CHAPTE 5 GAVITATIO 5 Chagoass aio and agni ons In Chap a bif ovviw of h Dynai Thoy was psnd Th fundanal pinipls of h Dynai Thoy w psnd in Chap o hs fundanal laws h onsany of h spd of ligh was divd h quid goy was obaind lassial and spial laivisi quaions of oion w divd and h ondiions quiing quanu hanis w displayd Th quins of h fundanal laws w aid fuh in Chap by looking a h gaug filds of h suling fivdinsional goy whn ass is onsidd as an indpndn vaiabl Whn quanizaion ondiions a onsidd in fivdinsions w found xpinal faus of pail physis quid by hs nw laws o xapl w saw ha os a quid fundanal sas inisn of Gllan's ighfold way; h allowd filds fo fundanal pails w shown o b quaniizd in li hag; and h adial fild dpndn o display a shoang nonsingula bhavio whih allowd i o pdi nula asss fo i's dviaion fo h Coulobi adial dpndn and nula day (ba day fo h asyy of whil pail fos Thus in haps and w hav shown how h Dynai Thoy podus by using h appopia siiv assupions h fundanals of all h un banhs of physis xp gaviaion In Chap h adial fild dpndn was divd and h longang dpndn quid ha h nw fild oponns b inpd as h gaviaional fild and h gaviaional ponial In his hap w will xplo a fw asps of his inpaion In paiula w will look a so of h pdiions of h Dynai Thoy in opaison wih Einsin's Gnal Thoy of laiviy Bfo w plung ino h divaion of a pdiion o opa wih h Gnal Thoy of laiviy l us fis onsid a qusion whih aiss fo h nssiy of kping h unis saigh aong h fild quaniis E B V and V whn hy a all o b onsidd as oponns of h fivdinsional gaug fild By onsiding h unis of hs fild oponns i is soon found ha a hagoass aio is ndd in od ha h unis of h gaviaional fild oponns ay b opad o pu in h sa quaion wih h li and agni oponns L us fis s if w an din his aio In h Chap h divaion of h filds allowd fo fundanal pails was psnd Ths fild xpssions giv is o h spifiaion of a hagoass aio whih allows onvsion of lassial gaviaion fild unis o loagni fild unis Th gaviaional fild oponn in h sys of fild quaions wih h li and agni oponns bings up h quin fo a 57
gaviaionoloagni uni onvsion This nd ay b sn by looking a h diffn fild quaniis is onsid h li as Th fild unis a givn by [E] vol/ whil h xpssion fo h li fo dnsiy is ρe wih unis of nwon/ o h gaviaional fild in h Dynai Thoy h unis a [V] wbs/ squad whil h gaviaional "un" dnsiy has unis givn by [J ] ap/ Thus h gaviaional fo dnsiy is givn by a (J /V wh again h unis a nwon/ In od o opa his sys o h lassial gaviaional sys w nd o b abl o go fo a gaviaional fild wih unis of nwon/kiloga o unis of vol/ ow n kg vol oul kg vol oul kg Thus if ß is a quaniy wih unis of oul/kg hn (/ß is h onvsion fao w sk Siilaly w nd o onv h gaviaional ass dnsiy (J / wih unis of oul/ o unis of kg/ Obviously ß will also b h onvsion fao fo his also Th qusion is; How do w din his hagoass aio and is i uniqu? If w onsid h filds h Dynai Thoy givs fo fundanal pails w ay din ß Thus l us look a h soluion of h gaug funion fo fundanal pails givn pviously ha is ln f f θ f φ f γ f W showd ha his ba ln f f f γ (5 fo fundanal pails Th funional dpndn of h gaug funion upon i o ass dnsiy was no dind hn alling fo pas ading ha asuns of a i dpndn of h ah's gaviaional fild hav bn pod ( W ay pod o ak h sipls possibl assupion abou h funional fo fo f γ and f naly lina dpndn If h funions hav only wak dpndn upon i o ass dnsiy hn his dpndn ould asily b askd in xpinaion no spifially dsignd o look fo i Thus ls' onsid h fo ln f (a b(s wγ (5 58
wh a b s and w a onsans o b valuad using known infoaion abou h poon and i dpndn asuns of h gaviaional fild Th gaug ponials a hn (ln f φ b (s wγ (i i (ln f φ d (a b(s wγ φ φ (5 and (ln f a a w(a b φ γ Using hs ponials h fild quaniis bo E ( b(s wγ Eθ Eφ B Bθ Bφ V ( a (a b w V θ V φ ( and ( a V (bw In Eqns (5 w ay s h ff of h i and ass dnsiy dpndn of h filds is noi ha h li fild E vanishs as h quaniy b vanishs o h xpssion fo h adial oponn of h gaviaional fild V w s ha b is h i dpndn of h gaviaional fild Thus in od fo an li fild o xis h us b a i dpndn of h gaviaional fild Siilaly on ay s ha h li fild us dpnd upon h ass dnsiy in od fo h o b a gaviaional fild o a gaviaional ponial V o xis no only us h gaviaional fild dpnd upon i bu also h li fild us dpnd upon h ass dnsiy This is a ah xaodinay vlaion! 59
