TENSOR ANALYSIS. Hence the above equation ( It is a special case ) can be written in generalised form as

Transcription

1 OCCURRENCE OF TENSORS IN PHYSICS TENSOR NLYSIS We are famlar wth elementary Physcal laws such as that acceleraton of a body s proportonal to the Force actng on t or that the electrc current n a medum s proportonal to appled E F m a Ie F a m J σ E J E It should be understand these laws are specal cases and apply strctly only to sotropc meda ( r) or to meda possessng hgh symmetry. In realty many medas are ansotropc ( eg. Glass water ) and as a result the acceleraton s not necessarly parallel to the appled force. Hence the above equaton ( It s a specal case ) can be wrtten n generalsed form as J E yey zez Jy ye yyey yzez J ze zye zze z y z Smlarly F m a equaton can also be obtaned n the above format as F ma myay mzaz Fy mya myyay myzaz F mza mzya mzza z y z Here m s actng as the couplng constant between called mass tensor of the partcle n the medum. F and a. Ths Eg: Ths stuaton arse n the study of the moton of an electron n a crystallne solds m s Tensors are ndependent under co-ordnate Transformatons NOTTIONS & CONVENTIONS CO ORDINTE TRNSFORMTIONS From the fgure Cosθ & y Snθ In the New ( Barred ) Co ordnate Cos y Sn Cos (θ-φ) Cos φ + y Sn φ y Sn (θ-φ) - Sn φ + y Cos φ Ie : ( y ) and y y ( y)

2 Consder an N dmensonal space and let 1.. N be any set of co-ordnate n ths space. Smlarly 1 N... be another set of co-ordnate n the same space. e s the old coordnate system where (1 N) and (1 N) be another set of co ordnate n the same space. Then from above concept we can wrte N (... ) 1 N (... ) ( N ) 1 1 ( N ) In general 1 ( N... ) (1 N) ( N ) (1 N) The above two equatons defnes the coordnate transformatons. On dfferentatng equaton (1) N d d d... d 1 N 1 d d d... d 1 N N 1 d d d d d 1 3 d d d d... d 1 3 N N N N d N d (1 N) 1 d EINSTEIN SUMMTION CONVENTION N d (1 N) 1 If any nde s repeated n a term then a summaton w.r.t that nde over the range 13 N s mpled. Ths conventon s called Ensten s Summaton Coneventon. ccordng to ths conventon nstead of epressng equaton () can be wrtten as n 1 a we merely wrte a hence d d (1 N) d d (1 N)

3 Thus the summaton conventon means the drop of sgn for the nde appearng twce n a gven term. In other words the Summaton Conventon mples the sum of the term for the nde appearng twce n that term over a defne range. The nde occurs twce on the RHS of equaton (3a) whle I occurs twce on the RHS (3b) summaton over these from 1 to N s therefore mpled n the respectve equatons. If an nde appears only once n any term t has a defnte value any value between 1 and N such an nde s called a free nde Hereafter the specfcaton such as 1 N wll be dropped and hence (3a) becomes d d d d n nde whch s repeated and over whch summaton s mpled s called a dummy nde. dummy nde can be replaced by any other nde whch doesn t appear n the same term. SPECIL CSE It s clear that from the above theory On squarng of a s (a )( a ) wrong concept a. a.. True concept Here LHS (a a ) (a a ) a 1 1. a a 1 1. a + a. a a. a KRONECKER DELT a. a a. a We defne the Kronecer delta as δ or δ 1 for 0 for hence the Kronecer delta can be consder as a dot product of two unt vector. (1). let us consder two vector a a ˆ & b b ˆ a. b a ˆ. b ˆ ab ˆˆ. ab When a b a.b 0 e here 0 for When a b a.b ab e here 1for () The Kronecer delta can be wrtten as 1 when & 0 when (3) Kronecer delta s a tensor quantty e n 3 Dmenson 3 & n Phase Space 6N (4) Kronecer delta s Symmetrc e snce..

