ABSTRACT T it le? of Dissertation: CONTI UOUS IJ\1.-\GINARY TUITE HISTORIES REPRESENTING BL.-\Ch: HOLE . rucLEATION IN DESITTER SPA ETil'vlE Paul M. Bra.noff, Doctor of Philosophy, 2000 Dissr rtation d irected b:v: Professor Dieter R. Bril l Depart111c11t of PhYsics We address the issues involved in fin ling and coustru cting continuous im agi- nary t im histories (CITHs) represent ing black hole nucl<~at ion in a background ci r Sitter spaceti111e. Such rates are often calcula.ted bv adop t ing; t he insta uton methods used to calculate ordinary particle-ant iparticle product ion rates in back- ground fi elds . Unlike the particle prod uctio11 case, there are certain instances of bl ack hole nucleation described by two separate and distinct solutions to the E u- cl idean Einstein 's equations, i.e., t he instanton is d isconnected. Hence, one must _justify including such histories in a path iu tegral. We first discuss t l1 e existence of cont inuous im aginary t ime histori es for bl ack hole nucleati on in theori es consisting of modifi catious to Einstein 's equ ations. F irst, we ?onsider adding powers of the Ricci scala r Lo E instein-Hilber t gravity wiLl1 a cos!llological constant. \\ ' hc11 the l1iglwr C'U1Tat 11rr co11pli11g co 11sta11ts arr negative . \\'(' find co11ti1111011 s i11stanto11s describing a b,ickgrn11ud dr Sit t<'r to dC' Sitter transition charac teri zed by a peri odi c. non-singular sca le factor a(r) . .\'rgative coupling constants imply a11 eq11ivalc11t theon- of Ei11stei11 gra\?ity cou- pled to a negative energy density sc:a l,u? fie ld. T his rnoti,?ates our cxplon1tion of Einstein gravity coupled to Narlikar 's n<'gative (' 11Ng,v density C-fic lcl. \,Ve again find c1 continuous background instanto11 . lrn t such a so l11tio11 exists onl :v wh en small violaLions of t li <' Hamiltonian constrain t a rr allowed. Because of the unattract i vc feat ures of the a bm?c sol 11 tious , we explore hmY Cll1e can constrn ct CIT Hs b_\? sm gicaJl ,v altering tlw disconnected insta11 to11 . In t he spirit of t he path in tegral, we claim that one should sum over all possib le geometries which ?an conn ect the instant 11. \Vr limi t a.ttcntiou to connections with topology 5 3 and 5 1 x 5 2 We fine! that the 5 3 . connect ion is preferred iu the context of "no- boundary" quautum cosmolog:s, ?. However , we beli eve that the 5 1 2 x 5 2 1 connec tion may bC' more preferred for t\vo reasons. First , the 5 x 5 co unection allows two of its dimensions to-be hugr , impl ving via holography, that in format ion from Lhe initi al state can "survive" the J1 ea.r-a1111ihilation, recrcatiou ized perturbations 011 the 5 2 process . Srcond , Planck s portion of the connect ion gi,? ri se to more histori es over which to sum in the path in tegral. ONTINUOUS Il\IA ,INAHY Tll\IE HI. TOfUES REPHESE:'\TJKG BLAC'I( I-JOL '.\UCLEXfIO:'\ IN DESITTER SP CETil\lE Paul .f\1. Bra11off DisscrUttio11 s11bmittcd to the Facult_\? or the Grnd11atc School of t he C 11ivcrsit_? of .t\laryland. Collcg<' Park in partial fnlfill111e11t of the rC'q11irements for tiH' dC'grc<' of Doctor of Plii losoplt~, 2000 .-\dvisor_,. Couunittee: Professor Dieter R. Brill , C hairrnau / Ach?isor Professor Bei- lok Hu Professor C harles \tV. Mis11 er Prnfcssor Sylvester J. Gates, .Jr. Professor \t\' illi arn tI. Gold man DEDICATION To rny parrnts and sister. JI ACKNOWLEDGEfvfE TS Although I have declicat 1 d this thrsis to m~? familv, I would likC' to acknowl edge tli em agai n ; without th 'ir support, t his work would not have been possible. I would a l o like to thank Dall , Santh. i\ Iaggie , udrew, a lld Daphn fo r their fri endship an_d support during both good and difficult times. F in a lly, I would like to thank m_v a,eh ?isor Dietu R. Brill for hi s limi tlrss patience a lld help . 111 TABLE OF CONTENTS List of Figures VIII 1 Introduction 1 1.1 Motivation and Overview . 1 1.2 Ou t line of the Dissertation 11 2 Particle Pair Production Rates 14 2.1 In trod uction . . .... . ... . 14 2.2 T unneling, Particle Production , aud Decay Rates 17 2.3 Instanton Approach To Calculating B . 21 2.4 Euclidean Path Integral Formalism . . 25 2.4 .1 Path Integral Formulation of Quant um Mechanics 25 2.4.2 Path In tegral Formulat ion of Quantum F ield Th<-)O r_v 27 2.5 Vacuum Decay in the Semiclassical Limi t . 28 2. 5 .1 T he Semiclassical Limit 28 2.5.2 T he Decay Rate . ... . 31 2.5.3 Part icle Motion and Evaluation of the Partit ion Function ... . . . . . . . . . . . . . . . . . . . 32 3 Black Hole Nucleation Rates 42 I V 3. 1 In trod uct ion ... -12 -17 3._ Top olog:v C hange 50 3 ..3 E uclidean Qua ntum Gnffi ty 52 3.3. 1 T he Action .. . .. .3.2 Iu defiui teness of the G n1\?ita t ion a1 Action . 55 3 . 58 3.-l T li e "No-Boundary" Proposal . .. .. 3.5 Com plex CoIJ tours a11d t he Semi classical 59 Approximation . . . . G2 3.6 l\!Ii11i superspace f-.fo dels T he de Sitter Minisuperspacc I\/Iodcl 67 3.G.1 Conditions and Uniqueness of Conto u rs 3.6.2 Boundary 72 Pair Pro E11 clidean pffrct iw poteutial for real (imaginary) k. 114 4.7 Phase plot fo r E11 cl idr.an (solid lines) and Lorentzian (clas lH-~d lines) solutions to the C-fie ld eq uat ions of motio11 . Thr separatric:es are ind icated b)' bold lines. 118 S.l Hyperboloid of one sheet with the y-coordiuate suppressed. T he interior of the h~rperboloicl represents the z > 0 region of a domain wall spacetime. To obtain the full spacetime, ouc glues together two copies of t his hyperboloid along the domain wall. . . . . . . . 141 5.2 Conformal representation of the z > 0 domain ?wall spacetime. T he heavy line represents the Lime evolu tion of the domain wall. At each instant of time, the domain ?wall is a two-sphere . For T < 0, the wall accelerates toward an in er tial observer un til it reaches its minimum radius of 1/"' at T = 0. It then rc-<~xpands away from th e observer. . . . . . . . . . . . . IX .J .3 'ot1formal cl iagTa111 rrprrsrnting tlit' C'lltir<' domain \\?a ll spacC'ti 1m?. T he regions to the left and right of t I)(' cl omai11 wall arr :diukowski spac with spacclike infini ty cut away. . . . . . . . . . . . . . . . . 1-12 5.-1 The '?lens'? in tan ton. obtained bv gl11ing togethC'r two ca ps of a four-ball along t heir 5 :3 bo1111dari<'s. T lwre is a ridge' of curvature located along the' domaill wall. . .. 1-17 5.5 The '?_vo-_yo'' instanton for a ,?irtual domain wall. 150 G. l on formal diagram for a flat FRW uHivcrsr. The surfaces B 1, B 2 . and B3 correspond to situatiolls whC're fl < R 11 , fl = Rn, and fl > R 1.1, respectively. The con e ?ponding past-ingoillg light rays arr dC'sig11atcd by L 1? L2 , and L:1, rrspC'cti,?cly ..... . 179 G.2 TIH' diagonal li11 r, represeuts t i!(' past <'H'JJt. hor izon of an iu crtial observrr located at x = 0 at past infinity. An_y surface B located in this regioll sati ?fi es the FS proposal. . . . . . . . . . . . 181 6.3 Part (a) is a conformal diagram for de Sitter spacctime. The clashC'd lines arc nu ll surfaces indicating cosmological horizons. The w-dges arc rlrawn ill accorda 11 cc with thr prescription de- scribed iu tlw text. Part(b) shows !tow the spac:rtirnc' can b<' prn- j<'ctC'cl onto two screens of infinit r area , namely 1+ ;-wd 1- . In part (c ) , the spacelike projection theorem is used to project half of de Sitter space onto E , the CYent horizon of an observer at x = 0.184 X shows a co11 for 111 aJ diagn:1111 fo r t] l(' l\" a ri ai spac-rt irn <'. T he G.-l Part (a) tl1 c dot-da shed lin es jagged line denotes the black hole sin gnl arity, <' cosrn ological l1 ori- th e black hole hori zons, and tlw cl ott ed Jines th zons. T he wedges are drawu iu acc ordance \\?it. Ji th<' pr<:'scrip tion given iu the text. Part (b) shows th e projec ti on of the spacetimr nto 1- . Part (c) shows the projectio n of t li e spacetimc outo the o cosmological hori zon, each of fini te size. 185 black hole horizon an Ort :1111 1vork by Langer [4], . howed how i11 stantoll method. could u used t O rn Im I ,1t ca,? procC'ss. th<' s,?stc'm is in c111 initial st.at(' characteriz<'c! I)\ '-,() J1 1r? lciwrst c'n crg~?- Th is initia l stat<' is S<'parated fro m other lower r11<'rg~? s1,11r?--, 111? 1 ote11tia l barr iNs. If the rnrrgy of the system is lowrr than thr e11 crg~? i ) f 1111' 110 1 <'ntia l bmri<'rs . thrn the' illi t ial state is a classica l rquili briu rn statr. I ll)\', r?11?1 , q uant um mechanicall y, the system has some fi ni t<' probabili ty to t11m1d 1!11r 111,c' !1 tJt,, por.c 11 tial barrier to a state of lower energy. Hence. t lw in it ial state is q1 1:1 11111 111 111 ccha11icall y unstable, and is sometimes referred to as a metastable r)1? f': 1J--,1? .-a , c:onLi11 uo11. ? history i11 imagill ary time that connects the ini tial and liii: ll :-: 1 ;1t1's. Iusta ntons a r<' solu tions to the Euclidean fie ld equations that describe 1 l1i~ l1i~t1)rv in im ag-i11 arv time. As 11 otccl above. the decay ra.tc is proport ional 1\ ) 1l i1? i111aginary part of the energy of the mrtastable state. One can calculate 111i s r?11t' r!s.,- fr om a Eucl idean path in tegral. Ju a pa.th integral. one considers all hio11. T l1<' expo11e11tial t<'l'lll iu tl1<' abow 1?cp1atio11 is often , ,1\!,?,i : i ,e classical co ntribut ion Lo th(' cl<'ca~? ratr . T hr qua 11 ti t_,. l\?, oftr11 ca lled 1 l11 ? -- i: 1, ? :?ac tor .. . is calculatC'cl frorn quantu rn mcchani ?a l flu ctuations about t he 111 "1: i1 1: , ,11. In casrs where t he prefactor bas beP 11 calculated . it bas been sho"'n k11 11 !,{s a11 algC'braic c.lepell d<'nce 011 the backgrou ll cl fir lds. 111, ?1, , <'xists a simila.rit.Y betwecll the fla t space p,a ng<' fip}cl th<~o ries desc ribing p; 1 1i , I,? production and the gravitat ional theorY clesnibing black hole nu cleation . 111?111 1? . 11 src1 111s ml1 ura l to st ud,v grav itat io1rn l i11sta.11 to11s. It \\'clS fi r t realized 1, , ( ;1l,l ,,rns [G] and Garfin kle a.nd Strorni11 ge r [7] that the analog of monopole 1>1 1, ii 111 11011 in a background magnet ic fie ld was the prod uction of magnetic:all .Y 1 i1;ir '..2 1?d black l1 olc>s in a background magnet ic fi<' ld. A great deal of work OJI black l 11 .J 1? p:111 prod uction in ot.hPr contexts .-0011 fo llowed , some of whi ?h we brief!~, 1I , '"1?11 l 11 ? in ha pt.c?r 3. 111 some of t hese cases, wha t is known as the "bounce" ;:p p111;:1 I to calcul at ing black hole nucleat ion was used . In this fo rm alism , 011 e l111d~ ;111 insta11ton that mediates the uuclca.tion process ; i.e. , a sing! , continuous ~?il 111i,1 11 to t li e E 11 c:l idean E instein 's equations t hat in terpola te. between ini t ia l ;ii;d li11; 1I slate. . Ouc the11 calcul ates the actiou of this instant.on. T he nucleatio11 r; 11 1? h t hen given b~- Eq. (1.1 ), where calc11 la. t.i o11 of the prefactor is iuvaria bl.v 111 ?L!. l1 '1?t, ?,I. T wo rC'asons fo r ignoring t he prcfactor a.re often c: itC'cl . F irst, its . we? il l11 strn tc an i11 tnesting foat m c of 3 some black !tole 11uclcat ion processes tha t cl cH's not appear to b(' present in ordi- nary particl<' prod uction. As we show in Chapter .3. in case's where there exists a cos111 ological constant, or some other rn attrr source whi ch ca11 ses the backgrou nd spac:ct irn c to close, black hole nucl cat iou is cl cscribccl l)y t \rn sC'parr1te and di s- t inct solu t ions to the class ical E11clid<'a11 E i11 stein 's C'q uations, i.e .. the instanton is di scontinuous. If one adopts a path integral fra rnework in which to calculate nucleation rates , it seems that nucleation processes described bv a disconnected instant.on would have a vanishi11g nucleat ion rate, at leas t scrniclassically. How- <'vcr. one would expect black hole? to form i11 a cosmological universe , . incc the cosmological expansiou would provid e the nccessarv force to separate the virtual black hole pa ir. One way arou11 d t hi s problem is to implement the "no-bound a rv" fo rmul a tion of quantum cosmology, cir.scribed more full y in Chapter 3. In this approach. one calcu lates t he wavefun ction for two universes: one conta ining black holes . and one describing thr backgro1111d spa.er.t ime in which t hese bla ck holes a re m1cleatcd . The wavefun ction of the universe is irl entifiecl with a gravitational path i11tegral. The rate of pair creat ion is then dc-)erncd proportional to the ratio of the modulus squared of the wavefunct ion for t he black hole universe to the background universe. Once again , however , we are faced with the problem of calculating a gravitational path integral. In the "no-boundary" formulation of qu antum cosmology, the semiclassical approximation to t he wa.vefun ction is often implemented . In this a pproximation , the wavefunction takes t he relatively simple form \fl ~ L A1,; e- 'k. ( 1. 2) k Here, A1r is prefactor , and h are the actions of instanton solu t ions to the Eu- cl id ean fi eld equa tions. Th e prefactor is typi cally neglected , and it is assumed rate 0 11 c1 i11 sta11 tou ovC' r \\'hich 10 f, ttJJJ. He1H ?c-, the? pair crf'at ioJJ lli;-it tliC'n' is only 1s g i,?e11 by ( 1.3) form as E q . (I.I). Howrver, th is method of calcul a t- T his eq uation is of t he same asons. pa ir production rates has !Jc?r11 crit icized for a 111.11 11 brr of re ing black hole 11 cleated from a11:? initi a l stat e. Iii 11 be n First , it seems that any fina l s tate ca ca me. A second r sens'. t 11 c fiu a l state has 11 0 "m emory" of fro111 w11 crc it som uous ? it ism is t hat t he instanton 11 secl iu the calculat ion is discont in ) rr latPd cri t ic it h only cont inuous histori es in factory if onP could deal w "'011 ld SPC' III more satis s to ask how can ina ry t im e. In light of t hesr cr iticisms , a natural question i im ag tion rate? s ure' that Eq . (1.3) givrs t he "corre ct '' c111 swer for t li r nuclea Ott e b e' ravi- s back to some of the diffiC'lllt. i cs involved in defining the g T ltis brings u n rt.hod is used to calculate bla ck s that " haten~r r tat iona l path integral. It seem ach or the "no- bound ary" qu antum unce a ppro hole nucleat ion , namely the bo one must evaluate a path i11te gral , or an approximation cosmologica l a pproach , tional path integral that is com mon to e grav ita t.o it. One ill-defin ed asp ect o f th r of integration. It is claimed that t he validity of t li e both methods is t he contou uHdary approximation to th e waxef1111 ction rests 0 11 t l1 c choice of bo scm iclassical this reason that wr give a fair ly egrat ion . It is for condi tions a nd conto ur of in t the boundar,v COlldi t ions sp ec- cl e t?a iled account of t he contou r of integration , and t ndary" proposal. \,Ve will see th a t a unique resul t for t he patl ifi d by the "Ho- bou n is not possible. Hence, a lth ough some itio in tegra l for a given boundary cond li rvr.d to be t he correct bound ary condi t ions and con tours do give wha t is be wavefu11 ct ion , there is uo rcaso11 for singling Sellli classical a pproximatiou t o the m s. Hence, we go by couvent ion aml o11t t hese houudary condi t ion s and cont.o 5 simply adopt q. (1.3). rn ocl ulo the' prcfn ctor A. :\<'ar thr end of Chapter 3. ,w? illustrate both nlC' thods of cakulatiJ1g black !J oi<' nu cleatio11 rntrs. \\'r ill11strare t hC' bou11 cr 11wtl1 ocl b_v calrnlat i11 g th(' nucl C'a t ioll rat.<' for mag11c tic:a ll .,? chmgrcl Reissner-:\ordstrcj111 black li olC's ill a backgro und 1mif'onn rnagn in hapter 4 and begill 0 111? clisrnssio11 of the possibl e cxis- tr nc?c' of continu ous instantons in tlwori<'s containing !llocl ifications to Einstein 's <'q 11 a tio11 s. T hr q11c>stio11 to ans\\'er is thc?n. gin'11 a pa.rti cular modification to Ei nstein s rquations, docs th is theory contain continuous instanton ? that dc- scri br 1n1 ?leat ion procr. ?srs similar to tliosr drscribecl b_,. disconnected ins tan tons in ordin ary genera l relativity? As meutionecl , wr will be consid ering black hole 1111cleation in a backgro uI1cl de Sitter spacet illlr . Hence, we arr looking fo r so- 1u t.i ons to modified Einstein 's tquations which describe a conti11uou transition from a de Sitter ?pacctirne (appropriateh? g<' 11 rrali zcd to these t lt coric.) t.o a spacetime containing a pair of black holes . ...\.s a first step , oue would like to determine the exist '11 cc of a connected background instanton. In this c:asc, the E 11 c: lidean de Sitter-likes ?ale factor a(7 ) wo 11 ld begin at maximum rad ius, shrink to sornr small but 11O11-z ro radius, allcl the11 re-expand to large radius. Since thr sca lr factor will take Oll small va lues at thr ??tum-around" point , we expect t he scalar c: urvatmc to brcomc large there. lt, is this idea that motivates our decision to consid r higher curvature correct ions to the Einstein-Hilbert act ion wi t li a cosmological consta nt. as a possi bk c:ancliclatc for possessing connected i11st.antons. \I\ rrst?ri ct attention to t}woric's \\'IIicl1 arc pol:vnornial i11 the Ricci G st look at t11c g<' IH'ra J casf' wli<'re t.l1< ' R icc i sca lar is a l- scalar cur\'atur(' . \\'(' fir exist i11 t li c Ri cci sca lar. but lowed to be rnriable. \1\'e fiud that b ounc(' sol11 tious ytically determine wh ether t he scale factor can be synchroni zed we cann ot ana l - i s bounce. Num ?rical solu tions indi c a tl' that such syuchronization is pos with t h to ini t ia l conditions. something th a t nsiti\'C si bl e, but t l1 ese resul ts are highly se el is natural in thi s context . We then 11 se the metli oc.l of Fuku ta ka , WC' do 11 ot fe with a cosmological constant with co r- et al. [8] to study Einstein-Hilbert tlieo rv certain ct ions quadratic in the Ri cci scalar. Their method reveals that under re Ricci ndi tions , !Jounce solu tions do exist. \ iVe then look at the case wh ere the co rrn turc is constrained to be constan t. Herc, we find that periodic, non- sca lar cu o exist when the coupling constants t o thC' s ingular solu t ions for the scale factor d negative. To intorpret this result , W( ' show that higher liig li cr curvature terms arc t to ordinary Einstein-Hilbert gravit y irnlen curvatur theori es of gravity are equ er d to non-standard fi elds. The nega tive coupling constants in the high couple s motiva tes curvature theory imply scalar fi elds with negative energy density. Thi Einstein-Hilbert theory with a cos- our exploration of connected instan tons in . Once ological constant coupled to Narlika r 's negative energy density C -field m ound instautons, modulo small violati ons of the again , we find connected backgr icatioll for how thi s small violation Hamiltoni a n constraint . We then g ive justif can occur. sed to cre- Chapters 5 and 6 contain our discuss ion of how "surgery" can be u ies that describe black hole pair prod uction. Such a surgery ate cont inuous histor iven s with th e di sconnected instanton . T hen , small , identical volumes of a g start illstanton. The discon- topology are excised from each part of the disconnec trd se removed regions. nrc:ted manifold is then identifi ed acr oss the boundaries of the 7 is 11 o t rw~r.v" hC'n' a solu tioJJ to Ei 11steiJ1 ?s ('q twti ons. and ThC' r<'s ulLing m a11ifold . However , one n.1 11 regard such a co 11 f:iguratio n as so is no t stri c tly an in tantcm is limi ting t of a smoo t h history connecti ng in itial a lld final statrs. T h i li e limi was shown by Bousso and hist t. it ory can br considered a 11 car in sta11ton. In fac [9] onr docs not ueecl to consider exac t insta ntons wh en ernluati ug hamblin that restricts t he path integral to co ntin- ne a path integral. They sli ow that w11 cn o s history, th C' result is only a uous hist.o ri 'S whi ch approxim ate thr disco11 t i11uou . T he classical term is ident ical f t he patl1 integral small cha nge ill t he prcfactor o m cosmology. This to that predi c ted by thr 11 0-L>o un dary fornwlat'ion of quantu he ?'proof'' t hat 11 0-bou11cl ary qu antum cosmolog_v does give t is rlairnccl to be stantons. 