Spacetime, geometry, gravitation

Transcription

1 Spetime, geometry, grvittion Ides nd some peulir spets Bertrnd Chuvineu Observtoire de l Côte d Azur Lgrnge lortory NieSophi Antipolis University Nie, Frne

2 Aims / ontents / topis Spetime & geometry Grvittion & spetime geometry Generl Reltivity, the lrge sle Universe & L Generl Reltivity & wys to lterntives Metri theories, f(r) theories, SlrTensor theories

3 Some keys ides relted to spetime & to grvittion («lssil» ides) : Newton (687) : neessity of preise definition of the spetime (spe & time) properties before doing physis Newton spetime grvity s «fore» phenomenon Mxwell (~865) : eletromgnetism equtions versus the glilen reltivity priniple The MihelsonMorley experiment (88) & the speil reltivity theory (95) Minkowski (98) : the reltivity theory versus Minkowski s spetime drsti hnge in the wy of thinking in theoretil physis Einstein (95) : grvity s spetime geometry effet ( Generl reltivity) (& lter : lterntives to Generl Reltivity) 3

4 I Newton versus Minkowski spetimes (in nutshell) nd grvity : Invrint quntity between two lose events (fundmentl) onstnt Spetime geometry? Inertil motions (just determined by the spetime hrteristis) Newton s spetime dt (or dt² ) (solute time) none NO (but spe geometry euliden) Retiliner & uniform Minkowski s spetime ²dt²+dx²+dy²+dz² (reltivity of time) ( priori nothing to do with light!) YES (pseudoeuliden) Retiliner & uniform in both, the grvity phenomenon requires «something else» thn the spetime s properties (something like «fore») Newton s spetime hs no spetime geometry (unlike Minkowski s), but its properties re (to some extent) s preise while different s the ones of Minkowski s spetime. onsider physil problems like simultneity, uslity, 4

5 flt spe & uniform time or flt spetime & no fore inertil motions = retiliner nd uniform Newton : the use of plnetry motions is speifi fore (spetime properties unhnged) Mtter genertes fore field : Newton s universl grvity theory But nother possibility ould be to rejet the ide of grvittionl fore (Riemnn) Mtter «modifies» the spetime properties Motion under grvity = inertil motion in nonnewtonin spetime Strting from the newtonin spetime : how hnging the spe geometry to reover the Newton s grvity theory equtions? (Riemnn) FAILS!!! 5

6 Strting from the minkowski spetime : the geodesis (extreml «length» urves inertil motions) of the spetime of metri U ds N dt dx (Newton's metri) re funtions x(t), y(t), z(t) tht stisfy, t lowest order (in U nd v²), the Newtonin s lw of motion under grvity potentil U, ie i d x iu dt ds Remrk : the «time prt», nd only it (t lowest order) of the Minkowski spetime is ffeted Riemnn ould not suess!!! N Obviously : the previous metri is not «geometri grvity theory», but just geometril reinterprettion of Newton s theory! dy d x dt dz i 4 U O U, Uv, v i A «geometri grvity theory» is expeted to give diretly the link between the spetime geometril objets nd its mtter (ie non grvittionl) ontent lolly minkowskin? (lol sptil isotropy?) 6

7 The metri tensor in (very) brief Riemnnin vriety : ds ds ds ( ds ds X ds, b intervl dx dx dx dy x dy x os y, Y sin dt dx dudv dy g xdy euliden plne (rt.oord.) xsin dz b x dx dx g x dx dx g x dx dx... g x (lolly) dy sphere (dim) y ds dz euliden dx plne (pol. oord.) dy Minkowski spetime ( ) ( d sin d ) ( dr, r Minkowski spetime (rt.oord.) u t x d v t x Flt spetime/vriety (metri onnexion mening) : one n find oordintes in whih ll the metri tensor omponents re onstnt (or : RiemnnChristoffel urvture tensor = ) ) ) Q( x i dx i ) P( x i ) dx dx ( lose points)... metri tensor omponents (symmetri) should define n invertible mtrix g & g g sin sin g & g g & g... g r / / Plne (no lol urvture) : euliden plne, Minkowski spetime, U Lol urvture (not plne) : sphere, Newton s metri ds N dt dx dy dz,... 7

