Notes on Differential Geometry

Transcription

1 Nots from phz 6607, Spcial an Gnral Rlativity Univrsity of Floria, Fall 2004, Dtwilr Nots on Diffrntial Gomtry Ths nots ar not a substitut in any mannr for class lcturs. Plas lt m know if you fin rrors. I. CHRISTOFFEL SYMBOLS A covariant rivativ oprator which is compatibl with a mtric ncssarily satisfis so that a g bc = 0 = a g bc abg c acg b, 1 a g bc = abg c + acg b, b g ca = bcg a + bag c, c g ab = cag b + cbg a. 2 Th scon two of ths follow from th first with cyclic prmutations of th inics. A th first two an subtract th thir to obtain a g bc + b g ca c g ab = 2 abg c, 3 whr th symmtry of th Christoffl symbols is us. Now, raising th c inx rsults in ab = 1 2 gc a g bc + b g ca c g ab. 4 A. Intitis involving th Christoffl symbols an covariant rivativs a ab = 1 g x b g 5 a ξ a = 1 g x a gξ a 6 [a ξ b] = x ξ [a b] 7 a a ψ = 1 gg ab ψ 8 g x a x b a F [ab] = 1 g x a gf [ab] 9 for A ab = A ba : a A ab = gba c gaa gbc g a g x c 2 x c Aa 10 1

2 In a gnric coorinat systm 2 φ = a a φ = g ab a b φ g ab c ab b φ = 1 g a gg ab b φ. 11 It is oftn convnint to fin c g ab c ab = 1 g a gg ab, 12 so that 2 φ = g ab a b φ c c φ. 13 II. THE RICCI IDENTITY AND THE RIEMANN TENSOR Lt ξ a b an arbitrary vctor fil, an consir a b ξ c b a ξ c = a b ξ c bcξ ab ξ c cξ ac b ξ bξ b a ξ c acξ + ba ξ c cξ + bc a ξ aξ = a bc ξ ab ξ c cξ ac b ξ bξ + b ac ξ + ba ξ c cξ + bc a ξ aξ = a bc ξ ac b ξ bξ + b ac ξ + bc a ξ aξ = a bc ξ + ac bξ + b ac ξ bc aξ = a bc ξ + b ac ξ bc aξ + ac bξ, 14 whr th first quality follows from th scription of th covariant rivativ in trms of th Christoffl symbols, th scon from th commutation of partial rivativs, th thir from th symmtry of th Christoffl symbol, th fourth from th Libnitz rul for iffrntiation, an th fifth by rarranging trms. From th Ricci intity w also hav With ξ a bing arbitrary, it is ncssary that aftr a rarrangmnt of trms an inics. a b ξ c b a ξ c = R abc ξ. 15 R abc = a bc + b ac a bc + b ac, 16 A. Algbraic intitis of th Rimann tnsor Consir [a b c] ψ for an arbitrary scalar fil ψ. From th Ricci intity [a b] c ψ = 1 2 R abc ψ 17 2

3 Antisymmtrizing ovr a, b, an c givs [a b c] ψ = 1 2 R [abc] ψ 18 but [b c] ψ = 0 bcaus th connction bc is symmtric. Bcaus ψ is arbitrary, it follows that R [abc] = 0, 19 for any Rimann tnsor. A scon way of stating this sam intity rlis upon th antisymmtry of th first two inics of th Rimann tnsor an is R abc + R cab + R bca = Consir a b ξ c λ c, for arbitrary vctor fils ξ c an λ c. It follows that a b ξ c λ c = a λ c b ξ c + ξ c b λ c = λ c a b ξ c + a λ c b ξ c + ξ c a b λ c + a ξ c b λ c. 21 Antisymmtriz ovr a an b: th lft han si is zro bcaus th connction ab is symmtric; th scon an fourth trms on th right han si cancl ach othr. Us th Ricci intity on th thir trm on th right to obtain Th vctor λ c is arbitrary, an w conclu that 0 = λ c [a b] ξ c + ξ c R abc λ. 22 [a b] ξ c = R abc ξ c. 23 Officially, this is th Ricci intity for contravariant vctors. Our intrst is usually on a rivativ oprator which is compatibl with th spactim mtric g ab, in which cas w can rais an lowr inics on ithr si of a rivativ oprator. Thus, it also follows that From a straight application of th Ricci intity W conclu that At this point w hav shown that an that W may now conclu that [a b] ξ c = R ab c ξ c. 24 [a b] ξ c = R ab c ξ c. 25 R abc = R ab[c]. 26 R abc = R [ab][c] 27 R abc + R cab + R bca = R abc = R cab R bca = R cab + R bca = R cab R acb R bca R cba = R cba + R abc + R bac + R cab = R cba R bac R bac + R bac + R cab 29 3

