SPACETIME METRIC DEFORMATIONS C O S I M O S T O R N A I O L O I N F N - S E Z I O N E D I N A P O L I I T A L Y

Transcription

1 SPETIME METRI DEFORMTIONS O S I M O S T O R N I O L O I N F N - S E Z I O N E D I N P O L I I T L Y

2 Lvori D. Puglis, Dformzioni i mtrich spziotmporli, tsi i lur qurinnl, rltori S. pozzillo. Storniolo S. pozzillo. Storniolo, Spc-Tim formtions s xtn conforml trnsformtions Intrntionl Journl of Gomtric Mthos in Morn Physics 5, (2008)

3 Gnrl Rltivity Introuction Exct solutions: cosmology, blck hols, solr systm pproximt solutions: grvittionl wvs, grvitomgntic ffcts, cosmologicl prturbtions, post-nwtonin prmtriztion

4 Spctim mtric Th spctim mtric gnrlizs th notion of mtric introuc with Spcil Rltivity. In Gnrl Rltivity th mtric mks sns only s n infinitsiml istnc btwn two vnts. From it ll th proprtis of spctim, gosics, curvtur cn b obtin. Grvittionl phnomn cn b intrprt s u to th spctim gomtry. It tks th following form

5 Spctim mtric s 2 = g x x b Whr th Einstin summtion convntion hs bn us x = ( ct, x, y, z) r th spctim coorints

6 FRIEDMNN-LEMITRE-ROERTSON- WLKER METRI Th FLRW mtric is riv imposing th homognity n isotropy of spctim it hs th form 2 s 2 = c 2 t 2 ( t) ( ) ( x + y + z ) 2 2 k 1 r (t) stisfis th Einstin qutions 2 ( ρ) 4 & k + = 2 && 8πG 2 = 3 P= P 8πG 3 ρ ( ρ+ 3P)

7 SHWRZSHILD METRI sphricl symmtric sttic fil is scrib by th Schwrzschil mtric, which ruls with vry goo pproximtion th motions in th solr systm. s = GM GM Ω 2 ct 2 r r c r c r Th motion of th prihlion of Mrcury s orbit ws xplin compltly only pplying this solution

8 Prsnt problms Drk mttr Drk nrgy: cclrtion of th univrs Pionr nomly Rltion btwn th iffrnt scls vrg problms in Gnrl Rltivity pproximt symmtris

9 Our ignornc of th rl structur of spctim ll ths problms rprsnt our ignornc of th rl structur of spctim ounry conitions Initil conitions pproximt symmtris or complt lck of symmtry Unknown istribution of rk mttr n rk nrgy (bsis its ntur) ltrntiv thoris of grvity

10 Exmpl: Dick s problm Is th Sun oblt? If ys thn prt of th prcssion of 43 scons of rc pr cntury of th Mrcury prihlion cn b xplin by clssicl til ffcts Proxiclly th xct xplntion foun in GR, woul tll us tht GR is wrong! ut t th sm tim on shoul lso consir th corrctions to th Schwrzschil solution! Th problm is still controvrsil, vn if mny vincs hv bn foun ginst th obltnss of th Sun. positiv nswr woul imply to form th Schwrzschil solution to tk in ccount th lck of sphricl symmtry ut o not forgt th Pionr nomly problm..

11 How cn mtric scrib our ignornc? Givn n xct mtric w woul lik to form it in orr to cptur th most importnt fturs of spctim, not prsnt in such mtric

12 Gomtricl formtions in 2D: Th surfc of th rth Lt us giv n xmpl of how w cn l with form gomtry If w tk th surfc of th rth, w cn consir it s (rotting) sphr ut this is n pproximtion

13 Th Erth is n oblt llipsoi Msurmnts inict tht it is bttr scrib by n oblt llipsoi Mn rius 6,371.0 km Equtoril rius 6,378.1 km Polr rius 6,356.8 km Flttning ut this is still n pproximtion

14 Th shp of th Goi W cn improv our msurmnts n fin out tht th shp of th rth is not xctly scrib by ny rgulr soli. W cll th gomtricl soli rprsnting th Erth goi 1. Ocn 2. Ellipsoi 3. Locl plumb 4. ontinnt 5. Goi

15 Goi in 3D

16 nothr img of th goi

17 omprison btwn th vitions of th goi from n iliz oblt llipsoi n th vitions of th M from th homognity

18 Dformtions in 2D It is wll known tht ll th two imnsionl mtrics r rlt by conforml trnsformtions, n r ll loclly conform to th flt mtric g ~ =Ω 2 ( x) g

19 gnrliztion? Th qustion is if thr xists n intrinsic n covrint wy to rlt similrly mtrics in imnsions n>2