'c t rat 'S for nucl eat ion processe s described by disconnected in co rr< blin surgery, the excised volume l Cham In the' spe ?ific cont 'xt of t he Bouss o a 11c excised region has 4 is a four-Gall ( topo logy D ) of radi us 11 ; t llf' bo1mdar_v of this 3 t he subseq uent identification i s ca rried ou t , tlie geom etry of to pology S . W hen ied wi t h a ncgativr energv dens ity dom a in ntif thp region abou t t he junction is id e storv, 011e adius TJ. Hence, to calculate th e action of tllE' con t inuo us hi wall of r wall as a correction to the action for ll inust include t 11 e act ion of this domai nec ted histo ry isconnected instantou. The re sulting total act ion of the con <1 d tion of the ini t ia l state , the ac t ion of the fin a l cons ists of tbrer terms: the a c in wall. Bousso and C hamblin m a ke the oma state, a nd the act ion of the vir tual d sign in the total la im that the action of t he fin al state appears with a minus c es the domain wa.11 in goiH g fro m the iuitial state ac t ion because as one travers This then a l state, the normal vector und ergoes an orientation change. to t he fin E' exponent of Eq. (I. I) . They ex pla ins the a ppa rent s ubtract ion of actions in t h ca11 celing a ny volume a lso ?]a im t hat this cha nge of o ri enta tion has the effect of 8 1< ?1111s 1!1;11 \\'0 11lcl otiH'rwis<' 1H' pr<'S<'llt in thl' action . \\?c? l"r?c?i I l1-1t t l1C'rc ar<' a 11umb<'r or diffirnlt ics \\'ith tlH? a bow proposal. and we' c? 11 1111 1c? rn 1c ? t !1C' lll in haptrr 5. One such cliffic11lty is their rationale fo r sub tracting ,H ?t io11 s. F1 1r1 hrrrn or<'. Llw Bousso am! Charnblin surgery relics ou the nucleation or a 1lC'g,11 i\?c? 1'nerg~? dorn ,li n wall , but their t l1 cor_v docs not comaiu the reqnisite :--:po 11 c' 11 t of "q. (1.3). Such a normali zation scheme is more in line wi th t ,1k i 11 g rn l ios uf probabil i t~? measures, as do11c ,1,fter the first equali ty of Eq. ( 1.3). ( '01did <' 11 cc' i11 our choicr of normalization rests i11 the fact that it :vields physi- C'a l I_,. rc ' ,1 s01 1ri! Jle ;-u1swers for nucleation rates. Fmthermore, it gives reasonabl e 1111 clc'<1t io11 rntrs for rrnd t iplr bou11 ces. I 11 t IH? spir it of the path in tegral forma.lism, we' claim that 011 cting topol - og)? of D2 x 5'2 . For t his tvpe of connrct ion. the excised rnlurn e is sma ll even wli c11 the radius of t he 5 2 is allowed to be la rg-c, sa~- Oil t he' seal<' of the instanton radius. \\'c calc ulat<' t li c action fur t he 1111c:lC'ation of a l\'ariai spacctimc using th is con nect ing topolog_v. and compare this resul t to the act ion of uucleat ion pro- cess where t he co11necting topology is D?1. We find that in t he context of t l1 e 11 0-boundar>? fo rmulation of quantum cosmology, t li ere is a slight preference for nucleation processes which arc co1rncctcd vvi t h a manifold of topo logy D '1? How- evrr , wr fee l t hat there are other reasons wh)? the D 2 x 5 2 rn11n ecting topology m a_,, be pref rred . F irst. sucli a connecting topology can facili tate the passage of information from one spacetim ' to another , and \\'f' argue t his using rnrious liolograpl1ic proposals. In genera l, such hologra phic proposals clairn that the bulk info rm ation of a spacetime can be encoded on two-dimc11sional screens located somewhere in the spacetime. We find t hat the size of the 5'2 port ion of t he cou- necting manifold can be as la rge as t hese sc reens, thus indicating the possibili t:v for t he bulk inform ation of thr spacctimc to survi ve thr nucleation process . A second reason why we believe the D 2 x 52 topology may be more preferred is t ha t t he path integral for such a conn ecting manifold contains many more states over which one can sum . For example , if t he 52 port ion of the ma llifold is divid ed up in to Planck sized cells, onr can imaginr pla ?ing small Planck sized "bumps" in various cells. By considering all possible combinations for pl acillg such bumps on t he two-sphere, one can argue that there i. an exponential co11tribution to the path integral no t present when t he connecting ma11ifold is D 4 . Furthermore, the size of t his exponential contr ibution is of the same order as t he expo11enti a l con- tr ibut ion of the saddle poin t act ion. However , si11 ce t he 11 or111 alizing background 10 111 1,, 111" ?? ?. ?. ,ii al so l1 an ' ~11 cl1 a co11t ribu tio11 . th <' r<'sulting 11uclC'at io1J ra tC' will 11 01 IH? :1!! ,?11 , i ! i<' ll CC' . we haw 11 0 ph,?,sical \\"H.\ ' t.o cli st,inp; uis!t alt ernati vr conu ('ct ing I . 2 0 11 tline of the Dissertation I 11 ( ?1i;i p1 1?1 .! . iVC cl isc11 ss parLi clt> pair product ion rates in the semiclassic:al approx- 1111 :111 , J1 1 11-, 111,.? Lh r E11 c-l icl ea11 path iu tegral formalism. \ Ve start with a general d1-,, 11 -,, 1" 11 ,ii :hC' r<' laLio11 ship bctwten part iclr prnduc:tion, tunneling, and vac- 111 111 1 iii?, :1\?. . 11,d show. from a heuristic: poiut of ,?icw. that the rate of decay of a 1111?1: 1 -, 1: ii 1'1 .., 1; _tc is pro portional to th<' irn ap; inar_,? part of the ?;round state energy. \ \ ?:i :--, (? d t1J 1 ;1 l<11 d idcan grav itational path in tegral. Alth ough we will not calculate 111(? I 111'!; 1(' 1, ,1 !or a black hole nucleation process, evalu ation of the cl assical part <,/. 1111? 111w i,?: 11i on rat C' docs i11,?olve a se1ni c:lassic:al approximat ion of t he grnvita- 1i ? JJ i; i/ p;11! 1 i111 cgra l. T he' crucial aspect uecded to .iusti fy such an approximation 11 : ,our of integra tioll . and appro pri ate boundarY condi tions. T he boun 1- . 1: ,. , , ,1, , .ition on which we foc us is the '? 110-bou11cl arv '' proposal of Hart le a11d I l,1\' i. 11,, . To illustrate the difficul ties inrn lnxl in definin g the path in tegral. we , ! 1. ,:, tl1 c path in tegral in the simplifi ed arena of a dr Sit ter minisuperspace :1 , ,, i- I \\ '<' show that a 11niq11c fo rm of the path in tegral cannot be specified . I 11?!1,, 11r a rc fo rcrd to bow to convention aud irnpln ne11 t the semiclassical form , : 111,? 11 ;1VC fun ction given in Eq. (1. 2), wi thout full justificati on. V1/e illustrate : 1 , , :1, ,, ,1oachcs to rnlculating black hole nucleation, m1.111cl y the bounce approach ., 11, i I i1 , 10-bound ary qmw tum cosmology approach. I1 1 t haptN ..J . we cl iscuss the existence of cont inu ous histories in theories 1? ?- 1: 11111,?. from moclificatious to Einstein 's equ ations. The first such modification 11,1. . ! 1 , ?, higher curn1ture corrcctious to th e Einstein-Hilbert action. '\file fo c:us 1, 11 , , Jt 1,?1:tions which are polynomi al in th e Ricci scalar. \Ve find that for both Ii' ? ,11 1rl i{ 3 corre ?ti ons, t herr exist pr. riodic, non-siugular solu tious describing a 111 1111111 1?1) background instanton. However, we discuss reasons why we are not , ii ii ,? 1" 1x tend this analysis to more general topology changing processes. \Ve 1I 1 , 11 ,I 1~ 1:uss co upling Einstein gravi t~r wi th positive cosmological constant to .1 111' ,, ,11 1vc <-' Jl erg_v density C-fi eld . \Ne find continuous histories describing th e I1 ,1 1 L,?11,1md insta11to11 , but. for this to occ:ur , there must be a small violation of 1111 ? I L1 111.ltoni an constra in t . I1 1 1 i1c)p ter 5, we? begin our discussion of constru c: Ling ontinuous histories via ,111, ., ., \ \\?r foc us here on the formali sm introduced by Bousso and Chamblin 111, 1 1111 1,: vir tua l domain walls. We show how evaluating the path integral over ,i i i l11, 1111i cs cxc<'pt th e disconnec ted one results in only a small change in the i> 1d :11 111 1. Since the Bousso and Chamblin procedure utilizes domain walls, we 12 g in? an fairly d eta il ed discussion of t li c prop<' r t i<'s or such topo logical defects. 1~i nall~-, wr discuss what ,n, fC'c l arc Haws a11d lirnit a ti ons of their pro posal. T his leads us to o ur a lt<'rnatin? proposa l in Clrnpt<' r G. In C hapter 6, we discuss our alt ern a ti\' , forrnalisrn for co 11st rncting contin- t1 0 11 s hi stories. \ \"(' define t li e action fo r such a 11 is t O l"_\". and use' it to c-alcula te n11c-l ei-1t io 11 rates for cl<' itter to de Sitter trn11sitions as well as cic' Sit ter to >Jariai tra ns itions. vVc show t hat t he D 4 con11 ccti 11 g topo logy is slightlv prefe rred in t lH' no- boundary formu lation of quantum cosm ology. HowevC' r , we t hen d iscuss reasons why we bcli<1 vr t he D 2 x S 2 connc ting topo log~? may be more preferred. F irst, we evaluate t he path integral for tlie D 'l. x 2 conn ecting topology with t he in cl us ion of Planck sized bumps on the S'2 portion of thr connecting m a nifold. F in a ll y, we d iscuss l1 ow s 11 cl1 a connecting topology c.-i n facilitate inforrn atioll ca n s 11rviw t he ncar-a1111ihilatio11 , recreati on proc<'ss b,? CXiU!liuing t hree ho lographic proposals. \Ne find two-s mfaces onto whicl1 t he hulk information of d<' Sitter aud Nari a i spacet irnes can be encoded. and argue? t hat th rse two-surfaces have area th a t can be accom 111 odated by such a com1cctiJ1 g topology. 13 Chapter 2 Particle Pair Production Rates 2 .1 Introduction T he su bject of pa ir creation of pnr t icles i11 backgrouud fi elds has beeu t he foc us of in t.ens research for rn a n:v _vears. T h(' top ic elates back to Schwinger [10] "?ho caku lat' par t , indicat ing an instabili ty in the vacuum . Schwi11 ger attribu ted this instabili ty to par t icle p roduction , and re- la ted the irn agiua ry part of t he Lagrangian to thr pa.ir production ra te per uni t vo lume. Ext nding d1winger 's formalism to 1J011- t ri via.l background electrom ag- netic fie lds is d iffi cult?, sin ce ob ta.ining effcc:t ivr Lagrangia ns and then evalua ting t hem for a rbi t ra ry Y<'cto r p itentia ls is 11 t an easy task. In t.lws<' cases , pa rt i- clr p rod uction has been studi Pd b>? t he method of normal mode ana lysis [11]. For background clrctromagnctic fi elds represented b>r a t ime-cl pendent gauge, norm a l mode aualysis results in a non-zero Bogoliubm? coeffi cient fl, implying pa r t icle p roduction (c.f. [12, 13, 14]). \\'li en thr background elcctrom aguctic 1-1 ffrld is rc?prc?s<'nted b_,. a tirn e- indep<'11d c? 11t gc111 gc . ,-J is tr i,?iall _,. zero. implying no particle produnio11 . A.t first gla ll(;e. t his n?s11lt seems to co11trndic t Sdl\Yi11 gt?r?s conclusions. Furthermore , consider the case of all clcctrornaguetic field capable of bring desnibccl by both a t ime-dependent and time-i11clc?pc'ndent gauge. ~ornial llloclc ,rnalvsis will show particlr product ion in t li c tirne-clcpenclc11 t gauge am! no particle production in tlte t ime-i11dcpn1clr11t gauge; particle produ ct. ion sho11 lcl lw ga uge iuvariant. To explain particle product ion in a tirnc-indepcnclc11t gauge. a "tuuneling intcrprctatiou" is inrnked (C' .f. [15. lG , 17]). In this formalism, onr decomposes t ltc? quantum fi eld into normal lllOdrs and then obtains an effective Schrod in gcr ?quat ion that describes t 11n11eli11g through a, poteutial barrier , as in ordiuar_,? quantum mechanics. It bas been shown that for electromagnetic fields which ?au be reprrsented b_v both ti111 e-dep<)11dent and time-independent gauges, t.bc? corresponding 111 ct l1 ods to compute part icle production rates agrc<' [18]. TbC' work of Lang<)r [4], and later Colcrn a11 [1. 2, 3, 19], showed how instan- ton methods could be used to treat t1mueling aucl vacuum deca~,. As a resu lt , instanton methods wcr<' appl ied to particle production. For example, Affleck alld I\Ianton [5] used instanton methods to ca n v 011 t a. sr.miclassical calculation for the rate of pair production of oppositelv chargrd monopoles in a uniform hackgrou11d rnagnet ic fi eld . 111 the weak fi 0 ld limit , their rcs11lt for the pair prn- d 11 ction of monopoles was analogous to Schwinger's result. Hence, they suspected Schwinger 's results c uld be duplicatE'd using instanton methods , as they later . howed [20]. Particle production has been shown to occur in a number of other contexts. For example, particles are produced iu expandiug universes [2 1, 22 , 23] . In such C'ascs , the gnwitatioua.l fi eld is ti111e-dep<'11deut and the? mrthod of Bogoliubm? co- 15 <'11t s mil I)(' 11 sC'd to cl<'tC'rmi11c particle produc ti on ml.C's. Particle' producti on <'ffi ci s those of' was also slimn1 to t akC' placC' in spacerirnC's possC's sing horizons, such a a black l1olf' or 11011-zern cos lllologica l co!lst ant [2 -1. '.?5 ]. In tliC'sC' cc1s<'s. norm a l s thr pr<'SC'nc:c of a thC'rnrnl spPctrnrn of particles . niodC' analysis rrvcal An a na logous process to partic le-auti part iclc pa ir CrC'ation in quantum field l relati,?it_v: in each case' , tli eor_v is thC' production of blr1d: hole pairs in ge1 wra e present to separate the black l1 olc pairs. othe rwise the then, must he a forc pair would annihil ate. HowevN, as poimr cl 01 1t in [2G. 27], t here are impor- air productiou. F irst, unlike tant d ifferences hC'twrcn bl ack holC' and particle p t li c pair production of parti cles , the p<1 ir prod11c t ioJJ of black l10les is a to pology cha nging process. T he spat ial topology of r1 blac k hole spacetirn c is d iffer nt d or w1,c:u um spr1cr t-.i111 c: fur therm ore. one spacet irn c t han t hat of t l1 e backgroun topology canuot be colltinuous!y tra llsforrncd iJJ t o the other. Second , the fund a- ud electromagnetic fi elds mental t h('Or_v of. say, monopole production in ba ckgrou is kn ow11 , namely Yang- l'viiJls-Higgs t.!1 eor_v. T l1 c c:oHespondiug qu antum theorv own. Despite this seco.JJd cliffcrC1Jce, it has beeu r ealized that of gr avity is not kn the instautou approxirnatious used in t he cak: 11 lc 1tion of particle pair production hole pair creation rates in the rate. can be adopted in the calculation of black e su111-ov r- lzi stories frame1vvo rk for quantum gravi ty. It is for this reason that w ntiparticle pair production via instanton a1 proxir nations here. review particle-a :\ot<' th at in this chapter , we !eave in explicit depen cle11 ce on n, unless other- wise' 11o tcd. lG ? ) ? > ? I'll nneling, Particle Production, and Decay H a tes \ , ._., , 111, 1i11, ,: ed i 11 t IH' pn,,?ious section. t li<'r<' is a close analogy between part iclc 1,,1 11 i,1,11i: 1, 11, ,n and the' co11cc' pts of t 111111 c' lin ?, and nH? 11urn dcca\'. Iu t he cas<' ,d ;i.111 : ,111d 1:,tio11 of 111 011 opoks. the constant background rnag11 ct i(' fiC' lcl is co11- ,1,\ , ?1 ??? ,,, 1.. . 1 fal se ,?ac1111rn: t he' fidcl e11 erp;_Y ca n bC' lowC'r!:'c! b~? iuscrt ing a pair , ,I --. 11111, i1 ?111\ \ ~epanit.cd monopoles. \1\'hc11 t.ll<' rno11 opoles move apart , the)' gain 1? 11, ?1~\ !1?11 1 11c fide!. thus all o\\'ing t he Yarn um to circa~? bv quantum mechani- r :1\ 1)1 ?111 ?11:1!1 ?,n t hro ugh an cfkcti Ye potential barrier. T lie height oftli r barriN i, ,:pp1, ,'.1 11 1:1 1Ply t.wicc' t hr rnonopole mass. Sin e<' C' IWrgy is co11 scn ecl during a 111111wl11 1c? 1,111 ,?css, L1 1crC' is a n it-ica l separnt.ion di sta 11C'C' bct.wer11 th<' monopoks lr ,i 1\ i1 1, 11 1111 tota l cl1angc? in C' 11 ergy is zero . It is thi s ni t ical separation dista11cc' 11 Ii i, 11 , l, ?1, ., 1: ; uc's the width of the potential lmrrier [:-]. i, , I 11 ? 1 1 ??1 .111dr rstancl dcca~? processes. IC't us consic! Pr some simple exarnplcs. ( 1111? < ;i 11 :-, \ ;irt by considering a particl!:' of unit mass rn m ing in a one-dimensional 1" " 1?111 i:il f 1.1) shown in F igure 2.1. Classically, t"lw system possesses a stable r?,1111li l11i11 111 i11 which a particle' is a t rest. at .r = :i:0 . Qmwtum nwdianica ll v, a 1,: 111i ,?J,, I, " ;1 ! ,,cl i11 rC'p;io11 I is ill a metastable state . chantcteri z<'d by an euergy 1) I. ,, ? I max ? T hr par tic le' lt as a fini te probahilit:v to tunnel through the l ,: 111w1 :111rl 1rnpagate to infini ty. Hence, the state is uustable and subj ect to rl, ,,?:11 : i1 1., :1 fa lse ground stat.e. To make the c-onHc t io11 to particle production, 11111 ' r :i 11 1! 1i1il, of t he coordin at.c' .T - x 0 as t he sepanttion betwPe11 ,t particlC' a nd it :-, :1 111 i- p:111i rlc. T he initi,tl state is that of a fal se Ya cuum wi th HO parti cles 1' 11 ?:-, with potential function U( ) shown in Figure 2.2. T he 1 1?11 1i :1I / ? possessps two relative minima, U( 9)? )- the energy difference of which J 1 ,i 111? tl, ?:?>1c:, 11: 1?.e b_\? f. T he state = + is a state of classical Pquilibrium and is 1111 :-; 1: 1lii 1? di (' Lo barrier penet ration ; it is a fa lse Yacnum. The state c/) = _ is the 1111 iq 11 1 q11:1 :1t11m mechanical vacuum state, or t.nH\ vacuum. Consider the system ? 1, 1 I)(' i11 1I :(' ini t ial sta te = 1: 1 ? 1?.111 calcul a te the lifet ime asso ?iated with the decay of an unstable states. 4>+ {J ('? \i\'e will see t?hat t. hr Lorentzian path int egral is . in g<' 11 era l, not w,,Jl defin ed; ,-u1alytic cont inuati on of Lorentzia n t ime is a wr1y i11 ,d1i c:h 011 c uw obtai n a better-defined form of the path in tegral. 2.4.1 Path Integral Formulation of Quantum Mechanics Let us start by considering ordinary qua nt um mechanics where we will fo llow t he discussion in [30]. O ne oftc,n wishes to sp r)c ?fy th(' state of a quantum m echa nica l s_vstcm , given that the svstcm started i11 so11H' iuitial configuration. There a rc, in general, ma n:v pat hs which counec:t t he initi al and final confi gurations. In the context of classical mechanics , t he path along which thE. sy tern evolves from the ini t ia l to final configuration can be determined by t he principle of least act ion. T he act ion , S, is omputed for each possible pat h . The class ical path is that which extremizes S. Q uantum mecha nicall_v sp eaking, a svstem sampl es a ll paths t hat connect t he initia l and fin a l configurat ions. Wl1en suitahl>' mc)asured. each path cont ributes an eq ua l amount to t he total a mpli t ude to evolve from the ini t ia l to final configuration , but each co11tributes b>' a different phase. The a mplitude for a particular path :z:(t) is given b_y (2 .22) T he total a mpli t ude is then given by su1n1niug over a ll such J aths, i.e., ( :Z. J, t,?J I :Z,.i , t,_?i ) -_ ; ? 'vTl. l- -( /, ) C iS [.r (t )]//1 , (2.23) 25 wlierc D.r(t) d 'notes a s ui table measure oYcr a ll paths from (.t,, f ;) t.o (:r , t ) . 1 1 n ct ion of T his quan t it_y is knmrn as a path integral. One can dcfi,w t h e' w,-n-r fu the qu11n t 11m mechani cal s tritr (.if ti) as '11(x? t ? t / 1? t ) = N / D r(t ) ( .iS/ i ( t.))/ li . ':I: f, f,?? '1??1? , t ?) = \ T ( '?1 ' ?.u j, ?J ? ? 1. ? IC ? .le . (2.24) Herc, N is a normalization constaut , and C denotes a class of path s which intersect at tim e t and which a re weighted iu a w;-ff that rPftects t11 c preparat i on of .TJ 1 t he s_vstPm. 2 hen attempting to evaluate the p;-1th integral. one notices th at the action vV hat the in tegra nd in Eq. (2. 24) will oscillate and t he sum ma y is real, implying t not converge. In a n attempt to make the path iu tegral convergen t and h ence rand. An n1 ore well defin ed , one can introduce a clamping factor in to t he integ alternative m ethod involves rotating the real tinw nxis to ima ginary t ime, t --+ iT. As a co11 scq 11 ence, the ac tion transforms as iS -1 - 1, where I is t he non-ncga t ivc E uclidean action . T he wavefun ction can then be defin ed by \JI (1: T ? x.- T. ?) = N- 1D t /:1(rlJ/ 11 f, f , .T (7 ) C' - ? (2 .25) ! ) 7. . c back Once th e wavefun ction is evaluated , the result can bf~ aual_vticall _v cont inu ed to Lorentzian time. 2 ri:iclinger An a lterna tive but. equiva.l ent way t.o calrnla.te qua utum dyn arnirs is via the Sch he systc111. The connec tion equation i ~;' = HIJ! , where H is the Hamiltonian operator of t in [30) . betwee11 the Schri:idinger equatio11 am! the path integral is cli i.ic usse cl more full y 2G 2.4.2 Path Integral Fonnulation of Quantun1 Field The- ory \\ ?e wi ll ll0\\' consider the pat.h iutegral forn111lat ion of a qnantu m fiPld theorY. On<:> reprrsclltatio11 of t li <' wavefunctioll as a firn ct iollal of th(' fi <' l cl configuratiou on a giwn spacelikc surface of co11 sta11t time is \J., [?(x), t] = IV .i. V cp(x )c'S[t/>( x), t.]/h. (2.2G) where ? is the quantum fi eld . Let us con ?ider the specific example of a real scalar fi eld ? of mas paramrter /l = m,c/ Ii and action (2.27) Tlie equatioll of motion is (2.28) The actioll is real, so the integrand of Eq. (2.2G) will oscilla.te aud th e path integral may not converge. Furthermore, findiu g a field coufiguratiou wh ich extremizes the action between two points in time iu vo lvrs solving the hyperbo li c equation of motioll (2.28) witli boundary condi t io11s, an ill-posed problem. T here could be no solu tions , or an infini te numbrr of solu tions; if there is a solu t ion , it will not ciC'pend srnoothly on th e boundary values [31]. As was the case for quantum meclrnni cs , th ere arc two method s by which ucing one can make the path in tegral better defined . T l1 e first involv es introd scction, a damping factor into the in tegrand. Altcrn ativel_v. as i11 th<' pr evious -, - I , one ma_,? rnake the anal.vtic continuation t --+ iT [32]. As a res ult , iS - 27 ucli dean a,ct ion and is uorrn c?gatin- fo r fields that a n' r<'a] where I is the E 011 space defiu ed by T, ~i: , y, z . Fm tl1 r rrn o1c. to find cx t renw of t li r t he Eucl id 'a n uclidean action , onr must so!Ye an rlli ptic rq1wti oJJ o f ruo1 iou with boirn clan? E d it ions. a well-posed pro bl em. Once t!Je cr1 lcula t iuJJ has been carri ed 0 11 t iJJ con the Eucliclcau regirn<~, the results can be anal_vtical l_v co ntinu ed back to Lorrutzia I1 t im e. 2.5 Vacuum Decay in the Semiclassical Limit ll uow deri ve tl1 r fo rms of thr quantities /\." alld Io in the semiclass icaJ limi t VVC' wi the formalism of Euclidean functioual in tegrals, as dc ?,?cloprd in [1, 3} . V\'e using will consider spin less particles of unit mass rnoviug in a one-dime11sioual potential U(:r) where the Hamiltonian is given by ?) H = J-r + U(.1:). (2.29) 2 T he form of the potential fun ction we will considf~r is s hown in Figure 2 .1. 2.5.1 The Semiclassical Limit The Eucl idean version of the path integral is given by (2.30) The qu anti t ies /.1:i) and /.1:.r) a re the ini tial and final position eigensta tes, H is at the HarniltoniaH, and T is time. If we deuote e1J ergy eige nstates by /n) such th H /n) = En /n) , we can write (.1:1 /e- llT/h/ ];i) = L <' - E,,T/h (:r:1 /n) (n/ .1;,:) (2.31) 71 28 On tli c rig1Jt -11 ,u1d-sicle of Eq. (2 . .30) . .\' 1s a 1wnwil iza tion co 11 s ta11t, I is the classical Eucl idean action given b_,. I = !-'T / 2' dT [l- (d 2 d_?T ) + U(:r) ] (2.32) . - f / 2 2 ,T s x( T) satisf:ving t he bouHd- and D .-r;( T) denotes a sui table measure over all fun ction ary condit ions 1:(- T / 2) = .i: a11d .i; (T / 2) = :r.r . 1 Let i? (T) be a fun ct ion satis?:vi 11g the above bo,rndar.,? co 11cli t ious. T hen, auy fuuction obeying these boundary conditions can be wr itten as (2.33) II ma l fun ctions mi the in terval wh "re :1: 11 (T) are a complete set of real, or thonor 1 aluate t li c [-T/ 2. T/ 2] at va nish ,tt t he bonndaries of t his interval. \ \1< ' cr1 n evth proxima- ri ght hand side of Eq. (2.30) in the semiclassical lim it , or saddle point ap tegra l com es from t io11. In t his limi t, t he 11iajor co11 t ribution to the funct iona l iH statioHary p oints of the E uc:lidea11 action . W<' will ass ume for now that there is only one such stationary poiut , .i( T). The saddle point app roximation iuvolves expa nding the act ion about .i(T) to second order. yieldin g . c5I I [:i:] = I 0 + !dT[.1:(T) - .f (T)] ~() + u.L T :T:(-r) -I j ? j ? c5 I dTdT'[x(T) - .i (T)] 2 , .( ' ) I [.'.i;(T') - .f(T')] + (2.34) 2 15:.c(T) c5.r, T :r(-r ) point. Since wl1ere Io is t he cla sical E uclidean actio11 evaluated at the s tationa ry iou evaluated i (T ) is a stationa ry point of the act ion , the first variation of the act at thC' stationary point will vanish: r5J I = - -r/2 :+T: -- rlU(:f: ) -_ () . (2 ..3 5) c5x(T) _ dT2 d:1: . x( -r) 29 at,io11 of motion for a particle of 1111it niass rn o, ?i11g in a pote!ltial T lii~ is t li 0 <'q11 - '(x). He,wc? , , _- -l(d- i) 2 - , ?