaces n the manfold. 5. A vector space s a manfold wth the topology of R n. 6. A Le group s a C manfold so that the group operatons are C maps of the manfold nto tself. n

5 .4. Global Consderatons Snce every manfold s locally lke R n for some n, all manfolds of the same dmenson are locally ndstngushable. Hence, manfolds are classfed by ther global propertes, e.g., S and R. To be more precse, two manfolds are equvalent (belong to the same class) f there s a dffeomorphsm (b- C becton) between them.

6 .5. Curves A curve s a dfferentable mappng C from an open set of R nto M,.e., C : R M wth P x, 1,, n where s the parameter of the curve. Dfferentablty here means that x are dfferentable functons of. For convenence, we shall assume each mappng represents a unque curve. Thus, curves represented by dfferent mappngs are consdered as dfferent even though they may gve the same set of mage ponts n M.

7 .6. Functons on M A functon f on M s an assgnment of a real number f(p) to each pont P n M. Ths s denoted by f : M R wth P f P If a regon U M s mapped dfferentably onto some regon of R n wth, 1,, x, we can wrte 1,, n f P f x x f x P x n so that f s a functon on R n. If f s dfferentable n R n, we say f s dfferentable n M.

8 .7. Vectors and Vector Felds.7.1. Vectors as Tangents to Curves.7.. Vector Space at a Pont

9 .7.1. Vectors as Tangents to Curves Consder a curve C descrbed by x x n M. Let f x be a functon on M. For ponts on the curve, f can be taken as a functon of through g f x. Thus, dg f dx (.) d x d Snce ths holds for arbtrary g, we can wrte d dx d d x (.3) Now, n Eucldean space, Thus, f we treat e x dx d vector to the curve C() at pont are the components of a vector tangent to the curve C. as bass vectors, P. d d defned as the tangent to some curve n the manfold. can be dentfed as the tangent In fact, n dfferental geometry, a vector s Note that the essence of a vector s ts drecton, not the curve to whch t s tangent. Indeed, a vector s tangent to an nfnte number of curves passng through ts pont due to dfferent reasons. Frst, for a manfold of dmenson hgher than 1, there are nfntely many paths that can pass through a gven pont whle beng tangent to each other. Secondly, a gven path represents an nfnte number of curves va dfferent parametrzaton. As an example, consder the curve C() wth x a, where a are constants. Obvously, C passes through the orgn O when 0. Its tangent s d d a. Another curve C'() wth x b a x O when 0 Hence, d d. Its tangent s b a d d also passes through x, whch equals to d d at O. s tangent to dfferent curves at O. Next, we can re-parametrzed the 1 st 3 curve as x a, whch passes through O at 0 wth tangent

10 dx d 0 1 a 0 a.

11 .7.. Vector Space at a Pont It s straghtforward to verfy that the set of all tangent vectors at a pont P forms a vector space called the tangent space to M at P and denoted by T. For example, closure under addton and scalar multplcaton s proved as follows. Let d dx d d x d dx d d x P a b a b d d dx dx d d d d x ( a, b scalars ) Settng we have dx dx dx a b d d d d d dx a b d d d x d d so that the lnear combnaton of tangents s another tangent. Q.E.D. Note that the vector space defned above conssts only of tangents at the same pont n M. Thus, vectors at dfferent ponts are n dfferent vector spaces. In partcular, the subtracton of vectors located at dfferent ponts n M has no geometrcal meanng, whch s the root cause of complcatons n tensor analyss on a manfold. Fnally, we defne a vector feld as a rule for assgnng a vector at each pont of M.