20 Rimnn thorm In n n-imnsionl mnifol with mtric th mtric hs f = n( n 1) 2 grs of from

21 Dformtion in thr imnsions In 2002 oll, Llos n Solr (Gnrl Rltivity n Grvittion, Vol. 34, 269, 2002) prov th thorm which stts tht ny mtric in 3D spc(tim) is rlt to constnt curvtur mtric by th following rltion g Ωh = 2 +εσ σ b

22 Gnrliztion to n rbitrry numbr N of imnsions It ws conjuctr by oll (grvittion s univrsl formtion lw) n show by Llos n Solr (lss. Quntum Grv. 22 (2005) ) tht similr rltion cn b xtn to n N- imnsionl spctim

23 Our finition of mtric formtion Lt us s if w cn gnrliz th prcing rsult possibly xprssing th formtions in trms of sclr fils s for conforml trnsformtions. Wht o w mn by mtric formtion? Lt us first consir th composition of mtric in ttr vctors g = η D D g =η Λ ( x) Λ ( x) η Λ ( x) Λ ( x) = η D D

24 b g η = b b b b g = = η

25 g ~ Th forming mtrix g g~ Φ ( x) Φ =η D ( x) D b Φ ( x) Λ( x)

26 Φ = ~ ~ ~ ~ b b b b g ~ ~ ~ ~ ~ ~ ~ ~ ~ = b g ~ ~ ~ η =

27 Rol of th formtion mtrics Th formtion mtrics r th corrctions introuc in known xct mtrics to consir rlistic spctim Th invrs problm is inst to fin n vrg solution which is solution of th Einstin qutions stisfying th givn symmtris Thy r th unknown of our problm Thy cn b obtin from phnomnolgy Or r solutions of st of iffrntil qutions Lt us giv n xmpl in trms of Nwtonin grvity

28 Dformtion of th grvittionl potntil

29 Φ Proprtis of th forming mtrics (x) r mtrics of sclr fils in spctim, thy r sclrs with rspct to coorint trnsformtions, thy r fin within Lorntz trnsformtion. Thy fin n quivlnt clss Φ ( x) Λ ( x) Φ ( x)

30 Intity Th intity is obviously th Kronckr lt δ but s sn bfor lso th Lorntz mtrics bhv s intitis, so w consir s intity th (infinit) th st I { Λ } = δ,

31 Dformtions n th Lorntz group Th formtion mtrics form right cost for th Lorntz group, i.. ny lmnt is n quivlnt clss fin by th rltion Φ Λ Φ Λ

32 onforml trnsformtions prticulr clss of formtions is givn by Φ ( x) = Ω( x ) δ which rprsnt th conforml trnsformtions g~ =η Φ Φ D =Ω 2 ( D This is on of th first xmpls of formtions known from litrtur. For this rson w cn consir formtions s n xtnsion of conforml trnsformtions. x) g

33 Mor prcis finition of formtion If th mtric tnsors of two spcs ~ M r rlt by th rltion Φ : g = η g~ = M Φ n ~ w sy tht M is th formtion of M (cfr. L.P. Eisnhrt, Rimnnin Gomtry, pg. 89) η Φ D ~ M D

34 Othr proprtis of th forming mtrics Thy r not ncssrily rl Thy r not ncssrily continuos (my ssocit spctim with iffrnt topologis) Thy r not coorint trnsformtions (on shoul trnsform corrsponingly ll th covrint n contrvrint tnsors), i.. thy r not iffomorphisms of spctim M to itslf Thy cn b compos to giv succssiv formtions Thy my b singulr in som point, if w xpct tht thy l to solution of th Einstin qutions

35 Scon forming mtrix g ~ =η Φ ( x) Φ D g ( ) ( x ) D G ( x)

36 W not simply tht Gnrl Rltivity n ll th othr mtric thoris cn b r-xprss in trms of ttrs (ttr-grvity) s vrils G (x) Th mtrix cn b n ltrntiv pproch in trms of vrils. W shll cll this mtrix th scon forming mtrix n Φ (x) th first forming mtrix.

37 Φ thir pproch to fin spctim formtions (x) cn b writtn s spctim tnsor by contrcting it with th ttr c n using th intitygcd = b writtn s η Φ D b = Φ (x) thn mtric cn b g~ = g c Φ c Φ b

38 y lowring its inx with Minkowski mtrix w cn compos th first forming mtrix η Φ Φ Φ =Φ =Φ( ) +Φ [ ] =Ω =Ω δ η +Θ +Θ + + ω ω

39 Th forming functions n th gr of from Ω is th trc of th first forming mtrix Θ is th symmtric prt is th ntisymmtric prt ω Rimnn thorm: mtric in n-imnsionl spctim hs ( n 1n/ ) 2 grs of from th componnts of th first forming mtrix r runnt