( - ?) E L,, .I (2.3G) 2 d, ? motion. T he sccoud var iat ional deri,?atiw of I at .T? is is a c-o ustant of the U" (: r)] c5 ( T - , ' ) . (2.37) he iuce the operator i11 squar , brackets is Herrni ti an on fun ctions defin ed over t pl ete set 111 (<'rval [-T/2, T/2] which va nish at the endpoints. it possc?sses a com of eigc11fu11c:t ions .1.: : 11 (2.38) the orthonorma li ty of the eigenfunctions, Eq. ( 2.34) ca11 be evaluated , wi th sing the result 1[.T] = 1 1 '"' '2 0 + - 0 c,,>,.11 + ? ? ? ? (2.39) 2 II ue of the sadd le point approxirnation. Lli c meas ure of in tegration V.T can By Yir t <' changed to th<' measure' V (:r-.i:). W<' ca11 make a further change of variables to h i11Legration over the coeffi cients c,,. Because the :z:11 a re orthonormal, the .Jaco bian [l]. we in trod uce into the of this cha11ge of measure is unity. Fol lowing olernan eau fu11 ctio11al ll1<'asur<' a factor of 1 / J2-if,, for each normal rnode. The Euclid int?cgrn l can then bP writ tc11 as (J? /c- llT/h/ .1:i) = NjIT drn exp (- Io/h) exp ( - L c;,>-,,./2) [l + O(tL)]. 1 ,,/2i!i, 7l (2.40) 30 If al l rig<' nYalu rs nre positi\"C' . ,Yh icli. as ,n , will !Mer sc?c is uot ahY;-n?s t h<' c;1s<'. tli cu Olll' can writr 1 ( .?i ?? I(' - I IT/ h I ' ) l\1 J ,1, , exp (- 10 / !i.) IT I\ [1 + 0 (/i )] . 11 v>-11 cxp(-10 /n) [ N---== ==== 1 0 ( Ii,)] . (2. --!l) j det [-D; + U"( :T?) ] 2.5.2 The Decay Rate \Ve can determine t he grouud state energ~? by rnnfi niu g attention to the condition :r; = .r 1 = :r. Hence , we will be interested in '?period ic" so lu tions to t he E uclidean eq uations of m otion. N-1 m ely, if we integrate Eq. (2.31) over all :r, we obtaiu a s tate sum L r,- 1:;,,-r/h / [(:r I n)[ 2 rLr 'II I: e- C,.T/h . 71 " ?here we have used the normalization o f the m crgy eigenfunctions 111 the last <'q uality. For large T , the sum over r1. ?will be domiuated by t he lowest energy lrvcl. Thus , we ca n obtain t he ground state' <-'nerg~? bv w herr = . ,J - / [T(T) 7']/1, Z N . V :r 11.:r e ? ' (2 .. .J2) / is the pa rtiti on function and is writ ten a bove as a functi onal integral for whicli the i11i t ia l a nd final confi gurations a re icl c11 t ical a nd l:dl ?ossibk illitial configurati ons 31 an? s11rnnwcl o,?er. Thr dc?c;.1_,. rate is t li r 11 g in'Jl b,? -? f = lirn ~ Irn (l n Z). (2.-43) T -HX; T .3 Particle Motion and Evaluation of t he Partition 2.5 Function ll make an additional restriction on the peri odic motion ; we fix the ini- We wi ; ; (-T/2) = :r (T/2) = .r = :i:0 . vVit li these boundary t ial configuration , i. P- .. . 1 1 d itions, we now discuss the ?os sihle sol11 t io ns to t li e classical Euclidcau fi eld con equation (2.35). dr = The first class of motions we will discuss is co nstaut solu tions, that is, d.r/ other O. 'vVit hin this class, 011 e solu tion is i(r) = :c0 . If BU/ch: = 0 at some d- valuc(s) of x , then other nearly constant sol u tions exist which satis(y the boun ary couditions. However, the action of these solutions differs by tP-m1s of order T the act ion of the solu tion .i = .1: , and can be neglected in Eq. (2.42) in the from 0 limit. T-* 00 [33). wn A second class of solu tions are thosP that, b egin at .r:(-T/2) = .1:0, roll do d then roll back to x(T/ 2) = .'to in a the potential hill and arrive at :1:(0) = a, an time-symmetric manner , ""here we take t11c lirni t T-* oo. This motion is called a For this particular motion, bounce , and is schemat ically illustrated in F igure 2.3. E = 0 and we have (2.44) The abovr equation impli es (2 .45) 32 X ,, ' / \ I \ \ (J / ,, / T / I \ I \ ,, ' rve F'igurc 2.3: T he so lid curve denotes t l1 c m otion for the bounce. The dashed cu derivati ve of the boun ce, to whi ch the eigenfun ctiou of zero represents the t ime eigeIJvalue is proportion al. ., the center where T 1 is an integration ronsta11t co rresponding to d.1;/rlT = 0, i. e i frorn :i; to a, is called an of t li e boun ce. Half of a bounce, i.e., par tic le rn oti01 0 ) , we seC' 111stanton. fo stantoJJs are highly localized in spac:ctirnc. From Eq . (2.44 that the size of an instanton is approxi matelv 1/ /U"(.1:0) (3]. third and final class of motions cons ists of multiple bounces, 1. e., thosP A otioIJs that make many trips away from and back to .1:0 . m aluate the partition fun ction for these motions 111 the followiug We can ev hat the partition fun ction co11sists of a s11m over a ll possible ini tial way. Recall t st OIJfigurations. However, because we a rc interested in deterrni11ing the lowe C e relevant ini t ia l conditions are then energy, we take t he limi t T -t 00 . Th == == x Thus, the sum over all initial c01ifig urations red uces to one term , .1:i .i?1 0 . and the partition function becomes z = (:co /e- !IT/h/.1:0) = N _/V.1:c- I/h . (2.4 6) se where there exists only O JJ C sadd le po int of the This is just Eq. (2.40) for th e ca multiple saddle poin ts E uclid an act ion . However , we have s een that there arc 33 pa rtit ion fun ct ion C' i-lll lw \\"ritt c11 a s to th c E uclid(?a11 ac ti ou . Thus. the ( z, z.) Zo l +- +- (2 .. .J 7) 2 {) 2Cl (} t so lu t ions. Z is U1c partition fuu c-ons ta n 1 Herc , Zo is th' partitio11 fu11 ctiou for c z. r t it ion functiou for mul t iple ho11nces . tio11 for onc ho 1111 ce , and is the pa now evalu ate t lie partition functio n Z . \;\Te dcuote the saddle point We will 0 .32) a nd ction for the solu t ion .f = :i: 0 by 1J 0J; it ca n be ernl uated usiug Eq. (2 a 1J0 t li eu th hat U(:r = 0. It fo llows t ha t ! = 0. If w<' d enote U"(.i:) = w:2 . e fact t 0 ) 2 r2 in Eq. (2.38) is a posi t ive opera tor. Hence . a l] >.." > w for a ll n, s ince - rf'.2 / d e ar t iti on function fo ll ows from Eq. (2 .-11 ): c igenntlucs a re posit iYc and th p - 1/'2 Zo = !V (c!C'L [-D; + U" (.r: ]) [l + O(!i,)]. (2.48) n t can be evaluated exactly a nd , in a a fo fact , for t he case at ha nd , t h r deter min particul a r norma lizat ion , is given by [3] 112 2 N (c!et [-cJ2 + U''Uf') ]) - = v(~w e - wT/ ? (2 .49) T - igcuvalues, it follows that the motio n .i = Xo i. a local From the posi t ivity of the e but not necessarily global minimum of the E uclidean action. . im e translation rn vanance of the T We uow turn to t he evalu ation of z, 38) possesses a n eigenfunction bounce irnplie that t li e eigenrnlu e equation (2. o. ith e igenva J 1e .,\ = T his can be seeu bv differentiat ing E q. (2.35) w :r, with 1 1 respec t to T t hus o btaining ] . (2.5 0) .1:, = /fo .i . bouuce motion , aucl the norm a li za - Here, 1 is the saddle p oint act ion for OJJ C 0 ketched as t he dashed cm ve t ion fo llows from Eq. (2.32) . T his eigeu function is s 3-1 gur<' 2.:3 . .:-\s OJH' ca n S<'<'. it poss<'ssc's a 11 ocl1'. il llplYi11g that tltcr<' C'Xists 111 F i will 1w,,? discuss the' 0 11 r cigenfun ct io11 .r0 ,,?ith a 11 egc1 tiw <'ig-o_ :i \\ .<' sign ifi cance of thi s mode. consta11t mo tion .i = .i: is a loca l lrnt uot global min imum of t'l1 c action. T he 0 cling Concern ing the one bo1111 cc rn otiou. there arC' mod es for n?hich the con cspou c, zero, or posit'i,?c. T his irn?li cs that th<' one-' bounce <'igc'm?al11 es arc ncgatiY 111 ot io11 is a sadd le point of the Euclidean actiou . To u11d crsta11d the be11avior and restrict attention to of t hC' act ion in hlll ction space , we' will first sim plif\' 1 t li c behavior of the action along a one-paramete r fa mil.v of paths? in function plified space [2]. T he path will be parameterized by the real rnri abJc z ; the '?sim partition fun ction" can be written as . d ::: - t (::)I" (= --e ' (2.31) -/ ~ ) t he Euclidean action along the path . Tl1c path we J1 ave chosen will where I (:: is i11clude the constant motion i: = :r;0 parameterized by z = 0, and the one bounce which satisfy the bound- motion ?arameterized by z = I. Other possible motions ary conditions arc shown in Figure 2.4 . Consider the motions parameterized by :: > ] . As :: increases beyond the value one, th e particl e makes larger excursions g amou11t of tim e ill this bc'yond the turning point l. For z > 1, the in tegrand of t he simplified partition fun ction becomes unbounded rtnd ( d iverges. To evaluate such all in tegral, one can apply the method of steepest descent. Oue distorts the contour of in tegratiou a long the real z-axis to a contour which goes along t he z-axis from -oo to the saddle point z = 1, and then is rotated to complex z . As a resul t of t his analytic cont inuation, the integral acquires an imaginary part. Let us now extend th is analysis to all of function space. T here are mall)' paths in fun ction space whose ac tion start at a minimum va lue, ri se, and then des ?end to large negative values. However , there is one path whose action rises the least amount. It is a long this path t hat the saddle poin t act ion corresponding to the one bounce motion is a maximum. T his path corresponds to the mode 36 nch? UH' mock c \Yit li <'igr11,?aluc' ?\ 0 . ln din'c-lio11s with 11 <'gatin' c?ige ll\?alue. 11,u 0 t ion doc's no t d<'ncase fro m its c?o rr<'sponcli ng to lll oclcs r? \\' ith 1 1 2'. 0. t Ii<' ac 11 n. \\ir ca n clrscribe t he significance ' of t l1 c' ncgat iw ,?aluc, for the one bounce motio from 111 lll orc ph~?s ical term ?. If a s_ystr111 i 11i tiall_v at .1:0 fluctuates awa~- oclc- in he dir<'ct io11 b~- \\' hi c-h th is is mos t c's of .I'. T this nllu r. it \\'i ll f,tll t o larger nl iu Tl1C' rn od<' c is au 1111st.a bJc, mock for mode c . 0 <'asily ?1c-c-0111plished is along the 0 sition t, h poc bou nc mot ion. T lw pa rticle conl c l either fa ll hack to its ini t ial r Oll ' boun cc' motion is t li c? main b a rri er hus. thC ?0 11 <' at :ro, or propagate to infi ni ty. T h pass to decay fro 111 the 111 t>tastabl e state :r . 0 11 e t. ro ngh which the particle must 0 <' lifetime of t he can then look lo t he in tegration over t. hC' mod<' c idc' t h0 to prm? inetastab!r state [33]. ificcl par tit ion f1111 ctio11 , W<' can c v,tlna te t he T hus , as we d id for t he ::;i mpl est dc?scc'ut apprnximation. I t. is i11 tcgrat io11 t c'Pp iutcgral over thr mode r- ~s 0 in th tart bv is mo IP whi ?h will I rm? idr the i1m tgi 11 ar~? comril ut ion to z,. vVe s ovc'r t h wr it i1tg z, as e,, - -~' ;, - lo/ h I --clc j IT d ,..-,,,_ I. p0 co., h ; ? de, - '1 -~"'1 ;, < e i>-o i/ --c I --e ' ./2ifi J2iFi 11 > 1 ./2ifi (2.52) eveals that the contour of in tegrat. io11 along thr real A ste0pesL descent aualysis r giuar:? co axis at thr sa ddle poin t to thr ima co axis is rotated by an angle of 1r /2 peak, = Furt li crmor , t he limi ts of in tegration cover on l y half the Gaussiau co 0. part icle propagates to from zero to 'i , sin ?e t his is t he d irection along whicl1 the other half iu \'Olves the par t icle fall ing back to iu fi nit_v. T he ?ont ribu t ion from th e tio11 of Zo [33] . T hu , the metastable state, aud ha bPen included in the en ilua we have (2.53) 37 L< ?t 11s now c?onsid<'r rhr i11trgrnl owr thl' mode' CJ .?- \ n i11 fi nitrsi 111al i11CT('J1H'nt i11 .r in tc?nns or thr rn odc c1 ca n be wri t ten as d. r = :r 1dr?J. T hr same i11 crC'mc11 t i11 t <'t'111s of lhr cr ntc? r of' t hr one boun ce 111 otio11 TJ is p;1,?c' J1 lw dT = f dTJ. T hus. (2.0-1) Tl1u~. t hC' int ,gral owr CJ becomes (I; ;?T/'l dTJ - (I; T Vh ll . (2.55) - T / 2 - Vh l . \\ 'C' can 110w writ(' Z 1 as (2.56) WIH'rc' the prime indicatrs Lha,t the zern c' igr1m-d1l(' is omit ted from computation or the? dcten :1iJ1ant. ;; \\'c, now turn to th r rva l11 at i n of Z*. ffrntll t liat i11stanto11s are highl:v local- izl'd i1J spacrtimc. As a resul t, the class of mot ions described by 111ul t i-instantons can he' thought of as a dilu tr gas of idell t ica l, no11 -i11t?era.ct ing excitat ion. [2, 3]. Frorn a sta tistical mccl11rnics poill t of vi ew, thC' partiti on functi o11 for a sys tem of '11 id 11 t ical, non-in tcntcting part icles ca11 hC' written in terms of the one p,uticl r part it ion funct ion a. [36] Z11 Z " =-J. (2 .57) n ! \\'<' ca.11 1rnclrrstancl this rrs11lt in terms of th e dil11t <' gas a,pproxi111a tion. T he ac- t io n for 'II bounces is simpl y n10 . Since thf' acti on is add itive, partition fuuctions ''Tlw rcadet should note t ha t zero eigenn dues rx ist only in th r T ~ limit. For fi nite T. th 'SC' l'igenvalues a re llOll -zcro , and can be incl uded in t he eva luat ion of the dctnminallt?. For a disrnss ioll of how this is aceomplishcrl, see [33]. 38 co11- ' ll'(' 11111 I . li cnnore. thC' zern modes o f' tli c n1ulti -ho1111 cC' rn ot ions < t ipliC'al i\'C' . Furt ?r ng. tribute -1 lf'r?ii1 y1, ; ? ? ? ? 1 factor1c-1I is rnclu c I C'd to pn'n'nt o,?erco 11 11t n 1., " .1 1 ere the 11' . . . n aclcl1- ,,s in the st?,t ? t? I . .. . al pr1 rt1cles. HO\n?,?cr, there JS a is 1ca mechanics of Jcl ent.1c? n cs. In the limit of tribu t ion to the parti t ion f oncti on for urnltiple bomic l iona] con e. T lrns. laro-c, T ? stant for large arnouJ! ls of tim 1 b ' rnu t ip1 C' bounce solu tions arc co 11 \\'e divide to Pn',?c'nt o . , . .0 . . 11n tJ .1 1 g tlie contnb 11 t1ons f .rom tl1c co11staJ1t. rn otion. ' c I c b,? 11 e fac tors of Zo. Thus , we hav Z Z;' ) .58) = + - 1 J (2 Z Z l + - l -Z( + ... + - - 0 ( Zo 2! Zd 11! Zl/ Th? ? ? for JS JS ies .JUSt. aJJ C'xpon<'ntiaJ ser (2.59) ate is given by By E q . (2.43). the decay r r li111 --2 Im (Z- 1) T T Zo )] 1-1/2[l + O(!i.)]. 0) Ji;" (2.6e - Io/ h I det.'[-8; + U"(.i Vh l clet[-8; + U"(.'ro)] re WC' h . hat Zo is real. It sl1011Jcl be no ted for later discussion wli e ave usc?c 1 th<' fact t ver. for the casf' at haucl , we showed tbat. the exponent is rC'a JJ _v -( Io - Id?\ Howe earlier that I Jo) = 0. ion of the abov<' to the theor y of a si11gle scalar Let us now consider the ext ens d the o ur-d imensional space time, an fi ?Jd ?(x) , where cp(x) is a fu n ction of a. f T he bounce?; was determin ed in n in Figure 2.2 . Potential U(?) is like that s how e is in variant 0( 4)- invariant function of x. Furthermore, the bo nJJ c [I ] and is an ing tl1 e existence of fo ur cigc nfunctious slations, imply ll llcier fo ur independeut tran ribution to the fu nctional i1J tegra l will be (Io/21r!i,)2. or %Oro eigenvalue. T heir cont 1 r' ues gives a fa ctor or T' = l 'T, w lte Furth n noP, integration ove r t li e zero mo 30 '-spacr? rnlurn e of tli e fo ur dirn<'nsioual spacc t iru r. If foll ows t'liat i ? is the t lirc?c t hr uurcnorrnal iz cl 1111 clrat ion rate per un i t ,?olurn e is gi n '11 h." [2] i_ = __!J_ O ti ,- l u/ l1I c lrt'[- 'v2 + U" (~)] - 1/ '2 l ( ,.)] . (2.Gl) I ' --!-,r'2 r,,2 ( det f- v7 2 + U" (+ )] tr for later discussion that tl1e expon ent i11 Eq. (2.G l) is really Oner agaiu , we no t? ssum ed that ! (<;)+ )= 0. of t,l re o rm -(l o - 1(?> )). However, " ?r !ran ? a ficul t to calculate the prcfactor in t lr c e xpressioJl for the It is notoriously dif decay rate. However, there arc cases wh ere it ca n be c:;:i lculated . One vacu um JJ [5] iJJ volving the calculation of t li e Sll ch case was thr work hy \ffteck and J'vTanto n rate of monopoles in a uackground !llag netic fi eld using an insta11- pair prod 11 ct io ith E uclidean ton approxim ation. T hey star ted wit l1 t li c S0(3) gauge theory w Lagrangian L = }_ [~ F 1' " F 11 + ~ (D1' ,iJ) (D A;) 11 + I '( )] . (2.G3) e2 4 a /1_,// 2 .., . /I 11'/' a 1, 2, 3. The where e is the gauge coupling constan t, ;;, , 1/ = l , 2, 3, 4, and at in Figure 2.2, except the minima are Potential fun ction has t he form of th 2 enerate. We make two assumptions: a weak external fi eld (g3 B / l\1 ? l , deg ated to t he gauge where y is the magnetic charge of the rnoJl opole a11cl is rel 2 1 ;:ik co11pliJlg (c ? 1). ext , coHsider a -o npling co11stant by = 41r/e), and wcc 9 0 11sta!l t magn etic fi eld B in some arbitrary dircc:tion . In these approximations, C senting a p in tlike the fi eld equations of the a bove tl, cory have a solution repre circle of radius R whose plane is perpen dicular to the 11lonopole traveling in a ion rate is given direct ion of the magnetic fi eld . They sh owed tliat tli c pair creat .JO _ (g B 7r J1!d .rl) r - (2 7r ))'2 . ( -3 ex P B + -1) i , (2 .Gg "i ass, all(/ we luwe set t,, = c = wh ere ll / R is the one-loop renormalized monopole m 1. 41 Chapter 3 Black Hole Nucleation Rates 3 -I Introduction cle pairs. In the last chapter, we studied the producti on rates of pa rticle-antiparti ly ly forbi dden bu t quan tum mechan ica l vVc s;:iw that such a process 1v as classic al c?o t ? l c 1e pro d uct1?0 n rates , we 11 t,?11 1?z e d t 1r E uc1 I? d ean pat 11 a llowrd ? 'TJ.,' o mpu e part1 tegral was integral formalism . fo the semiclassica l approximation, the path iH minated by stationary points of the E u clidean act ion. Assuming shown to be do n , the pair production rate per 11ni t rn lume was given by r / \' ~ only oue instanto d fi d t h e s?o 1? ? u tw? n to /\?e - lo/h rr, ? J.O evalu ate the exponential, one si mply necde to n t ion th lassical Euclidean equations of motio n and compute the corresponding ac e c 10 s computed from flu ctuations about t h e instanto11 and ? T hr prefactor X wa was given by Eq. (2.62) . -ant ipart icle Li this chaptrr, we will study the gravitatio nal analog of par ticle of bla ck holes Oll a background Pair product ion , namely the pair p roductioll 5 e. I t was reali zed by Gibbons [6 ] alld Ga rfinkle and Stromiuger [7] r>acetim fi eld was the th etic at the analog of monopole productio11 in a background magn nd magnetic fi eld . Prod uction of magnetically charged black holes in a backgrou 42 dtzality, one Ci:111 ca lculate the pair prod t1crion of ('1<-ct ri ca lh? d1argcd bl,1 ck l's i!lg roduction holes i11 a backgroun d electri c fi eld [37]. Iii t li c case of' u1 011 opole p t the end of Sect ion 2.5, the u11d erlvi ng quantum tb eory is Yang-JVI ill s- discussed a Higgs. The analogous charged black lwlcs are d escribed by EinsteiIJ-Maxwe11 c tromagnetic fi eld t li eor,v. In lJ ot li ?ases, pair creation is possible becaus e the ele rO\?icl es the necessa ry force to separat.c the rnJCl ea tC'd objerts. Black hole pair p productioIJ bas been studied in a variety of ot11 er c ontexts. For example, it was cosmological backgro u11d ; la ter shown that black holes can be pair created in a a posit ive cosmologi cal constant provides the fo rce necessary to separate the hla ck holes. Bousso [38]. and Mann and Ross [39] have st udied the cosmological Booth and Mann [40] prod uction of both neutral and charged black ho le pairs; st the cosmological prorluction of charged aucl rot at ing black hole pairs. udi ed Black hole pair ?rodu ctio11 has al. o beeu :-;t,1died iu more exotic contexts . For tion of black holes on example, Hawking and Ross [41] stt1died the pair prnduc ic string. Caldwell , et al. [42] have studied Llack hole pai r production in a cosm t he presence of a domain wall. bove examples, the black hole nucleation process is mediated In all of the a by a corresponding gravitatioHa1 instanton. Analogo us to the results of th e last clea tioIJ chaptr.r , the semiclassical rate per unit vol111ne f or some black hole nu Processes can be calculated using the expression r = A exp [-(Ii.,1, - li.,g)] ? (3 .1 ) re, h1i the bounce action , which is twice the action of the gravitat ional i11stan- He kground alone, ton mediating thr. nucleation process; hg is th<' ac:tioll of the bac a dimenstonful prefactor calculated from quan tum flu ctuations about aud A is the instantons. Typi cally, onlv the classical term o f Eq. (3. 1) is calcu lat r.d , since 43 orrections for 11 011 -tri ,?ial black hole rn1 clC'ation proC('SSC'S is calculating qt1amum c nd not well defined ..- \nother reason th at is oft en cit ed fo r neglect- complicated a nrgligible ing t he pr factor rl is t hat i ts con tribu t ion to thr rn1 clrat ion ratr is n the classical trrrn. compared th (' exponent ia l contributio n fror o thi ? author 's kno\\'ledge, only t\\'o ana lysrs of the prcfa.ctor have been T ation . One was U]J(l ertakcn by Gross , made in t l1 e context of black hole nu cle les in hot fl at Perry, and Yaffe [43] for the nucleation of Schw arzschild black ho li other was the recent effort b,v Volk ov and Wipf [-!4] for nucleation of space. T und. In each c1:1.?e. a partition fun ct ion gro a Nariai spacetimc in a de Sitter back lculated a nd found to p ossess oue ucg at ive rn oclr. indi cating instability of was ca was t li r case i11 the last chapter , the th r. system to black hole nucleation . A s he rtition function can be identifi ed with a gravitat ional patl1 in tegral. Unlike t pa the last cha pter , the gravitational pat h in tegral is au ill-defin ed quantity. case of r ti tio JJ fuJJ ctions to calculate nucleation T his the ?is does not fo cus on using pa at the one- loop leV<' l. fo stead, wr focu s 011 a different approach , ,.vhicl1 we rates to systems 110 describe. As we sa\\' in the last chap ter, Eq. (3. 1) was applied "" the nucleation process was described b_v a continuous history in imagin ary where lack hole nucleation processes enumera ted t ime, i.e., a bounce. In some of the b cont inuous his tory does not exist , i. e., t he manifold represe11ti11g the a bove, such a then is, how can one extend or modi fy instanton is disconn cted . The questi on One the "bounce" method of calculating nucl eation rates to include these cases? ouJJ dary" formu lation of quantum cosm ology. Briefl y such proposal is the "no-b st d , t he nucleation rate, mod ulo a dirne nsionfo l prefactor, is given by the ra- ate ce time from '?nothi11g" t io of t he proba bi li ty measure for creat ing a black hole spa e to "nothing" . The t O t he p robability for a nnihilating the backgro uIJd spacetirn pro babili ty measure for a spacetim r is giw 11 In? t l1 r Illod ul11s sq 11 a,1 ?pd of the \\ '];1ck hole 1111clcatiou 11 c? tg?1 1c t 1? c fw' c. _-;.s \H' \\?1 polltt 0 111. 