4 (5) We now that or Or here n statement no (5) s a dummy nde and I & are free ndces. Hence or When we undergo transformaton the component of a tensor changes by two dfferent ways 1. Contravarant Tensors. Covarant Tensors CONTRVRINT VECTORS Let us consder a dsplacement vector ( N ) N 1 d d d d d N d d system. are the component of vectors le component n the new coordnates s the transformaton coeffent. Let a physcal entty charactersed by the N functons when epressed n the coordnate system.when we measure the same entty n coordnate system charactersed by the components a.the are sad to be the component of a contravarant vector f they transform under coordnate transformaton. Inverse transformaton s obtaned by multplyng J J J { for multplcaton we use LHS & RHS )} ( s free nde n LHS & RHS) ( s dummy nde n J δ

5 J { The ndces must shows the same propertes n LHS & RHS ) COVRINT VECTORS set of N quanttes whch are functons of coordnate system are sad to be the component of a covarant tensor f they transform Under a change of coordnate from vector n the barred coordnate system. Consder a scalar gradent n two coordnate system to where are the component of a N 1... N (snce s scalar ) N N 1 In general Inverse transformaton obtaned by Multplyng

6 TENSOR OF RNK TWO set of N functons are sad to be the component of a contravarant tensor of ran f they transform. In general no. of component N r [ N dmenson r ran ] MIXED TENSOR OF RNK TWO Eg: ronecer delta It can be represented as n Cartesan coordnate system.snce there s no dstncton between covarant and contravarant tensors n rectangular coordnate system. But ts true value s Concept s true n rectangular coordnate but false n general theory of relatvty and Remanan space TENSORS OF HIGHER RNK The ran of a tensor only ndcate the number of ndces attached to t per component.eg: are the component of a med tensor of ran 3 havng contravarant ran and covarant ran 1.f they transform accordng to equaton The number of component of ran s gven by N r SYMMETRIC TENSORS If two contravarant or covarant ndces can be nterchanged wthout alterng the tensor then the tensor s sad to symmetrc wth respect to these two ndces.

7 For a tensor of hgher ran ndces only. l f l l s sad to be symmetrc w.r.t the symmetrc property s ndependent of the coordnate system used. symmetrc tensor of ran n N-dmensonal space has ndependent component N( N 1) Eg : moment of nerta about XY as s equal to YX as. I I I y I I y I I yy I I z yz z zy zz It s a second ran tensor wth n (3 9 ) components out of whch n (3) components are dagonal and rest n -n (9-36) wll be n equal pars due to symmetry. Total number of component n( dagonal ) + n n n + n n n n n nn ( 1) In general number of component of a ran symmetrc tensor wll be ( nr1)! r!( n1)! Case 1 When r ( n 1)!!( n 1)! ( n 1)! ( n 1)! nn ( 1) Ie.the above result SKEW SYMMETRIC TENSORS tensor satsfy or s sad to be sew symmetrc (ant symmetrc) f ts element For a tensor of hgher ran l l then the tensor l s sew symmetrc w.r.t ndces and f a tensor s such that two contravarant or covarant ndces of t when nterchanged the component of the tensor alter n sgn not n magntude the tensor s sew symmetrc the property of symmetrc s an ntrnsc property of a tensor and s ndependent of the choce of coordnate system

8 symmetry and antsymmetry can only be defned for a par of smlar ndces Symmetrc n the frst two contravarant ndces. It s mportant to specfy the poston of the ndces rather than the ndces themselves Symmetry and antsymmetry can be defned only for smlar ndces not when one nde s covarant and the other s contravarant. TENSOR NLYSIS 1. EQULITY ND NULL TENSOR Two Tensors are sad to be equal ff they have the same ( number& type ) contravarant and covarant ran and every component of one s equal to the correspondng component of the other. 1 3 p p B p 1 3 p If all the N r components of a tensor of total Ran r dentcally vansh It s sad to be a null tensor.. DDITION & SUBTRCTION The sum and dfference of two or more tensors of the same ran results n a thrd tensor of the same ran.moreover f F λµ G λµ and are tensors of the same ran the s also a tensor of same ran. B C To add or subtract any two tensors correspondng elements are added or subtracted B C a a a a b b + b b a b a b a b a b ( B ) ( )( B ) C ( ) C Statement ny tensor havng ether two contravarant or covarant ndces can be epessed as a sum of two parts one symmetrc and the other s sew symmetrc Let beng any tensor then we can wrte