40 Expnsion of th scon forming mtrix Substituting in th xprssion for formtion th scon forming mtrix tks th form G + η 2 = Ω η + 2ΩΘ D ( D D) D Θ ω + ω Θ + η ω ω Insrting th ttr vctors to obtin th mtric it follows tht (nxt sli) + η D D Θ Θ D

41 Tnsoril finition of th formtions Rconstructing form mtric ls to g~ = Ω 2 g This is th most gnrl rltion btwn two mtrics. This is th thir wy to fin formtion + γ

42 Dforming th contrvrint mtric To complt th finition of formtion w n to fin th formtion of th corrsponing contrvrint tnsor g ~ ( ) ( ) D Φ 1 Φ 1 b g~ =η D

43 thir wy to fin th controvrint mtric form In tnsoril trms w cn proc s w i for th covrint tnsor. W cn mk th nstz g~ = α 2 g + 1 If λ =0, thn α =Ω, n w cn suppos this conition tru vn for λ 0 λ

44 Thir wy to fin th contrvrint mtric formtions λ Rgring, it is rlt to by th following rltion, γ λ b = Ω ( 2 1 b δ +Ω γ) γ c 4 c whr inics r ris n lowr with th unform mtrics.

45 Dform connctions W r now l to fin th connctions whr is tnsor Γ ~ c = Γ c + c c = ~ ~ g~ 2 c 2 1 2g( b) Ω gg Ω + 2 c ( γ + γ γ ) b b

46 Dform gosic qutions Th form connction suggsts tht in form spctim th gosic qutions corrspon to vition of th gosic motion in th unform spctim 2 x s ~ x s x s 2 x s λ µ ν λ λ + Γµν = Γ λ µν x s µ x s ν = λ µν x s µ x s ν

47 Dform gosic motion Thrfor th form gosic motion tks into ccount of our ignornc of th rl spctim mtric, inpnntly of th origin of th formtion Exmpls: Pionr nomly, unxpct lnsing proprtis, rk mttr istribution, vition from th suppos spctim symmtris, iffrnt mtric thoris, n so on

48 Th Pionr nomly Pionr nomly From Wikipi, th fr ncyclopi Unsolv problms in physics: Wht cuss th pprnt rsiul sunwr cclrtion of th Pionr spccrft? Th Pionr nomly or Pionr ffct is th obsrv vition from xpcttions of th trjctoris of vrious unmnn spccrft visiting th outr solr systm, notly Pionr 10 n Pionr 11. oth spccrft r scping from th solr systm, n r slowing own unr th influnc of th Sun's grvity. Upon vry clos xmintion, howvr, thy r slowing own slightly mor thn xpct from influnc of ll known sourcs. Th ffct cn b moll s slight itionl cclrtion towrs th Sun. t prsnt, thr is no univrslly ccpt xplntion for this phnomnon; whil it is possibl tht th xplntion will b munn such s thrust from gs lkg th possibility of ntirly nw physics is lso bing consir.

49 Dform urvtur tnsors Finlly w cn fin how th curvtur tnsors r form ~ R c ~ R ~ R = = R ~ R c + + b = g~ R= g~ R+ g~ c b bc + + c b b bc [ ] b+ b

50 Einstin qutions for th form spctim in th vcuum Th qutions in th vcuum tk th form ~ R + = 0 R + b = 0 b

51 Th form Einstin qutions in prsnc of form mttr sourcs In prsnc of mttr sourcs th qutions for th form mtric r of th form ~ R ~ T = 1 2 g~ ~ T ~ R = R + b + b ~ 1 ~ b+ b = T 2g ~ T 2 1 ( T Tg )

52 Stnr grvittionl thoris History of grvittionl thory Nwtonin grvity (NG) lssicl mchnics Gnrl rltivity (GR) History Mthmtics Rsourcs Tsts Twistors

53 ltrntivs to GR lssicl thoris of grvittion onforml grvity Sclr thoris Norström Yilmz Yilmz Sclr-tnsor thoris rns Dick Slf-crtion cosmology imtric thoris f(r) thoris

54 Einstin rtn rtn connction Whith Nonsymmtric grvittion Sclr-tnsor-vctor Tnsor-vctor-sclr f(r) thoris with torsion Othr ltrntivs

55 Tlprlllism Gomtroynmics Quntum grvity Smiclssicl grvity Discrt Lorntzin QG Euclin QG Inuc grvity usl sts Loop quntum grvity Whlr Witt qn Unifi fil thoris

56 Thory of vrything Suprgrvity M-thory Omg Point quntum grvity TOE Suprstrings String thory String thory topics

57 Highr-imnsionl GR Kluz Klin DGP mol ltrntivs to NG ristotl Mchnicl xplntions Ftio L Sg MOND Unclssifi omposit grvity Mssiv grvity Elctrogrvitics Grvitomgntism nti-grvity Lvittion Othr

58 onforml trnsformtions onforml trnsformtions hv oftn bn thought s mthmticl vic to fin solutions of th Einstin qutions Thy wr us to fin rltion btwn th solutions of ltrntiv grvittionl thoris (.g. rns-dick, f(r) thoris) n Einstin s gnrl rltivity

59 Dformtion n bimtric thoris imtric thory rfrs to clss of moifi thoris of grvity in which two mtric tnsors r us inst of on. Oftn th scon mtric is introuc t high nrgis, with th impliction tht th sp of light my b nrgy pnnt. In gnrl rltivity, it is ssum tht th istnc btwn two points in spctim is givn by th mtric tnsor. Einstin's fil qutions r thn us to clcult th form of th mtric bs on th istribution of nrgy.