4 whr w us Eq. 28 in th first quality, intrchang th first two inics of vry trm to obtain th scon quality, us Eq. 28 on both trms to obtain th thir quality, intrchang th last two inics of vry trm to obtain th fourth quality, an us Eq. 28 on only th scon trm to obtain th fifth quality. In th fifth lin th thir an fourth trms cancl, an w ar lft with R abc = R cba R bac + R cab = R cab R abc + R cab ; 30 th scon lin follows from intrchanging ach pair of inics on th right han si. This asily simplifis, an w finally hav R [ab][c] = R abc = R cab = R [c][ab]. 31 To simplify th unrstaning of th inpnnc of ths algbraic intitis, assum that Eq. 31 hols for a 4-inx covariant tnsor, R abc, an s how much aitional information is larn if Eq. 28 also hols. Eq. 31 immiatly implis that Eq. 28 hols if any two of th inics a, b or c ar qual. So nothing nw is larn from Eq. 28 unlss a, b an c ar all iffrnt. But, Eq. 31 also immiatly implis that Eq. 28 hols if is qual to any of a, b or c. W may conclu that Eq. 28 givs nw information, byon that contain in Eq. 31, only if a, b, c an ar all istinct. Now w ar in position to count up th numbr of algbraically inpnnt componnts of th Rimann tnsor of a four imnsional manifol. An antisymmtric pair of inics [ab] may b chosn in 6 iffrnt ways, so thr ar 6 ways to chos th [ab] in R abc an 6 ways to pick th [c]. Also, R abc = R [c][ab], so R abc looks lik a 6 6 symmtric matrix, which has 21 algbraically inpnnt componnts. Eq. 28 provis prcisly on mor algbraically inpnnt rlationship whn all four inics ar istinct, an w conclu that th Rimann tnsor has twnty algbraically inpnnt componnts on a four imnsional manifol. an B. Diffrntial intitis of th Rimann tnsor Th Bianchi intity is a iffrntial intity of th Rimann tnsor. Start with a [b c] µ = 1 2 a R bc m µ m = 1 2 µ m a R bc m R bc m a µ m, 32 [a b] c µ = 1 2 R abc m m µ R ab m c µ m. 33 Now, antisymmtriz ach of ths ovr [abc]. Th righthan sis of th antisymmtriz vrsions ach of ths quations ar qual, µ m [a R bc] m + R [bc m a] µ m = R [abc] m m µ + R [ab m c] µ m 34 4

5 Th algbraic intitis of th Rimann tnsor implis that th first trm on th right han si is zro an that two of th othr trms cancl with th rsult that Bcaus µ m is arbitrary, w conclu that µ m [a R bc] m = R [bc m a] µ m + R [ab m c] µ m = [a R bc] m = 0, 36 which is th Bianchi intity. Th contract form of th Bianchi intity follows from first contracting ovr a an m, Now contract ovr b an, 0 = 3 [a R bc] a = a R bc a + c R ab a + b R ca a = a R bc a c R b + b R c = a R bc ba c R b + b R c = a R c a c R + a R c a, 38 which may b writtn as a R c a 1 2 g c a R = This last rsult is also rfrr to as th Bianchi intity, or somtims th contract Bianchi intity. III. THE ALTERNATING TENSOR In four imnsional spac-tim ɛ abc = ɛ [abc] = ±1, 0 g 40 pning upon whthr a, b, c, is an vn, o or no prmutation of t, x, y, z. Also ɛ abc = ɛ [abc] 1 = ±1, g Th contractions of th prouct of th two ɛ s hav simpl xprssions. First, with no contractions ɛ abc ɛ fgh = 4!δ a [ δ f = 4! δ 4 aδ [f δf aδ [ δg aδ [f b δ cδ h] δh aδ [f b δg c δ ]. 42 For th scon quality, not that th right han si is xplicitly antisymmtric in [, f, g, h]; th factor of 1 is th rquir normalization. A similar stp is prform at th n of 4 th following quations. 5

