( ) 2 a b ab. To do this, we are to use the Ricci identity (which we use to evaluate the RHS) and the properties of the Lie derivative.

Transcription

1 Exercise [9.6] This exercise sks s o show h he ccelerion of n (infiniesiml volme mesre V long he worlline he volme s cener e o he effecs of spceime crvre is given by: D V = R V ( b b To o his, we re o se he Ricci ieniy (which we se o evle he RHS n he properies of he Lie erivive. Now, recll h he Lie erivive of ensor fiel T (of ny rnk is he ifference beween T s efine ech poin, n h sme fiel T rgge long n infiniesiml isnce by vecor fiel V. If we ke T o be he 3-form represening he volmeric elemen of spce (i.e. T xÿyÿz in some priclr observer s reference frme, n V o be he normlize ngens o he se of geoesics (worllines of pricles iniilly res in h frme (so h V, where τ is proper ime, hen - T represens n infiniesiml volme of spce V evolve hrogh ime (rgge V long he geoesics by n infiniesiml mon, s compre o he nevolve V he sme poin: i.e. i s he re of chnge of he 3-form V per ni proper-ime, e o spceime crvre. More formlly, le p be spceime poin, n ni imelike vecor efine p, efining proper ime for some reference frme p. The irecions perpeniclr o p efine spce ron p. Prllel rnspor of rilly o from p long hese perpeniclr irecions yiels srfce of simlneiy S in locl region ron p, wih he vecor fiel on his srfce now represening he irecion of proper ime for ll poins on his srfce (in he given reference frme. There is corresponing imelike geoesic hrogh ech sch poin, wih s is ngen. We cn hs frher exen o complee vecor fiel in he spceime ron p by efining everywhere s ngen o his se of geoesics: Th is o sy, we efine s ngen o he se of proper-ime prmeerize worllines of (infiniesiml pricles in volme of spce srroning p n iniilly res ( he insn of p in he given reference frme. Since is normlize, prllel-rnspore long, n ngen o (se of geoesics, we hve: = g = = ( b = = 0 (3 Also, since ws efine o be prllel o ( p in he spce S ron p, we hve (for spil coorine vecors x, y, z = = = 0 he poin p. Combine wih (3 bove, his yiels: x y z b ( p = poin p = 0 (4

2 If represens he irecion of ime, hen he corresponing 3-form represening spce (perpeniclr o is he Hoge l of, : = xÿyÿz The re-of-chnge (velociy of his qniy (w.r.. proper ime s i is rgge long he bovemenione geoesics is given by - bc. This erivive is iself 3-form. In generl his migh represen boh he volme chnge (he prely spcelike pr n lso Lorenz roion (boosing, escribing ny chnge in ngle of he srfce-of-simlneiy over ime (he imelike prs. Aclly in his cse he ler re zero (see clclions below. Since we re only inerese in he volmeric chnge, n we wn i s sclr vle, we nee o exrc he coefficien of he prely spcelike pr (he xÿyÿz componen. One wy o o his is o evle Ÿ-. Anoher more-or-less eqivlen wy is o ke he Hoge l, so rning he 3-form bck ino vecor. The xÿyÿz pr hen becomes he componen of he vecor, which we cn exrc (s sclr in he sl wy for vecors, by king he inner (o proc wih. Ths we ge: D ( V = -V g This gives s he velociy (re-of-chnge of he volme. We clly wn is ccelerion, D ( V. However, since D( V s efine bove is js sclr qniy, we nee only ifferenie gin w.r.. in he norml wy for sclrs (sing or i.e.: ( V = -V ( g D (5 Evling his expression will reqire he following five ieniies: Hoge l of vecor: n 3-form: v = e v (6 bc bc 3! bc Q = e Q bc (7 Lie erivive of 3-form (see exercise [4.]: Q = Q + Q + Q + Q (8 bc bc bc c b b c The clclions n resls re he sme excep for n iionl fcor of e bc (i.e. he 4-volme ensor ŸxŸyŸ z in he firs cse. (This is rivilly esy o verify sing he igrm noion.

3 bc bc Conrcion of Levi-Civi ensors (see cpion of fig..8; noe lso Ú =-e : p q }} KceKf e f e!! {{ KcKw = pq[ Kw] p q Ú (9 Re-nisymmerizion of n nisymmeric expression hs no effec: T efgh ( T = e Ú (0 [ bc] 4 bc [ efgh] Now, from (6, = e bc bc ; sbsiing his for Qbc in (8 yiels: = ebc + ebc + ec b + eb c bc 4443 = ebc + ebc = 0 The firs erm vnishes becse e = 0 (clrificion in Appenix C n by (3 lso = 0. bc Also by (3 we hve e = 0, so we cn freely his expression o mke he [ ] erm bc compleely nisymmeric. Applying (0 followe by (9 o i hen gives s: ( rce = e = bc bc bc For ny vecor v, we hve bc e v = e e bce v = -v (sing (6, (7, n (9. So: 3! ( rce ( rce = = - = - = - g g rce = (from( { ( rce Finlly sbsiing ino (5 we ge: ( rce c D V = V = V c (

4 Now, compring his o (, n noing h he ron he V merely serves o inice h i is (posiive sclr qniy (rher hn 3-form, sy, we observe h i remins only o show h: R = b c b c he poin of ineres, p. On pge 30, he Ricci ieniy (which hols in he bsence of orsion is given s: R x = - x c bc b b Exchnging some symbols gives he sme ieniy expresse s: Then: R = - b c b Rb Rcb b cb c c = c - c = - c c c c c = + - c c c c c c This is re for ny rbirry vecor fiel. If we se he we efine erlier, hen we cn pply (3 c o he finl erm o elimine i (i.e. we hve ( 0 secon erm ( c c =. Frhermore, we noe h he c is proc of firs erivives of, no n erivive, so from (4 his erm will vnish p (or inee ny poin where neighboring geoesics h efine re ll prllel; where his is no he cse, we re no longer consiering V h s iniilly res n so Newon s -4p G M forml for volme ccelerion oesn pply nywy. Therefore, we hve shown h, he poin p: D V = V = VR c b c b n he exercise is complee!