12 .8. Bass Vectors and Bass Vector Felds As can be seen from the defnton (.3), the dmenson of M. Gven a coordnate system the coordnate bass of TP s the same as that of x for a neghborhood U of M, we call x T for all ponts n U. In general, we shall use an overbar P to denote a vector. For example, a general bass for TP s denoted by e. Thus, V V x V e where V s the component of V along e. If V s a vector feld, the components V and V are functons on M. In whch case, V s dfferentable f V are. In callng the set of vectors x a bass, we have mplctly assumed they are lnearly ndependent at a gven pont P. We shall show below that ths requres x to be a set of good coordnates,.e., t provdes a 1-1 map of some neghborhood U of P onto a regon of R n. To begn, consder a set of good coordnates can be wrtten as 1 n y y x,, x 1,, n y on U. The map from x By the nverse functon theorem, ths map s 1-1 and hence nvertble n U ff the to U Jacoban J det y x s non-vanshng. Ths means the set of n vectors V 1 n 1 n y y,, x x,, y y V,, n n n x x are lnearly ndependent. On the other hand, the chan rule gves n y x x y 1

13 so that V s ust the vector x expressed n the y bass. Hence, x are lnearly ndependent. Fnally, snce a 1-1 map of a set of good coordnates s another set of good coordnates, our asserton s proved.

14 .9. Fber Bundles The exact defnton of fbre bundles s rather nvolved [see, e.g., Y. Choquet-Bruhat et al, "Analyss, Manfolds and Physcs", nd ed., North-Holland (198)] and won't be ntroduced untl.11. Here, the basc concepts are presented usng as example the smple varety called tangent fbre bundles. Loosely speakng, a tangent fbre bundle s ust a manfold TM obtaned by attachng a tangent space T P to each pont of a manfold M. Here, M s called the base manfold and T P a typcal fbre. By defnton, dm(tm) dm(m) + dm(t P ).

15 .10. Examples of Fber Bundles 1. The tangent bundle TM. For an n-d manfold M, ts TM s n-d.. Tensor bundles wth tensor spaces as fbres. 3. Isospn bundles wth spacetme as manfold and sospn space as fbres. 4. Gallean spacetme wth tme as manfold and Eucldean space as fbres.

16 .11. A Deeper Look at Fber Bundles Global Propertes.11.. Formal Defnton of A Fbre Bundle

17 Global Propertes 1. Gven spaces M and N, there s a (Cartesan) product space MN consstng of all ordered pars a, b where a M and b N.. If M, N are topologcal spaces, so s MN. 3. An open set U M wth coordnates x ; 1,, m, taken together wth a n open set V N wth coordnates y ; 1,, n, becomes an open set U, V M N 1 1 wth m + n coordnates,, m n x x, y,, y. 4. In the formal defnton of a fbre bundle, what dstngushes the base M and the fbre T P s the exstence of a proectve (many-1) mappng T P M that maps every pont n T P to the pont P n M to whch T P s attached. 5. A fbre bundle s locally a product space U F : P where U s an open subset of the base manfold B and F s a typcal fbre. Ths means a fbre bundle s locally trval. 6. To be globally trval means that the entre bundle s a product space B F. Or, more generally, there exsts a C 1-1 map (dffeomorphsm) of the bundle onto B F. Wth a non-zero F, the nverse of ths map gves a no-where zero cross secton of the bundle. Hence, a necessary condton for global trvalty s the exstence of a no-where zero vector feld. 7. The tangent bundle TS of the -sphere S s globally non-trval. Accordng to the fxed pont theorem of the sphere, every C 1-1 map (dffeomorphsm) of S onto tself must leave at least 1 pont fxed. A no-where zero vector feld wll generate a map that has no fxed pont. Hence, TS cannot be globally trval. 8. The tangent bundle TS 1 of the crcle S 1 s the product space cylnder. Hence, t s globally trval. 1 S R, whch s a 9. The mobus strp s also a tangent bundle usng S 1 as base manfold and R as the 10.. typcal fbre. Locally, t s trval but globally, not.