60 imtric thoris (N. Rosn) Th first bimtric thory ws introuc by Nthn Rosn (rmmbr EPR) in th rly 40s Grvittion is intrprt s physicl fil scrib by tnsor (physicl mtric) on gomtricl bckgroun with gomtricl mtric Th composition is lik foun by us with Ω 2 =1 g =η + γ Th fil qutions for th physicl mtric r prcisly th ons writtn prviously

61 Einstin qutions for th form in physicl mtric of th bimtric thory in th vcuum Th qutions in th vcuum tk th form ~ R + = 0 R + b = 0 b

62 Krr-Schil formtion

63 Krr-Schil mtrics Eington: Schwrzschil mtric (1924) Trutmn: Grvittionl wvs trnsport informtion (1962) Krr: rotting sphricl boy (rotting blck hol) (1963) Nwmnn n Rin complx coorint trnsformtion from Schwrzschil to Krr mtric (1966) Krr n Schil: th Krr-Schil nstz oll, Hilbrnt n Snovill: Krr-Schil symmtris (2000)

64 Eington mtric Eington foun tht th Schwrzschil mtric in coorint systm cn b writtn in th following form s u = u 2 2 GM + 2 ur r Ω u 2 cr 2GM 2GM ln 2 r 2 c c = ct r 2

65 Propgtion of informtion by Grvittionl Wvs Trutmn show tht grvittionl wvs propgt informtion. H stui th mtric g = gˆ + F ( σ) l ( x) l b ( x) g = gˆ F ( σ) l ( x) l b ( x)

66 Th Krr-Schil formtion mtrix W cn rformult Krr-Schil s nstz introucing suitl first formtion mtrix V V H V V V + ± =

67 pplictions: Grvittionl wvs n prturbtions Th grvittionl wvs solutions r consir s from linrizing th mtric, prturbing th Minkowski spctim, g + η h h <<1 with th conition n rstricting th coorint trnsformtions to th Lorntz ons.

68 Grvittionl wvs Imposing th gug conitions h =0 = 0 h th linriz Einstin qutions r h bc =0

69 Grvittionl wvs In our pproch th pproximtion is givn by th conitions Φ ( x) ( x) ϕ ( x) << 1 δ +ϕ This conitions r covrint for thr rsons: 1) w r using sclr fils; 2) this objcts r imnsionl; 3) thy r subjct only to Lorntz trnsformtions; 4) Thy r tnsors with rspct to Lorntz trnsformtions of rnk (1,1)

70 It follows tht Grvittionl Wvs ( ) Φ 1 =δ ϕ Th mtric composition w hv foun os not n to b linriztion Howvr pplying th usul pproch to grvittionl wvs w hv to consir th following qutions in th vcuum

71 Grvittionl wvs n prturbtions 0 ~ = + + = b b R R =0 + b b = 0 b 2 =0 c b bc R γ γ

72 Th qutions for th sclr potntils

73 Grvittionl wvs n prturbtions W hv us th gug conitions which r no mor coorint conitions. Thy ppr s rstrictions on th forming mtrics. Th linriz Einstin qutions cn b trnslt in qutions on th forming mtrics.

74 Discussion n onclusions W hv prsnt th formtion of spctim mtrics s th corrctions on hs to introuc in th mtric in orr to l with our ignornc of th fin spctim structur. oncptully this cn simplify mny clcultions, but lso suggsts us mor mbitious gols To this im w hv introuc th in 4 imnsionl spctim 6 inpnnt sclr grvittionl potntils Th conitions to b impos on ths potntils r not rstrictiv s rsult w show tht w cn consir th grvittionl thory s thory of ths potntil givn in bckgroun gomtry W show tht this sclr fil r suitl for covrint finition of prturbtions n grvittionl wvs ut lso to stuy th rltion btwn GR n ltrntiv grvittionl thoris Thy cn connct spctim with iffrnt topologis W foun tht th cn suggst wy to show th prouction of th r ntropy in lck Hols It coul b intrsting to consir th quntiztion of ths potntils in orr to fin quntum grvity thory grt l of work in th futur yrs!!!!

75 Thnks for coming!