5 Appenix A: Comping ( V D s 3-form inse of s sclr. I ve one his exercise by comping sclr volme ccelerion, becse h seeme simples, n i is ll h is reqire in orer o pply Newon s -4p G M forml. However, yo migh hve inerpree he book o men h V is he cl 3-form rher hn js n infiniesiml mesre of 3-volme ; n if yo hen ke D, yo ge D ( V, which is lso 3-form rher hn sclr. From erlier we h: ( rce = = ( rce + ( rce = ( rce + ( rce So D ( = = ( rce + ( rce V V n, he poin p, where = 0 (n hence lso ( rce = 0: As reqire. D ( rce V = V = VR b b Noe h µ mens here is no Lorenz roion (boosing of V s i is rgge long is geoesics, since hs no ŸK componens. This cn be expece, becse is lwys perpeniclr o, n iself oes no chnge irecion s i is rgge long is geoesic, or s ime evolves i.e = 0.

6 Appenix B: Comping ( V D s 4-form. We migh lso sk wh hppens if we ke V o be he 4-volme ensor spil 3-volme. Do we ge similr resls? e bc, rher hn js We hve: bc ε = e + e + e + e = e bc c b b c bc bc ( rce ε = ε ε = ε ( rce + ( rce ε = ε ( rce + ( rce So D ( = ε = ( rce + ( rce V V n he resl is excly he sme!!! Iniively, he componen of ŸxŸyŸ z oes no chnge s i is rgge long he geoesic fiel, whils he remining spil pr chnges js s clcle previosly in Appenix A.

7 Appenix C: Why oes ε = 0? In my clclions bove I hve me se of he fc h e bc = 0. Penrose escribes he Levi- Civi ensor, e bc s hving olly nisymmeric componens ebc ( e[ bc ] = n normlize so h e 03 =. If his were lwys he cse, hen he componens of e bc wol be consns, inepenen of coorine sysem. Le e% bc be his (so efine consn objec (clle he Levi- Civi symbol. Obviosly e% = 0 (where ; however, e% 0 becse of he exr bc erms involving Chrisoffel symbols ( G bc. In fc, e% bc is no even ensor, which cn reily be m seen simply by consiering (for exmple coorine rescling in Cresin coorines: x m = x. This shol mliply he componens of ny Í b he componens of e% bc re he sme in ll coorine sysems. x bc rnk ensor by fcor of b - b by efiniion The Levi-Civi ensor e is clly: e bc = e g e% bc Wih inverse (efine so h e Ú bc = 4! : bc bc Ú = Ú % e g bc (See Lecre Noes on Generl Reliviy by Sen M. Crroll, chper ; his is online hp://ne.ipc.clech.e/level5/mrch0/crroll3/crroll.hml, n he relevn pr is ner he boom of he pge. Penrose hs glosse over he isincion beween e n ε% pprenly becse in snr, orhonorml Minkowski coorines, he eerminn of g is -, so heir componens re he sme in hese coorines. However, i is e, obviosly, which ms be se for nisymmerizion n clclion of he Hoge sr, ec., so h he resl remins ensor.

8 We cn show explicily h ε = 0 : e = e -G e -G e -G e -G e s s s s bc bc sbc b sc c bs bcs (See exercise 4.6. The ls for erms re mnifesly nisymmeric in, b, c,, so we cn re-wrie: e = e -G e s bc bc s bc s ( eg e g = - G e% s bc Conrcing g ( g g g s sr G = + - (see exercise 4.6, we ge G = g g : bc c b b c cb s sr sr ( eg e g g g e = - e% ( bc sr bc Now look he forml for he eerminn of mrix Levi-Civi ensors ε n Ú, b col eqlly well se ε% n %Ú, since he T b he boom of p. 60. I ses he e g n e g - cncel o: i.e. he eerminn is lwys js sm of procs of mrix elemens. The eerminn of g follows excly he sme forml, componen-wise, b we fce he sligh problem h boh inices of g re on he boom (n I isn hr o see h he eqivlen forml in his cse is: g b hs ifferen mening logeher: b. e g bc efgh = Ú% Ú % 4! g g g g e bf cg h Is erivive is: bc efgh bc efgh ( e g 4! Ú % Ú% ( g g g g L Ú% Ú% g g g ( g bc efgh = Ú% Ú% ( g g g g = + + e bf cg eh e bf cg h 6 e bf cg h (by noing h ech of he erms in he [ ] re he sme js perme he inices.

9 So: sgn( e g eg = e g ( e g sgn( e g bc efgh = Ú% Ú% e g ( g g g g e bf cg h (3 sr b Now we noe h g is in fc efine s he inverse of g sr, so h g g bc = c = I. Fig 3.7 on p. 59 gives he igrmmic forml for mrix inverse. Trnsling his igrm bck ino ensor noion, n once gin compensing for he fc h boh inices of g re on he boom, we ge: g = Ú% Ú % e bc efgh 6e ( g g g g bf cg h e bc efgh = Ú% Ú % ( g g g g g g e 6e g e bf cg h Noing h e g sgn( e g e g = : e sgn( e g bc efgh e g g ge = % % e g ge gbfgcggh Ú Ú (4 Compring (3 n (4 we see h: eg = e g g g e e n plgging his ino ( gives s: e = bc 0 s reqire.