18 .11.. Formal Defnton of A Fbre Bundle The global propertes of a fbre bundle are descrbed by ts structure group. a fbre bundle s formally defned as the quartet E, B,, G consstng of a base manfold B, a proecton : E B, a typcal fbre F, a structural group G of homeomorphsms of F onto tself, and a famly U that In fact, of open sets coverng B, such 1. E s locally trval. Ths means the bundle over any set U,.e., the nverse 1 mage U, s homeomorphc to the product space U 1 homeomorphsm : F. Ths U U F s of the form p p, h x, where p E, x p U B and : h x F F x whch maps the fbre F at x U to the typcal fbre F. x. The global propertes of the bundle s gven by the structural group G whose elements are the homeomorphsms k 1 : h x h x F F where x U Uk. 3. The nduced mappngs g : U U G k k by x g x h x h x k k are contnuous. They are called transton functons and satsfy g x g x g x k kl l

19 .1. Vector Felds and Integral Curves 1. A vector feld s a rule that selects a vector from the tangent vector space at each pont of M.. Consder a vector feld V P for P M we have V P v x dx d. Gven a coordnate system. The tangent vector to a curve v x (.5) x s gven by whch s ust a set of 1 st order ordnary dfferental equatons so that a unque soluton always exsts n some neghborhood around any gven pont. 3. Hence, gven a vector feld x, v x, a soluton, called an ntegral curve, of (.5) s a curve whose tangent s everywhere equal to the vector feld. 4. Unqueness of solutons means that dfferent ntegral curves never cross except possbly at ponts where v Thus, by udcal choce of ntal condtons, one can fnd a set of ntegral curves that flls up M. Such a set of curves s called a congruence.

20 .13. Exponentaton of the Operator d d Consder an analytc (C ) manfold M. The coordnates x of ponts along an ntegral curve of Y d are analytc functons of. Hence, ponts wth d parameters 0 and 0 + are related by the Taylor seres x dx 1 d x 0 x 0 d! d 0 0 d d 1! d d 1 x 0 d exp x d 0 (.6) d d The operator exp expy s called the exponentaton of the operator d d. As d d denotes an nfntesmal 'moton' along the curve, exp d d denotes a fnte moton.

21 .14. Le Brackets and Noncoordnate Bases Non-Coordnate Bass.14.. Exercse Le Brackets Le Algebras

22 Non-Coordnate Bass Gven a coordnate system x, the bass x s called a coordnate bass for the vector felds. One mportant characterstcs of a coordnate bass s that ts members commute,.e.,, 0 x x x x x x On the other hand, arbtrary vector feld need not commute. For example, let d V and d d W, we have d d d d d V, W d d d d, V W W V x x x x W V V V W W W V, x x x x x x x x, V x x x x W V W W V V W, x x x (.7) where U x U W V U V W x x Thus, the commutator of vectors are n general another vector. of non-commutatng vectors s called a non-coordnate bass. A bass consstng

23 .14.. Exercse.1 Consder the 'unt' bass vector felds for polar coordnates n the Eucldean plane defned by r cos x sn y cos x sn y θ sn x cos y sn cos x y Show that they form a non-coordnate bass.

24 Le Brackets When appled to vectors, the commutator s called a Le bracket. 1 Consder a -D subspace S of a manfold M descrbed by coordnates x, x. By defnton, x 1 s constant along the lnes of x, whch are ntegral curves of. x Ths s the reason why, 0 1 x x. Next, consder other arbtrary vector felds, d V and d d W, n S [see d Fg..0]. In general, can vary along an ntegral curve C() of V and can vary along an ntegral curve C() of W. Hence, V, W 0. As wll be show below, ths means and are not coordnates. Consder a pont P at the ntersecton of C() and C() wth respectve parameters and. [see Fg..1] If we move along C() by, we reach pont R wth coordnates d x R exp x P d Movng further along C() by, we reach pont A wth coordnates d d x A exp exp x P d d (.9) On the other hand, f we move frst along C() by, then along C() by, we reach pont B wth coordnates d d x B exp exp x P d d The dfference between the coordnates of A and B s (.10) d d d d exp, exp x B x A x P (.11) For nfntesmal, we have d d exp, exp d d d 1 d d 1 d 1,1 d d d d

25 d d, d d so that, x B x A V W x P (.1) Thus, the Le bracket V, W s proportonal to the dfference of the end ponts when one moves along the ntegral curves by the same amount but n dfferent order. Obvously, n order for and to be coordnates, we must have x B x A or V, W 0.