1101 a l r-, procc?ss ,,?s 1 . b l I) Ya si ngle solu tion to thC' E ucl idca11 E i11 st ein 's eq uatiolls. ?' arc? c escn ec cosmology and thC' NBP ca 11 be \ Ve wil l t hen discuss li ow a t heory of qu ant 11rn used to calcu late pair production rates in tliesc cases. l . l'\otc that, in t hi s chapter, we set ti,= G = r? = 3 .2 Topology Change ogy of spacetirnC' can fi11 ct 11 atc was col l ccivecl b,v vVheeler T he id ea that, t he topol 47 measuring the [ ]. T he reasoning behind this idea w as as fo llows [48]: Consider ~ ature of ave?r age l t . fi ld . s?acetimc volume L i11 a vacu 11111 at a tem per e ec n c e Ill a bu- O I( . In general, one will not meas ure zero ele ctri c field . but a stat istical distri eaked aro und a value of 1/ L'2 . To sec? t hi s, Jct 6[ he th e flu ctuation t ion liighl y p cli is (()[)'1 L3 . From a11 application of the t h l ergy of wlii ill c electric fi eld , t he Cl llows. In a similar ma n- energy-time unccr taintv relat ion ' 6.E .6t > l the result fo . - ' ement of ner , consider the quan tization of the li nearized theory of gravity. Measur ? t? 1? d. . L 1.1 1 vacuum t he Rie rn erir 11n ens 1on mann tensor averaged over a space tun e o = l is the at O K will not yield zero , but a value of m agni t ude ).p/ L:1, where Ap length . To see this , consider flu ctna tio ns in the metri c of size 6_q ~ .6 P lanck io , the Lagrangian is of order (6./ L)2 , over t his s?acctime region. fo this rcg 11 flu ctuations to contribu te and hence , the act ion is of order (L.6 )2. For these n t ly to a Feynman sum , (L6)2 ~ l , implying 6. ~ >..p /L . T he natu re signifi ca om fl at11ess until one examines its of the vacuum docs not depart signifi cantly fr d gravity is behavior over regioll s of linear dimens io1J s L ~ ).p. Although linearize 11 0 of t he P lanck length , on<' can still argu e that flu ctuations longer valid at sea.Jes ace can "pin ch off" or in t li c curvature ca n become so violen t that portioJ1s of sp 47 bC'come rnul tip ly co1me ted. \tVe saw in the last chap ter that particle procl u ct io11 was a class ically forbidden <' ahl<' t o connect t he clas- but quantum mechanicallv allowed prncrss. \\'<' \\'C' r ic-a lly fo rbidd en region into whicl1 a syst<'lll c;rn flu ct 11 a te to classicc1 1ly a ll owed s instantons. Can 011e do the s;:i,rn c in quantu m regions by cont inuous paths, i.e. , g rav ity, say fo r black l101e pair creat ion '! A s mc11 tioned a bove, black hol e p air creation is a topology changing process. A n ini t ia l background spacetime con- tain i11 g no bl ack holes ernlves into a final s pacetirne containing two black hol es e topology of t l1 e spatial sections of immersed in the backgro und spacetime. Th t li e initial spacetirn e is different from the to pology of the spatial sections of the hna l spacet im e. T hus, we must find a way to connect these different topologies 1s. However. diffi culties arise when via a continu ous history of fi eld configurati o1 assical general relativity. Consider one consid er this problem in the context of cl S' with differc11 t topology. Geroc:11 [4 9] showed th at two spacelike surfaces S a11d independent of a ny fi eld equations , the Lo rentzi an s?acet ime that interpolates ave eitlier closed timelike cnrves or between the two spacelike surfaces must, h singulariti es. If one does assume Einstein 's equat ions (or simply the weak cn erg? nP), Tipler [50] showed that generic topology condition T?vn/Ln 11 > O for al] null changing spacetimes are s ingular. ents indicate three avenues of approach to the problem of The a bove statem topology clrnnge [5 1] : 1) G ive up the local euergy co nd ition and try to make erate metri cs srnsc of negative energy and causali tv vio lation, 2) accept degen nd "s trength" of tl1 e singul arities rnqu ired by topology and determine the uature a of change, or 3) give up Loren tzian metrics and cousider a Euclicl eau forrn ulatioJJ topology change. 48 about opt iou 1. Howen- r. lr1 11 s br iefh? consid('r \ \'r \\'ill haw, notl1i11g to sav tv theo1T. OII C restricts attc' IJ - op tion 2. In the context of classica l gene ra l reJat i,?i ch are non-degenerate everywhere 011 t l1 e spacetim c manifo ld . ti on to metrics whi ivity can lw derived from som e qu an- If one ass um es that classical general r elat he quantum um theory of gravit~', t hen thi s res t ri ct ion should he derivable from t t th rnc a fun ctional integral formu lat ion of qu a11tulll gravity; ,n, eory. Let us ass u T he e a m ore detailed account of th is formu lat ion later in the cha pter. wil l giv J eiS(c,w), where V e is a measure over as V eV w fu1J c t ional in tegral ma,v be written tzia n action ll tetracls, D w is a measme over a ll connections. and S is the Loren a ivity. O11e may try to exclude dege n- of t he first order formu lation of ge neral relat nerate m etri cs erate m etrics if one can show that they are far away from non-dege iddings [52] has shown that this is 11 ot th e case. He in fie ld space. However , G re the space of tetn1ds can be ob tained from t l1 c notes tl1at a "nat m a l" m easu 0 11 norm 4 (3.3) //oe// 2 = I oe;~oet e~e; 1Ja.b'T/cd(det e)d :E. vanishes as ei s- -t 0, ensuring that a degeuerate te trad is a finite di T his norm nother ,vay one cal! try to exclude tance away from a non-degenera te one. A only in- enerate metrics is by restri cting the fun ctional in tegra l of gravi ty to deg been slwwn bv Horowitz [5 1] that eve ry clude invert ible tetrads. However, it has equations is gauge equivalen t to a solu tion non-degenerate solu t ion to E instein 's fun ctional at is degenerate somewh ere, impl ying that this restri ction on the th ant . T he conclusion is that degen erate metrics a re in tegra l is not gauge invari ioll to ant , a ucl t hat if one expands the class of metri cs under considerat i111port auifold , one can show that tliose wh ich a re degenerate at som e point(s) on t l1 e m a1 relat ivi ty that de- therC' exist smooth solutions to t he first order form of geHer 49 [5l ]. Since topology cl1a11gc can occur clas sically in this scribe? topology chaIJg<' ll_v. by the corresp011- context , one would expect it to o ccur qua11tu1u lll ecl1auica clcnce ? .? l . ument ass umes uHiquc cvolu tioll of i11i t ial data. but ? pnnc1p e. This arg it has th l1crmore, is docs no t hold wli en metrics b ecom e dcgcncratC' {j l]. Fmt ? opo log_,. lweu s110\ , t l . ? ? ases , whell sca lar fl elcls arc propagat,ed in t 'n 1at 111 ccrtarn c ing backgro11ncls. iIJfini te particle prncluctio11 res ul ts [--18]. chang the context \t\le wi ll have no more to say abo ut classical topology change in n a lternative of th l relativity. fo stcad , we will foc us oe first order form of genera 3 re possible we ana1-v tica11v conti nu e the Lorentzian manifold to a ? Herc , whe ' . eed not become singula r a.t points e metric n Euc lidean manifold . In this case, t h 1?d f' orrna 1i?s m t urns where s pa? .n t? 1 ? E uc 1 ea 1a sect io ns undergo topology chang e. T l1 e ]. a topology changing process into a t unn eling problem [48 3 .3 Euclidean Quantum Gravity antum theory of gravity will allow for t he Having adopted the idea that a qu ory of of topology change, what formula tion of the as yet incom plete the process nonical um gravity will we adopt? There a.rr. two main formulations: the ca quant d the path integral formalism. Let us first briefly discuss the c:anon- forrnalism an , one introduces a family of spaceli ke surfaces ical approach [53 , 54, 55, 56]. Here - and canonical equal-time commut a and uses them to construct a Ha miltoni an wo obvious difficulti es arise. First , to form such a Hamiltonia.11 , tion relat ions. T , in classical general relativity, tl1 ere one must speci(y a time variable. However ke surfaces. Iii 18 f spacel i no preferred time variable and he nce no preferred set o th spacetimes, one can choose time as measured m e case of asymptotically flat onia11 quantum mechani cs th is asymptotic region. If one wishes to form a Harnilt 50 clear choice or I irn C' niri,il>lc. a lt hough 1rn rnY or a cl osed c-os in ology, t lier<' is no IkgardlC'ss of tl1 e cl1oicC' of tim e i38]). s iiggest ioJJ s haw lwcJJ put fo rth (c .f. [5 7. c? dC'pell dent , implying that th ere' riable. t l1e resultin g Hamilton ian may hr ti111 va ? thcorv . . 1 sc?co11 c l ci ?ffi 1 cu Jty stf'ms from il l llot ex1?st ? ground state for tlw --1. w ? a llluqu C' n ensiou. This th t of spacetirnc in to three spati al clirne11 sions and 011r time dir r spli 3 orne three-manifold. u?st 1 h<'rr l\[ is s ricts the topology of' spacetim r to R x J\[l. w of spacet irne can flu ctuate, 0 11 c would opologv In ght of the suggestion tha t thr t li gies. hopr to br able to COllsider sp acetirn es of. all topolo . the path integral forrnulatioll of quantum gravity (c .f. [3 1]) Hence, we t urn to for a spacrtimc to f'volvc fro m an he ampli tude Oll c approach is to represent t elds 2, S2 / .r./l , 1, S1) = L J) / V [g , ?]eiSf.g,,p,1'!J, (3.4) 1/(l\ I\ I mallifolcl l\ J, V [.q, ] is a measur e on for t]ie where 1/(J\1) is a we ighting fact or [9 , etri es Iike surface. all snch four-g<'om cliffrrent four-gcomrtrirs. ./\?c ar this span' Hence. plc, bC' rxprcssecl ill tcrrn. of G a 11ss ia11 11 or111 al comdi11 atcs. can, for rxam 1 i e po, siblf' fom-metr ic is t li e tli rrc- rnetri c li, i11 tri11sic to the li e, onlv freedom in t l 1 n sp ('cifies the four- pacc?likc s11rfacc. Speci(vi ng th is three-metri c is how orJC' t lie s l1us, a niorc specific for11111l at ion of Eq. (3 .. J) is gcomrtry on a spacclikc surfac e. T s_vs t?crn to go frorn tl1 r initial configurat ion as fo ll ows: t he total a mplitud e for a (1 i:J' d.' ) ,? 'P to the final configuration (h;J. ?") is givc ll h 1 9 ' ,J\f ) . (3.5) (1i;j , .IJ. rp] c sf ,ff ?/', J\1" / h; , q/, J\1') = L 1/(1\J) / V [1 J\f . anics and quan- logous to our discuss ion of the path integral of qu antum mecli Ana fun ctioJJ of a quantum gravita tiomd sys tem can be t um fie ld theor,v, tlH' wave h in tegral, as ""e will more folly discuss in the context of the identified with a pat no-boundary proposal. 3.3.I The Action n. For a fixed spacetirne rnanifo lu J\ J, we take t he We must now specify the actio Lorentzian act ion to he (3.G) mological constant , Lm is tlic where R is the Ricci scal,u curva ture, A is tli e co aries ian, and is the determiuaut o f the fom-111 etric !/?11 ? If one v inattcr Lagrang 9 h riation of the metri c and the v ariation of the normal t c action such that t he va y of tli e mani fold 81\1, the11 the derivative' of the metri c vanish on the boundar S2 action is 8 t a t i? onary if and 0111_,. if ? (3.7) rr . . i , wli erc th<' st1?e s r? I -energy tensor o t JC mat ter fi e l cls , .1 1111 , JS g1v<'11 uy ? ? ) yiw = _I_ rYLrn (3.8 2H r5,r;, //I ? ? ? t? 1 es tl1? t l I ' van?a t1?0 J1 o1 ? t J1 c 11J ctn? c a!IC I no t, t,i H ' van at10n o t 1P ff one sti J)tdat? , , c1 Ol! y t I< ' bou11d arv, then va riation o f tl1 c nonua] derivatives of the m etri c vanislJ cH1 th< . . er varn . s 11c?s , as 1t ?hould f .o r action in Ec1 (3? ?G ) w .i ? t 11 rrspect to tlJ <-" 111ctn c 11 0 lon g tegral an xtrcrr ? ? ll a1 to the in ium; van at1011 of Eq. (3.G) will yield a term proportio ? th i c Oll the boulldary. If wr of c normal derivative of the variatioll of tJJ c metr (3.7), n Lo bP stationary for solt1 tio ns of Einstein ?s equations want this nu-iatio l ? ? rm to the act1011 of ? E, q. (3.G). T he res ul t ing 1 hr n we? m11st a, cl c au appropnau? te actiou is 2 C, , ?] = / \/-g rl1.T [fl - 1\ + Lm(r; , ?)] + ~ / ft,, rl1.1; A' +S[g ,1, 161r 871 .lr')/11 l (3.0) . c l\ i .i OJJ the boundary 8!11, and C Where F ?\ is t 11 e trace of the ex trinsic curva t m metn?c I7 , an d not on t h c Wt Ju cs rT JS a, term 111 ?c 1 he bo1 mda w 1 cI epend. only 011 t lly fl at. metrics. or terior poin ts of the manifold . In t li <' case of asymptotica .0 ) ?' cl If' erencC' ba S ,: [59 t io11 ?, ]. \\'e Cr:1.11 tl 1 1 cJ e fi sical ac e1 nc' a qua11 tity known as tli e phy (3.11) l ? ) J1 c phys1?c ' fil J ? (3.9 . T a 1 act i?o n 1? s fi nr ? te for a er whcrcS[g t? 'A+,' ] i ,s 1c action 111 Eq. ini ty, if wf' fix a boun dary ....., oo ne ar inf configuration (g. ?>) under the condition t li rit . the, fie lds ( ) ? 1I I CC' t li e samC' fJe lcls on t l1 c IJ 0 1111 dary as (r;o. Ill < fi I Is , 1w act1?0 n S'[ C' c t g. ",p] w1?1 1 Iw tn For rc>?'1 ! Lor?c' n t -zi? an rnetn ?c s and real 111 a t gral will , in real? te J . . oscillate' and the pa.th in ' lCnce the rn tegra11 d in Eq . (3.5) will deal wi?t i 1 ? 1 n 1I t 1u ?s ci 1?f f cu ty 111 t 1e genf'ra1 uot c one ca? ,onvC'rgc. we have> seen that nalytically continuing f quantum mechanics and ci uantum fi eld theory hy a C"lsc o l . J . . . . . .pplv t us aua ytic contrnu at 10n reaJ time to . . . unagrn ary time. The idea, then, is to a tion of the geometry n ly to the act ion of the rC'al matter fi elds bu t to t !Je ac not o to S2, both of 31]. Consider the evoJ11 tion fro m a spacelike surface S1 ;-1.s weJJ [ l t d fi ned e interva e at Which a rc asymptot ically fl at and are separatrd uy a tim r? 1 1 engt 1 infini ty A rli. . . aud S by a. tirnelike tube o rn S 2 ? ? s noted ea er, one c an Jo 1 al to imaginary t ime rge rad ius. O ne t lieu v\ ic k rotates this tim e interv t at la ? ? d efi n ? t r e. via t ~ ? duced metri c the timelike t ube becomes pos1t1ve- 17 ? T he in 0 11 e C'-dcfini te metrics !J whi ch ill duc Bence', the path in tegral w ould be over positiv H { T l uri?c 1 ( t ? f ary !:ound u 1,1, ? 1e v, ro t, a ,,on ? th e g?1?,,e 1 the b1 posi? t1? vc-defiui tc? metric h 0 11 c actio11 to IJeconw -i/g; he nce, ch appears in tl1 ti n1 e causes the facto r ~ whi 1? l t' 1. T l ie pa t l1 ' t' rms as iS ~ - I , wh ere J i s the E uc 1c ean ac ,101 th <'l c ion t ransfo int0g ral of Eq . (.3.5) ii o w becornC's 1 , ( 1i;~ ? ", U"/; ,;i , ?' . AI') = L 11(11! ) / V{.r;. rb]c- f., ; ,1,? .,\J ) (3 .12) ? Al ! j ( }. f. ) : . I [.r;. 9] = _ l -R - 2-.\ + m ] - - I 1? - L V" r1 .1: \ - \ 0fl .//1/ [ LG1r 81r . a1i1 (.3. 13) gral approacli to quan-ea n path i11 te Equations (.3. 12) a11d (3. 13) specif\? a E uclid tum gravi ty. tion 3 3 ss of the Gravitational Ac . .2 Indefinitene ill not. in gen0rnl. lead to a convergent path alytic con tinuation of t he action w An it ive- s is tha t t l1 e action of Eq. (.3.13) is not p os in tegra l. O ll e rca. Oil for thi ?r o pos1?t ? cJ cf i ? ? r r rn te rnetn cs. J.O see that 1n'- de?f in ite , Pven I one r st ricts atten tioll t ? t? 1 ? tran?J _v negative. cons1 ?c 1 er? tl1 c con orrn a the g-rav1?t a t?1 0naJ act1 ? 01I ca ll be made arb1 form al factor D 2 is a 2g /, where the con transformation of th e m etric Y,w = D 111 cf' transformation of the R icci scala r and t he tra conformal Positive fu nction . T he _v {60} t h x t rins ic curvature, i11 n-c limensions, arf' given b of e e R = n-2 [fl - 2(n - I )_q'll/ 'v/V,/ Ju n 'vv lu DJ , (3.1 4) -(11 - 2)(n - l)_r/'v('v,, Ju D ) (3. 15) 2 J< = n- 1p. . ? + (n - l )D D,1,n'' ? In fo 111 ? c1 ? . 1men s10 11s, (3.16) R (3.17) 55 I l D 11 {? d we ass 11 me th at wlic?r0 n'' is tlzc, 11111?t ou t, w;u 110rma ] to t l H' )01111c a1T 1 . I t lwre are . ? no matt er fir lds. the action brco111 <'S 1 I[!]] = - ;1r i, 1 U? + G_r;1 1fii c1 .1: R " V1,n-v,1D - 2.,n? ) 1 (3 .18) - ~ / J;; l 1.T ff (k - l\"0 ). 81r lo"' . . It is clear ti t J[ -] . . nade arbnranl y 11 <'gat'1vr by d10osmg a large a11cl/o r 1a g can be r ra l of Eq. (3. 12) may rapidly vary ing conformal facto r D. Hence, the path ill teg not converge. s problem has becll pu t fort. Ji by Gibbons, H,nd ;: iHg A Propo ?aJ to deal with thi li e11 .'\ = 0 a11 cl thr spacetime alld Perry [61]. will first consid er the casr wW0 811!. Let 9 he the spr1c<' of all rn Ptri cs _r; 011 /I f whi ch induce Possess a boundary the .? 1. The metrics in 9 can be partitio ned !1 given metri c h 0 11 the boundary 8 . c tor in to eq ? ,. . t ions. T he conformal fa un efo rc Ee (3 2 ) . . . d urnquely cl r frn ed J. ? 0 is CO !ls1dcred a?propna tel_v an f (2) . . r o n scliemc, (3) a measur e, and ( 4) a contou fix illg t erm s, a regul an zat1o on will be most in tegration. F purposes, sp<'ci(ving the conto11r of i11tegrati or om 1111 Por tant . 3. 5 plex Contours and th e Semiclassical Com Approx imation . f' . . 1 " . ave uuct10n 111 t 1e case . . tor the "' There is d fi . . m te prescnp t1011 tor th e contour no e ed prol>Jem has been prop os of los I t to address this ec space times. One a ttemp in restri ctions tha t ?l1 0uld be They suggest certa by Halliwell and Hartle [G3]. : tion gives sensible phy sical fun c 1111 sed on a contour so tl i at the resulting wavc Po , although not all of t l 1 cm s. \t\ e will enum erate these res trictions 11 crc Predict ion Will be, 0 f central interest to 11s. 59 path illtcgraJ c-onn?rg<'s. \\'e se11 s11cli tJ1at the (l ) The? contonr sho11lcl b<' clio for rea l met ri cs. Thus. n? ha,?c sce ,1 tl . t 1 . unbo111J ded bclC'c-aus a ,.ria e contour of rn . to c-011\'r rgr. the mffeflln ct ion wo11ld rcsulti11g 1?1 1 t . ..egra 1 wo1rld dn?ergr. If it were 11 sistent wi tli rcst ri c- not i wo11ld be i11 c o ? n grnera l. lw osci ll a to ry a11d . hrnce. (3) l on of classical spacr tirnc whe11 the universe is tio11 0 J\\? , ll a!llclv the? predictie < ? bl f ? ? contour o rntegrat10n Slllta c large. Thus ? t o cl e fi ne a convergent path i ntegral. a he methods used in slio l J b lex. This idea is not to o far rc1110vPd from t ? u c c comp . are mptof .. 11 . uss ed. complex contours as.v i sc ica .Y fl at spacctun e, w here as w<' lian? d . . 11 <'Pded to ? . ? , . . . d rt Ive d t cl c:011,?crgcnt w ave fu11ct1 0 11. tour of integration shou l d 2 e wavefun ction generate d from tJ1e chose!l con ( ) T li . F 1 . . . a. t 1c01v . ?at isfv the c t .. . . . . . plcrnentrng d1ffcornorpJ 11 sm Jiffanance. or 0 11 ? s urnts im . . , . . the \V!JeP.l er-De\V1tt equ ation described l , E? . ? . trarnts arc J_\ rnstel!I ~ action, thes e cons and the momentum constrain ts. erse; to will define a possi bJe \\ ?m?cfun ction of the uuiv Th<' a bo,?0 two criteria . . . or pbys1caJ pred1 cl1011, o uc' ca n Pi 'k h . i ch arc sensibl<' fout t ose wavefun ct ions wh 1 . . 0lpos, tl1 f' . e oJJ owrng restn ct10ns: acetime OJJ familiar scale s wheu 3 classical sp ( ) The wavefun ction sh ould imply J f ? ?ave mic- ? s the ca e i quantum co smology wlien t 1e w the univ 11 erse 1s large. S11 cli i approximation. Such tio11 ? l described by a certai u type of semiclassica l is wel l ? t ? approx1rna 1011 scmicJa ? ? J r.o xi. . . \,V]icn . in a stPepcst c es cent a ss1ca app mation ar i ses . J I t are comp ex auc repres en to t]1p , . f' . Nave unctwu, the dom inating saddle porn ts - a11 aJvtic? 11 , . - a .) coutrnu0cl clr1ss ical spacetirn<'s. timcs) co,1- 4 s of the fuuct ioual illt,eg ral (i.e. classical space ( ) At saddle point ? ? . vered rn1 es Ill those spacet Vcnr d be r('c o 1 0 na! quantum fi eld tli cory for