ow w us hk hs fild quaniis in h igh fild quaions Baus B hn is saisfid x iplis B B x E whih is saisfid by h sphial syy of h E fild Thn is saisfid if h un dnsiy vo J vanishs as i should fo pails Looking a w find ha his is saisfid by a spaial hag disibuion of x E µε E µ π B J a x πρ E a ε V γ V γ ρ ( b(s wγ ε π (55 Th oninuiy quaion ρ J a J γ quis J γ (56 Thn B xv a γ iplis 6
whih is also saisfid by h sphial syy of V Whn w onsid h adial oponn of w find ha i is saisfid if xv V V E a γ whil h oponns in h and diions a saisfid idnially Th las quaion is (57 µε V π µ J V This quaion quis ha a w J ( (a b πµ (58 o ha h xpssion fo J saisfis Equaion (56 W also find ha h adial dpndn of J is idnial o h adial dpndn of ρ Thus w ay wi Eqn (58 as TC Van landn of h aval Obsvaoy has pod a asud vy sall i a of das in h gaviaional fild givn by hi o b appoxialy 6pas in p ya If w dsigna his a of das by dg/d hn o h xpinal asun a (a bw J µεb(s wγ ρ b G & a (59 & 6 x /y 9 x 8 G s hn a b a Thus h nonzo fild quaniis a 6
6 wh Z W should no in Eqns (5 ha h gaviaional fild dpnds upon h univsal onsan a o bing nonzo This iplis a linkag bwn h axiu ass onvsion a and h gaviaional fild in sowha a siila way as h li fild dpnds upon h spd of ligh uh i should b nod ha if h li fild dos no dpnd upon h ass hn h ould b no gaviaional fild ow h oal hag is givn by Thfo using Eqn (55 and h sphial ln of volu dv sinθ ddθdϕ w hav wh is h adius of physial xn of h pail If h ass dnsiy in h pail is a onsan γ hn h hag is givn by ρ γ µε π γ ε ρ γ b(s w aw a J Ga(s w Z wa Z a w(a b Z a V agz(s w E & & (5 ds q vol ρ φ θ θ γ π ε φ θ θ ωγ π ε d dd (s w ag Z d dd ( (s ag Z q vol sin sin & &
6 Siilaly w ay dno h gaviaional "hag" as Thn using Eqn (58 w hav ow h lons' fo is givn by o sin s >> εγ If w opa his wih h lassial xpssion fo >> γ hn w us hav o Thus (s w ag Z d (s w G a Z q o γ ε γ ε & & (5 dvol J vol wa Z a ds wa Z a vol µ πµ (5 ( ( s w ZaG s w ag Z q E γ γ ε & & s G G a Z & πε ε Z Gs Za & G s a & π ε (5
as ± πε G& whih givs h podu of a and s in s of h xpinal quaniis ε and dg/d L us now un o h gaviaional fo whih is givn by By opaing his wih h lassial xpssion fo h gaviaional fo w find w us hav o Z a wa µ g V (Z a wa (Z a wa µ (Z a wa µ G aw Z a Gµ ( If w onsid h pail o b a poon hn Z and h aio of h agniud of h li fo o h gaviaional fo is givn by πεg hus G πε and µ ( a wa πε fo Eqn (5 Thn w hav 6
aw a µ πε ε a π (55 ow hoos a (56 ε so ha by Eqn (5 b G& ε (57 and fo Eqn (55 εg w Zao (58 o Eqn (5 w find s π G& (59 Ths valus poin ou h xly wak i dpndn of h gaviaional fild and h vy wak ass dpndn of h li fild This ass dpndn ay b b sn if w wi azg(s & wγ E hn subsiu fo h fild paas o aiv a 65
66 ow l us un o h sah fo h hagoass aio sin w hav all h nssay infoaion Th quaniy w dfind as h gaviaional "hag" is givn by Eqn (5 If w divid his by w hav bu by Eqn (5 his bos o wiing Thus h hagoass aio w sk is givn by o G a G Z G a G Z a G ZG E ε πγ πε ε π γ πε π γ π ε & & & & wa Z a µ G G G Z a Z a ε µ µ µ µ G ε ( G / ε β (5 oul/kg x G 96 ε β
Th supising and plasing hing abou his sul is ha i is fod as h podu of wo known physial quaniis ah han dpnding upon nw quaniis suh as a and whos valus a no wll known uh in osp i appas o b h siplis if no h only obinaion of an loagni paa and a gaviaional paa whos unis a oul/kg I is wohwhil o poin ou ha h dpndn of h filds upon i and ass dnsiy (i b and w is xly sall bu is ssnial in sablishing ß and h induiv oupling bwn h loagni and gaviaional filds Th hagoass aio bings up h noion ha a oaing gaviaional lially nual body should hav a agni on sing fo h ffiv li hag assoiad wih h gaviaional ass Givn h hagoass aio w ay quikly look a is pdiion fo h ah's agni on Using ß h "ffiv" hag assoiad wih a gaviaional ass is givn by q ff β o h ah his ffiv hag would b qff 5 x oul Thus if h agni on of h ah is givn by µ ( q / A ff wh A is h ah's angula onu hn w hav qff µ I ω 7 5 (5 x (97 x (79 x (598 x o µ 86 x ap This pdid valu of h ah's agni on opas vy wll wih h xpinal valu of 8 x ap 5 Pihlion Advan o sious suggsion ha h addiional vo fild in h fivdinsional gaug quaions of h Dynai Thoy b h gaviaional fild an b ad wihou giving du onsidaion o h xplanaion of h planay pihlion advan povidd by Einsin's Gnal Thoy of laiviy Though sval aps hav bn ad o xplain h pihlion advan by oh ans non has sudd in asing uh doub on Einsin's xplanaion L us all so of h ain faus of h lassial pobl of planay obis Kpl's fis law sas ha a plan dsibs a losd llipial obi wih h sun a a foal poin Howv h psn of suh 67