6 an For on contraction, ɛ abc ɛ afgh = 3! For two contractions, ɛ abc ɛ abgh = 2! 1! δaδ a [f δf aδ [a δg aδ [f b δa c δ h] δh aδ [f b δg c δ a] = 3! 4 3 δ [f = 2! 1! δ f b δ[g c δ h] = 3! 1! δ[f δg b δ[f c δ h] δh b δ c [g δ f] δbδ b c [g δ h] δg b δ[b c δ h] δh b δ c [g δ b] 43 = 2! 1! 4 2δ c [g δ h] = 2! 2! δ[g c δ h] = 1! 2! δc g δ h δc h δ g.44 For thr contractions, ɛ abc ɛ abch = 1! 2! δ c cδ h δ h c δ c = 1! 2! 4 1δ h = 1! 3! δ h. 45 Finally for all four pairs of inics contract, ɛ abc ɛ abc = 3! δ = 4! 46 Summarizing ths formula, with iffrnt labling of th inics, w hav ɛ abc ɛ fgh = 4!δ [ a δ f 47 ɛ abc ɛ fg = 3! 1! δ [ a δ f b δg] c 48 ɛ abc ɛ fc = 2! 2! δ a [ δ f] b 49 ɛ abc ɛ bc = 1! 3! δa. 50 ɛ abc ɛ abc = 4!. 51 IV. THE PROJECTION OPERATOR An obsrvr with four-vlocity u a may construct a spcial spatial altrnating tnsor ɛ abc = ɛ [abc] ɛ abc u = ±1, 0u t or u t g 52 An an obsrvr may us a projction oprator h a b to projct tnsor inics prpnicular to his four-vlocity, h a b g a b + u a u b. 53 Not that h a bu b = g a b + u a u b u b = u b u b = 0, 54 whr th scon quality follows from th normalization of th four-vlocity, u b u b = 1. It asily follows that h a bh b c = h a c, 55 as woul b xpct for a projction oprator. Any tnsor inx which is prpnicular to u a may b rais or lowr by ithr g ab an g ab or by h ab an h ab. If a normaliz, timlik vctor fil u a is hyprsurfac orthogonal, thn th projction oprator h ab also plays th rol of th mtric of th thr imnsional, spatial hypr-surfac which is prpnicular to u a. 6

7 V. ELECTRICITY AND MAGNETISM an Maxwll s quations for th lctromagntic fil F ab = F [ab] with a sourc J a : [a F bc] = 0 56 b F ab = 4πJ a. 57 For a givn obsrvr with four-vlocity u a, th lctromagntic fil may b compos into its lctric E a an magntic B a parts by projcting F ab paralll an prpnicular to u a, E a = F ab u b 58 an B a = 1 2 ɛ abcf bc u. 59 It is asy to show that E a u a = 0 an that B a u a = 0 so th lctric an magntic fils ar spatial vctors to th obsrvr, u a. W may irctly writ F ab in trms of its componnts as F ab = 2u [a E b] + ɛ abc B c u ; 60 this may b vrifi by substituting this xprssion into th abov quations for E a an B a. Th forc on a charg particl of charg q an mass m moving with four-vlocity v a is th right han si of mv b b v a = qf ab v b, 61 which is th quation of motion of a charg particl in fr-fall through an lctromagntic fil. VI. MAXWELL S EQUATIONS an Maxwll s quations for th lctromagntic fil F ab = F [ab] with a sourc J a : [a F bc] = 0 62 b F ab = 4πJ a. 63 Imagin a clou of charg ust moving through spactim with four-vlocity v a an co-moving numbr nsity n. Each bit of ust has a mass m an charg q. Th consrvation of ust implis that a nv a = An obsrvr with four-vlocity u a ss a charg nsity qnu a v a an currnt nsity qnh a bv b. To s that Maxwll s quations rquir th consrvation of charg, valuat a b F c b a F c = R abc F + R ab F c 65 7