0. Tliis ir ? ia t IC' domi11aLi of rest ri ct ion (' 1) arr rposes, restriction (3) am! crrtain aspects For our pu . )Ort ? t? 1 . . grcatrr detail. Fir st, cousider in ost iniJ th('rn 1 11 an , 1ence. we mll dis cuss n ac tion fPst ri ct io11 (3) 1 . . unes saddle points to IJ <' c-ornplrx. The Euclidea w nc-h rcq . f such sol t. . . plex and can b<' wn ttc n as o 11 10 11s is also com . ?, 1ll] - 111 J, .'11 ] = I ,'.tc[li. . A. DA I ; _l; . ~' .11] ? iS [ Ii. \ . DM J; 11 [h, \ , 8.U; _l; , (3.21) . . . . lcpclldrncc 0 11 the boun dary c Here. the a t' 1 . . s of ,ts c ion 1as becu wntten rn term . . . , fJ 1/) izing solution ([; . r/>, ill ). The subscnpt n 1s data ( h ' /\_? aud on the ex trcm . such saddle pornt n1 ereh? a l? J" . . _ _ hat there could be man y - a 1) e rng index, rnd, catmg t Hamilton-Jacobi equati on of the form actions. Tl1 e action will satis(y a Eu clidean (3.22) _ _ . nce the real aud imagi- Wh ere, 'v i . .. . . t ro h;_; - Si s a ,,tnat ,onaJ dcrm1-,t1 ve with respec ? ? ? S d ot f J ?1 ? ral. scpar::1trl_v satisfy th i?s eq11at1011 , " OPS n nary Parts 0 ? ? n "'' 1 not , 1 11 gene . . fi elds. However consid er . r defi ne an b . an I m atte ensem le of classical g eometnes . the real . cl . . .? . twn: an 11nag111ary parts of the above equa 0. (3.2 3) ? the S w1?1 1 IJe d , e11 t ot ? JHn e , n n If thc g?radient o f' S a 1 11 is mucl1 larger than tl1c gr I 1 . n t 11s l - . the Lorentzian Hamil ton-Jacobi equation. an ap )! Oximate solution to s a e of classical geometrie s aIJd matter fi eld s. A bl c:ai:;e , it docs defin e an ensem l1 e n11?c Ja ss1? ? t?o to t in the se ca J approx1ma 1 n , ?onsequen 1nc- becomes important ce, n Wavefun ct? . 1 wl11ch can be written a s 011. , X- fJ11J] ~ L 1/(A I ) L (3.24) .;i,,,,- ,,',1??r;s,, ' \J.i 0[h /If 71 61 and thC' pr<'fa ctor ~ " dr.pcncl 0 11 the hou11 da1T n-li ere it is un de rstood tha t J,'. 1_1?? S,,. t he rea l pc1 tt of the Euc1icl ca 11 cl ,:lta aucl cxtrcmi zi11g condit-i ons. As 0 1H' can sec, ant cont ribntion to tli r \\?m?r- c1ction is exponentiated and m ay be the mos t signifi c fu11 ctio11 . er n to illdicatc that tl1crr rna,? ' more than 0 11<' sadd lr poin t . IH \Vr a lso sum ov r 011 e or tffo saddle ough sli o11 Id note tha t it i s only appropriate t o sum ove a l th Oil (-' fi nd complex ts of comparable act ion [GG }. He nce, onr must have a way to poin e validity of the semid assicaJ thermore, th solu t ions to Einstein 's r.qua t ions. Fur contom . It is t_vpi caJly thoug ht t l1 ;.:1t he i11 tegrat io11 Approximation depends on t nt? wl10sr. act ion has tbe t hr pa.th integral is always dom inated bv U1r sadd le poi one can distor t the co11 tom in to a ast real part . Thi s is not tl i e case 1rnless le rn [45}. st escent contour along \\?hich tl iis ac tion is a global rn aximu eepes t d 3.6 Minisuperspace Mod els l quantum mechanical path in tegral by cval- One can attem pt to simplify the ful ls. How<'ver , the sense in whi ch pace mode ua ting it i11 t he context of m inisupers erties of the full th mpli fi ed path in tegral can be hoped to refl ect all of the prop is si , sucl1 an opeJJ question. T hey do sh are some pro l>lems, however Path integral is erspace of the contour of integration . In this seJJse, the minisup as specificat ion is d a remt in which to study suc h issues. T his Path i11 t cgral prnvides a simp lifie e preseJJ tat ioJJ giveJJ in [45]. wha t wr hope to do Ji ere, wli ere we cJosrlv follow th cc models ffi t l1 metri c of thr fo rm We will be interes ted in mini superspa (3. 25) o be homogeneous and the th ree- fo r which t he laps . function N ( 7 ) is res tricted t G2 lllctric c-ompo 11cnts h;.1 a nd matter fi eld components are clcscribed IA a fi11ite number of fu nctions r/1 (,) , 0 , = 1, 2, .. . , n. Iu tcnns of t lt c'se fun ctions . the action of Eq. (3.1.3) ca 11 be wri tten as 1 NJ = T" ( -1. d(/ dqfj ) I[q~(,) , l d,N f a/3 _? __,_ + U(q) (3.26) r' 2N2 ciT rlT where fn tj is a met ri c 0 11 minisuperspace of indefinite signature (-. +, +, +, .. . ). \ \ .<' 11 0 w t urn to coustrnction of the mi11isuperspa.ce pat h integral between fix ed initia l a.nd final {q ~} fo r a s_vstern cl<'scribccl bv t li c action (3.2G). For convcui eucc , t he gauge iV = O is chosen. T he form of t his path integral was rigorously deri ved by Halliwell [67]; it is a simplifi ed version of Eq . (3 .12) aud is given by (3.27) Sub_j cct to t he boundary conclitious II Cl 11 c/'(1)= q. (3.28) I t should be noted that the path integral can also be specifi ed for fixed ini- ti a l JJ a,nd fix ed fin a l q. Vve will consider both forms in the sp ecific contex t of t he de Sitter min isuperspace model below, bu t for now we restrict attention to 0 Eq. (3.27). Saddle points of the action (3.26) arc coufigura tious (q (,), N) for which c5 ] ??ex 1 au (j r 0 -(} ? , - r(3- (3 .29) (5(jc, = 0 = N2 + N2 fJ, act (3.32) s the time-depernJe11t Euclidean Here, 1 ie . TliC' I" arC' actions of tJ, c c % JJi zC'd on'r (/1 ne 11 as no" ? rxt rm a11d N. t li e' rn ns t rai11t (3.30). sin ce ' o erate the sterpest descent contours for ,rhi ch the T he next s tep is to enum oin ts dominate along eacl1 conto11r. t to sec whicl1 sadd le' p Pa li integral converges r form r?- '" and wiJl be sim ilar Eq uat ion (3 ..3 5) will then b e a su m of tcrllls of th of contours which link sadc l]r points. . (.3.2-1). \Vlrnt we' now h,f fc is a se t? t?o Eq be di stor ted in to one n then decide whrtli er or not a giv<'H contour call \re, ca ? I l nd thus whet 1cr t 'lat sadd le ? t , a whi ch 1?5 ? cl om m? a ted by a parti cul ar saclcl]{, po111 1 for a given contom. point contributes to Lhr sc 111i classical apprnxi111r1tio1 which dominate the pat11 in tegral will generally not T hr complex saddle points E 1?d t . J r rcaI L oren zian haw 1.nv 1. mme d. tc m? ? a ? uc I ea11 o ia tcrprPtatiou in term s of n~ c . stricted cl.-1ss of so111 tious which may be s a ccrtai 11 re inrtri cs. However, there i nd ma ] Lorentzian so lution s [63]. ea l E ucliclcau a regarded as a co111bi11ation of r 0 x solu t ion q (r) with a pose the action of ECJ. (.3.26 ) is eva l1ia ted a.ta colllple Sup oluti on Note that N and 7 a lwa_vs occur as Nr i11 both the s complex va lue of N. on variable 0 ction . One can then in tro duce a complex integrati q (r ) and the a alent to the complex integ ral T === N(r - r '). Equation (3.26) is then eq uiv /7' [I dq" dr/1 ] (3.36) J = Jo dT 2./~.;1 dT r!T + U((J) ' T l 1 ? ai?g ht lr?n c f? f\T( " - ' ) ie co11i:our cr?e t i T our the str row O to T- -- T T ? wli 1e cont 1s the real axis from O to Re(T) and e running along niay be distorted into on long the first allel to t he imaginary axis up to T. The integrand a from there par au four-m etri c. 0 11 the ion is purely real, corresp onding to a real Euclidc sec t imaginary resu lt t ion, und er suitable condi t ioJJs, one obtains a purely second sec tri c. corresponding to a real Lo rentzian me ler the fo llowi ng [6 3] . To n11d erstazid the distortioll o f the ?on tour better, coHsic 65 T <';-1l. Oil th<' secollcl S<'ctioll , 011 c can wri te L<'t j = i1 iT'J.. whrrC' 71 a!ld 1 n rr r = iT t , t l1e c1ctioll will be pmel_,. dT d T = + , sin cr 2 T1 ilT2. wli 'H' 0 S t S 1. T hen s 11ding fo ur-metric wi ll be r eal Lorentz ian. if r/' (T) i o lln agin arv. and the corrrsp n?al. E l o (T) I < (J in f about the poi n t t = 0. oht.ai 11illg XJM I (3.3 7) es the second-order differe ntial O. Since q0 0 satisfi ext, suppose that q (T1) = fo llows tliat all the other uation (3.29), whi h is u11 changed b_v T -t -T. it eq ish at T a11d r/' (T) is t11 en real. T he in tcgra11 d i11 odd derivatives of q 0 also van 1, the co11 to 11r will t11 rreforc be re al, and Eq. (,3.36) along t lic scco 11 d sectio11 of tll(' 11 under which the complex solu tion rv. The collcli t. io actiou ?..vill be purely imagi na cs is, combill ation of' real Buclid ca11 all(J Lorentziau met ri in ay be regarded as a on is just the th (Ti) = 0. In terms of th e Uuee-rnc tri c, this condi ti ereforr, rjo: T T- vanishing of t} 1e ex t,n..n s1.c curvature at = ,. . Vv'c are joining together purely other wa_,? We can see the above resu lt in an t ? ?11 11) e ? uch?d ca11 so Ju r ?, 1? 011 s?. T l1 c resu l t?m g me ,n c w1 Lorclltz1? ?t1J so 1u t?i ons with purely Ec iou for such a solu tion will be finite, makillg real and discontinuous, hu t the act in tegral. The actions will be finit e th table candidates for the path em accep across the e-metric hi.i and the extriu sic curvature A";.7 match Provided the thre audard od of t11e surface cau be wri tten in st Surfacc. The metri c in the ll eighborho iding surface . We th en dcf inr. 3 + 1 decomposition with i cating tlir div 7 = o ind the 0,x t'1 ??1? us1?c curvature as -N ? ) (3.3 8) . = -1 ( -Dh-,:.i + Di," (t J ) ) l\. .1 2N OT ziau fo r botl1 Euclid eau a11d Lo rcnt wh 're D; is the derivative 011 t.hr three-surface, 66 Sj)clC'('t illl(:'S? ? 1~] 1us, I , ?' J ?1 s purelv real f?o r spae<' 11?1 , c s11 r 1?a crs 1. n E uc1 1? cI can spacr- t i lll <'S? c?r n d pur?r .1~ ? 1? magm? 1u-Y ?m Lorf'ntz1?a n spacct 1? mcs . .- -\1. rf'a l E uc 11? cl r;-111 a11cl rea l Lorrntz iau met ric ?a n bt? matclwd across a spacclikr surface onh? " ?hen (3.39) 3 .5 .1 The de Sitter Minisuperspace l\1odel ow t hat 1vve have a somewhat. firmer fo und atiou for the rninisuperspace path in tr.gral and its serniclassical approximation , we can a,pplv it to the de Sitter miniSupcrspace model, a metric fo r which is (3.40) \Vr will actua llv c nsid<-> r the cousequences of two different forms of the path integral. T he path integral discussed in the previous sectiou is that for which initial and final q are fixed . One can also fonnulate a path integral for which ini tial JJ and fi nal q are fi txed. We wi ll discuss the form er firS . Fixed Initial q T hr actio appropriate for 1 t1 h is form of the path integra l is given by (3.41) where >, = 2A/91r is a resca led cosrnological constant . The field equation and coustr?c,. i? n t , arc, respectively 51 (:3.42) _ -q + 2>- === o , JV"2 ?2 + (3.43) ..J._ /\(j - 1 === 0 . 5N - 4J\ T'2 67 The ?s o l 11 ,L ?I O !I to t 11 c' f.i eld equat ion s11hject t o the boulldary c-011di t io11s (3 .2 ) is ln ?s rrt ?m g t i1 1?s so l11t1.o n in to the act ion (3. -Jl ). " "<' haw - >,."2y:i 1, + [->.-.(r'/-' +- -r/-) - -ll (..r/' > 1. For the first case. ,vc have fo ur saddle points. Two are locat cl at real, po ?it.i\"e values off. (3.46) Wi th negat1. v e action 1 = - ~ 0 [1 ? (1 - >..r 1 ( / 3'/ . 4T) "2 ] 3).. Two othe 1? , are Io cated at real, negative values o t? 1, , (3.4S) Wit h I) Os1.t.1 vc actioll (3.49) For th e case )..q" > l , wP also have four saddle poi11ts. Two are located at T- = ~1 [l ? .( >-q 11 - l ) I /2] (3.50) i , GS wi I It ac- 1i on (3. 51) The? orhe 1? t, wo sa d d ie points a re located at (3.52) wi th action (3.53) J 0 = 3,l, \ [1 =i= /. ( ' l) v2]/l ("f - . It should be noted that for each saddle poin t located at Re(T) > 0, there . these saddle existed ?m ot]1 er sau,.J d le poin t at -T -. T he metric is icl e11ti cal for r bu t they corresp nd to different signs of ,/ii through N . poin ts. ? IIOtt'd c?trl i?c , r?, we arc restri cted Lo Ik( jg) > O; tlrns we rest ri ?c t attention to r R - ore, we have assumed thn t oue is in terestrd onl_v in co rnpJcx c(T ) > 0. Furtherm - saddle' ? Hence, thi s furth er restricts us to tJ1e saddle por.n t located at T porn ts. given in E q. (3 ?5 0) wi. th act.ion given ill Eq. (3.5 1). I pPci t?y t h c \Ve Inv bt ? e cl the action for thi s saddlr pom? must a so s c e o arn t. One ? ll [G3] describe tli e cou tour w J1 1. cJ 1 passes t J1 roug l I t 11 c contou? r ? H?a 11rwe and Hartle Eq. (3.50). \ ,Vp will bricfl _v discuss their result . Given thP value sadd le poin t of of T in Eq. (3.50). th e so lutiou q(T) is given b:v (3 .54) tor t t l1 e One can sec tha t q(T ) = o a t T = l/ A, i. e., at T = R.r(T) . One can dis contour of integration for T to one which nms p arallel to thP real axis from 0 I 1 1 1 e o r? to 1/ ,\ , a? d t? ? ? t 1e comp ex va 1 n r om t?h ere, parallel to t !J C' irnagrnar_Y ;-ix1s t o l C'al T gi,' en m? s t w purel:v r Eq. (3.50). The first segrneu t of t l1 C' cont.o m _vield 69 ;-tct ion - 1/3 >.. w hich is th e correct resul t fo r ha lf a E ucli.)(>.q" - J)3/ 2 _ T his is t he correct act ion fo r a Lorentz ian dr Sit ter spa.cctime. Fixed Initial p In t his se-ct?i on , we s J1 a 1I use t he f?o rm of? t he ?at li ?m tegra l stn? te d f.o r? fi x?e d 1.1 11. ti. a l JJ and fixed fin a l q. G(r/' Iv') = ./ (3.55) dN V(/ DJJ (' - i[q (r),p(r).N] . fixed iui t ial p and Herc. j is t he Hamil tonian fo rm of the E uclidean act ion fo r fi xed fi na l q, - TII (3.56) I = I I 1 dr( p.q + 1) . 2 l ? T he c,l a ssi.c..a 1 fi eld equat1. 0ns are then ,\ (3 .58) p = N -2 (3.59) (j - -4Np . 7 A 7s in the last section, we will make the coordinate cbang..1/' < 1. a nd ,\r/' > I . I1 1 the fonnrr case, there are t\\'O sadd le poin ts wh ich occ ur at (3.69) at w!ti ?h t lw action is givc?n by (3.70) For tl1 e ca e >..r/' > l , t\\'o sad dl e poi!lt s oc-c11r ;1 1 - 1 [ T = ~ - 2/J' i ( ).If'' - I ) 1 '.! (3. 71 ) where (3. 72) I--IaYing extremized over JJ , rJ , a11cl T. t li C'sc s;iddlc point s are now solu tious to the fi eld equatious and t he constraint . 0 11 <' 11111st now find steepest descent contours for whicl1 Irn (io) is a constant . :\s slimrn iu [65], there arc rnrious co ntours which pass through rnr ious sndcllc poi11 Ls. each of which leads to a par ticula r asymptotic form of the pat h intcg rnl. 1, 1 !1upe of fixin g a coutour , one can impose t he n -bouudary proposal. Hm\?e \?c' r. as shown in [65], t his will not uniquely specify a contour , ;-u1d hence will not sp<'('ih? ,1 unique wavefunction . We will briefly discu. s t hese difficulties in t lH' ll t'X I st?< ?1i o11. 3.6.2 Boundary Conditions and Uniqueness of Contours As stated a bove , both boundar:v condi t ions ancl a rn11t (ll!r of integration must he specifi ed to uniquel.v determine the wwcf'1111 ct in11 . L<'t ns first discus ? our results 72 for tlic? hw1.. . . < lllllial q forlllalisrn. H<'r<'. \\ 'C' calrnlat<'d G(q" / q') \\?it l1 the "110- bouncIJr ,?? ? . . . . . } Initial cond1t 1on q' = O. WP haw see11 tlrn t th is stipulation corresponds to a I num )er of saddle points. By implemcnti11 g thC' contour rest ri ctions proposed b,r Halliw 11 d . . . e an Hartle [63) . we were abk to pick out thr sadc!JC' po111 t gwe 11 by Eq. (.3.50) with t? ac (3 1011 .51 ) . In addition. 11 ?t? w<' ,Yen? a) e to speu y a contour. How0,?er . ? it was shown in [65) that this contour is 11 01 1111 ique. fo fact, they showe I th at the re? arc many contours whicl1 pass through this saddle poin t, not all of which ? ,? . . . . . . . rece1v0 their domrn ant contn b11tion from this pomt. st A fir attempt to e,?aJ1iate the wavefunction for the de Sitter minisuperspace n1odeJ w?:1.s d b . . c ? ma e .\" Hart le a.ud Hawking [46) . This was done rn the context of a steepest d ? esccnt approximation. On tJ1e basis of a part icularly simple contour of int0grati on, they ohta ined (3.73) In their m . . ore n gorous analysis. Halli well and Lo 11ko did find a contour which \VouJd r eproduce this re ul t. However , they could fin d no justification fo r singling out th i . s part icular contour. One co u ld at.tern pt to reproduce the resul ts of ? Hartle and Haw l,?m ? g? bY t ? , . ? 1xrng the in itial momenta. ' 1 1 . T he idea is to choose the ini tial momentum JJ sue 1 t 1at, via the H- . am J1 tonian ?on train t, q' = 0. This implies, that p' = ?l/2. One must now decide on tl . . . . . . le sign. There 1s no clear choice of s1gn . However, the Hartle-Hawking resul t fo ll ows from choosing p' = -l/2. . It seems as thougli a rigorous analysis of the saddle poin ts and contours of in tcgrar . ion of the de Sitter minisuperspacc model does , in some circumstances, reproduce tl ie on?g m- al result of Hartle a11d Haw1 o ?n g?. Ho,,,e ve?1?.? there is no clear reason f or singling out these particular contours. Hence, there rernai11s an am- 73 bi g ui?t :, I? ll th' form of Lhc wavC'fuu ct.i ou. In light of' this fac t . we resort t.o the SPm iclassical a? pp1.0 xm? iat1? on ol? t he waYcf.1 1nrt1. ou. albc1. t w1. thou t n.g orous j ustificn- t ion ? ? s we wi. sec , this approximation d ocs rcproclu c:e the Hartle aucl I-fo"?kiug 11 result of Eq . (3 . 73. 3.7 Black Hole Pair Production sec , we discuss the bounce approach to ca lculating black hole uu- lu this t?1 011 ra s and hmv it can be extended t.o cases where there does not exist H cl<'atio11 L cont i11u ous I . L0 1.., . 1.1 1 1. magm. ar,v t .u ne cl escn.b l.l lg the black hole nucleat1.0 n p-rocess. 115 3.7.1 Bounce Approach ertain inS ances of black hole nucleation cau be described by a single Euclidean t solu tion of th e Em. stem. equ at1.0 ns, i.e. , a bounce. Such a solu t1. 0n m. terpolates be- tween a n u.n.t.i al spacelike section without black holes and a final spacelike section cont, a1? ning? bl ac". holes, bouncm. g back to the .J) IJ. tt .a l secti.o n w. a tn. ne-symmet1.1? c 1 ssuming u011-compact spacetimes, the 11lan11e? r '? a n instanton is half a bo unce. A for s~ h . s is deternu?u ed ? the f?o JI ? ? F. ? t act' uc a nucleat10n proces 111 owmg way. u s ,, 10 11 one calcul ates the act.1. 0n of . the instanton med1. at.w g the nucleati.o n process.. The n <' act' 11 1?s bou then twice this result. Text , oue ca Jc u 1a tes tl1 e b ounce ac - 10 t ion of the bac.k g.r ound spaceti. me, Js, . Both actions are typically in fi nite, b nt their t1 ?Jf fe rence can be rna . -,.? st r" I> l a? ck holes 1. n a barkgrolln cl wag11 ct1.c h. C ' !d. Thr background ? .I -. , ord 0111 acetimr is . k.n own as a \ l r h ?in uni\'C'rsc. T J1i s parti rular black 1101<' pair cre- sp ation . cons1derrd to lie t li c analog f the Schwinger pair produc tion of 1) l OC'eSS is ? .? rn agn t?1 ( . lll 0 110J)O I<) s. 1. 11 a hackgro1111cl J11 ag11 ctJ.C fi. eld. . Lrt us st 11 ch? ? ti n.s. pa. n. . nc?at1.0 11 process .,n cktai.I . T he black lwk ? pacct1. rne 1?s, g1v cn by t he' rn ?t metric Thp gaug potent ial is givc'n b,v (3. , 5) Whcr(' k . IS a CO i 1stau t, and 2 (3. 76) 1 :2 B ,,\ = (1 + -Bq.?; ) + - 2 4A.2 (.r - y-)-:-G(T). 2 The f uncti.o n G.1 s gi. ven b,v (3.77) 1 11 W,. her e we note t?h at, the char~e q is specified by q2 ~ r +I'-, a cI th e mass n' ~ t(i +r - )/ 2. For later convenience, we set~, ~ - 1/(r _A) and let 6 <:; I;, < (, be ree ro t ) h b ? tte 1e tl1 . o s of the cubic factor in G. The function G(( can t en e wn n a s (3.78) In ord er l?o r t he metric to have Lorentzian sigoa turc , one restn?c ts ?a t,t, en t ,011 :S J; <: c T . . t . ti1 a t spatial infini tv is t 10(3 - " ? he fo r111 of the coufonnal fact.or Jll ( ica es ' ? ? " 4 75 ... ' \ I I I I I I I Acceleration Hori zon Event y = ~3 Hori zon l =O Figure 3.1: C nstan t ti me slice (t = 0) of the E rnst metri c. ,,?i t h th<' roordi,rn tc ? s upprC'ssed. rC'acl1 ccl when .1: , y --+ (i, whilf' '.I/ --t .1" for .r i= ( 1 conc?sponds to mill or t i111 clike infi11i tv. T he range of t he coordinate y is tltcrd"o r<' - < y < .r. T he axis .r = ( :1 p oi11ts towards spatial infini t~- and the axi~ :1: = (4 points towards t h<' other black J1 ole. T he s urface y = (1 is tlw inner black hol<' hori zon, y = 6 is tli c black hole even t hori zo n, and y = 6 is tlic accelerah o11 horizo n (s<)e Figure 3. 1). To ensure that t he black hole is free of s ingularities a t the po les.?: = 6, ( 4 , 011 c must impose t he condi t ion (3. 79) For later convenience, we Jet L = ,,\(.1: = 6) - WIJe11 rnndition (3 .79) is satisfi ed , t li e b la ?k hole event horizons are regular if OIH' identifi es ? with peri od 2 .6? = 4nL _ (.3. 0) G' (6) 7G 11 ctcr fl" = flG' (E.:i) / 2L'lf' 2 . "?hicJ, 011 (' c-an d<'fi ne a JJh.,?sic-a l 11 iag11 cri C' fie ld parn 1 fi ? 1' J l . l l of ' lt.,?. 1<: p 1vs1u1 c imge is th e, , ??1 !11 c' O1 ? tJ < IC' magnetic- fi eld on t he axis ;-i t in ll thc hlac-k hole is t? heu (3 .81) pan-111H'tns: the 11 iagnrt iC' fi e ld T lw sol11tion (.3. 74) aud (:J. Tei ) d epC'nds 011 fom black hole .-1. a11d th<' bl ack lw k mass BE . tlic charge !j, the accc? l<'n i ti ou of' tll(' m. ? ? I ? T Ji (' h?,ckg?r?o d , or I\ I e Jn ? ? ? n n ? s mag11 rt1c u111n?rse, 1s g 1n.!IJ i., n u (3 .82) whcr, t he .. .e . . gaug potent ia l 1s ' ' ) B .~1/J~ (3 .83) ?- 1 ,1, = -;:\; -- ) - / 1\ 1 and ( 3.84) xi ? p = O aud dccrrases to zero The Maxwell fi eld is F 2 ; it is a maxirnurn ou thr a at infini ty. n contains t li <' Mrlvin so lu tion .i n t lw limir utio \,VC' 11 0 w show that the E rnst sol C ? i 1 f. ,1?1- ? lr?n g to .1:, ,I/ --t (:i. 011s1c er t 1ie c 1ange o ofJarg?es pace 1, r d1stan ?es, con espom ate from (~r , Y, t , ?) to (p, (, 'I] ,? ): coordin 2 4L 2 P G1(6)A 2 (p2 + (2f ' 4L2 (2 G'(6) rF (p2 + (2)2 ' 211 t G'(6) ' 2L2 - --r/> . G'(fJ) 77 . . . For largc '2 -2 . nc rcclt1ces to f he :\ Jclnn forr n /J + ~ , t Ii(' Ernst rn et (3 .80) IY 1. +--i r/ I ' IYG' (E.:i f (3.8G) -J.Dl of t11 c Ernst rn ctric whicli \\ 'i!l ioll 1\ow we can tur11 f o f !1 (' E u cli dC'an scct ? ? l ?i zr (?orrcs1)0 d t O l ? stanto 11 of ? the bl ack holf' ut1clC'at10n proce ss. To E11clJc e,rn ? ll t IC' rn ? ? ? E 1? ! t? t1? mr to " ll C 1c eau .1rnc . the 111 etr ic 1?1 1 (3? ?- -') ?' f'-1. ? we a11al_vt ?J cally cont1111 w LorentzJarJ y is taken to 1 ---? For 11 ou-extrc'J11 c black holes. f. < 6 _ am! the range of 1. e., iT. 1 it ive rl efinit c' metri c. To avoid couical siHgularities bc 6 S .I/ S 6 to obtain a pos th bl ack hole horizons, we req uir e the period of T to I)(' at e ac:celeratio11 a nd 41r (.3 .87) /J = ~ T = G'((;) ' T l l . . n t, 011 11 as and G'(t::2) = -G' (t"::_,. , :; . JC resu t r n g 11Jsta " ) . T his gives 6_ - f.1 = f. 1 - ( T l ? -. ?1 . . . JJ.s 1 1sta11 ton topolog?y 52 5 2 { f.:J. ?? x - pt} , "vh erc t he pomt removed 1s :i: = .'/ = o oppo, itely charged black is interpreted as representing the pair crcatiou of tw holes c . ?o nnectcd by a worrn holr. r rrns is Thr E uclidean act ion in cludi ng bou11dar,v t L:J . "I} - (3.88) I c =- - l ; ? rl, , 1? . TJij(R - F"2) - -I r.1,vn, . IG1r . "' ? Sn . iJA t The vol . of R is zrro , ?ince R = 0. The volumr in tPgra l of the ? Ulll e mtegra.l 1? t I u1.m? e d WJ?t,1 1 Maxwell Lag raugi? an p2 can br. converted r ? nto a s111? ace :erm a11 c corn - thr ext.11?1 1 ? 1 rvature. T li c act ion can then l w writtcH as ? ' s c cu l f. 3, "I ,- tS n ( r -tS ,1 1 '' ) , (.3. 9) l e = - - d .1: v 11< v 1, Sn. BM 78 e? 011t warcl pointing norn ia l to rllC' ho11 m la n ?. aud r - n whc?n? 11'' is th in trnd uc<'S r1 l>o1111clan ? thrce-smfacc at 1arg<' ra- To en1lu atc the act ion, 011 c For the E rnst so lll t io11 . one ca11 take thi s surface to lie a l .r - :i; = f 1~: for the cl iiis. . \Iclvin sol11ti on. 0 11 <' can ta ke thi s surfac e to Ii <' at :r - :i; = fi\ 1? 0 11 thi s surface one? can show t l1 <1t [G ] (3 .90) n of the Ernst sol;it io11 is 1w-1.de of two p ar ts: t11 e bound- T!w action of the regio contribu tion from the thrC'e-surface ern l wclcl r cl in the Ernst solu tion. and a ary s t1 ed contribu tion from the three-surfac:<' C' r nli eclcled in the l\Iclvin solu t ion . htract t ribu t ion to the action from t.J1i s surface in the Ernst solu tion is T he con (.3.91) e-surfric:e. wh ere c-" = etri c: o n tli e t hre >.. (u - ( 1) / (:1; - 6) , and h is the incl uccd m T he contribu tion from the surface in the 1'd clvin solll tion is (3 .92) - 2 G'(6)2 2 (3. 93) ..-1 =----A . 2L 2G" (6) n Loth 1 and h , are infin ite i11 t he limit Er:;, EM --+ 0. However, the actio ate that 8 of thc Ernst solution is t heir d ifference, wl1 ich is fini te and given by / Ern st I r:; - h1 21r L2 (3.9-l) A'.lG' (6)(6 - (,) Tlw Pair creation rate i. th en proportion al to exp (-2h:rnst )- 79 3.7.2 antum Cos1nology Approach Qu k hole nuc!C'a ti on pnJc?c?sscs to , ,?J1icl1 lliC' ahovr forn1alisu1 Thc?rc an, some blac ere is single E 11 cJidc:mi sol11iio 11 to s. th JJO ca ,inot he appli C'cl. In t]l('se prncrss c ect ions st C'quat ions tha t describ e.'> a ro 11 ti nuo11s hi storv frorn spacelike s Ein ein 's case rne with black holes to 011C' witli /;Jack holes. This is Ll1c of a spacet i 110 ses tJ c uni vrrsc to close such a s a posit"ivc when matter is pn'sc11t which cau 1 will show this latrr in tlw chap ter wall. \Ve cosniologic-al constant or a dorn a in II] th(' C'l S(' f?'. . . . c , o r1 pos1tn?e cos111olog1cal co nstant. these cases us111 g the ih con ? of T he' nuclc'atio11 rates cr111 I)(' ca Jrnlated i11 orks wi th the ,,?,wc- quantum cosmology and t li c -- 1w - bouncl ar_, ... proposa l. One w s ? 1?tt? ?a Ju es f'o r? a? u111? , ?e rsr fuzi ction ot? tl 1c uni?v erse . uch a wavcf ?u n ct1011 takes c r ercIJ t , . T his wavefon ction is c:,.ilcu latecl wi th boundary con- Wi i; b and without black holc!s ? ? . 11 ? x1-?m r1te d ditions ?s pcc??if ?w e l I> ? ? 1ca .,? appzo ? .V the NBP. The wa vdu11 c:t10n is sern1class to hr1ve t l t? ? ? 2) . Tvpically, tlir prefo ctor .-11.