sall influns as oh plans oving in h suns' fild auss a pubaion in h oion of a givn plan and h suling obi is no pisly llipi Indd on ay hink of h aual obi as a slighly bupy llips whih ay pss in h plan of oion; ha is h pihlion shifs abou and dos no always ou a h sa angula posiion Th fa ha h idalizd lassial obi is a losd llips is a sul pulia o h wonian invssqua law; in fa won hislf found ha if h fo of gaviy w popoional o / (δ insad of / hn a planay obi would no b losd and a pihli shif of od δ would ou Indd his sul was akn o india ha sin planay obis a vy naly losd h wonian invssqua law us b vy aua as in fa i is L us now ask w ay h b oo fo diffns bwn h pdiions of lassial lsal hanis and h lsial hanis of h Gnal Thoy of laiviy o h Dynai Thoy psnd h Sin Kpl's fis law is xpinally vifid o b o o a high auay w igh xp ha nonwonian Thois ay ly add a fw bups o h naly llipi obis and onibu sowha o pihli oion Sin angls a uh o onvninly asud in asonoy han a disans i is naual o onna on pihli oion Convninly nough h is in fa a wllknown dispany in lassial hanis onning h pihli oion of h plan uy Baus of uy's high vloiy and ni obi h pihlion posiion an b aualy dind by obsvaion Th diffn bwn h lassially pdid pihli shif (du o pubaions by oh plans and h obsvd pihli shif is sonds of a p nuy Evn hough his is a vy sall diffn i is abou a hundd is h pobabl obsvaional o and psns a u dispany fo h vy pis pdiions of lsial hanis whih has bohd asonos sin h iddl of h las nuy Th fis ap o xplain his dispany onsisd in hypohsizing h xisn of a nw plan Vulan insid h obi of uy and uh hoial wok was don o pdi h posiion of Vulan using h known pubaion on uy's obi Howv aful obsvaion faild o disov h hypohial plan and h hypohsis was finally abandond in 95 whn Einsin usd gnal laiviy hoy o xplain h obsvd ff ow l us look a wha h Dynai Thoy offs as an xplanaion fo h pihli advan and hn opa i o h pdiions of h gnal laiviy hoy Th lassial quaions of oion a && θ& ( (5 θ&& & θ& Th sond of Eqns (5 has h soluion 68
wh L is h angula onu Using Eqn (5 h fis of Eqns (5 ay b win o wh L θ & (5 L & ( W a sking h pdiion of h Dynai Thoy wih sp o h pihlion advan This ay b found by opaing h fquny of sall adial osillaions abou sady iula oion fo h ffiv ponial givn by Eqn (55 fo h nonsingula ponial of h Dynai Thoy wih h fquny of voluion By onsiding h nonsingula ponial of h Dynai Thoy Eqn (55 bos & & [ v (] (5 L v ( v( (55 K v ( & L (56 wih K G wh G is h gaviaion onsan is h ass of h sun and is h ass of h plan of ins Equaion (5 givs h fquny of voluion To din h fquny of sall adial osillaions abou sady iula oion w nd o valua h sond divaiv of h ffiv ponial v h adius fo whih h fis divaiv is zo Th fis divaiv of h ffiv ponial is obaind by diffniaing Eqn (56 wih sp o This ay b found o b K L [ v (] (57 Th sond od divaiv of h ffiv ponial ay b found o b appoxialy v ( K L (58 69
7 whn s involving / a onsidd ngligibl wih sp o s involving / W ay din fo h ondiion Th adius is h adius of na iula obi and h ff of h xponnial fao and (/ fao will b ngligibl fo << Thus w ay appoxia Eqn (59 by so ha If w appoxia h xponnial fao in Eqn (58 by is pow sis xpansion and aining only hos s whos dpndn upon / a lina o lss hn Eqn (58 is appoxiad by ow h fquny of sall adial osillaions abou sady iula oion ay b found fo Thus w hav An appoxiaion fo h fquny of sall adial osillaions abou sady iula oion ay now b ad by aking h squa oo of Eqn L G (] [v (59 L G G L (5 L K L K ( v (5 ( v ω L G 6 L G L G L G L G L G L L G L G G 6 6 6 6 6 6 ω (5
(5 and onsiding h sond s of h sond fao as sall opad o on Thus w hav Th pihlion advan p voluion ay now b found as h diffn bwn Eqns (5 and Eqn (5 valuad a dividd by h obial fquny o so ha G π L G G ω (5 L L ω θ& δθ π θ& G G L L L G G δθ π L Th pihlion advan pdid by Einsin's Gnal Thoy of laiviy is givn by δ θ Gi G π L If w o b suh as o povid an idnial pdiion as h Gnal Thoy hn would hav o saisfy o G 67 x 8 g /s 98 x g and x /s o G 8 (67 x (98 x ( x 7 x 5 7 x This is an xly sall valu opad o h adius of h sun bu is suffiin wihin h Dynai Thoy o povid h sa pdiion of pihlion advan as h Gnal Thoy of laiviy 7