26 Le Algebras A Le algebra of vector felds on a regon U of a manfold M s a set A of vector felds on U such that 1. It s a vector space under addton.. It s closed under the Le bracket. Note that condton (1) means A s closed under lnear combnatons of vector felds wth constant coeffcents. Condton () means the Le bracket of vector felds s another vector feld.

27 .15. When s a Bass a Coordnate Bass? In.14, we have shown that the necessary condton for a bass to be a coordnate bass s that the Le brackets of every par of ts member vectors vansh. show n the followng that ths condton s also suffcent. To begn, consder a -D regon U of M. Startng from a pont P n U wth 1 coordnates x, x, we move frst a parameter dstance 1 along pont R, then 1 along d d Ths equaton defnes a map exp x Q 1 x R We shall d A to a d d B to a pont Q. The correspondng coordnates are d d d x d d by exp 1 exp1 U d d d d, x, exp exp x P Thus,, forms a coordnate system n U f the map (a) s 1-1,.e., f t has an nverse. It wll be shown later that ths requres A and B to be lnearly ndependent. Usng P (a) n n d d d d exp d d d n! d n n0 n n1 n d n 1 n 1! d n n1 n1 n1 n d d 1 1! d d exp d d we can dfferentate (a) to get x d d dx exp exp d d d d d P n n d n n d n0! d d x d d d exp exp x P d d d d d where P s the orgn of the coordnate system,. If, 0 d d, the nd equaton becomes x d d dx exp exp d d d P

28 Consder now the bass vectors we have x x and x x d d dx exp exp d d d x exp d d d exp d d d d P d Q where Q s the pont wth coordnates,. Smlarly, P d d Q Snce, d d s a coordnate bass, so s, d d. Q.E.D. To complete the proof, we need to show that the map (a) has an nverse. Accordng to the nverse functon theorem, the necessary and suffcent condton for ths s J x, x 1, x det x Ths s the case f the vectors B are ndependent. x x 1 1 and 0 are lnearly ndependent,.e., A and

29 .16. One-Forms A one-form s a lnear, real- or complex- valued functon of vectors. Thus, a 1-form at P s a mappng : TP by V V such that av bw a V b W a, b (.15) Defnng the addton and scalar multplcaton of 1-forms by a b V a V b V (.16) we see that the set of all 1-forms at P s a lnear space * TP dual to T P. The dualty s easly seen by rewrtng (.16) as V a b av bv (.17a) so that, combnng wth (.15), we have V V, V V, (.18) Each expresson n eq(.18) s called the contracton of wth V. In tensor analyss, 1-forms are called covarant vectors.

30 .17. Examples of One-Forms 1. Gradent of a functon.. In matrx algebra, column vectors correspond to vectors and row vectors to 1-forms. Contracton corresponds to matrx multplcaton wth row vectors always on the left. 3. In Hlbert spaces used n quantum mechancs, the Drac kets are vectors whle bras are 1-forms so that. Note that the assocaton of each vector wth another 1-form, called ts conugate or transpose, s equvalent to ntroducng a metrc, or nner product, to the vector space [see.9]

31 .18. The Drac Delta Functon As an example of functon spaces, consder the set C 1,1 of all C real-valued functons defned on the nterval x 1,1. Ths set s a group under addton and a lnear space wth multplcaton by real numbers as scalar multplcaton. The 1-forms n ts dual space are called dstrbutons. the Drac delta functon x wth any C functon f C 1,1 s a number f 0 x, f x f 0 (.19) One example of a dstrbuton s whch s defned as the 1-form whose contracton,.e., Thus, a dstrbuton s a "functon" whch maps functons to numbers. However, ths s not what Drac meant when he called x a delta "functon". To see the dstncton, we frst note that for any functon g C 1,1 g such that ts contracton wth any functon f C 1,1 1 1 g, f dx g x f x (.0), one can defne a 1-form s gven by The proof that g defned ths way s ndeed a 1-form s straghtforward. Now, what Drac meant as a delta functon s really the "functon" satsfes 1, f dx x f x f 0 1 x whose 1-form The problem s that one cannot defne x rgorously as a functon R R. 1 1 The specal rule used by Drac that x gves meanngful results only nsde an ntegral s ust an mplct statement that t s really a 1-form. "dervatve" of x defned by For example, the 1 1 dx ' x f x dx x f ' x f ' 0 1 1