- is JJeglccted and Eq. (3. r ? ? le orrn grvcn rn ? e 1 oul ?v O il<' ? - sac l dl c pornt is assurned to exist. H cJJcc , w 1aYe (3.95) One can tl ien defin e a probabi lity measure for a universe: (3 .96) ? ? .. t e. .At, fir?st gfa11 ce, One znu ?s t? no w I ?c Ja te pro l:. >abd ? i? t_v measures to a pa.Ir crcat1011 1 a , ? rsr it seem tl ty urnve - s ' iat one can ou]y compare probabi lity measures of a11 emp J 11 ru t o and a u 111? 1-s?c wi? th a sr ? ngle black hole pair. This rcsu t wou c not sec VP- have, . . ally growing universe. An 1 . ? c ny cosmolog1cal importance in an exponenti ? [ ?d ? ? a t' g the situation wr1s proposed rn 6 9 ]. St art by cons1 cnng al t,e rn ive way of viewin t-er atial OJH~ Hubble time, its sp an arbitrary Hubble volume of d e Sitter spacr. Af 80 rnl 11nH' will hav<' inn<'rlS<'cl by r1 factor of c?:1 :::::: 10. BY the d<' Sit te r 11 0 ]1air t IH'OH' lll . OJH' c;-w regard (?ad1 of the 20 H II bbl <' n>l111J1 cs as hm?i11 g lwrn 1111dcatpd incl rpcndent l:v, eit?li cr by t?lw background or bl ack hole i11 sta 11to11 . Hence , siu c<' sonic' of the new Hubble volumes \\'ill contain black holes, on<' has black hol, , 1111 clC'ation. O ne can assign a probabili ty n1casmc to each t,?pc of n>lurnc: Phi, is th C' probability measure for a Hubbl<' n>l1111H' 1?011t a i11i11 ?; a black ho!C'. and Pi,g is t. li l' probability mcasur<:' for a H11bblc volume of background spacet irn c. T ll(' rat io o f' tlicsC' measures is tl1C' 11 t he number of bl ack hole pairs procl ucC'cl per Hubble' vo lt1111 e, per Hubble Lirne: (3.97) From Eq. (3.96) , we ca n write r :::::: rx, 1) [- (211l(? - ?) 1111!,Ji - 1,g ?) ] . (3. 98) Hence, for t he particul ar type of black hole 1111clcation prou?ss one is consideri11g , Oll C' must calculate the real part of the Euclidcau action describing t li e black hole spacrtime and thC' background spacetimc. \i\ 'e should note. however, that t.l1C' a hove' argument for calculatin g nucleation rates has no rigorous _j11stification, but is a,t best, heuri stic. Neverthel ess, we adopt Eq. (3.98) and illustrate' how it can h~ 11sed to calculate black hole nucleation rates. \iVC' will work in the co 11 text of minisuperspace models. One starts with the E ucli dean fi eld e(Juations for the metric g,,.11 and studies their solu t ions. Presum- ;-1 hl y, these solu t ions will be the black hole and background spac0times in which oue is i11teres ted. One then finds manifolds 0 11 which these metri cs can he defined as ,naxirn a lly extend ed r ?gular so lutions to t he Einstein 's equatio11s . Finally, 011 c would invest igate wh ether an:v of the c manifolds a11d the classical solutions on 81 1 th e bo1llldan ? coJJdi t ions of the 110-h ou11daiT proposa l. lic?,n arC' compatible' \\'itl1 . then tl1C'S<' arC' precise ly the H tlicre an? sud1 111a11ifolcls and c lass ical soJ11ti ous rgral a11cl arc its clomim1J1t contri lm t ions. sad dl e points of t l1 e 11 0-ooundary in t i11 a we ar' i11t?crested in st ll clY iug bla ck liolC' pair creat ion procC'sses Si ucC' tions will be concerned with W(' cosmologica l background , the E uc lidean fi eld eq ua can be \\'ri tten as (3.99) 1 Witl1 J\ th the ro m1c l 111 ctric:, 0. So l11 Lio 11 s " ?liicl1 are knO\\"II are [70]: (1) .5" wi 2 2 2 h the product of two ro u11 d metri cs . (3) C' p "l. with t l1 r Fubini-Stucly ( ) S x S wit 2 metric, (S) C P2#nCP '2. 3 ~ 11 ~ 8, with a inrtri c, (4) C P #CP "l. with the Pa ge 4]. Of these solu t ions, It1"s f1'cd ?1t T = 0 witl1 1c ? ? ? " ? ? c , ? a = 0 and i1 = l. since it is req uired tl1at t l1 er<' is 110 boundaiy at T = 0. Let us first consid er an in tegration path a long the 1T tc axis. from T = 0 to T = 27f / H - 1 T his solu t ion descri bes ha lf of the E uclidean de Sit ter i11 sta nt011 of topology S' ? 1111 T he in tegration path can now br rotated parallel to the T axis where o ( T 1r n) = -l cosli H lrn (3. 111 ) T . H T his describes half of a ordin ary Loren tzian de Sitter universe. T l1 c real pa r t of the action for tlii s saddle p oin t comes from evalua ting the action along the real par t of the T axis, wi tb tlw res ult (3 .11 2) 1 2 Minisuperspace Models with Spacelike Sections of Topology 5 x 5 \1/e will now consider the same procedme as a bove, bu t now wi th a spacetime con- taining a pair of neutral black holes immersed in a cosmological background . This spacetime wi ll have spacelikc sec tions of topology 5 1 x 5 2 . The Schwarzschild- de Sitter so lu tion is such a spacetimc. Its Lorentzian metric is given by (3. 113) with (3 .114) Herc, J1, is th e m ass of th e black hole; we see that this metri c reduces to that of the de Sitter space timc fo r 11, = 0. Th e equ a tion V(1) = 0 wi ll have two roots; the larger roo t corresp onds to th e radius of the cosmological horizon and the smaller 85 e horizon. It can be shmrn that there do es no t Pxist a E uclid ean to t he black hol i n 's C'quations tliat rn 11 be a naly t ically ccrntirnr cd t.o contain so lu ticm to E i11 str t h ni cal is spacct imc, in gcnentl , because one' can uot sirnultancouslv rc?rnm?r t he co zo us . Tl1e d egc?uerat.e si ngulariti es at bot h th e black hole and co snr ological hori s is an excrptio!l , and we case of equa l black hole a11d cosm ologic al hori zon radiu How discuss i ts drrivation a nd act ion . logy 5 1 x 5 2 T he ansatz for the E uclidean m etric wit h 'J)atia l sections of topo is the Kan towski -Sadis m etri c (3.115) ccl in t he computatiou of the il ct ioll a re Tli<' terms 11 eed i?? 2i/ (3. 116) (J. '2 J'\ T'2 + b'2J\T'2 , ci, 2b (3.117) aN + bN' 2 (3. 11 8) b2 . Th e action theu b 'Cornes I ;? ( i/ 2ahiJ J\l.l ) id/ 2abb) I = - 7f -1r aN11d it io11s 011 r; a nd 11! o f _r; has nega t ive r1? gc'nval11C'S so111 C'\\" J1 e1? C ' 0 11 ? ? ? . ? ? t?, ? 01?< I ? t?o ]1-, , ,c, t !1(' iu trr1)rr tat io 11 as ar<' t IH ' con d I CIOIIS a gco r1 1 r t r_\? must sa t 1 ? :? Il l <' t ? ? a gn:ivitat iona l ins tan to n describing topologv cha uge in t li r as,?mptotica ll_r fl at t. h context . For t he' case of gravity miui111 a ll_,? coupled to a rea l scala r fi eld ? mi11imall>' coupled to a held theory. \\'c will t hr 11 disrnss thr condit ions unc!N whicl1 t he inscantons for \Yhich wr are looking ca 11 e.?ist ill thcsC' highrr cmvaturr theori es of grnvit_, ?. ;\"?c?xt. ill thr cout<'xt of gn-wit,? tlil'orics t hat a rc' p ol_v11omial in thr Ricci sca lar. wr di. cuss thr existcucr of con11 rctccl backgroun d alld bla ?k hole nu cleation instantons. Final lv, we discuss t he r.x istcncc of co11tiu- 11ous i11stanto11s in C -field theory. 4.2 Higher Curvature Gravity Theories 4.2.1 Background Higher curvature gravi ty (HCG) theories have a long history and have been pro- posed in severa l different contexts. For example, t hev were first introduced b_v Wc:y l [77] in an attempt to unify gravity and electromagnrti sm. It has a lso been s hown that the addition to the ord inary E i11s tciu-Hilbcrt action of trrms quad ratic in t he Ricci tensor leads to a renornializaulP th eor_v [78, 79]. Attempts have beeu made to use higher curvature terms in the context of cosmology to avoid the iui t ia l siugularity [80, 81]. The presence of higher curvature terms may solw prolJlerns associated ?with the uon-positiveness of the ordiuary E instein-Hiluert action for Euclidean quantum cosmology [82]. Quantum wormhole effects have b een shown to give ri e to quadrati c correctious [83] . Las tly, hi gher ordm- La- graugians arise naturally in higher dimensional theories like Kaluza-Klein and st ring models [84 , 85 , 86 , 87, 88, 89]. We wil l explore what light higher curvature t l1 eori es can shed on connected instantons. Since we will be assuming that the radius of the connection is much smaller t han the radii of the universes it connects (irnply ing high curvatures a t t he connection ) , we expect higher order correct ions 92 to t It <' E i11stc i11 -Hill )(' r t act ion to hcconw irnporta 11t . A ction and Field Equations We will be limi t ing our consideration of liigl1<;r currn t urc theori es of gnn-i t,~- t.o t hose wh ich have a Lagra11 gia 11 dc11 sitY of t lic form ? = f( R)Jrj, a 11 d E ucl idean act ion I = - 1~71" / f 1(R ) jg d' (.J. 2) .T, wh ere f (fl ) is given by (-J. 3) Herc, R is the Ricci scalar curvature, J\ is t lt c cosrnological c:0 11 s ta llt ; and et . "/ , etc ., a rc coupling constants whose vaJut w<' ]eave uuspecifiecl fo r th e mom ent. Th e E ins tein-Hilbert action with cosmological constant is const ructed from the first t wo terms of f (R). Vari a tion of the ac tiou ( 4.2 ) with respect to the metric y ields the fi eld equ a tions f' R,,.,/ - if .(J/ lV + f" (v',, v',,R - !J,,.,/ v'2 R)+ f'" (v',,.R v',,R - !J?v v' ' R v' , R ) = 0 , (4.4) where a prime indicates differentiat ion with respec t to R . It can be shown tha t for quadrati ? correc tions to the acti on , it suffi ces to consider f of the form f (R ) = R - 2/\ + n:.R2 . O11c can start by considering f to cl cpeucl on R and tl1 c four qu adrat ic curvature invariants, i.e., .I . -_ .f '(R, , R, 2' R,2,. ,,, R,2,. ,,pu, C,2,. ,/()fl ) ' (4.5) 93 I " . tl ' ,, is tl1 e n1?c?c1? t, -. 11so1?. R . 1 I. r IS ? 1( IS t l<:-' , ) J.( 'JllaJl ll t C' IJ SO J'. all( I... 111,pr wh ere? R,, 0'- . 1/wpu ? 1 ? ll ? ? I t tensoi?. N. t t J1 at i11 fo nr-c inH'11sJ011 s, t l1 e t? opo ogJCa _,. 1m?an an W<'yl ? <'X . 11 ot<' Ga uss-Bo1111ct corn/ i1J atio11 2 l R2) (.J.G) \ = . . d I .r v1j (R 1111prr - -lfl111, + / rn11fon 11 al1y C' li1ni mit,e tPnns proportion al t.o Rj, pu [90, 91]. The e used to 1can b invariant express ion t i? ? t t 'nus whi cJ1 1?s ?t t O t J l f? fat1? ve in fo ur dirnensio11 s, a Jl ows 011e o C' m1rna ,<' < ? c a c 'r . ?) 1 C . ?1 n2 [ port JO ll al to c ,~// fJT' \\ ' ICJ'e /1.V/JT IS t lC J)roport? l ms proJ ona to ?11 92] . Lastly, ter . ll Le shown, b,v using Eq. ( 4 .6) , to be Wcrl t s these ca ? ensor, can Le dropped a Proportional to I: [93]. a Scalar Field Equivalence to Gravity C oupled to a general pol,r- terest to note that 11 011li11car gravitational theories with It is of in l . t? . . i?:i.] d - d . 1. - unction are nom n e11 c:e 011 the R1 cc scalar cu rTatme through a11 amt ,vt1c c cpe J fi eJ d JJ l us a rn as- conformal! Y equi ?v alent to Einstein gravity c oupled to a sea ar . . . . . . co11ta111111g sivr SJ)ir1 2 fi ld [ ] . 94 . Th is equivaleuce has also bern shown 111 theon es - e t t 1 ? ? ? ensor [95 , 9G]. \tVe wi ?l l cJ 1 0ose to ?11 1 us ,rate , us Powers of th c R? 1cc1 and Riemann t ? R o 1? ./ '( R) = - -?A+ n R?2 , fact by con 81?c t cu.?n g t l1 e act1?0 11 ( 4.2) 1 ?1 1 t li r srn1pl e case . t ? t? ? ? [9 ~] Van a wn o for Which J t J1 c sca lar fie ld is present in th<' eff?ective theory r . on .V . tli e fi r ld equations (which c: au also this a t? c ion with respect to t he me tric yi elds b .b e o tained from Eq. (4 .4)) 2 2 (1 + 20!R) R + q ( ~aR 2 2 + {R - 2a'v R - 1\) = 0 - 0!V1/V ,,R - ? v . ?.v 2 (4.8) conforr nal tra nsforrnat io11 of the metric :\c'xt. consider cli e (4.9) rmally tnrnsfonu ecl R icci t ellsor and the orig- T!ic rela tionship between tl1 c confo al Ricci tensor, in n-dimen sions, is givrn b_,. [GO] in . n = 4. w<' have For t l . t? an d le con orrnal transfo rmat i on under considerat10 11 R - R- + 2 o- \7 \7 ,I., _ iw - 1w -l +- - 2o.? / I l/ '-f/ ( ) I iow; of -1. 8 t 1e11 take th e eq 11 at which lea 1 ? ? li <' fi e] l .cs one t o ?d 1 with R . T forrn rd scalar s represent Einstci11 gravit y coupled to a nonstand a T hese fi eld equat ion Ir::" rp has mass 1/ v Ga, a vac1111rn fidd ? I t fi eld ? Call be shown [9 7] tliat tli c scalar ion value of 4A, and fi eld equ a t (4.13) Ne t? ga Ive Eigenvalues of R ' / LI/ l sur t? ? e , a necessa1y m t not 1 - As Previo us? l .Y stated, r ?n tltc asymp tot1call _v fi at c as particular t l1 eor_v is t li a t tli r o11 ? in a cicut coiidit ion fo r thr exi stence of illsta ll t I 011 t I1 c rn am . f.o l d . EucJidea R,?1 cc1? te11sor possess negati ? ve e?ig envalues somew 1ere ll 95 \ \ ?c? are 1 t . . . t? . . a st 1p11la t1011 on thC' Ricci r.cn sor for spacet irn<'s " ?lii cl1 1 0 , aware o such nt0,resting to uotc tlie co 11 cli- ay, asym ptotically de Sitt C'r. Howc'YC'r, it- is i an-, s n' eigcn\'alu es er which th e Euclidean Ricc i t<'nsor may possess ncgati t ions und se co11di tiow; arc prC'c iscJy t.lw sC' u11d C' r \\'hicJ1 hat t l1 c ill t-Jiis context. \\'e will sec t 1sta11to 11 s. WP fiu d con11 C'ctcd i1 btai11ccl from f(R) = R - 21\ + nR 2 . o ry o \\'(' start wit11 the simple HC G tlt c ent scalar fi eld theory. HcncC ' , ?a l It wilJ be more conveni ent to c onsider the equi , ding wr, take the trace\ of Eq. ('1. 1 2) ,vi ?J 2 2 ) - Gn, ( 1 2 (-1.l-.l) = (l + 2n rp )'2 "v'' "v,,rp + 3n rb + 3n 2 1\ R st is back into Eq. (4. 12) yi elds Sub itutiug th R _ Go i _ ( 1 r, ' 2 -lA ) (.J .15) (1 + 2arp)2 v,,.?"vv

O. For a rea1\ < 0, R,,., rnav have negative at for n in Eq. (4.15) are pos ir,ivc . It then fo llows th ?t? t J sea l ar fi c1 d JC eigenvaJu es somewhere 011 th e . pac:e timc manifold. Howrver, 1 values. cci tensor ca11 havr 11egativc eigen is iinaginary, i.e., ? ----?- 'irp , th en the Ri rc,g?,,rctJe o f ' r ss the sign of o?. , 4.2.2 Continuous Background I nstantons instanton solutions u1ay exis t under certain couditions, Baving established that r arn1i hi- th re exist continuous instan to11s wJ1i ch describe the 11 ea we now ask if latio11 . d aIJ subseq u 11 t rebirth of de Sitter space. 9G ...... de Sitter Space in HCG Theories It is interest ing to no te that the general rclativit\' de Sit ter spacetime is a so lu tio!l of H CG theories. To see t his , collsid er t l1 e followi11g . The de Sitter spaceti rn c' is a m aximally sym metric vacuum solu tion of the gravitatio1ial fi eld <~q11 at ions. For such spacetimcs . t he Ri emann tensor can !Jc, wri tten in t he form (4 .l G) where contract ion on ?, a nd () implies thr eq uivalent conditio11 ( 4. 17) Usi ng Eq. (4. 17) a nd the fact t hat ( Rl'fJ - ~gl'fJ R) = 0 , (4. 18) 2 ;(J we have R;p = 0, implying that R is covariantl y constant, i.e ., R = R0 . (4.19) From the field equa tions (4.8) and the constraints (4.16), (4 .17) and (4. 19), we have the existence condition Rof' (Ro) = 2f(Ro). ( 4.20) Hence, given any J(R) gravi ty theory, if t here exists a solution Ro of Eq . (4.20), then th e theory conta ins the general relativit:v de Sitter solu tion with cons tant curvature R0 and with metri c scale factor a(t ) identical to the form taken in geueral rela tivity (fo r EiHstc i11 gravity wi t li a cosmological constant , Ro = 4.t\). Howevm?, it should b e noted that t his solu tiou may not he unique. 97 Field Equations ttri c to haw the E uc:Jidcau Robcrts on-\1Va lkr r fo n11 apprnpri atc Wr wi ll take the m O tiv<' cmTat u re: t threc-dime11sional spat ial i-;Jices of co 11 stc111t posi (-l.21 ) -t ic continuation t ~ ir, 1\ '(7) Here 7 is im aginary t im e deterrnill ed from the anal_, th un ct ion, a(T) is the uni verse rad ius. and dDj is th e metric on tl1c is e lapse f i\/ alld o allows 11 s to obtain uni t? tliree-sphere. Having tli e rn0 tric depend 0 11 e cpcndcnt Einstein cquatious by v arving 0111_,. these fun ctions i11 th all the i11d riat ion with respec t to a gin?s 11s t he 0 11c indcpend cHt spaccJike act ion (4.2) . Va ment equation , and vari ation wi t li respect to N yields the timelike time develop ? 1 <.:onstra1?nt equat ? as r?n ore1 r? nary E.mstern t 1 eory: ' 1011 , -[ }!__ 3 ~ r_u,_:i ( f.No 3 )I 0 (-1. 22) oa) ( ./ NaJ) - ~ dT ro? n,( .! N a )I + dT:2 2 oh . , ?3 I . + -D?Nf a??i - -rl [ -D-. ( t ??N :i)] = 0 . (-1.23) cl" 0 ? fJN ciT DN . conformal chang<' of the rn etri c, ;:i A furth er Yariation that is easily p erformed is race of the fi eld equations, which is uot indepeudent of the other giving the t der the general case equations but involves only the fun ction f (R) . Wr wi ll consi .; the conformal transforrnatiou is writteu as in n-dim ension (-1.2.J) where D = 1 + t:, f. ? l . Variation of Jg gives /f; - ~ Jrj. (4 .25) b ?9 = Jrj nc from the relationship between the confor- lows The variation of the Ri ?ci sca lar fol i sca lar given b_v Eq. (3.14). rna!I,v trausformed Ricci sca lar and the original Ri cc 98 Tl11 1s. (-1. 26) Vari ation of t he act ion :,?iC'lds Lhc four-dimensiona l t race eq uation 2Rf' + G'\11..f' - ..Jf = 0 (-1.27) where ( 4.28) a cl ot denotes d/ rlT , and a prime denotes cl/ dR. Equation ( 4.22) is a fourth order o rdina ry diffcrc11 t ia l equation , a nd Eq. (4 .23) is a third ord er first in tegral of this equation. T hr trace eq uation (4.27) shows that we cau regard R as a n ind ependent variab lf' , sat is f\ring a second order ('Cjttatiou. In th is view Eq. ( 4.2 7) replaces Eq. (4.22) (Lo which ir is equiYalent) , and a also satisfi es a se ?ond-order differential equation ( 4. 29) In addition we still have t he const raint, Eq. ("1. 23) , a firs t ord er rela tion between a a lld R. Connected lnstantons in H 2 and R3 Theories 2 3 In t liis subsection we explore the existcn ?e of cormected instantons in R a11d R HCC t heories. Let us first, consider a Lagrangian wlti cli is quadratic in the Ri cci 2 scala r curvature, i.e ., let .f(R) = R - 2A + n,R . T he E uclidean trace equation is then .. a . 1 R +.3 - R =?- (R - -li\). ( 4.30) a Go 99 V 75 50 25 R -5 5 10 15 20 5 30 - 50 -75 F igure 4.1: Potential fuu ction 60,, }'(fl). in g(a) , where a is now t he independent vari ablr : + -J\a 2 + 60 [g'" g 1 - + -1() ~ + 3r/ c/ (JC/ ] ~ - q 11 - gg 11 - :_ - -?? _ = 0 , 3 4 a a o (/, ( 4.33) where a prime denotes d iffe rentiat ion with rcs pC'ct. LO a. T he problem 1s now red uced to solving the above second order equation and the ''conservation of energy" equation ( 4.32). T he solu t ion of t he latter equation is well understood , s ince we know t hat then' will exist a bo11 nC(' if we have two turn ing poi11 ts in t li r potent ia l g(a) . T l1 e second order different ia l eq uation for !J is derived and a ua l_yzecl incorrect ly by Fukutaka. et al. To get an idea for the possible solu t ions, we assume th at a solu t ion is of the fo rm g = g0 + g1, where' g0 = -(A/3)a.2 is t he unperturbed de Sitter solu t ion . If we assume (as did Fuku taka , et al.) that J\ ? 1, and that a, is large. we get 2 2 d g 1 d_q1 a -,ad2 + a-r ( 4.34) ai - \ 1\Te note t hat only for a < 0 do we get Bessel's equation . I t should a lso br noted that t he corresponding Eq . (5 ' ) of Fuku taka , et al . is in correct ]~; written . 101 l I , . l l S1. so I . e ttC'r ut10 11 pC'rtt1r I 'c )_, . a "C'uma1rn )C For IJ ('rra tiY(' n . one ms t J(' large-o d b 1 ? ? I nt?1? 011: ?m t I 11?s casC' . so utwns fll 11 q1r ?tion Fo 1? > () C' g<'t the modifi ed BC's se c ? n, . W a r solu tion. Hom'wr. ff (' OIH ' ca 11 st ill lu r-a dr it te for Di diverge from th e larg which th e i cli depends 011 the poi nt at larg<'-o t t1 ni ing point , the existe11cr of wh diver . , . I ? g?c nu s )ecomrs la rgr. snrnll a expli citly. T h is ca11 hr. clone b,? e ca11 analyze thC' bC'h avior fo r On ordr.r , we (a) = ,c + , ar+i + ? ? ?. To lowest 2 assuming a solu t io11 o f the form g 1 10 1 cl1 avior as the == 2 family of solu tions has the sa me small-a b fi !l cl E == ?2. Thr E v 2 c ted i11 sta11 to1Js. ror r = - dp Sitter ? c?o 1111 e 0 J t? hC' cl is ? u 1011 ; 11 c11 cc, \YC r< 'conir t ?) ? , , < 0. g ( o ) = - ( \ /3) o 2 i + , ,/ rr gives a 11 a d a .s ui t,a l) l e c1 1 01 ?c c' of? parameter nimum rad ius is small bounces 0 J t? ? I f the mi u 1011 with some minim um radiu.? arnin? dC' Sitter or r1 Nariai s pacc time, bounce to either a 'nough, then OJJ l' ca n l1av(' a onn ected de Sitter to aves locally like R,., _ H ence, i t seems that c since eac;J1 beh 'bl ? 1 ese tl . s are indeed possi e r n t 1 H~on es. de Sitter? a I d e s?i tter to Nariai bouncenc ? F J re, F u k t 1 ? ur t iermo ?u a 0. Th 'SC' res 0 1 ti11uatio11 can take pl ace a t an t ic co lytic continua ti on . ~ ot,c t li at thi s analy 11 aua l <' e u rn nucl eate c)ither a large-sea a rbi tra r ] f a. Hence' , on Y va ue o the sca le f a cto r or baby universe. tl,e possible solu tions . wP. caJJ posit an ore detailed picture of To get a m <' ffoctive L? . ? ag rang1a11 of tir e form 2/\ .5 0) . )2 ,J rr = 2(aa. - -a + ? 2 (4 ~a. . Le 3 a sense, found an equ i v- h ion of motion ( 4.43). \I\ r have, in Whic yields the equ a t ns for the case a lent lo d . heory that res tri cts a t tention to solutio wer enva ti ve t ? l ' . to the ca11 on1 ca coo rd .Ill ich(' a 1s f :::: f' === 0. Tl 1E' momentum conjug ate fJL ,, . ( 4.Gl ) JJ = - = --10 -0 . a Dri d olutio!ls i11 Figures 4 .4 an We g1?a 11 n and Loren tzian s P Pa vs. a for tli c E uclidea 4-5 ? res? Pect1 ? vely. 106 Cl F igure 4.4: p hase plot of solu tions to the E uclidean equation of motion ( 4.43). Solu t ions wl11. c? h a r.e p. en. odi.c and non-sm. gula r occm whc11 0 < ao < 1. Solu t ions which a. r.e .sm? g.u l ar occur for ao = o, and a, 2". 1. T he "dot" on the a-axis occurs 0 at a= J3/2A. {I Figure 4 ? r.: .? Phase p lot of t he solu tions to the Lorentzi. an equati. ons of. moti. on. 0 Th e separat n? x i? s denoted by the bold line aud crosses t 11e a axi?s a? t a = Vr,o;-;;; ;L;.3. lA /2?? 107 1 a ls ? t? ? all(! Euclidca 11 so utions <'as ily nTe T he J)h?ls rna ll r, the existru are cu111 berso111 e and . n and , from the sc ? l ? . f' = o fo r genera l COile cond1 t1011 J = e the art ifi cial res tri ction rr? 11 ?11 er, if we cl1 oose to ru ak not te 1 ) .V I umin ating. Ho\\'ev Ct - - Q, we have 1 (-1 .52) 1' = - 108. \ 2 ' . form in1p lying? .? 1? lw P<' u oc 1c. non-collap sing solu tions of t r/(T) = l_ + (oi _l _) cos (V2!\ ). (-1.53)T A .\ ? ? I f' ? to that of the quadra tic t wory o The Pliase ? ? space of these solu t ions is analogous / r:.A . . . s located a t a= l v 1L ure 4. 