5 dshifs Einsin's Gnal Thoy of laiviy pdid h advan of h pihlion of planay obis by using h full ff of h goial quaions W saw in h pvious sion ha h Dynai Thoy pdis a siila plan obi pihlion advan Anoh of h pdiions of Einsin's Thoy onns h dshifs assoiad wih ligh ivd fo disan ligh iing objs o whn ligh avls hough a hanging gaviaional fild Th Dynai Thoy should also pdi fquny shifs ha a xpinally asuabl If i dos no hn i dosn' hav h sa sngh of pdiabiliy as Einsin's Gnal Thoy Th a wo yps of dshifs suling fo h hoial appoah of h Dynai Thoy is h is an xpansion d shif du o h inasing "nopy" of h univs Sondly h is a fquny shif ausd by a diffn in h gaviaional sngh bwn h poin of ission of h ligh and h poin of is pion Boh of hs yps of fquny shifs a h sul of a diffn in h ffiv uni of aion a h ission poin and h pion poin Boh of h abov yps of fquny shifs ay b fd o as of goial in oigin in ha hy boh o fo h gaug funion Howv ah oiginas fo a diffn vaiabl hang in h gaug funion o insan h xpansion shif involvs onsiding h univs as an isolad sys suling in h nopy pinipl quiing sall fquny shifs owad h d This os fo h gaug funion bing dpndan upon i Th sond yp of fquny shif os fo h gaug funions dpndn upon spa and ass W ay fis onsid hs yps of fquny shifs o b indpndn and look a ah in un Thn w shall onsid h ogh is w will nd o onsid h loal syss wh a phoon is fis id and hn wh i is ivd In boh syss h ngy of a phoon is givn by hv wh v is h fquny and h is h ffiv uni of aion Th ffiv uni of aion is h podu of h gaug funion and Plank's onsan if a loally fla i is onsidd A h ha of boh yps of fquny shifs is h gaug funion whih has pviously bn givn as and found o b ln f f f f θ f φ f γ (5 ln f f 7 f γ
wh f and f γ indias funions of i and ass dnsiy W nd o din o abou h gaug funion han w pviously hav Th squa of h a lnghs diff by h ulipliaiv gaug funion as ( dq f(dσ o ( dq xp f ε f γ (dσ o his w s ha h diffnial hang in nopy is givn by dq xp f f γ dσ (55 alling ha h an b no das in h nopy fo an isolad sys w us hn onsid h possibl ff of h nopy pinipl upon h univs as an isolad sys W an s fo his lin of hinking ha f in Eqn (55 is h on o fous on fo h on Th sipls funion is of ous h lina funion and his lina dpndn appad pviously in sion 5 Suppos hn w onsid h ffiv uni of aion fo an isolad univs a so i whih w will s a W find h h f whih an b win h h xp [] (56 a H h valu of h ffiv uni of aion osponds idnially wih Plank's onsan h A so la i T h ffiv uni of aion would b givn by h T H xp [AT] wh A is a onsan L us fuh onsid a hang of vaiabls using h disan ligh will avl in f spa insad of h i T Sin T L/C L is h disan vaiabl w sk W now hav AL h H xp T (57 If a phoon w id a i would hav bn id wih an ffiv uni of aion givn by Eqn (56 If ha phoon is ivd a 7
h la i T a a disan of L hn h univs's ffiv uni of aion would b givn by Eqn (57 a pion If h ngy of h phoon whn id is givn by h ν 85 hn no loss of ngy by h phoon unil pion would qui ha Subsiuing fo Eqn (56 and Eqn (57 ino Eqn (58 w find Th fquny shif would b givn by o h ν ht ν (58 AL ν ν xp (59 ν ν ν ν ν AL AL ν AL ν (5 Th qusion aiss whh o no h fquny shif givn by Eqn (5 is d o blu? o Eqn (5 i ay b sn ha h fquny shif is ngaiv if A > hus h shif is d o blu as A is posiiv o ngaiv Going bak o Eqn (55 and using h gaug funion of Eqn (57 w s ha dq This indias ha a givn ln of a lngh dσ yilds a lag hang in nopy dq o a h i T L/C han bfo a i Thus h nopy hang is inasing and ou univs is xpanding Th xpansion d shif givn by Eqn (5 ay also b xpssd in s of wavlngh as AL dσ Equaion (5 ay b xpandd as ν ν AL (5 AL! AL! AL (5 whih ay b appoxiad by 7
AL (5 whn AL << Expinally i has bn dind ha HL wh H is h Hubbl onsan whih is givn by 7 H (56 _ 6 x s Thn w find h pdid fquny shif givn by Eqn (5 is sn in nau whn h onsan A is akn as h Hubbl onsan o Eqn (5 w s ha xpinally found d shifs an b usd as asonoial aks fo h xpssion L H ln xp (5 o sall xpinal d shifs opad o on h asonoial disans L givn by Eqn (5 a no signifianly diffn fo h lina aks givn by h appoxiaion of Eqn (5 Howv as h d shif bgins appoahing uniy h diffn bos signifian Tuning o h sond yp of fquny shif and uning o Eqn (5 o again wi Th ffiv uni of aion is xpssd by ln f f f ε γ (55 kl j s h h f gˆ x (56 s l jk i Bu whn in a loally Eulidan goy h g kl δ kl hn Eqn (56 bos o h xpssions fo h gaug ponials h h f h xp f f γ (57 75
(lnf (ln f φ j (58 j x x and Using Eqn (59 givs h adial oponn of h gaviaional fild o Eqn (58 w find and (ln f φ ij φ φ (59 f i j f γ j i df ao f d γ γ (55 Th xpssion fo h gaviaional fild givn in Eqn (55 is in s of a fild dnsiy By ingaion ov h volu oupid by h gaviaional ass dnsiy w hav h gaviaional fild V ao df f d (55 wh is h oal gaviaing ass o any wak i vaiaion in h fild w an igno h i dpndn Thus w hav df V ao a d (55 o b opad wih h xpinal fild G xp (55 V Cainly fo >> V in Eqn (55 is appoxiad by h xpinal xpssion of Eqn (55 if A G and df d o Eqn (55 w find ha 76