32 s really the result of the dervatve of the 1-form.

33 .19. The Gradent and the Pctoral Representaton of a One-Form A feld of 1-forms s a rule assgnng a 1-form to each pont n M. The contracton wth a vector feld s ust the applcaton of (.16) to every pont n M. Furthermore, the coeffcents of lnear combnatons can be a functon on M. Treatng the contracton V as a functon n M, the dfferentablty of can be defned n terms of that of V and V. For example, f V and V are both C, so s. As wth vector felds, there s a fbre bundle called the cotangent bundle TM * and conssts of M as base and 1-form felds. * TP as typcal fbre. Cross-sectons of TM * are The best known example of 1-form feld s the gradent df of a functon f. It s defned as the 1-form whose contracton wth a tangent vector s the dervatve of f along the ntegral curve of that tangent,.e., d df df (.1) d d To see f df thus defned s ndeed a 1-form, we need only check on ts lnearty: d d d d df a b a b f d d d d [ by (.1) ] df df a b d d [lnearty of dfferentaton] d d a df b df d d For ponts wth nfntesmal coordnate dfferences f f x df x x x, we have Hence, a 1-form can be vewed as a seres of parallel planes whose separaton s nversely proportonal to the magntude of. The contracton V s the number of planes perced by V. Note that ths nterpretaton does not requre the noton of lengths or metrcs.

34 .0. Bass One-Forms and Components of One-Forms A bass e for P T nduces a dual bass for * TP so that Thus, e (.3) V V e V V (.) The lnear ndependence of follows mmedately from (.3). Alternatvely, t can be proved as follows. Consder an arbtrary 1-form q so that q V q V e V q e V qe (.4) where q q V (.7) q e (.5) are called the components of q on the bass dual to e. Note that (.4) can be wrtten as q V q V Snce ths s vald for arbtrary V, we have q q (.6) Snce q s arbtrary and the number of 1-forms n s equal to the dmenson of * T P, s ndeed a bass of * T P. Extenson of the foregong to 1-form felds s straghtforward. To summarze, gven a coordnate patch x on a regon U M, there s a natural bass x for vector felds and dx for 1-form felds wth

35 x x x dx (.8)

36 .1. Index Notaton Hereafter, we shall adopt the followng conventons for the use of ndces: 1. Components. Members e V for vectors V have superscrpt ndces. of a vector bass e are denoted by subscrpt ndces. 3. Components q for 1-forms q have superscrpt ndces. 4. Members e of a vector bass e are denoted by superscrpt ndces. 5. Ensten's summaton conventon: a par of repeated super- and sub-scrpts ndcates summaton, e.g. dx V V x V V dx V x V 6. Note that there s no summaton for q p. V W or

37 .. Tensors and Tensor Felds Consder a pont P n M. A tensor T of type (rank) lnear functon T : * N TP TP M M by 1 T N 1 N so that N M 1 1,, ;,,, ;, M V V V V 1 1 N 1 M T a b,,, ; V, V M N T N,, ; 1,,, ; 1, 1 M at P s defned to be a at V V b V V (.9) and smlarly for each argument. Here, super- and sub-scrpts n parentheses denote dfferent vectors and 1-forms, respectvely. to a tensor wthout specfyng ts arguments, e.g., S,, ; 3 1. A tensor feld s a rule to assgn a tensor to each pont n M. Blanks can be used f one wshes to refer denotes a tensor of type Lnearty (.9) then apples to each pont n M whle the coeffcents a and b can be functons on M. By 0 conventon, scalar functons are tensors of type 0. [see.8] Note that the expresson ; T denotes a 1-form and T ;V a vector. Hence, T ; can be thought of as a lnear vector-valued functon of vectors or a lnear 1-form-valued functon of 1-forms.