4 1 gravitv 1?u t d . t of Fig 5 iscussed . In this cas e, the do ? ? . theory. In this case, . he q11 adrat1c The Lore t . analogous to t n zian solu trnns arc also r:. t a. = l / v A. the separ t . os. the a-ax is a , a nx of t he phas e plane cr . ?t t ? es1?d <'s t. Ji r one c l esc1-1? b?m g ? tl1c ms an 0 11 ere ar ~ h ever, other solu twn s b Th ? e, ow om tJ1 e aracteristics, at leas t fr we sou ?h . rable di g t , these solu tions possess undesi n for all of these so- Point of ? r example, becaus e the actio view we take here. Fo . . . es some v . te over a ll vaJ11 es of a , wh1rl1 mclud0 lt1 t ion1:, gra anishes, we should in te b,? these solu tions. ided disconne . lelll i not re a lly avo cted instautons, so this prob 108 prncl11ct.ion of bab_,. Ulll - hich \\?01Ild allO\ r ist solut ioJJs w Furrlicrnwn?. t h rrc ex ,?crscs . tC'rrsting iJJs tanto 11s haw ber11 ill t , i t. is worthwhi le to notC' th;:it .-\t tliis poi1J t? E. cm. l d to a grant._ ,. coup e d ? ? rnst 1vc1le nt t hcones o ? ? found for HCG tl 1eon ? cs an eq u ological constant . is cou- o cosm fi l exarnplr , wlicJJ g ravit-v with n e cl theory. For tion rxists which cl .-1,,,, , aJJ i11stanto 11 solu sor firl ed to an a nt i-sy mmetric ten [73]. Tl1is work pl niverse Jy E uclidean rrg ioJJ to a bah,? u tical COllnccts a 11 asy mpto gica l co11sta11t a nd arbitrary t :l yers [100] to i11clud r a cosmolo was ex ,enc cd by l\J solu tio 11 could at an as_vrnptoti cally de Sitter 'timc dirneusi011s . He found tli spac< d esult was found by Halliwel l an a baby uniwrse . A similar r b COllnectPd to nstant con- in gravitv with cosmological co nste Laflamnw [IOI] i for t he case of E J s could be rein terpreted as J a scalar fie ld . Tl1C'se solu t io fo rmally couple d to rS<'. Instantons were Y 1111 ivc o de Sittrr uJJiv erses via a bab 1 tunueli11 g bctwr rn tw 11plcx sea l ar fi fl ld wit 1 s coupled to a c o1 gravity i a lso fo un d m? t J1 c case where f102J. In each case. the peculiar r field osed on t he sca la special conditio ns imp egativr eigenvalu es of 1e existence of JJ f t he fields are r esponsiole for tl Properties o the Ricci tensor . k Hole Nucleati on s for Blac 4.2.3 nto n Connected Insta where the ous do exist in HCG t heories nu ectcd insta11 tco vVe have seen tha t mological consta nt arr cos tein-Hilbert actio n with positive EiJJs corrections to th e nta in pathologie s ve mentioned , th ese solutions co 2 1 er, as we ha R and [{ . Howev conta in . l\Iost importa n tly, they still iw th beli eve makes them unattract at we a.l \\'a.,? to <'xrludc such there seems to b e no 11at ur th nstanto11 , ande disconnected i sol ? 1text. ? u t?i ons m this co1 .109 To cl etP rn1inc the C'x istcnce or rn111iected i11stanto11s clescribi11g black hole nu- cleat ion. one would IH'ed to solw Einstein "s <'Cp1atious to find a solution that ini t ial]~, hebaved de Sitter-like , reached a small radius, t urned around , and lw- liavecl Schwarzscltild -de Sitter-like. Presputl,?. it is unclra r how tl1is could be do n<', si nce a topol gy ch:-u1 p;c would n<'<'d Lo occur at t he t?mn around point . One could ass um a chwarzschild-de Sitter 11wtric and attempt to find solutions that begin at small radius and evolve into a chwarzschild-de Sit te r spacetime. This solu t i n could then be matched to the corrc'spo11ding solu t ion for a de Sitter spacctime. However one still faces the problem of a topology change at the point wh ere the solutions are matched. \Ve haxc not attempted such a cakulatio11 here. However , a similar calculation is carried out in the next section in the context of Eins tein gravity coupled to Nar!ikar 's C-fi ld . Furthermore, we consider such a ?alculation to be s imila r in .-pirit to surgical procedures used to construct con- tinuous imaginary time hi t.ories. These t.cdmiqurs a rc discussed in haptcrs 5 and 6. 4.3 Continuous Instantons in C-field Theory In this section , we wi ll look for the existence of continuous instantons representing tile creation o f black hole pairs when Einstein 's equations are modified by the pr sence of a negative-energy density "creation fi eld" or C-fic ld [103 104]. O ur motivation for doing so stem.- from the fact that a negative energy density scalar field coupled to gravity is equivalent to HCG th eo ries\\ here, for example, coupling co nstants like a a re negative. llO 4.3.1 reation-field Cosm ology C m ology that in rnlv r~s a singular ori - 1 ? n ?osn10logy, or auy mod el of cos Friedm a . could dea of barrnn cons crvat1on. Ouc g 111 to t h ? . ith the i e universe , 1s consist ent w I d ? C conserve , part1 cu - a,? not > then take t l1 ? por? nt o f n ew that baryon uu mber m ? e . - ? on uon- vitat10na l fie ld . To encompass ba rv larh ? in t l . ic presence of a str ong gra ? i11 trndu c:ecl creatio 11- f cosm ology, Hoyle and Narlikar theory o couservation in a ? rgati? ? J a n w C'llerg, ?v cI ens1ty sea a.r 1 ol ? [Jo ]. T l11?s corpora tcs ficld cos? 11 ogy . 5 th Pory m ? cl . . f' creat1.0 n or ? , which cl etcrmrne s t l1 e con 1t10n o field C ' call ec I t l 1r creat10n fi eld- ater paper b,v Nar Jikar In a l st ba ryons at any p o in t in spaccti rn c. cle ruction of s presented as a th eory 03, 104] , creation-fi eld cosrn ologv wa [1 . allcl Padmanabhan 1oulems as- h ? ?what t hese author s t l10ugh t were pr of cosm 0 J . ?h could remedy ? ogy w 1 existence of an . y, muuely : (1) Tl1c sociat d ? l , ory of cosmolog e wit 1 the big- ba ng the I ernat ic:a inconsiste nc.,? . . h i a l a11d .bl fi nguJa n t,r, i11d ici-1t in g a mat init ? oss1 y a na l sr ) ? J w? n o ?' co11 servat1 ?0 11 o t? theory. (2 The v10 at and phvsic 1 ? comp 1 e tenes. in thern v a ter without violat-mat ositive-energy dens ity energy. To expla iu the creation of p acts ust have some degree of freedom which th ing e conservation of energy, oue m 11 ? ce o f? sma p arti ?c Ie l 1onzons ? . (3) The exis ten as a 11eg?a t i? ve energy densi ty mode oll problem " . ( 4) T lie Jack ?'horiz th universe, sometim es knmrn as tl1<' in e early stics of structures in the r he origin , evolutio n , a nd ?J1 aracteri of expla 11 ation fo t ry of ess problem ', i.e. , the big-baug theo e "fl atn e at small scales. (5) Th . Univers served large-scale fl atness of the s l pl anation for the ob co ? m o ogy has no na tural ex llniverse. erg_,. density fi eld , ca ll ed a t ive-en mpt to remedy the se pro blems, a ncg In an a tte nce of this negative- sical leve l. The pre se the d at the cla ? -field can be introd uce ? tl ? ? t? pos1? ? ergy rn a tt. er w1 1ou,. l expJaiJJ t he creat 10n o t1ve-eu cnerg?y zn o c e can now 111 and for the g<'omctn?. (1.57) where (<1.58) 1 erg_v de11sit_v (for I > 0). iarJik ar st S gives a IJ egative c1 The ress-energ_v t nsor T, energy coIJdition is not a n 3 the , 104] ha given reasons why this violation of [I0 Einstei11 grrivity of aIJ exp anding universe. when the C-field is coup led to objection n-Walker ansatz For Lo t ? reu z1an cosmology we ma ke a Robertso - 2 2 + 2( t ) r.i n 2 ( 4.59) 2 = N (t ) dt Cl H :3 ? d ? . . ? re o/ t l11 s geometrv we ass ume that C 1s ~agr cnt with the homogeneous natu ecm d equations, derived homogen . nds on ly on t. . T he fiel eous rn space and hence depe .. en setting N = l , by f the action with respect to a, 'V, and C, and th variation o are ii, a2 + I (4 .60) 2 -+---!\ a a2 . 2 ?) 41r_ .l c??- a 2 (4 .61) a + !\ __ l - -a- 3 3 d(a:ic) (4 .62) = 0. a3 dt first (time development) egral of the The second equation , as usual , is a first int ? l ? - s t . The t ion . cl ? ? ? he latter except for extra neous so utwns a - con equa , an i t implies t third 0. . ,quation has the in tegra l . I< (4 .G.3) C = -:--I (/,' 113 Vcrr I , 0 ll+ a F igurC' 4.6: Effect ive potent ia ls fo r a dr Sitter- li ke universe. T he solid li ne rcp- rese11 ts the Lorentzian effective potent ial of Eq. (4.6-1). T he das hed (dotted) line represents the E uclidean effective potential fo r real (im aginary) k. w!J erc I< is a real constant. T he reali ty of the consta ut /\. fo llows from the fact t ha t t he C-field is real. \ t\'e can obtain an equ ation of thr. "con ?ervation of energy" t_vpP fo r a bv elim inating C in Eq. (4.G l ): -ii:2 + T ?L i:i.2 A ? 21rfJ{"2 1 11 rr = - - - a- + --- ( 4.64) 2 ' 2 6 3a'1 2 T his is the usual de Sitter equa tion supplemented by a te rm in 1/a4, whi ch is uuimpor tant at late t imes when a is la rge and docs not change the qua li tati ve Loren tzian t ime development at any t ime. For later conveni ence, we have chosen to a bsorb t he factor - 1/2 on t he right- hand-side of Eq. (4.64) in to the effect ive potent ia l, the graph of which is given by the solid liue in F igure 4.6. 11 -1 4.3.3 Sourceless C-field in Euclidean Cos1nology To a rriw a,t th e Euclidean equations of mot ion , we' will star t fro 111 scratch and a na l>?t icall y con tin ue t li C' action of Eq . (4 .54). Acl'ording to the prescrip t ion of Hawking [31] givell in Chap ter 3. t he action fo r both the geometry and t he fie lds s ho uld be anal>?ticall :v cont inued . Thus. w<' hav<' l e= - Jg rt 1: c (f-l --2-.\ + -?/ -C? 2 ) (-1 .65) / 1G7r 2 2 Var iation of thi ? act ion with resp ec t to a, N , and C yield , ill t he gauge N = l , a a.2 - 1 2- + 2 + J\ .11rf C2 , ( 4.G6) a Cl ?2 + J\. '2 -hrf ?2 2 <1 - l - n ---Ca ' (4.67) 3 3 d(a3C:) (l . ( 4.68) (J''>rfT Solving Eq . (4.68), W<' have ( 4.69) E liminating C in Eq. (4 .67) , we obta in ( 4. 70) Vvc HOW t urn to th possible motions for t lii ? system and study the LorcJ1tzian a ud Euclidean effective potent ials in Eqs. ( 4.6-1 ) and ( 4. 70) . As we did in t he Lor ntzia n case, we wil l ab ?orb the constant factor on t he ri ght hand side of Eq. (-1.70) in to tl1e Euclidean effective potential. Furthermore, we will as. ume t !J at J{ = /k/ . For convenience we write t hr effect ive potenti al below: ? L J\. 2 + 21!' 'l J\'2 + -l \ (' (f --(l (4.71) 6 3a4 2 ' J\. 2 211'./ /,?2 1 -(l +--- -. ( 4. 72) 6 3u4 2 115 BoLh t hr Lorrntzian r1nd E uclidean syste1us can 11mr be co11sidered to ban' total r ncrgy E = 0. WP ha ,?e schema t ica lly illm;trat.rd these potrntials in Figure 4.6 . . s w, ha, ?e a lrea d:v m entioned , I< must be real. In a quantum rn ccha nical contrxt , however. both real and imaginary k have ;-111 i11trrpretation . Imaginary k, LcL us take k to be imaginary, i. e. , k -+ ik. The resulting E uclidcall effective potential is given by the dotted line in Figure 4.6. T his potential is , of course, the 11egativc of t he Lorentzian effective potential for real J\?. For .f /k/'..! ? 1, t h<' C'ffective potential crosses the a-axis at ~ 1 + 21rf /k /2 /3. Th is potential does 11ot possess wormholr solutions, 1.c. , 111 t he E uclidean regime, the scale fr, ctor doc? not decrcaS<' from a n initi a lly large radius to a. mall rad ius , turn around , a nd grow again for a ll later E uclidean time. For t hi s to occ ur , a(r) > 0, which does not occur for a(r) > 0. In. tead , a may initially start ge radius in Lorentzian t ime arid shrink to a rad ius of ~ 1 + 21rfit2 at a lar /3 . . t t hi s point , the momentum conjugate t.o a, namel.Y p(L ~an,= 0, and one may a na ly ti cally continue to Euclidean time \\'here the scale factor would reach a = 0 in a finite Euclidean t im e. This is a geom etrical singularity if k =J. 0 because, for example , it fo llows from Eqs. (4.66) and (4.69) that R = 4J\ + 81r.f(/k/:2/o.6) . Another possible motion exists: in the Euclidean regime, a may start at. zero a.11 I grow to a radius of ~ 1 + 21r.f /k/ 2 /3. At this poiut , the momentum conjugate to a is zero and one may anal_vt ically ?ontin11e to Lorentzian time , where t he t,;Cale 1':=1ct.or grows for a ll real time t. llG Real k ? 1 I 1 t? rrn o t? t1 1 e dashed ? n e nse of - I 1 t 1 E ucl1? ,,;, c dea11 effect ?l n~ potentia 1as t 1e o I th , < ? ? rea 1 aw lidean solutions to exist . we rnust li lin e' in Fig ure -J.G. For E uc ?) l (4. 73) f k- < -.. - 1r,\2 h turning poin ts at t hat for .f/ ,;2 ? 1, bounce solu tio ns ex ist wit It is eas ily seen zian 4 2 1 1 __, l-21rjk2/3. How('ver, J ct us consider thC' Lorent ,.. a_~ ( 1rfk / 3) l awl a ely k2 it has a turning poin t locate d at a pproximat , effective potential. For small J 1 t? ( ) t 1a rge f 2 C ?d 1? rn?t 1? a11y large n1Jue of t l1C' sea e actor a. t a l + 21r ? k /3 ? 011s1 er an + 21rjk2 /3 at some e t. The scale factor ca n shrink to a radius l Loren tzian t im ? 11 ? t E uc1 ?d I ean ime t At th? ? nt , I? ? ? alyt1ca .Y contmue o fini te t ? 1s p or f one wishes to an H ? ? ? am1? 1 tornan constra rnt. on t im e one 1n us t ma k 0 a re1at in,1v small viol ation of thr ' ? th 2 nce can resul t from the thermal flu ctuatious of rd rgy differe c o er of fk . T his ene r-like, kgro und is not exactly d e Sitte de Sitter spacetime; or it may occur if the bac . ? . . . . t can supply the neces sa1y but con t ? ams som e grav1tat10na l wave exc1tat10n tha the black h o1 e w1? 11 t?o rm. Th?is n 'vvhere smaJJ ene ? l regio rgy d ?ff?e rence rn the locaI f a_, a+? tunnel to a smaller radius o Would all ow the scale fa ctor to reach a size n relati vely small violati on of the Hamiltoni a ck to a+, make anotherbounce ha in crease to tic contin11ation to Lor entzian time, aud then constraint upon analy b b ? ? seems less J1'k e JY f' or a a Y arbitrar?1l Y Ja enano, 1t rge values as t-+ oo. In t his sc: ? ? ? L t ? en zrnn rse t t? ? s would require analytic cont muat10n to or unive 0 orm, srnce thi larger equiring a considera.uly hen the scale factor rea ches a radius of a_, r time w ~- Tl t 1J e- C-? .d e1 . of l - v J A 2-. ms, violatio ?f Hamd? .n the torna n constramt, on the or . . . . . l l . -. se. , . 1t avoids . t?01mrng a )a ).Y urn ver fi eld mak ble, b 1 es a contr . nuous rnstanton possi (i f X were rea l) cluri11g t he 1 rked j11 [106], a real cJuw ge in C . . . We a lso note that, as rema . t after the pair creation , stan insta nto ha nge iu the gravitatio 11al con 11 could be interpreted as a c 11 7 a F igure 4.7: Phase plot for E uclidean (solid lines) and Lorentzian (dashed lines) so lu tions to the C- fi eld equations of motion. The ?eparatriccs are indi ?ated by bold lines. To furth er ill ust rnte the possi blc a llowed motions , we have drawn th ? phase space for the Euclidean and Lorentzian solutions ill F igure 4.7. A lt hough we have assum d that t he constant k can be either real or imagillary, there is an argument by Lee [102] which would constrain it to I r imaginary. He considers th e toy quantum mechanical s_ystrm of a particl e of unit mass moving on a plane with a potential \ '(r) and real angular momentum Q = r 2(J. T he Lor ?ntzian Lagrangian is given by (-1.74) Aualytically continuiug thP Lorentziall Lagraugian via the pre. cript ion Lr, = - L L(t--+ iT), we have i?2 ,, .2 ;p LE= 2 + - - + \l(r). (4.75) 2 which would be undcsirabl . 11 8 The E 11 clidc'?t ? cq1 1;u1ons of ? ll JOtion ar(' < I) ) i: + riJ2 - T''(r) 0 , (.J.76 d 2 ? . ( 4. 77) -(r fl) () {IT oh ?ing for iJ, we ha ,?e .. where a .? rent iatio11 with respec t to r. S Pllme denotes d1fle . (j (-1. 78) 0 =-2 ? 1' tes tha t the correc t sta Note t l1 t ? ) corresponds to our / ,;_ Lee 1 , a 1 11 Lee 's m ode Euclic.lea11 ? rnotion i give11 by eq uation of 2 T? = 0 . (4.7 9) r.. +Q 1 -- ( r ) r-3 . ) p rovided q = iQ, d ( 4 7 T his eq uat? t? . (-1. 76) a n ion o motwn will fo ll o w from Eqs. ? I ?t1 l l t 11 e , ? ? ? ? ? T l s IS 1 ? magrn a1y. 11 ?S ? l S lll i:ICCorc ance W Which , Per our ass um pLJOns, implie (j ?. . . _ _ 1g11Ja r momen t um 1s . and that a 1 fact t hat Q ? . . . is a cycli c van abl(' rn the Lagrn JJ grnn rgumen t rved ? th . at ion . T l1 e variables Q and q in Lee 's a ?onse 111 - e a na lyt ic co nt inu , we have een that corresp ond t . 0 a nd k, resp ect ively. For imagin ary k our /\ . ave seen that bounc e solu t10ns bounces 0 1 . e h u t ions are not possi ble. For real k, w lation of the Hamil to11ia 11 constrain t s a sm all vio a re Possible provid ed there i ? ? o f' 1 H ?1 ? v10lat10n t 1e 1-u111 torn a11 lii s at the m om t f t ion . T en o the a na lyt ic con t inu a . ngular mo- , the corresp ondrng " a constra in t . 1 that for real kis re ated to the fact e transit ion from re al to th r t h ory is not cons erved wheu ma king ment um" in ou 1m .? ag1na ry t ime. imaginary k . For ces of real and t us fur ther inves ti gate the consequeu Le + 1r.f/k/ 2 /a0 = 4A 8 , For i cci scalar is give11 b_v R imagina ry k th e E uc lidean R fi . ale fa ctor a(r) ca11 be orne zero at rn te imagin a k ~ e have ?een that t he sc ry w ? ? ? fi ? t l e l.r 111 rn te 1crc. 8 11 a r - g that the Ri cci scalc1 r becom es pos1t1v clide 11 une, implyin 11 9 - l ? J ? fi ?t \\ "I1 c'n rn1 l ua tm? g t 1r ct1?0 Hence>, the Euc11?dea11 a 11 brcomes negat1n\ y 111 m c. f urat1?0 n \\?1?1 1 I pos1? t1? YeJ ~ ? ? J JE' - 111 fi ni te y g Path in tegral 1 , ~ h c co 11 tn ? 1) ution from this coll i ?11 h ny other fillit e action coJ J tribu tio11 . For exam- large 'lI1 d "11 eJm a ' c WI ence ovcn\ ? 1 r. aucI R . . a 1s fi 111. te, nnp l _n.n g srngu ple ' for re al k 1 ? becomes , t11 c sea e factor urver . 1 . 1 . a- i11 a sem1 c ass1ca appr oxun that th,, co1? . 1 ? ell ' respo11c 111g action will br finite. Ev imaginary /,- will domin ate. This configuration for tion to the path integra l, the clidean actions are posi t ive, as tl1ey are, distinction to the cas e when Eu is in ories. There, infini te ac - anical sys tems or fi eld t he say, in simple quantum mech ce, s wo 11 .? g iu thC' semiclassicaJ a pproximat ion. and hen tion u c contri bu te uothin are certain tention to solu t ions wi th finite action. T l1 crc one would restri ct at 1? d ? 1 pect t o u be 110r111a 1ze ""' t 1 re s circurnst1nc? 1 ? f i ? e w 1en Ill 1111 te Euclide an act ions ca c ctio11 is typically infinite itself. . This background a some appropriate hrtckg round background , if any, wou ld apply. In t l ic present case, it is 11 o t clear what Hamiltonian constrain t e assumed that 1111der s mall violations of the We hav ? t? ? ?r om Loreutzian to E uc1 1?d ean 1me, th eling fe scale? fact or may b e thought of as tunn field can br considered nd e C - ack again . \tVe should a lso address whether th a b ave been assuming ng with the scale facto r a. In general, we h to tunn el alo .d d rov1 e an hat a E 1?d ti. lu t ion p t I n solu on can be joine d to the Lorentzian so c uc ea continuation. A s we hav e te mom entum vanishes a t th e point. of analytic ttppropria 1 1 ? 1 ? d ulo the sma v10 at10n wn tl to r a, mo sho 11?8 1?s ' i ? nde .e d possible for the scale fac ? fi l I l ? ? wever. 1?s ? ?t) 1 C e that thr - e c can w poss1 of the Ha mi? 1t 011?1 an co11stnunt . H o 1t ? fi to t 1 C - c ld 1. s Pc = a? ? ic? , JC C'Onsidp-r?ed t co1qugate o t u11n el ,! The mom entum . 1 1, 1 1 . tant. In ge11 craJ, tl1i s c onstan t does not varns Which we 1ave s 10wn 1s a cons ifiable to interpret the C nd justth does not seem a us, from this poi nt of view, it fi eld as t 1? le factor. unne 111g along with the sca 120 Th<' van i. bin g of t he conjugate 111 ou1 cnturn at the point of aual.,?t.ic co11 ti11- 11 a t io 11 is one criteri on by which to judge a so lu t ion to represent. a u instanton. Brown , et al. [107] have a somewhat different in terpretation . T hey claim t hat a so l11ti on can b e considered Lo be an i11stanL011 if one c;-rn match t he normal derin1: - tivcs o f the metric a nd field smoothly at t h<' transition point. As indicated a how, t l1i s can be done for t he scale factor. Let us cousici <' r t he C-ficld. T he matching condition requires t hat the norm al derirntive of Lhe field have the same ntluc wh<'Lher approached frorn th E uclidean side or t h<' Minkowski side. T he normal d erivative, n/'o?, must be chosen with the n-ctor 11/' normali zed in t he same way on either side of t he transition. They choose to uormalize in the Lorentzian sense. ( 4.80) At, t = T = 0, v = D/ Eh 0 11 tli c E uclidean side and u = 8 / at ou tli e Lorentzia n s ide. Hence, n = - ?i,o /fh Oll t he Euclid a.11 side and 11, = 0 jot Oil t he Lorentzian sid 0. Let us now calculate t he normal derivat ive of the C -field on each side of t he t ra nsition. On t he Lorentzian side, we have (4.81) where a0 is the s ize of the scale factor at the transition point. On t he Eucl idean size , we have ,, u~,, C 11 = .dC . k -1,- = -'l,1' ( 4.82) riT ao For real k = K , the normal derivatives of th, C-fi elcl cannot be match d and li e11 ce , it is not appropriate to consider t he -field to tunnel along with the scale factor. However , for imaginary k, the normal derivatives can be ma tched. 121 4 3 Black Holes in C-fi eld Cos1nology . .4 stall ton an a fin al st s an e11dstatc of the particle creation i11 As ep W<' exhibi t a , 52 This describes expanding? ? ? - fi cld theory of spatia l topology S x . um verse Ill C . . a t t l1 r 'ariai unin,rse ?1?t l a) black hole paJJ' 1n t l1 C' sa me seJJsc th a " ? J a11 extrem ins tein 's theory. T he metric E [108, 109, G9] describe s such a uni,?crse in solu tion has th h e 0 mogeueous form ( 4.83) coordin ates on S 2 , and odicitv appropriatr t o S J. (} and -- lrns peri fun ction only oft and d . The C-fi eld Jike" ?ise is a a au b are fun ctions only oft b h . to Eq. ( -l .G3) , therefore 0 eys t r conservation Jaw analogous ? . A' (-1.8-l ) C = b2 . (l The fi eJ I ? 1ous then take the fo rm c equat ii + 41r .f ]\'2 J\ _ 2ah -1 A (4 .8S) Gtt + ---- - + 4 b2 0 20 ab 2 f x 2 .. ? 41r .86) ex 2 b b + l ( 4\ +A= -- - -- + 12 a2lJ4 X b lJ f x 2 41r ( 4.87) Goo + J\ ii - -ab \ =-- -- b+ 1 a2lJ '1 a ab b olution , the effects th e C- If the u ? se volume expands simi lar to the . ariai s niver . the fore reasonal>Je to sol ve fi c]d ' ?11 I at Jatr times. It i s there vi )ecome negligible tic to the Na ri ai e condition that the solution be asympto fi eld equations wi th t h oment of b(t) = 1/ JA. We also req11ir e a m Universe, a(t) = (1/ v'A) co sh v'At, ry time) . The solu tio n to na etry (to enable the tr ansition from imagi tirne-syrnm 122 first ord er i11 c = 41rf f t-'-.f3l 2 is a(t) l v IJA\t , - -E 1A IA cosh ln(2 cos h v At) - -? c - 2v.?\ t + ? ? ? ( 4.88) v.'\. 