77 Th onvsion o a fild dnsiy ay b don by dividing by h ass so ha h gaug funion in Eqn (57 bos o wh has bn usd o obain a unilss quaniy whih us b h as fo f ow using h uni of aion givn by Eqn (555 suppos a phoon is id fo on body wih a gaviaional fild and is ivd in anoh gaviaional fild h onsvaion of phoon ngy would hn qui o so ha Th fquny shif would hn b onsan f onsan G H h xp G h h xp (555 G G ν ν h h G G ν ν xp xp G G ν ν xp (556
ν ν ν ν ν G G xp (557 o >> and >> hn Eqn (557 ay b appoxiad by ν ν G (558 Th appoxiaion in Eqn (558 shows ha if / > / hn his fquny shif givn by Eqn (557 is ngaiv o owads h d nd of h spu W an ak h fuh siplifiaion of assuing ha boh G << and G << hn w would hav h appoxiaion ha ν ν G _ G (559 In s of wavlnghs w hav ν ν G xp G (56 wih h abov assupions Eqn (56 is appoxiad by G G _ (56 Suppos w look a his d shif fo a phoon id fo h sufa of h sun and ivd a h ah's sufa Th ndd nubs a: G 667x n /kg sun 99(598x kg ah 598x kg sun 695x 8 ah 67x 6 x 8 /s Thus fo Eqn (56 78
6 x o 6 in k/s This is h sa as pdid by Einsin's Gnal Thoy of laiviy o a sial s of h d shif h pdiion would b G G _ If 7 f 95 hn / x 5 Sin h appoxiaion givn by Eqn (559 ay b xpssd as ν ν d _ wh ϕ G[( / ( / ] hn i ay b sn ha h d shif givn by Eqns (557 and (56 podu h d shifs pdid by Einsin's Gnal Thoy of laiviy if >> >> G << and G << Howv if hs ondiions of appoxiaion a no hn on us so bak o Eqn's (557 and (56 fo h pdid d shifs Suppos on onsidd a phoon whih ay hav bn id on a dns sa suh ha G /C is oo lag o allow a siplifiaion of h xponnial xpssion If his phoon w ivd on ah hn by Eqn (56 w would hav xp G xp ah ah Baus of h sall valu of (G ah /(C ah 7 x hn h appoxiaion bos G _ xp (56 wh G/ is h gaviaional fild a phoon ission o Eqn (56 w ay lan sohing of a sas' dnsiy by h d shif in h ligh ivd fo i o insan Eqn (56 has G _ ln (56 xp oi ha vn h lag d shifs displayd by quasas a allowd by Eqn (56 wihou quiing h o b a h fa ahs of ou univs 79
W hav lookd a h pdid fquny shifs as if hy w indpndn phnona In h sns ha hy s fo h sa gaug funion hn boh yps of dshifs ay b psn in any ligh ivd Thus w should look a h xpssion fo h ni d shif Suppos w l ln f k (a b f f f (s w z γ γ γ wh s w z a b and k a o b valuad Th ffiv uni of aion bos k h h xp (s wγ zγ (a b (56 fo o h abov l us ak s w hn Eqn (565 bos kz(ab h h xp (565 whn ingad ov h ni gaviaing ass as bfo and h subsip fs o h uni of aion a h pla and i of ission of h phoon A siila xpssion is found a h pla and i of pion o kz(a b h h xp (566 Equaions (565 and (566 ay b usd o xpss h fquny shif quid o onsv phoon ngy sin hν hν Thus w hav zk (a b ν ν xp zk (a b hn 8
ν ν (a b xp zk (a b Siilaly h xpssion fo h wavlngh shif bos (a b (a b xp zk (567 If w wi Eqn (567 as a pow sis and ak h appoxiaion of kping only h fis w find zk (a b (a (568 wh w'v also l By ling L/ Eqn (568 bos bl a zk a (569 W wan o valua h unknowns a k z and b in s of h pviously dind quaniis suh as G and H Thfo suppos ha h gaviaional fild a h i of ission of h phoon is h sa as h on ah a is pion hn w find Eqn (569 bos zkbl C ah ah (57 Expinally w hav found h xpansion d shif is givn by xp HL hus w should hav zkb ah H (57 ah 8
8 On h oh hand if h i bwn phoon ission and pion is suffiinly los hn ou appoxiaion o Eqn (569 an b win as wh >> and >> Thus in od o opa wih xpin w s If w now subsiu Eqns (57 and (57 ino Eqn (568 w hav wh / is h adius and ass of h ah as in Eqn (57 Whil h full xpssion Eqn (567 bos o Eqn (575 i ay b sn ha if h iving loaion is h ah hn h i dpndn in Eqn (575 is givn by so ha b of Eqn (567 is givn by H Thus h gaug funion dpndn upon i is also givn by b H all ha a i dpndn of h gaviaional fild has bn pod by T Van landn wih a pod valu of b 9 x 8 s Th asud Hubbl onsan is H (56 ± 6 x 7 s so ha x 8 s H 6 x 8 s Thus w s ha h Dynai Thoy pdis ha h xpansion dshif and a i das in h gaviaional fild sngh a boh h sul of h i vaiaion of h gaug funion uh i ss aazing ha wo suh diffn and diffiul yp asuns hav suh a good agn! uning o h wavlngh shif givn by Eqn (575 i ay b sn ha h onibuion of xpansion o gaviaional ponial o h oal d shif is onaind wihin his quaion Equaion (575 has h unknowns: h asonoial disan L h ass of h iing sa (o obj and h siz of h iing sa Givn only wo pis of zka _ (57 G zka (57 HL G (57 HL G xp (575 H HL