38 .3. Examples of Tensors 1. In matrx algebra, column vectors are vectors, row vectors are 1-forms, matrces 1 are 1 tensors, and smlarty transformatons of matrces are tensors. 1. In the functon space C 1,1, lnear dfferental operators are 1 tensors ) nto other vectors. snce they convert vectors (functons n C 1,1 3. A stress tensor gves the stress vector across a plane n a straned materal. Snce a plane (surface) s a 1-form, the stress tensor s a lnear vector-valued functon of 1-forms,.e., a 0 tensor.

39 .4. Components of Tensors and the Outer Product Gven vectors V two 1-forms p and q, and W, we can form a 0 tensor V W such that for any, V W p q V p W q (.30) Note that the lnearty of V W follows drectly from that of the vectors. The operaton s called the outer, the drect, or the tensor product. In general, the N N ' N N ' outer product of a M tensor wth a M ' tensor s a M M ' tensor. The components of a tensor are the values t takes when all the arguments are bass vectors and/or 1-forms. For example, n the bases of a type S 3 tensor S are S,, ; e, e (.31) k k lm l m and e, the components

40 .5. Contracton Accordng to.4, the drect product V V. The contracton V V Therefore, t s a scalar functon or s a 1 1 tensor wth components s a number ndependent of bass (see below). 0 0 tensor. In general, each contracton of a par of upper and lower ndces reduces the rank of the product tensor by 1 1. We shall show that the contracton of a 0 tensor A wth a 0 tensor B s ndependent of bass. To begn, let the componets of A and B be A and B n some bass. The contracton s then by ; k C V A B V k C V k A B k. Consder now the k 1 1 tensor defned Usng A, A, A k and Be, V Be, V ek B V k k we have ; V, e, V, e, V C A B A, B,V A B Thus, C s ndependent of bass, e used for A and B. Therefore, ts components C k A B k s also ndependent of, e.

41 .6. Bass Transformatons 1. Transformaton Matrx. Coordnate Transformatons

42 .6.1. Transformaton Matrx Consder the tensors defned at a pont P of M. The shft from one bass to another e ' ; ' 1,, n e ; 1,, n transformaton matrx so that e e (.3) ' ' can be accomplshed by a lnear Snce the mappng must be 1-1 onto, the matrx must be non-sngular. Note ' however that s not a tensor. The natural 1-form bass wth respect to e s gven by Thus, e (.3) e ' e e ' ' ' (.33) ' The nverse of ' s denoted as ' so that and ' ' (.34) k ' k ' ' ' Thus, (.33) gves ' (.35) ' ' ' e ' ' ' Therefore, the natural 1-form bass wth respect to e ' s (.36) ' ' Note that the transformaton matrx for 1-forms s the nverse of that for vectors. Some examples of the use of the transformaton matrx are V q ' k ' ' V ' V q ek ' q k ' e (.36) ' V k ' qe k ' q (.37)

43 .6.. Coordnate Transformatons Consder a regon U M wth coordnate system x ; 1,, n. Let ' y ; ' 1,, n be a new coordnaton system gven by ' f x ' ' y f x 1,, x n for ' 1,, n (.39) wth J y,, y 1' n ' 1 x,, x n ' y det 0 x By the chan rule, x y y x ' ' (.40) so that x ' (.41) ' y Smlarly, ' y x x y ' and ' ' x y x ' k ' k y k x x ' ' y (.4) x Note that the coordnate transformaton (.41-) satsfes x y y y ' k ' ' k ' k k ' (.43) ' y

44 .7. Tensor Operatons on Components Consder a tensor T wth components T n some bass. An operaton on T that result n another tensor s called a tensor operaton. Obvously, such operatons are ndependent of bass. Some examples are: 1. Addton:, A B A B wth A, B A B. Multplcaton by a constant: T at at 3. Outer product:,.. A B A B wth, k k A Bl A Bl 4. Contracton:, m m Ak Bl Ak Bl..