9 1 1 ( ) ( ) B + C {we wanted to prove that s B symmetrc and C s antsymmetrc so that can be represented as symmetrc tensor + antsymmetrc tensor } B 1 ( ) ---(1) On nterchangng the ndces B 1 ( ) whch s same as (1) hence B B C 1 ( ) ----() C 1 ( ) - 1 ( ) - C shows that t s sew symmetrc 3. OUTER PRODUCT The Outer product of the two tensor s a tensor whose ran s the sum of the rans of the gven tensor. Thus f r and q are the rans of two tensors thus the outer product wll be a tensor of ran (r+q). The tensors and transform accordng to followng equaton BP q & B p q B p q Multplyng the components q p B B q p Let B C p B q p C q C q p C p q C p q s a tensor of contravarant Ran 3 covarant Ran and total Ran 5. It has therefore N 5 components each of whch s the product of and B P q Ths s nown as the outer product or Kronecer product of two tensors. The concept of outer product of tensors can be easly etended to more than two tensors. The contravarant ran of the outer product tensor s the sum of the contravarant rans and the covarant ran s the sum of the covarant rans of the tensors whose outer product t s.

10 4. INNER PRODUCT OF TENSORS Let us consder two tensors and B P consder a set of functons q and B P where t has three free ndces there wll be N 3 functons. We wanted q to show that N 3 functons are the components of a tensor of Ran 3. For ths purpose put ρ γ n equaton (1) q p B B q p q p B B q p q p q B B Now P s a dummy nde. Owng to the appearance of the delta symbol t s clear that the summaton over P from 1 to N only that term wll survve for whch p q B B q Ths transformaton coeffcent shows that the tensor transforms le the components of a tensor of contravarant ran and covarant Ran 1. Now let us denotes the component of a New tensor as C B & C B q q C q Then the C q s sad to be the nner product of the two tensors and BP q. and B P & and B P are the nner product of and B P. In tang the q nner product of two tensors t s mportant that one contravarant nde of tensor should be equated to one covarant nde of the other. CONTRCTION OF TENSORS The algebrac operaton n whch the ran of a tensor s lowered ether by two or multple of s nown as contracton. In ths process of contracton one contra varant nde and one covarant nde of a med tensor are set equal repeated nde summed over the result s a tensor of ran lower by than the orgnal tensor. C q l m lm

11 Puttng ρ α n the above equaton and summng over α from 1 ton l m lm m l. lm m m Ths shows that s a tensor of contravarant ran and covarant ran 1. m Ths process s nown as the contracton of a tensor. When a tensor s contracted by equatng one of the contravarant ndces wth one of ts covarant ndces the resultng entty s a tensor whose ran contravarant and covarant rans reduced by one each. Thus reducng the total ran by two. l m are varous contracted forms of the tensor l. tensor m can be repeatedly contracted thus a tensor of total Ran 5 on contracton lm gves a tensor of total ran 3 whch can be further contracted to gce the m tensor or of contravarant Ran 1. QUOTIENT LW Quotent law states that f the nne product of an entty wth an arbtrary tensor s a tensor the entty s a tensor Let us consder the entty havng N 3 functons ( ). Suppose t s nownthat the nner product of ( ) wth an arbtrary tensor B s contravarant tensor of Ran 1 ( ) B C s the free ndces. Summaton over and on the LHS s mpled. Let (αβγ) be the N 3 functons n the barred co ordnate system then (αβγ) B βγ C α (αβγ) B C ()B { Snce B s a contra varant tensor wth ran & C s a contra varant } [ (αβγ) - () ] B 0 ths equaton must be true for an arbtrary tensor B epresson n the square bracet 0 (αβγ) RHS of the above equaton () then by tang nner product wth n LHS &

12 (αβγ) () (αβγ) () (αβγ) () whch shows ( ) s a tensor of contravarant ran 1 and covarant ran whch can be wrtten as It s mportant n the use of the quotent law that the tensor wth whch the nner product s taen should be an arbtrary tensor FUNDMENTL TENSOR Easer to handle Cartesan tensor than general tensor because n Cartesan there s no dstncton s made between covarant and contravarant. But n General theory of relatvty the presence of Gravtatonal feld produce curvature n the space tme contnuum and thus the Eucldean character of the space s destroyed. { Small space ds It s very small may be regarded as flat space If we loo from a longer dstance It s Curved } Consder a regon ds Cartesan Tensor f( General Tensor ) X X ( 1 3 ) ds dx (1) + dx () + dx (3) ( dx ) dx dx X X d d g αβ d d