xpinal daa suh as h dshif and h appan luinosiy w an din h asonoial disan L and h gaviaional dnsiy / by assuing a hanis fo h ligh poduion (i sunlik Givn anoh daa poin suh as ligh fluuaion piodiiy hn a liiing siz igh b obaind 5 "ifh" o Is a "fifh" fo ally nssay? Obviously a nw fo is uh o xiing han finding an xplanaion fo h asud ff wihin anoh fo dsipion A nw fo ay b o nabl baus i ay appa no o op wih xising fos A oion o xising fos ay lak xin and us ainly b shown o b opaibl wih xising fos wh hy a asud o ga auay A oion o an xising fo is usually diffiul o find and ay go agains h pfn of any Bu o assu h is an addiional fo is o assu is indpndn and would hn nssia y anoh fo o b "unifid" Aua asuns show ha h gaviaional fo of h Eah diffs fo won's Law a los ang o spifially h diffn in h Eah's gaviaional fild ov a diffn of high in a dp wll is no h sa as pdid by won's Law This lads o a sipl hoi; ih won's Law of gaviy nds o hav a oion o an indpndn fifh fo is ndd o xplain h diffn In h pas whn w found ha h poonpoon saing daa diffd fo h oulobi pdiions w opd fo an indpndn fo and hav had h fun of sahing fo a hod of unifying loagnis and h song fo vy sin W ould ak ha sa hoi h o w ould invsiga h diffn in h pdiion of h nonsingula ponial and won's law of gaviy To do his w nd o look a h gaviaional aaion on a ass in a wll dp down fo h sufa of h Eah shan xs on physis ypially show how o alula h gaviaional influn of a hin sphial shll on a ass boh insid and ousid of h shll This is h podu w nd h baus a ass in a dp wll fl h influn of boh h ass of h Eah inio o i and in h shll of h Eah xio o i wh h shll hiknss is h dph of h wll If w all h podu fo his using h wonian ponial hn w b ha fo h / ponial all of h ass inio o h s ass aas i as if h ass w load a h n of h Eah On h oh hand w all ha h is no gaviaional influn du o h ass in h ou shll whih is xio o h s ass in h dp wll o a ponial whih diffs fo h / wonian ponial hs onlusions ay no b u Indd ons fis suspiion is ha hy a 8
no h o onlusions Wha w now nd o do is o alula h influn on a s ass boh ousid and insid of gaviaing ass is suppos w alula h gaviaional influn on a ass xio o a hin sphial shll Using h nowonian ponial w find h ingal o b x πgρ x igu owonian Ponial x igu owonian o d dx f(x x x igu Gaviaional aaion of a sion ds of a sphial shll of a on wh by aking us of h ingal abls and a lo of algba w aiv a h soluion is an ipop ingal fo This ans ha s in h sis hav dnoinaos whih nd o zo as nds o W us show ha h sis onvgs baus his is h as whn ou s ass is a h boo of a dp wll I would hn b a h ou sufa of h inn ass I is asy o s ha as nds o zo h soluion nds o h lassial soluion If w now onsid ou s ass o b insid a hin shll and look a h fo πgρ πgρ as whih is h lassial sul Suppos now w look a a oupl of appoxiaions whih ay giv so insigh ino h influn of h xponnial in h ponial is l us onsid a ass ousid anoh ass fo whih >> hn w would hav h appoxiaion x( x dx d dx ( (! ( f(x x x dx dx log x ( G ;if >> 8
This sul shows ha h fo of aaion on a s ass ousid anoh ass is dud by h sond in h squa baks This is of ous wha w should hav xpd fo a ponial whih dvias fo h wonian ponial by uning aound and going bak o zo Th fis dviaion fo wonianlik haa would b o bo wak Th oh appoxiaion o onsid is ha fo h xpssion fo h fo an h s ass insid h shll o his w find ha insid h shll if >> hn G ( ( ;if >> This is a fo away fo h n of h shll and owad h insid of h shll Th Big Qusion is: Wha is h fo whn h s ass is on h idia xio o inio of a shll? Tha is do w hav onvgn of h infini sis in h soluions fo boh h insid and ousid fos on h s ass To addss his onsid h absolu valu of h aio of h n and h n h s in h fo fo a ass ousid of a shll of fini hiknss hn fo ou pvious suls w found ha πgρ log (! ( ( Th qusion aiss whh o no is fini fo >? If / hn / and / o / / ; / πgρ if w hav >>/ and >> hn w ay ak h following appoxiaion (! log 85
G log (! ( ( ow look a h aio of wo sussiv s in h sis a a n ( ( (! ( (! ( ( ( ( Bu >>/ so ha li a a li ( Thfo h sis onvgs absoluly! ow how abou an xpssion fo h fo fo a lag nub of shlls of hiknss suffiin o ak up a sph? If h a shlls aking up h sph hn h hiknss / Whil if w wan o onio h fo on a s ass a h sufa of h sph hn h liis on x in h ingaion vais wih ah shll o insan fo h ou shll h liis on x would b x bu o h nx shll and x 5 o h liis on x would b x 56 o 86