8 Wht is? How interpreting it? How nming it? Minkowski spetime s definition hs nothing to do with the light/mxwell theory! (despite the ft it ws disovered thnks to some «strnge» properties of the light, first reveled by the Mihelson & Morley experiment, motivted by Mxwell s equtions, ). is just the modulous of fmily of modulousinvrint speeds, tht exists from the mere ft tht the spetime is equipped with ( ( + + +) pseudoeuliden) geometry Requiring the vlidity of the Mxwell equtions (in their usul form, nd in the usul formultion of the theory), ie of the eletromgneti theory (tht is not inherent to speil reltivity, but tht n be «stik» in it) in ll glilen oordinte systems requires : Speed of eletromgneti wves in vuum (s soon s Mxwell equtions pply) Speed of grvittionl wves in generl reltivity & ST (in generl, relted to GW) inherent to the mere spetime definition, s soon s it is dynmil (t lest in Generl Reltivity & SlrTensor grvity) Refering to s the «speed of grvity» (rther thn «speed of light») : would it be better motivted hoie for the terminology? 8

9 II Generl Reltivity (GR) (& motivtions to go beyond?) : Newton : grvittionl potentil mtter ontent : Grvittionl potentil U4G (Poisson eqution) Mtter ontent Einstein (GR) : spetime geometry mtter ontent : GR Einstein s eqution : R Rg 8G... 4 T T & mtter field eqs where : (depends on) spetime s geometry Mtter ontent (& spetime s geometry) R g R e g g g g g g e R e b defined by g be g e e b b e e e be Rii urvture tensor (Rii) urvture slr metri/christoffel onnexion ontrvrint vs ovrint metri omponents T T... (onservtion eqs) 9

10 GR in the wek field se : wek (grvittionl) field n not generte veloity hnges of the order of the spetime s metri is lose to Minkowski g m h with h ( oord system) dig,,, Linerized GR s Einstein eqution : m e e h (usul Dlembertin) 6G T 4 Tm using relevnt onditions on the oordinte system Einstein eqution In vuum h x y z t Sttionrity (besides wek field) Poisson eqution = speed of grvity grvittionl wves (interpreted in reltivisti terms) bk to Newton s grvity ds N U dt dx dy dz is the lowest order (PN) solution of GR

11 GR nd the Universe (try to desribe the Universe s whole) : 97 (Einstein) : no sttionry dustfilled (finite, (hyper)spheril) Universe ds dt dr r R r d sin d with R onstnt (rdius of the Universe) but if the field eqution is ompleted by osmologil term R Rg Lg... withl 4 T 8G T onstnt thene sttionry dustfilled solution exists : R E L & E L 4G stisfies R Rg Lg E 8G 4 93 Slipher, Hubble, Lemître : Universe s expnsion omoving dust Nturl behviour of GR osmologil solutions without (or with) L L no longer required!!! (but my exist )

12 9989 Perlmutter, Riess, Shmidt dust filled Universe RW osmology ds GR elerted expnsion dt dr t r d sin d kr requires L!!! symptotilly Wht is L? How interpreting it? An ppeling point : R A new fundmentl onstnt? Something else? ~ ~ Rg Lg 8T R Rg 8T 8T P ~ the osmologil term is to perfet fluid with the vuum eq of stte!!! (QFT) As (QFT) field, the vuum should grvitte the presene of L is nturl!!! LQFT BUT ~!!!!!! L observed?????????

13 Going beyond GR? ould be worth looking for n lterntive «story»!!! The «equivlene» : R ~ ~ Rg Lg 8T R Rg 8T 8T P ~ suggests : don t hnge the Universe s mtter ontent, but the grvity theory lterntive grvity theories reminisent from Merury s perihelion s shift problem suggests : don t hnge the grvity theory, but the Universe s mtter ontent drk energy theories reminisent from Urnus orbit s nomlies problem other options : hnge neither theory nor mtter, but llow for voids (lol inhomogeneities effet) ; remove lrge sle symmetries ; The min topi of the following (just some lterntives) 3

14 GR lgrngin : GR 4 g 6G it suggests some possible wys for lterntives R L g L NG ; g T g L NG ; g T... GR grvity setor «mtter» setor 4 6 G g R L gl ; g NG Could the Newton s onstnt be upgrded s slr field? * More intrite dependene in the metri? * Metri s the lone geometril field? Does the mtter neessrily ouple with the metri only? pure geometri theories (metri &/or independent onnetion) pure metri theories slr(s)tensor theories others (slrvetortensor, bimetri, mssive grvity, ) 4

15 III Purely metri grvity theories (MGT) : Let us identify bsi geometri onepts : the metri g : defines intervls (generlizes the «length» onept) the onnetion : defines (lol) prllelism vetor t P The metri s onnexion displement dx onnetion vetor t Q Px ds g is just one peulir onnexion (tht nevertheless possesses some fir properties) i A e g g g g g be b e e B i i i A A Q x dx with A dx : dx b b A b dx But if these objets re onsidered s independent (metriffine pproh) vrying wrt the metri n eqution (the «Einstein s eqution» of the theory) vrying wrt n eqution tht links the onnexion to the metri metriffine GR s tion formlism g)!!! 5