45 .8. Functons and Scalars A scalar functon s a functon on M whose value s ndependent of the choce of bass. For example, the contracton V s a scalar but the vector feld component 1 V P s not. On the other hand, we can defne a scalar functon f so that for every bass, f P always equals to the numercal value of the 1 st 1 component V P of the vector feld V P n one partcular bass. Thus, whether a functon s a scalar depends on ts nterpretaton, not ts numercal value.

46 .9. The Metrc Tensor on a Vector Space.9.1. Inner Products and Metrc Tensors.9.. Canoncal Forms.9.3. Characterstcs of Metrc Tensors.9.4. Rasng and Lowerng Indces

47 .9.1. Inner Products and Metrc Tensors An nner (dot) product s an operaton that maps real vectors to a real number,.e., such that : TP TP by,, 1. symmetrc: U, V V, U U V U V U V (.44).. Gven a bass e, the matrx g wth components g e, e e e g (.45) has an nverse. 0 Snce g maps vectors to a scalar, t s a tensor called the metrc tensor. A manfold wth a metrc for ts tangent spaces s called a Remannan manfold. If there exsts a bass such that g then the bass s called a Cartesan bass, the metrc a Eucldean metrc and the manfold a Eucldean space. We menton n passng that a metrc space s a topologcal space X wth a dstance functon d : X X by x, y d x, y for all ponts x, y X.

48 .9.. Canoncal Forms Under a change of bass, the metrc tensor transforms as g g (.46) ' ' ' ' In matrx form, Λ g Λths becomes T g ' (.47) where s a matrx wth the, ' element equals to ' and T s the transpose operaton. Note that s the ', ' element of 1. Snce s arbtrary, we can wrte t as Λ OD (.48) where O s orthogonal ( T 1 Λ OD D O DO T T T and (.47) becomes 1 T O O ) and D dagonal. Thus, 1 g ' DO g OD (.49) Now, snce g s real and symmetrc, t s normal. Therefore, there exsts an orthogonal transform that dagonalzes t. Settng O to be ths transform, we have 1 O g O gd dag g,, 1 gn where g are the egenvalues of g. Eq(.49) thus becomes g ' D gd D dag d,, 1 g1 dn gn (.50) Snce d 0, we can choose d so that d g g 1 g sgn 1. [ Note that g s nvertble so that g 0] In other words, there always exsts a bass so that g s dagonal wth elements 1. If dag 1,, 1,1,,1 g, t s sad to be n the canoncal form and the correspondng bass s called orthonormal. The trace ( sum of the dagonal elements ) of the canoncal metrc tensor s called the sgnature of the metrc. Note that the sgnature s nvarant under any smlarty transformatons.

49 .9.3. Characterstcs of Metrc Tensors The sgnature s a fundamental characterstcs of a metrc. For example, f g s postve- defnte, ts sgnature s whch must necessarly be Eucldean. n, where n s the dmenson of the vector space, If the egenvalues of g are of mxed sgns, the metrc s called ndefnte. One example s the Mnkowsk space wth canoncal form dag 1,1,1,1 η and sgnature +. Another mportant characterstcs of the metrc s the symmetry goup of the bass transformatons that leave t nvarant. For the Eucldean metrc, ths s the orthogonal group O n. For the Mnkowsk space, t s the Lorentz group O n 1,1 L n.