Thus fo h h shll w hav liis givn by Thn w hav x G( ( (! x ( log ( ( ( ( ( [ ] ( ow looking a on h oh hand fo xp if _ xp ( if _ xp xp if if _ xp _ xp uh sin ( 6 ( πρ ( If w hoos suh ha >> 6 hn w ay wi 87
88 Suppos w look a h shlls suh ha hn w find o ow w an do h sa so of hing fo h "insid h shll" fo This as has h low liis on x hangd so ha wih [ ] ( (! ( ( ( ( ( ( G ( πρ log 6 5 if 98 G 5 98 5 G [ ]! 6 ( 5 6 G πρ πρ log 58 G ou x wh dx x f(x dx d G x x in ρ π
89 Thn fo a shll wih as h inn adius and a hiknss of hn / and w dsi o know wha h gaviaional ffo fo Thn w hav / and / hn if >> >>/ hn w hav h appoxiaion Exapl: Copa h hang in gaviaional fild as on gos fo h Eah's sufa down a dp wll o a dph of d Th wonian gaviaional sngh a h ah's sufa is o h nowonian fo w us s h sa fo hus w us s x! ( x x x ( x f(x x x log! ( G in ρ π (! ( G in log G 55 G G
o ow h wonian gaviaion a a dph of d o Th vaiaion in fo fo h sufa o h dph is hn o 55 G (d G G ( d ( d d G (d d G G d G G d G d d On h oh hand h gaviaion fo in h wll wih h nowonian ponial would b givn by G (d (! ( log 9
hn ( ( d G G ( ( d ( ( d! ( d d d log bu 8 so ha fo d 55 f 676 and 6 x 6 hn 55 d G 9 x 95 d 5 x (! 56x ( 7 (676 ow Thus w hav (! 56x ( 7 5966 x 9 x 7 7 (56 x 56 x (676 8 ( x 9 x 7 6 (67 x x d G 58 676 d G 58 d G (d d 9
Bu h nowonian gaviaional fo a h boo of a wll of dph d is givn by G (d 58 d G 58 d G (d 796 d whih ay b win (d (d 796 This givs h fis od appoxiaion of h dviaion fo wonian gaviaion pdid by h nowonian ponial and shows ha h pdid gaviaional fo of h Eah dass o apidly han wonian gaviaion dos 55 Inial and Gaviaional ass and hi Equivaln Th a h ways in whih ass appas in won's Sond Law whn gaviaional fos a onsidd Consid his gaviaional fo law whih ay b win G and his Sond Law whih is d d (v d d In hs quaions h a h inial ass and wo gaviaional asss and Th fo quaion os fo onsiding h fo on du o h h gaviaional fild of In his as is usually fd o as h aiv gaviaional ass whil is h passiv gaviaional ass Classially won's Thid Law is iposd in od o show ha h aios of aiv and passiv gaviaional asss us b qual Consid 9
so ha G a p G a p wh h subsips a and p f o h ass's ol as ih an aiv o passiv gaviaional ass This lads us o h quaion a a p p whih ans ha sin h aios us b qual h a and p ay b ad qual Th qualiy of inial and gaviaional ass is no pdiabl by won's laws ah i is akn as an assupion This assupion has bn subjd o inasingly aua xpinaion by Eovos in h 88's by Dik in 96 and by Baginski in 97 Th psn lii of opaision bwn gaviaional and inial ass in abou on pa in ow l's onsid hs sa h ass onps in h onx of h Dynai Thoy is h is h inial ass dnsiy I aks is appaan in Sion whn w ipos h pinipl of inasing nopy as a vaiaional pinipl Th i ln is givn in s of h spifi nopy whil h nopy pinipl is in s of h nopy dnsiy Th ff of his is o inodu h ass dnsiy as a podu of h alaion ino h quaions of h fo dnsiis (s Eqn (5 Th sa inial ass onp lads o h Einsin ngy and ass laion in Sion Th oh wo ass onps n fis hough h fild quaions givn by Eqns (5 and fo h h fo dnsiis in Eqns (7 In Sion 5 w wn hough h fild quaions o din h hagoass onvsion ndd o kp h unis onsisan H w found ha h passiv gaviaional ass givn by Eqn (5 was p εg whil h gaviaional fild assoiad wih a gaviaional ass is givn by Eqn (5 whn h valuad paas a usd as (576 9
G ε [ G & ] a V a a wh h ass in h gaviaional fild quaion is o b onsidd h aiv ass and hfo w'v usd h subsip a o dno his Th gaviaional fo du o h passiv ass bing in h gaviaional fild V is hn G By looking a Eqn (578 w ay s ha fis i is h aiv gaviaional ass ha has h i dpndn and no h passiv gaviaional ass uh only whn h aiv gaviaional asss a idnial wih hi 's h sa will h aiv and passiv gaviaional asss b qual W'v usd h subsips in h fo quaion o dno h fo on ass in h pssn of h fild of ass W ay onsid h fo on ass whn in h fild of ass and w find If w fo h aio of Eqns (578 and (579 w find ha (577 a a ( G & (578 p a G a a ( G & (579 p a p p a a a a a a (58 uh only fo idnial gaviaional asss will won's Thid Law b saisfid wihin h Dynai Thoy 56 Cosology Th ho big bang odl of h Univs is h odl whih is in vogu now Viually all h jounals pin nuous ails laing o so asp of h ho big bang odl Th odl is basd upon h 9