16 From now on, we do the following hoies : MGT : the grvittionl setor of the theory depends on the metri only the onnexion is priori the metri s one g) (seond order formlism) The most generl MGT s lgrngin one ould imgine : metri 4 6 G gf d R, g R R, g R, R R, R R,... gl ; g b b «ovrint» derivtive (inludes urvture effets) d Put ll the metri (only) dependent terms you n imgine NG where F is required to be slr (invrint in ll oordinte trnsforms) ensures the «ovrine» of the theory (reltivity priniple) The simplest hoie (for non trivil theory) : F( ) = R GR!!! GR is the simplest MGT! (the following step being GR with L ) 6

17 Wht is the Einstein s eqution of n MGT expeted to look like? Preliminries : some generlities on lgrngin systems rst order lgrngin L F q, q' The ssoited «energy» reds E F d dt L LF q' L q EulerLgrnge F LF q' q' F seond order eq : q' ' f q, q' nd one shows tht this energy my be bound, in whih se the theory is «stle» (Ostrogrdski stility) L S d order lgrngin q, q', q'' d dt LS q' d dt LS q'' L q nondegenery S fourth order eq : q' ''' f q, q', q'', q''' The ssoited «energy» reds E S L S q' LS q' d dt LS q'' LS q'' q'' nd one shows tht this energy is generilly unbound the theory is «unstle» (Ostrogrdski instility) R. Woodrd (7) 7

18 F(R) GR MGT lgrngins depend on R, thene on the metri s seond derivtives relted to the prior hypothesis g) expet the orresponding Einstein eqution is of fourth order ; Ostrogrdski instility of the theory!!! BUT : If : F( ) = R (or R+st ) : the seond derivtives terms surfe terms GR Einstein s eq is of d order no Ostrogrdskin instility First order effetive lgrngin In the other ses : fourth order Einstein s eqution... but NO Ostrogrdski instility if F( ) = f(r) nd in this se only!!! R. Woodrd (7) A strong rgument supporting the f(r) theories mong the whole MGT fmily 8

19 Field equtions : 8G f ' R fg f ' g f ' T 4 & the usul mtter equtions generte fourth order terms (s soon s f is not ffine) A result on RW osmology in this f(r) frmework : ds dt dr t r d sin d kr One n hoose (reonstrut) funtion f(r) in suh wy tht (t) fits ny prior Universe s history!!! R. Woodrd (7) f(r) theories would be of wek interest if limited to this physil problem (but lso : Solr System, stellr s struture, osmologil perturbtions, ) 9

20 IV (Usul) slrtensor (ST) theories : 4 6 Motivtions : Field equtions : R Rg & the usul g R g U b gl ; g some ttempts to quntize grvity (or unify with other intertions) BrnsDike (BD) like theory (BD = ST with onstnt) in some ses, lose to GR in some sense (see lter) OK solr Systems tests Ug 8T b g 4 d du 8 3 U T 4 d d mtter equtions NG g b miniml mttergrvity oupling Metri only (hoie) ST my be lolly interpreted s grvity theory with vrying effetive Newton s onstnt (in Cvendishlike experiment) : G eff 4 3

21 Remrks : d order field equtions no Ostrogrdskin instility, s soon s > 3/ f(r) versus ST : f R grvity : f ' R fg T R U BD grvity : R 3 g U U ' b T T f 'g The ressemblne of the (Einstein s) equtions is suggestive! n go further? f Let us define, from f(r) : R f ' R & U R f ' R (exludes RG) or b d f ' g U d ( R The f(r) theory n be seen s peulir BD (thene ST) theory f ' R ) f(r) Ostrogrdskin stility, this orrespondne requiring > 3/ P. Teyssndier, P. Tourren (983)

22 ST versus GR : The ST Einstein s eqution in the onstnt slr se : finite ie GR grvity. But the slr eq yields R Rg Ug T U ' U T 3 3 (with U onstnt) Consider (for simpliity) the se without potentil. The slr eqution then requires : T (if T ) 3 Thene, the onvergene of ST to GR generilly requires BUT : R Rg T b g g!!! OK limit = GR, but with n extr mtter term : mssless slr field b originting in the BD slr vnishing prt! (turns out to be = in sttionry ses )

23 Physil relevne : sk physil question : Consider flt RW dustfilled universe, with the observtionl onstrint tht H is known. Wht n be sid on its ge in the frmeworks of () GR ; () (infinite )BD? GR s nswer : BD s nswer : T GR T BD 3H / 3, H 4 3 H 4 3 Thene the (infinite )BD s nswer : T BD lim 3H, 3H different nswers (infinite )BD differs from GR but ]/(3H),/(3H)] orresponds to the GR s nswer got for dust + mssless slr filled flt RW universe (in ordne with ) BC (7) 3