13 g tells the nature of Space. Fundamental metrc Tensor s an epresson whch epresses the dfferences between two ponts metrc ds g d d If we change & ds g d d whch means g g e It s Symmetrc here ds s a scalar quantty formed by multplyng ( g d ) wth d ( g d ).d Scalar [ Vector. Vector Scalar ] ( g d ) Vector Covarant Vector hance g s a covarant tensor wth Ran. Eucldean space means flat space and Remannan Space means Curved Space. If all the coeffcent of g s ndependent of space becomes Eucldean Space. g d d gd d g d d g d d [ g g] d d 0 g g multplyng by we get g CONTRVRINT FORM OF FUNDMENTL TENSOR Let us ntroduce the contravarant form of fundamental tensor g g 1 g Cofactor of g Consder the nner product of fundamental metrc tensors g.g g 1.g 1 + g.g + g 3.g g { g 1.cofactor of g 1 + g.cofactor of g +.. But g 1 cofactor of g 1 g and g 1 cofactor of g 1 0 f Hence g.g 1 0 when g

14 1 g when g SSOCITE TENSORS Let be an arbtrary contravarant vector. The nner product of wth the covarant metrc tensor g wll be a covarant vector. ths nner product s denoted by g & g Showng that the relaton between & s recprocal. The Tensors & are called ssocate Tensors. The Rasng or lowerng of ndces of a tensor changes a covarant nde nto a contravarant one and vce versa. The Operaton depend on the nner product of the gven tensor wth fundamental tensor RISING & LOWERING OF INDICES When the nner product of a tensor s taen wth g one contravarant nde of the tensor s lowered to a covarant poston.. g -. g - The Reverse process n whch the nner product of a tensor s taen wth the contravarant metrc tensor g rases a covarant nde to a contravarant poston hence s nown as the Rasng of an nde. - g - g TENSOR CLCULUS The partal dervatve of a scalar feld w.r.t the co ordnates are the components of a covarant vector. The dfferentaton of a tensor ( ecept that of Ran 0 ) w.r.t the co ordnates doesn t gve a tensor. Let us ntroduce... l... l 1. DIFFERENTITION OF TENSOR Let us consder a covarant transformaton

15 Dfferentaton both sde of the above equaton wth { The second term on RHS has a tensoral character but the appearance of the I term shows that the functon do not transform le the component of a II Ran tensor The Co-ordnate dervatve of any tensor ( Ecludng the Scalar ) donot transform le the component of a tensor CHRISTOFFEL SYMBOLS In Cartesan the dfference of two vectors stuated a two dfferent pont of the space s a vector. But n Remannan space Transformaton equaton vary from pont to pont. Therefore the dfference of two vectors are consdered at the same pont of the space. Thus to fnd the dfference of the two vectors n Remmannan Space one of the two vector s to be parallel dsplaced to the pont of locaton of the other. In Cartesan system parallel dsplacement donot change any magntude. But n Remannan space ( curved ) components are changed due to parallel dsplacement. So consder a Physcal quantty μ at a pont wth co- ordnate μ. When t s dsplaced parallel to neghbourng pont wth co-ordnate μ + d μ μ + 0 n cartesan co-ordnate ( flat space ) 0 n curved space Where Thus d vansh n flat spce. But est n curved space. The Coeffcnt ( ) are non Tensors. Let us ntroduce the Chrstoffel three nde symbol or smply Chrstoffel Symbol of I & II Knd respectvely by 1 [ ] ( g g g ) m 1 m g [ m] g ( gm g m g m) PROPERTIES 1. [ ] [ ] 1 1 Snce ( g g g ) ( g g g ) But these two are equal g g & g g

16 . 3. m LHS g [ m] m RHS g [ m] but [ m ] [ m ] Hence g [ ] [ ] 1 1 RHS ( g g g ) ( g g g ) 1 1 ( ) (. g ) g LHS g g TRNSFORMTION OF CHRISTOFFEL SYMBOL I KIND e [ ] [ ] 1 ( ) [ ] g g g But we now that g g Consder g g [ ] g g [ ] g g Then smlarly we can wrte g & g g [ ] g g g [ ] g g In equaton (1) I term of RHS we put w.r.t and w.r.t β. Consder RHS the I term contan metrc tensor g Symmetrc. So t s not necessary to change 1 Then from LHS ( ) g g g Ie 1 {(3)+()-(1)} 1 1 [ ] g {..[ ]} 1 [ ] { g.[ ]} Ths s the transformaton of Chrstoffel symbol of the I nd from one coordnate system to another. The presence of the I term on the RHS of the above equaton shows that the Chrstoffel Symbol of I Knd s not a Tensor.