wonian Gaviaional and h noion of a sal of h univs ha is hanging wih i This noion is boowd fo Einsin's Gnal Thoy of laiviy howv Einsin's hoy is no usd in h ho big bang odl islf I would s a sha o disuss a nw gaviaional ponial suh as psnd in his book wihou so disussion of is possibl ipa upon h ho big bang odl I would hav bn pfabl o wai unil h ni soluion ould b psnd Howv his is no possibl now so his psnaion will inlud a disussion of how on igh xp h nw ponial o ipa h ho big bang odl and h pobls ha nd h soluion illusiv Th dvlopn of h sandad big bang odl bgins wih onsiding a sphial pi of h univs wih an obsv a h n This sph is onsidd o b filld wih "dus" of dnsiy ρ( wih a galaxy of ins plad a h ou bounday of h sph whih has a adius dnod by x Whn Gauss's law and won's laws of oion and gaviaion is usd on aivs a d x π g d x ρ(gg π Gρ(g x x (58 Bu onsid wha happns if on wishs o opa his wih h nonsingula ponial of h Dynai Thoy Thn Eqn (58 bos d x π g d x ρ(g g x x x π Gρ(x x x ow l us pla x wih h ooving oodina x( wh ( is h sal fao of h univs and is h ooving disan oodina as is don in h sandad odl Whn w also noaliz h dnsiy o is valu a h psn poh ρ o by ρ(ρ o ( w obain d πg ρ d o (58 W an bgin o s h nd o b xpd fo h univs fo Eqn (58 by noing ha should w look bak in i o h poin whn / hn w would hav a poin in i say T whn h alaion of h univs would hav bn zo A is bfo T h would hav bn an alaion ouwad whil fo is af T suh as h un poh h a of h xpansion of h univs is slowing down This is a vy diffn soy han is old fo by h sandad odl Bu how is i diffn? I is h sa as h sandad odl in ha fo Eqn (58 on ss ha h univs was fod ino xpansion a aly is and is now slowing down is a of xpansion On big diffn bwn h soy o b old by Eqn (58 and h sandad odl is ha Eqn (58 givs h ason fo h iniial xpansion and i dnis ha h univs 95
was v ollapsd o a singula poin as supposd by h sandad odl To b s h fis onnion w should pod a lil fuh If w uliply Eqn (58 by d/d and inga wih sp o 8πG ε( & ( k ρ (58 i w find In Eqn (58 w hav inludd h fo h adiaion fo oplnss ow l us valua h onsan of ingaion k by sing h valus of d/d ρ and ε a hi psn day valus of H o ρ o and ε o Thn Eqn (58 bos 8π ρ G o H o πg ε o k (585 If w now ak h dfiniions H o ρ 8πG and ρ o Ω ρ Eqn (585 ay b pu ino Eqn (58 o obain W ay now ak a look a so of h iplid dynais fo Eqn (586 is look a h dynais as nds o infiniy and h is no adiaion o his as w would hav H Ω o o & H ε o Ω (586 ρ & H o( Ω whih is h sa as in h sandad odl ow suppos w look bakwad in i o h i whn d/d was zo? Thn Eqn (586 bos ε o ρ ( o ρ o ρ (587 This is a ansndnal quaion whih ould b solvd fo if w knw and h dnsiy of h dus and adiaion a h un poh I ay b sn fo Eqn (587 ha if h is no adiaion and dos no qual zo hn 96
ρ ρ o (588 Th is also a ivial soluion a in Eqn (587 bu fo his as h alaion is also zo and hfo no dynais a allowd Whil a fis glan i ay appa ha w hav as good a dvlopd soluion as is aivd a in h sandad odl onsid h following poins Ou galaxy was onsidd o b on h ousid lii of a sph of dus o h un poh h dnsiy of h dus is vy sall loally o h galaxy and Gauss's law fo onsiding h oal ass of h sph of dus o b plad a h n should hold vy wll Bu wha abou whn w a looking bak in i whn h dnsiy was a lo ga A so dnsiy w a no long abl o appoxia sul usd in Eqn (58 bu us us h soluion dvlopd in h disussion of h ifh o Thn ou onlusions aivd a abov a only good in a gnal sns and a no quaniaivly aua A sond poin onns h fa ha w hav dvlopd h gaug funion in pio sions If his is h sal of h univs as h gaug funion is supposd o b hn why a w again ying o solv fo i h? If h sal of h univs is givn by h gaug funion hn h dynais ay b ov spifid if w pu h adiaion ino h quaion fo h alaion suh as Eqn (58 On h oh hand wha is h sou fo h adiaion? If h is no ho big bang fo h adiaion o o fo wh igh i oigina? Th Dynai Thoy displays an induiv oupling bwn h loagni and h gaviaional filds ould h adiaion b du o h xpansion of h gaviaing ass of h univs? If so hn a knowldg of h gaug funion igh un h quaion of oion fo ou galaxy ino a pdiion of h adiaion quid a h psn i This pdiion igh hn b opad o h asud adiaion Howv h is h nssiy o hav a fo h univs I ay b obaind fo h gaug funion also as G/ Bu how is dind? Phaps his is suffiin o poin ou ha h ovall piu of osology o b givn by h Dynai Thoy is no y opl bu in any vn will likly b vy diffn fo h ho big bang odl of h univs Will i allow fo high paus ndd fo aouning fo h abundans of h lns? Sin i allow fo h univs o b uh sall in h pas i would hav h assoiad high paus Y i should no hav h infini paus assoiad wih a singula univs 97