8 from th Ricci intity. Now contract Eq. 65 with g ac an g b an us th antisymmtry of F ab to s that th lft han si of th contract Eq. 65 is Th right han si of th contract Eq. 65 is a b F ab b a F ab = 2 a b F ab. 66 g ac g b R abc F + g ac g b R ab F c = R b F b R a F a = 0 67 whr th first quality follows from th finition of th Ricci tnsor, R b R a ba, an th scon follows from th symmtry of R ab an th anti-symmtry of F ab. Thus, from Eq. 63 4π a J a = a b F ab = 0, 68 an th four-currnt nsity must b consrv for th consistncy of Maxwll s quations. Eq. 62, [a F bc] = 0, is th intgrability conition for th local xistnc of a vctor potntial A a, such that a A b b A a = F ab. 69 This is vry similar to th Euclian gomtry thorm that if F = 0 thn thr xists a vctor A such that F = A. Lt us fin th quation govrning th vctor potntial. From Eq. 69 c F ab = c a A b b A a = R cab A + a c A b c b A a, 70 from th Ricci intity. Aftr contraction with g bc, this bcoms b F ab = 4πJ a = R a A + a b A b b b A a. 71 This is quivalnt to b b A a a b A b R a A = 4πJ a. 72 In th Lorntz gaug, whr b A b = 0, this simplifis to b b A a R a A = 4πJ a. 73 A. Altrnativ E&M Gnrally, whn w consir an ara of physics which is wll unrstoo in spcial rlativity, such as lctricity an magntism, an try to fin th gnralization of th rlvant quations to curv spactim, w o as littl as possibl to th quations an typically just rplac orinary rivativs with rspct to Minkowski coorinats by covariant rivativs with rspct to an arbitrary coorinat systm. Howvr, this procss is not unambiguous. With this in min consir th vctor potntial A a fin in trms of th lctromagntic fil by a A b b A a = F ab. 74 In flat spactim th vctor potntial satisfis b b A a a b A b = 4πJ a. 75 8

9 It might appar rasonabl to vlop curv-spactim lctricity an magntism, by starting irctly with ths quations for A a an F ab whil consiring th rivativs to b covariant rivativs of curv spactim. Show that this vrsion of curv-spactim lctricity an magntism has an unplasant fatur. Hint: Look at th nots to s th accpt vrsion of curv-spactim lctricity an magntism. From Eq. 75, w can valuat th ivrgnc of J a. Spcifically, Focus on th first trm on th right han si. 4π a J a = a b b A a a a b A b 76 a b b A a = b a b A a + R a b b A a + R a ba b A = b a b A a R a A a + R b b A = b a b A a = b b a A a + R ab a A = b b a A a + b R b A 77 whr th first quality follows from th Ricci intity aftr intrchanging th orr of th covariant rivativs, th scon follows from th finition of th Ricci tnsor in trms of th Rimann tnsor, th thir follows from th symmtry of th Ricci tnsor, th fourth follows from again intrchanging rivativs an using th Ricci intity, an th fifth from again using th finition of th Ricci tnsor in trms of th Rimann tnsor. Now substitut this final rsult back into Eq. 76 to obtain 4π a J a = b b a A a + b R b A a a b A b = b R b A. 78 Gnrally, th right han si of this last xprssion is not zro, an w s that in this othr vrsion of lctricity an magntism in curv spactim charg is not consrv. This is consir an unplasant fatur. VII. STRESS-ENERGY TENSOR Th strss-nrgy tnsor for th lctromagntic fil is Th consrvation of strss-nrgy thn implis 4πT ab = F a c F bc 1 4 g abf c F c. 79 4π a T ab = a F a c F bc + F ac a F bc 1 2 F c b F c = a F a c F bc F ac a F bc F ac c F ba F ac b F ac = a F a c F bc 1 2 f ac a F cb + c F ba + b F ac = a F a c F bc, = 4πJ a F ba = 4πJ a F ab 80 9

10 whr th pnultimat lin follows from Eq. 62 an th last lin from Eq. 63. Exprss T ab in trms of th vctor potntial as 4πT ab = a A c c A a b A c c A b 1 4 g ab c A A c c A A c = a A c c A a b A c c A b 1 2 g ab c A c A A c