50 .9.4. Rasng and Lowerng Indces One mportant role of the metrc tensor s to serve as the transformaton between vector (upper ndex) and 1-form (lower ndex) components of a tensor. for a gven vector V, V V, (.54) s a 1-form called the dual of V. Snce g s nvertble, V s unque. Its component s gven by V V e V, e V e, e V e, e For example, V g g V whch can be vewed as the lowerng of the vector ndex of denote the components of the nverse of g by so that g V by g. We now g g k (.55) k we can rase ndex by g V g g V k k k V V k (.56) To summarze, we have V g V (.57) V g V (.58) Other examples are A g A k k (.59) A g A g g A k kl k k l (.60) k l A g g A kl (.61) Thus, n a metrc space, vectors and 1-forms are closely related. Thus, t s possble, and preferable, to refer to all tensors of type N M, where N M L, smply as tensors of order L. For example, all tensors of types 1 0 0, 1 and are order tensors.

51 Note that n a Eucldean space wth Cartesan bass, g and g so that V V,.e., there s no numercal dfference between vectors and 1-forms. However, ths s no longer the case f we shft to curvlnear coordnates. Fnally, for completeness, one should prove that g s a 0 as an exercse. tensor. Ths s left

52 .30. The Metrc Tensor Feld on a Manfold Metrc Tensor Feld.30.. Locally Flat Lengths

53 Metrc Tensor Feld A metrc feld s a rule to assgn a metrc tensor to every pont P n M. Once gven, t serves as metrc on TP and * T P. Furthermore, t also nduces other geometrc propertes lke lengths and curvatures nto the manfold. Further dscusson of ths wll be deferred to Chap 6. Here, we'll explore a few smple concepts. Frst, the feld should at least be contnuous. Ths means ts canoncal form must be a constant everywhere snce ts dagonal elements are all ntegers that cannot change contnuously. Thus, t s meanngful to speak of the sgnature of a metrc tensor feld. In general, to make take the canoncal form everywhere means usng dfferent bass transformatons for dfferent ponts. Moreover, ths transformaton feld may not be coordnaton transformatons. A notable excepton s the Eucldean space wth Cartesan coordnates so that g everywhere. However, even here only the Cartesan coordnates generate an orthonormal bass. For example, the polar coordnates r, n r generate an orthogonal bass, normalzed. The normalzed verson s not a coordnate bass. but t s not n

54 .30.. Locally Flat A C metrc tensor feld s locally flat n the sense that there s always a coordnate bass x for a neghborhood U about any pont P such that g P (canoncal form at P) 1. g. 0 k x P (canoncal form good approxmaton near P) 3. x k g l x P not necessarly zero. (canoncal form not necessarly substaned away from P)

55 Lengths The norm V of a vector V s defne as V V, V gv V If s postve-defnte, V s real and non-negatve. If s ndefnte, V s called a pseudo-norm. A vector V s called space-lke f V 0, tme-lke f V 0, and null f V 0. Note that null vectors are not necessarly the zero vector V 0. Consder a curve C() wth tangent V d d dx d x so that dx dx V, V g d d The length dl of an nfntesmal segment on C s defned by, dl dl V V d (.6) gdx dx If s postve-defnte, we can wrte dl V, V d (.63) gdx dx where dl s real and postve. If s ndefnte, the curves are classfed by the sgn of dl the same way as for vectors wth norm dl.

56 .31. Specal Relatvty 4 The manfold equpped wth a metrc of sgnature + s called the Mnkowsk space and serves as the spacetme manfold n specal relatvty. The canoncal form of s dag 1,1,1,1 η. The correspondng orthonormal bass can be provded by a class of global coordnate systems called Lorentz frames. Curves n the manfold are called wordlnes. The arc length of an nfntesmal segment of wordlne s an event nterval and defned by s x x x x x x (.66) c t x y z Snce s ndefnte, the norm t nduces s a pseudo-norm. vectors s gven by Inner product between V W V W (.68) In a Lorentz frame, the 1-form V dual to V s gven by V 0 0 V 0 V (.69a) for 1,,3 V V V (.69b) The vector gradent df of a functon f s defned as the vector dual to the (1-form) f gradent df dx. Thus, x so that f x df df f x df 0 0 f for 1,,3 x df

57 .3. Bblography