17 TRNSFORMTION OF CHRISTOFFEL SYMBOL II KIND m g m [ ] g [ ] the LHS s obtaned by tang nner product of equaton () wth g g [ ] g { g.[ ]} g g. g [ ] to mae g n unbarred co-ordnate system lm no l m n o g g g [ ] lm no l m n o lm l g g g [ ] {Snce mg g g g est only when } n n g [ ] n n Due to the presence of I term on RHS ths s also not havng tensoral n character. COVRINT DERIVTIVE It has been ponted out that space dervatve of a scalar feld s a vector. Dervatve of a scalar feld transforms le Covarant Tensor. Eg : Gradent ( φ ) Dfferentaton both sde of the above equaton wth Whch s not a transformaton equaton as whch destroys the tensor character. However n Cartesan frame 0. bove equaton become If 0 e vector ( Tensor wth Ran 1) dervatve transformle a covarant tensor of Ran n Cartesan frame. In case of general curvlnear co ordnates ordnary dervatves gven by (1) doesn t transform as a tensor. So let us ntroduce a new nd of dervatve defned as covarant dervatve specally for tensors. (1) But from the transformaton of Chrstoffel symbol II Knd n n On tang the nner product wth n n { }

18 n n Substtutng (4) n { } ( ) ( ) ( ) ( ) The functon ; s called th covarant dervatve of the vector COVRINT DERIVTIVE OF CONTRVRINT TENSOR Let us consder a contravarant tensor The transformaton equaton gven by [Substtuton s made for convertng to ] from prevous sesson we proved that But we want so put -- - and β { here we put ρ - β and γ } From (b) & (c) { } By combnng I and III term Put & by [ ] ( ) ( ) RIEMNN CHRISTOFFEL TENSOR Consder an nfntesmal parallelogram PQRS as descrbed wth adacent sdes PQ d and PR δ. Imagne a contravarant vector P ( P )beng parallel dsplaced n the followng two ways. 1. Dsplace parallel from P to Q resultng n parallel from Q to S gvng the vector SQ defned at a pont Q & then dsplace Q

19 . Dsplace P parallel from P to R gvng from R to S resultng the vector SR Now to fnd the two vectors SQ and does the dstance between them depends. Dsplace R and then dsplace R parallel SR at S be equal. If not what P parallel from P to Q. d Q P P SQ Q Q d P P P P l m Q lmp Q Q The Chrstoffel symbol depend on the metrc tensor whch n turn s a functon of co ordnates. For a small dsplacements PQ d we can wrte Hence d n lmq lmp lmp n ( d ) n l m SQ Q lmp lmp n Q Q d ( d )( d ) n l l m P P P P lmp lmp n Now SR d d d d d l m l n m l m l n m lm lm n lm lm n SR s obtaned by ust replacng d by and by d lm d lm n d lm d lm n d l m l n m l m l n m The frst 3 terms n the RHS of (1) & () are equal. e n II and III term ust replace by m Hence ()- (1) SR - lm n d lm n d lm d lm d SQ l n m l n m l m l m m Now for tang d common changes to be made n I II & III terms { I l n II l m & III m & m } Then the bove Equaton becomes SR - m d m d l md lm d SQ m m l m l m [ m m l m lm ] d l l m R d m. m Where R. Snce l l m m m l m lm d and m are arbtrary vectors t follows from the Quotent law that R. m s a tensor of ran 4. It s nown as Remann Chrstoffel Curvature Tensor. It s ndependent of the vector tensor and ts I and II dervatves and depends only on the metrc RIEMNN CHRISTOFFEL CURVTURE Tensor Identcally vanshes at Eucldean Space