24 Rmks : the residul slr field is zero onsidering sttionry solutions/problems the sme onlusions essentilly work, in some sense, for generl ST Experiments/propgtion of light (Cssini) : ST pss Solr System tests if > 4 C. Will (4) Some lrge sle (osmology, ) studies re grounded on ST theories effets tht re by fr more importnt thn expeted from solr system onstrints Coniliting the two? A possibility ould be the so lled hmeleon mehnism The slr hs n effetive «mss» inresing with lol density (in brief) rnge mss lrge rnge in glti, osmologil, mediums wek rnge (then no effet) in plnetry, solr systems, mediums Yukw like (sptil) dmping J. Khoury, A. Weltmn (4) 4

25 V Slrtensor theories with n externl slr (EST) : The ST theories lgrngin 4 6 g R g b U gl ; g NG Vry wrt metri g Einstein s eqution Vry wrt slr field Slr eqution (fter ombining with Einstein s eqution) Vry wrt mtter fields H Mtter field equtions onservtion equtions (energy, Euler, ) However, onsidering tht : physil onsidertion sometimes led to slr fields tht re imposed in the theory (externl, ie not vried in the lgrngin) M. Reuter, H. Weyer (4) resorting to externl fields is sometimes required in physis (unimodulr grvity, bimetri/mssive grvity, ) S. Weinberg (989) ; C. de Rhm et l () ; C. Böhmer, N. Tmnini (3) it my be worth to tke lose look t the onsequenes if the slr field is not vried t the tion (lgrngin) level 5

26 Digression : the «role» of the geometri identities GR with independent mtter fields & R 8G T Rg... & & 4 T T T got vrying the tion w.r.t. the metri got thnks to vrition w.r.t. got thnks to vrition w.r.t. Tke the divergene nd get (thnks to some geometril identities) : T T (nothing new) The sme theory (ie sme tion), but with not to be vried (externl/nondynmil field) 8G R Rg... & nd tht's ll!!! 4 T T T got vrying the tion w.r.t. the metri got thnks to vrition w.r.t. Tke the divergene nd get : T T T is bk!!! Not new eqution, but mde expliit thnks to the geometril identities (showing tht Einstein + onservtion eqs re the sme in both theories) 6

27 not vried the slr eqution is lost Vry wrt metri g Einstein s eqution Vry wrt slr field Slr eqution Vry wrt mtter fields H Mtter field equtions Is the resulting theory «less onstrined»? One ould be tempted to lim : NO, beuse geometril identities eqution is bk & onservtion equtions ensure tht the slr Rii id : ontrted Binhi g b id : R Rg Let us hek!!! Two points : () onservtion equtions, (b) use geometril identities () Conservtion equtions : not trivil tsk, but OK here sine : T the mtter tion is slr ll mtter fields in the mtter setor does not enter the mtter setor no externl field in the mtter setor 7

28 (b) Use geometril identities : it yields 8 4 T ' U 3 U ' b in (usul) ST theories leds to GR like if solutions The EST theory equtions dmit : usul ST solutions (but lso) GR solutions (nd even!!!) some mixes of the two!!! An (unexpeted) osmologil solution : Flt ( k ) RW ds dt t dx dy dz ( = st & U = ) EST equtions in the dust filled se : ' 3 ' ' 8 3 ' '' '' ' ' ' ' ( X ' dx dt ) 3 ' 8

29 Let us remrk the form of the indued eqution : ' ' ' p C t t t t /3 At B t q q t t t p p q 3s / s / B... &... C A ontinuity of,, nd (nd ) A possible solution GR phse followed by BrnsDike one (with > 4/3) (or the onverse infltionrylike senrios?) Illustrtion with numeril vlues : 4 BD slr = mthing time = A = C = 3 GR slr disontinuity of seond derivtives, but ontinuity of '' / ''/ BC, D. Rodrigues, J. Fris (5) t 9

30 The EST grvity llows the oexistene in sme spetime of (ext) GR regions & (ext) ST regions some (new) kind of seening mehnism for ST theories? Some questions : externl slr : does it men it should be fixed priori? not the se here, but hppens in some theories with ext elements deterministi sttus of EST? In the previous osmologil exmple : the mthing time (GR BD) is rbitrry the «jump GR ST or ST GR» is not ensured to our more dt provided for the Cuhy problem? To be explored/in progress : spheril symmetry : stti, LTBlike, sreening mehnism? perturbed solutions in vuum grvittionl wves? Emission mehnisms? perturbing out RW solution revisiting osmologil perturbtions? «mthing»? other kinds of externl fields? 3

31 thnk you for your ttention!!! 3