3 8 1 b(t) = - , I -C e - 2./AI +?? ? . ( 4.89) Ji.. G T besf' fu nct ions do uot d iffer rnuch from those fo r t hf' Naria i solu tion fo r any t ime t. However, the differences wou ld bf'c:orne large iu t he cont inuation to i111ag- ina ry t ime, as the volume drcreases . In order to reach a minimum volume we again need a n im aginary A" (virtua l C- fi eld ). T hi s mi11irnurn volume, like all t = const . surfaces , has topo logy S 1 x S 2 and wo uld therefore not fit directly on t he minimum-a surface of a de Sitter-like met ric , Eq. (4.21); a solu tion with less symmetry in bot!J spaces wou ld be needed to make the match. 4.4 Summary In this ba?ter, we have considered two mod ifications to Einstein ?s equations. T he first was the addition of higher curvature terms to t he Eins te in-Hilbert act ion . For simplici ty, we chose to consider only te rms polynomial in the Ricci scala r curvature. For further ana lyt ic tractabi li ty, we chose t.o consider th C' sp ecifi c cases where f (R) = fl - 21\ + nR2 + , H3. In the case of a quadrat ic f(R) , we used t he method of Fukutaka, et al. to show that bouuce solu t ious do exist wlwn n, < 0 . Furthermore, for small turn-around radius, we argued that one could nucleate a Naria i spacetime, s ince for small rad ius, Naria i is locally like R1 . \Ne a lso fo und period ic, non-siugular instantons when R = R0 and o < 0. In the eq uivalent scalar fi eld theory, tl1e fi eld has 11egative energy density. We fo und an a logous resul ts for a t heory wit l, cubic R, with the a lbeit a rtifi cial restriction o f o = 0. Hovvever , t hese soluti ons d id co11 tai11 pathologies t hat d id not make 123 them attract ive as candida tes for connectf'cl insta ntons. For example. all allowed Lhf' existence of disco11nectecl instanto ns , and all allowed fo r the formation of baby universes. T he second modification to Einstein 's equations was the addition of a negat ive energy density C -field. Here again , we? found periodi c. non-sin gular iustautons. However , as shown , for such solu tions to exist , the Hamiltonian constraint must be violated to a small extent . Such a violat ion could occur as the result of thermal fluctuations or gravitational wave excitat ions resulti11g from black hole formation. Iu each case, instanton solutions were fo und when gravity was coupled to a negative energy density fi eld. Wt' showed earli er t hat i11 this circumstance, the E uclidean Ricci tensor vvould possess negative eigcnrnl11es somewh ere on t he ma nifold . However, unlike the case of instanton solutions occurring in spacetimes which are asymptotically fi a t , we know of no theorem relat ing the presence of instantons and the existence of negative eigenvalues of the Ricci tensor in t he case where the spacetime is asymptotically de Sitter. 124 Chapter 5 irtual structing Connected Instantons via V Con D . oma1n Wall Surgery 5 ? 1 Introduction r creatioll tli at arc not there are instances of black hole pai As we saw in Cha pter 3, t i e f'o r t i1 c ? ? ? J tu? ne. S uc 1 was 1c cas described bv -:i co r1t? arv ., ? muous Jl !?S torv rn imagui ? . . ck- f holes rn a cosmological ba creatio n ? ' ? dP Sitter bl ack max11nal Schwarzschi ld- ess describes the anni hilat ion of ; the d iscontinuous in stanton for this proc ground f' S 1 sc h 1'l d -d e s?1tter ent creati ?on o c nvarz de Sitters ? subsequ pace to nothrng and t he ? ? 1 ? g ? te pair crea t1011 rat es rn t 1ese cases usrn space from hnot mg. Oue can stil l calcula . h<' probabi li ty to cr eate the fin al state th, no b - oundary proposal; one diviclrs t initial state from (to) oba bilitv to create (or a nnihilate) th e r from nothing by the p 11 te, al though exponeu tia y ng? I . . mterest, the nucleatio n ra nothi 11 ? most cases of suppressed , is nonzer o. g the no-boundary fo rmu lation fee l that calculating n ucleation ra tes usin We r a rnriety of reasons. For exam?lr. in is unsatisfactory fo of quantum cosmology 11sal; a clisconncctecl th WKB approximation , tirn<' evoJ11 tion is ca e context of the 125 i11 stanto 11 1s acausal a nd li eu ?c would not lw obserYed. Sa? I ? ? ,lC 111 ,mot.b r r \\"ay . ' when calcul at ing nucleation rates from the path integral formalism of quantum gravity, one should sum over a ll conti nuous pa ths connecting initia l aI1 d fin a l states. [f one st ri ctly adheres to t liis formalism. nucleation prnc:c'sses described by a cli srn11ti11uous instant.on should have a vanishing pa ir creat ion rat<', at least, in t lw s?miclassical a pproximation. Fur thermore, the "creation from nothing" formalism seems to im ply that any final state could be created from any ini t ial state; t he fin a l state seems to have no "memory" of from where it came. In this chapter , a nd the 11 ext, we explore ways ill whi ch our can cont inuous imagin ary time hi scories (CIT Hs) by surgicall)' a ltering disconn<'cted instantons . vVc note that these histori es a re not everywhere solu tio ns to Einstein 's equations. Hence, we consider t he co11 sequ e11ces of considering only t hese coutinuo11s histories when c:alculat i11g path in tegrals and nucleation rates. T his chap ter is organized in the fo llowi11g way. We wi ll first discuss how one can define a cont inuous history in imaginary time and its corresponding action . This discussion will lead to an overvi ew of how Bousso and Chamblin [9] utili ze a geometr i1:a l object call ed a "vir t ua l domain wall" to facili tate connecting other- wise di sc:onn ec: tcd instantons. Since the proposed surgery involves virtual domain wall s , we will then give a fairly detailed treatment of domain walls, primaril y i11 the "thin-wall" approximation . \t\Te wi ll then discuss in detail how virtual domain walls can be used to join otherwise disconnected insta11tons. Finally, we will di s- cuss wha t we fr.el a re shortcomings of t hi s rnethod , which we ho? E' ?will rnotivate the di cussion of tl1 e fo llowing chapter 0 11 altr.rnative methods for constructing connec ted instantons. 126 5.2 Defining a CITH and its Action To st.art our ciiscussion , we reiterate what we showed in Chapter 3, namely tha t when a positive cosmological constant is present , there is no continuous instan- ton that call connect spatial hypersurfaces of Lorentzia11 de Sitter and Nari ai spac.:et irn es. From the poin t of view of t he path integral however, 011 c does 11 ot nf!ed a 11 exact , connected sadd le point. Such an argulll cnt was made by I30 11sso and C ham blin [9], wl1 erc they found t l1 at evaluating the path integral over a ll histories except the saddle point and small perturbations a bout the saddle p oin t resulted in only a sm all change in the "prefactor" of the path integral. A review of th is calculation will be given in t he next section. To define an act ion for a coHnected i11stanton , we first assume t hat there exist conti11u ous histories descri bing the nucleation process and an appropria te act ion fun ct iona l. We m ake thi s assum ption sim ply because it seems one shouldn 't try to cure one singularity (that associated with the discontinui ty of the instan ton) with a nother. One can then defin e a continuous, but non-smooth history as the limi t of continuous histories. Depending ou the "type" of non-smoothness, t he change in the act ion in going from a no11-smootl1 to a discounectPd instanton can be eit her con t inuous or di scontinuous, as we shall show in Chapter 6. To construct a CITH, one can star t with th e disconnected instanton , excise a sm a ll four-volum e, and identify along t he ident ical boundaries of the removed regions . As the size of this volume tends to zero , we recover the disconnected instanton . If we constrain the volume to be non-zero , we obtain a connected instan ton that is not smooth. However , this non-smooth instanton can be thought o f as the (one-sided) limit of a smooth instauton . T hu.? , we expect that the ac ti011 of this non-smooth ins ta11to11 is that of t he smooth i11stanto11 in somr 127 a pprop riate lim it. O ne way of fac ili tating thi s ro 11 st rnction , which we disrnss in t his cha p ter. is t h rough t hC' use of what i. k11 own as a 11egat iYe energy deusit_,. or ?'vir t 11al" d om a in wa ll. as was clone by Bousso a ll d C ham blin [9]. \\'e 11ote t.lrn t th en , a re pro bl r lll s wi t h t he Bousso and C ha mblin procedure. whi ch mot in1 trs a n a ll crnat in' co11strnct io11. discussed in hap tcr G. 5.3 Brief Overview of the Patching Argument 0 11 c method for crra.t ing a cont i11 uous hi story from a n ot herwise d iscont inuous Oll C in vo lws p rrfo rr ni11 g some ty pe of '?sm ger>?" on the ma11i fo ld representi ng t he d iscontinuous insta nro u . Dousso a nd Clia111bli11 ca rry out such a surgery using "vir t 11 a l do niai11 wa lls", i. l!., domai n wa lls wh ich possess negative energy density. , irn ply p11 t. a sma ll region is rem oved from each part of the d isconnect ed ins ta n- Lon ; t he disco1rnec tcd instanton is t hen co 1111 t' c: tecl a long t he bounda ry of t hese reni oYPcl regions , ?ia ;:. vir t ua l domain wa ll. O f course, t his 11 ew "connected insta n- ton" is no t a solu ti on to E inst ' in 's equat ions, hu t as B011sso and Cha m blin show, su ?It h istor ie ? sti ll dolll inate t he pa th in tegra l in t he semiclas. ical a pproxi ma tion and res ul t in onl y a sm a ll cha nge in t he p refac tor of t he pa t h integral. on. ider a path integral t ha t has one s ta t ionary poi11 t and ask what conse- quenc fo llow from integrat ing over a ll paths except the stat io11 a ry path and very small p er t ur hati ns a bout t hi path . T he ac t ion cau be wri tteu as (5. 1) where J0 is t he sadd le point ac t ion , p is the second derivat ive of t he act ion eval- uated at t he sadd le point , a nd r/ descr ibes t he sm a ll p 'rturbations a bou t the sacld l po in t. As will be descri bed la ter , 11 will correspond to t he rad ius of t he 128 Yirt11c1 l domain wall. Hence. we rrst rict 11 to hf' larger than the P la nck lcug th : fmtllC'rrnore. it ?hould lw sm a ller than the' racl i11s R of the instanton. T h11 s. 111 P la.nck uni ts , wf' h avC' R ? 17 > 1. Evalu ating thC' path integral, we ha\'e (5.2) wh ere, as we later show , fJ is of order om'. It is cla,imed thar. t he errors from t he approximations made in the cvaluation of the path integral result only in a small ?ha ng;e in the prefactor; the exponential, which is rnu ch rnorf' significant , doesn ' t cha nge. Hence , in the abse11 ?c of disco1rnccted geometri cs . con11 ?ctcd g<'omet ri s g ive t he no-boundary proposal result . A furth er claim of the Bousso and C ham- bli11 proposal concerns the apparent subtraction of actions i11 the exponenti a l of q . (3. 1). Bousso and 'ha mblill claim that this ca n be accounted for b~r t he cha nge' of o ri ntation of t he normal vector wlw11 travNsing the virt ua l domain wall. 5.4 Topological Defects: Domain Walls In this section , we will briefly di scuss topo logic:al defects , how t hey arc class ifi ed , a nd thC' physical processes by which the~? form. T hi s discussion wil l be fo llowed hy a m ore detailed ex posit ion of a particular type of topological defect. t he domain wall. The dom ain wall model in wh ich we will ultirnately be interested consists of a oue-component real scala r fi eld . a Lagrangian possessing a discret symmetry, a 11d a potential fu11 ction with a dis retc set of degenerate minima. \Ve ?will assume t hat t he thickn ess of the domain wall is Silla!! compared to a ll other length ?calcs i11 t he system ; this assumpt ion is knmrn as t lH' "t li i11-wall" approximation. T li e concept of a topological defect lias become useful iu dC'scribi11g phcnolll- 129 ter and cosmology. f t na iu a varich? of ph_,?sics s ucli a s condensed ma e o areas . ine l .? . stall t nde11sed 111 atter sv stern .s rnclude cry Examples of opo og1cal defects 111 co fer- . onductors. and dom ain .structures in dislocation . ti c flux lines in sup erc s , magne . . . . fec ts .? 1 systems. exampl<'.s of topo logical de romagnetic ? t In cosrnolog1cal ina er 1a s. . a l de- . lls. In general. topo logic i11clude mo 1 . . . and dorna111 \\'a uopo es, cosmic str1 11gs onopoles), line defe cts (stri ngs fects are 0 f 1 t? ts (m t 1c allowing types: ?oint defec . tures. Each type of defect ), a11d tex and vorf ) omain wallsices , surface defec ts ( d . l , str ings . . oles are zero-dime nsiona is charact ,? mono p e1Ized by its d1m ens1011 aJity: xtures can be a re one c1? . walls arc two-dim ensional, and te a - imcnsional , dom ain be classified by a f defect can variety f cl? . ch type o 0 1111 cnsions. Furtherm ore, ea ? d cl n o w I1 1 a d e t?e ct, b e 1 c o ngs epen s o T! e 1 topy group t i 1 homotopv group. 1 JOmoJ eory which gives ris e to th fiel d th of the vacuum ma nifold M of the e topology . is disconnected all(J t he homotopy t lt e clet M ect. In the case of d omain wall s, otopy groups are . roup is (Jvt) s, monopoles, and te xtures, the hom g 7fo ? For string . 1r1 (M ) r (Jvt ) tively {11 0] 1 2 , a11cl c 1r3 (M) , respe ' of sponta neolls sy m- re formed as a res lt of some type11 Topological defec ts a tates u-triviaJ set of deg enerate ground s reaking which giv es rise to a no l11etry b ? ? 1 t? o11 ? ' bed 11 t 1c owm . g? Wc:l.) ? 1 ? g can be descri 1 [ IO] ? Sp t b reakrn on aneous symme try ? ? Hsessrng cer- alar fields with a H amiltornan po Consider . cl system describe d by sc f the fi elds in the g round state may tain . es. The expectat ion value o symmetri s is the case, t hen f the Hamiltonian . If thi ot Possess a ll of t he _vmmetri es o n eous s_vmrn etry br eaking tan sy111 rn t . usly uroken . S pon ? e n es have b een spo11 taneo 1 a change in t 1e c undergoes a phase transition , i.e., can o cur w11en the syste m ? ? , ?e d l? try of the sys tem is 1 ucec . me vacuum . As a res ult , the sym d s tate of the system . . a11gc i11 the or er . escribed by a ch this red . . n rn symmetry is quantitatively d uct io 130 ii io11 . the order p a- P?=tra .-\ t t he> point of thr phase inrns of tlir system . . . c nirtc?r . aJ defects . . IJJ ts where i opoJog w rameter is t . 1s at thes<' J)OJ 10 ? ' urn quely d< 'fi ned. H lds. s_vmrnctries, a 11d . can form T l . defrct that forms depends 011 rhc fi e ? ie t,rpc> of topolog?y f tl e s.,?s tem. 0 1 the vacu u111 manifol d of 1 1 ? ? srn o I ? ? 11g 1t to t 1 ? fect ? 11 a co og 1? ca 1 SC'ttmg 1s t 10 The formatio 1 n of ' opo og1cal de For cx- ed rnatter systems . Parallel cl . ens 1 Y t he format ion o f defects in cond ose . ... model. the 1rn tial s tate aniple, let , . srno logv. In t l11 s us assu me a hot b ig-bang co nd possesses c1 par t icular group of of the u ? , . h ternperatmc a nivc r e has a very hig . thro ugh A 1 . ds and cools. its tem perature passes symmetries ? s t 1e urn verse ex pan ? r. At these points, ns occu various . ?r ph ase tran sitio en ical temperat ur es Tc at whi ch eously broken, leaT ing th sent in the uni vers e a.re spontan e init ia l symmetri es pre behind .? 11 . 112}. var ious topo logica l defects {11 0, 1 quent pl1 ase transition a ll(J the subse get a better idea fo r the concept of a . . . To . of an . . spacet1mc theo ry cons1st111g forrnatio11 0 f top0Jog1cal de fc~cts , consider ,1 Hat . angian co11pled to an O(N) invarian t Lagr N-co1n ith Ponent r ?al sca lar fie ld ?> w l ? mmetn?c ? , B {l 11] Ttn x ? 1e ma 11 t ( N - l) c t, or fi elds represented by th e ant1-s,v N ve ? t I t? ? ? heory has 1e orrn ? rng tl11 s t rnost gen ?a 1 . ~ li?z ble Lagrangian desc nb er 1enorma a (5.3) + e [B, B,,] , and e and .\ a rc , 1 , Where D ,.;.. _ a - a B JI _ fJ /1 B I/ /4 e B /t(,/I.). B /W - // 1 t? 1t 'P - ?

) , i .s the order par o O(N - 1); tli e va cuu m oken t ry of the theory is spontaneously br O(N) symmet ex . ? Pectatio . u va1 u e of ?> 1s givr n by (5.4) 131 , . ero tempera ture. the where th . has topology 5 N- l . For 11 on-z ? e vacuum manifo ld . . . vac u11lll ex ?t . ue of rp 1s gwcn by , pee ,at1on val . . Where the . ? ? . 1 perature 1;_. 1s g1w1 1 by ? cnt1 ca tem (N+ 2 N- 1(' 2) - l/'L (5.6) , - A, T ---c-'f'o --+12 4 ,,\ is ken and a disorder ed phase For T > Tc, tl1 e symmetry of the th eory is unbro an ordered is broken to O(N - 1) and ent. For T < T,., ry the O(N) symmet Pres N- 1 . . . l11cl1 has topolog_v S . Phase is J)r t esen , t 11 e vacuum mamfold of w . (S.3) provides the s implest model l case of the theory described by Eq A specia explicit form of th e of a dom . der the N = l case of Eq. (5.3). The ? am wall. Con si Lagrang?? ? ? I an is given by (5. 7) ossesses e-component rea l sc alar fi eld rp and p th an consists of a on o ? Bere, e Lagrangi . . has topology 5 con sist- the ref! . ? . etry rp -+ -o 0 11 011(' side in terpolates between an asymptotic va de. No\\' , consider th r effect of squeezing ptotic value ?0 on th e other si an asym ouble-well structure he minima in the dth oge er the potentia l fun ction so that t t ting the minima wi1l become taller and er. T he hump separaapproach one anoth . i t, the field wi ll beha ve like a thinner h . th is lim approac mg a delta functwn. In -rj; evrr,vwhere on one side of the st d will have the rnlu c 0 ep fonctiou ; the fi el In this limit , the domai n wall will be an and - fJU 0. dz2 8?; . 'hc first in t . l o f?' t l1 1 . . 1 egra s cquatw . n o t? moiwn 1s H!!)' - (5. 9) U(?) = 0 stress-energy tensor T,~' is defined b_v 1'he T'' = --B-L - fJ ''L ,1q; + (5,1 ? [) ( [)/I rp ) II 133 7). and using Eq. (5.9). gives ia11 in Eq. (5 . Evaluating Tt fo r the Lagra11g , 0) . (5 .11 ) Tt = 2U(cb) diag ( - 1, - 1, - 1 . thr stress-c11 ergy tP usor has a delta th type di scussed here For in walls of the ] wri -te fun ction s? 1 ? y 1o catecl at the wall [1 13. Thus, ff(' can ? rngu an t (5 .12) ' = ac5(z) cliag(- 1, - 1, - 1, 0 ) , T,~ - ? alJ , as well as its tcn sron iu the e w where ? h ? rface energy density o f th a is t e su m ions [11--l]. Hence, t li e pote11 tial has the for tangcIJ tia l direct (5 .13) 1 0] to describe a thin domain given i11 [1 \Ve will now fo llow th r presentation ? 1 ? t? t1 1 c wa 11 t 1) the spa tra section o e th a Wall in c m ur? ve d Spc.Jcetz ? me. \tVc will assu e the .:--axis at ;; = 0, wh ere z is a spacclik nar and located orth ogonal to is pla ible compared to its rd ckness of the domain wall is neglig i coo inate, 2) the th 1? ? ? t p1c r?n i ? ts wo spa c 1 ?k e other din1 ) hom ogcneou ? and 1sotro ens1? 0ns, 3 the wall is ? ? ? 1 mm etn c wi t 1 respect dimens? ) - 4 the spacetime geo metry is rcfl C'c:t IOn sy ions, and to the wa ll . wall will l>e described by l worldsheet of th C' do main The three-dimensioua (5. 1..J ) 1'hc L - . orentz1an action wiJl b e given b_v (5. 15) Wi th . matter Lagrangian (5. 16) 13-1 ? J t? ? n wi? ( ti o t lI a eal scalar firld. U d> ) 1? s a po tentia un c whrre r,) is ?1 ? c onr-cornponcJJt r ' . . . . nd . ut of the 4-mrtnc g w, a 1 disc:retr set . tenn111a ? ? 0 1? l ate 1111mrna , g 1s the de c cgencr om the ress-rnergy tensor T,"' fr ca n be drtermined R is t he Ricci scalar. Th<' st expression c5Lr nat + L (5.17) T. = -2-A-- 'J11.11 111af ? I.U / ugiw . For the b L . a ovc agraugian, we have (5 .18) sian norma l coordi-the Gaus icinity of the worJd sheet . we ,viJJ usr In the \? f the wa ll. In these z is the normal directi on to the smface o nates ((u, z) where coo d. r m ates, the me tri c h as t he form (5. 19) I 2 = I i,( " C/ (1; + ( I ,.~ 1 _ C 8 lab< Wh ere / ? t he worldshee t m etr ic 1,11,1; is (5.20) / I I/ I lab = !J,,,, :r 0?7; b. ' ' ? ? ? b iven Y he SJ)acet? J c element in the 11 ew coordrnates 1s g T un e vo um (5 .21) wlicrc h = det (h . ) a/; . he trace of t/1e vari at ion rm of the actioll Sdw ? T We can uow simpli fy tlie fo of S ? h resp ect to !J,w yield s dw wit (5.22) P lu .? . . rng th is mto Eq. (5 . 15) gives gg p)F9d4/ u(r .1; Sdw 1 / U(rp)H, r( (dz 135 , ? . . . . r fi elds appearing rn . The llext s ' ? te that rn the thrn wall lurnt. the sca la ? ? ~cp 1s to no infini te planar wall discussed abo,?c 5 G) are t l1 c sam e as f or the Hat space Eq. ( .I t it11ti11g th is as the form given i ll Eq. (5.13). Subs [llO]. Thus, the potent ia l h expression f U( ,") ? or rnto Sc1w gi ves 'f' (5.23) 1 ;? -,: :i Sdw = - (J V r - h r/ ( . 2 the domain wall will surface stre.ss-eJJ crg y tensor S? ,/ of For our purposes, t he l fi . . t? S r Frorn t 1e c Ic Ill tJOJJ o 11/ 1 oc more t1,~,? e~ t? 1 l ,v. u t Jan the stress-e nergy tensor T, (5.24) rgy tensor has the f o rm Eq. (5.12) that t he surface strrss-cne it fo llows from (5 .25) 51w = -ah,/.tll. 0Sv7 = 0 t liat w11 from the conser vation law h wh/V h 1t Fur crmore, i t can be sho events 0 11 the three -dim ensional lue at all is a constan t, i. e., has the same va a tin1eJike ' ld h s cet of the domain wall [113]. ~ or ric (5 .19) describing the to specifv the coordin ates ( of the met We would like . .s ? f? . . t?1 p 0ns , . at a coordrn ate systP m sat1 y 111g ass um domain 11? ,~ mg th wa ' ve star t by not at takes the form [11 3] one, three, and four has a geometry th (5.26) pa tib le with t he stre ss-energy tc 11 sor fo r and B that is co m 1,1 To find a sp ec ifi c for m the wall. Tlrns , vac uum rom (5 .1 2), we note tha t T,,v = o away f given iu Eq. 136 ipJds t J1 e four eq uati ons Eiust<'in ?s ? . t ions 1mplv R O This condition .,? ? C'(jua . /II/ = ? - B,11 + B ,:::: () 1 B = (S.2 8) ,B - - BI JJ - - ' JJ I 0 B I? - -2BB ?- ' ' ? ,- . ~ 2B 1 +B- 2 '2fl, l i t - 2B. J . () (5.29) + B ' ,~ - ? ' ,- ,- B .tt ' z? - ~ 2B ., 2 B B ,:: '2 0. (5.30) 4B - /J_ /1 + IJ. :::: + ..J.B2 - 2 . . E q. (o" '> oJ11 t10 n to .~ 7) 1.s . n c s T!ie mo.st' g?e ne~r. a I rC' f?l ection syrnrnc t (5 .31) B (t, /z /) = F (t - /.: /) + G ( t + /z /) , .28) . G . s. Substit 11 tion of Eq . (5.31) in to Eqs. (5 Where F d an are arbi trary fun ction d ? te m? to two 1 .s tm? ct ? s sepa ra (5.29) an I (" 30) yi ?eld equat10 ns whose solu t1011 ' c o. . 0, bu t not both, G' = classes CJ . I 1 . t10ns are charactcn ztd by F ' = 0 or ? ass- so u th respect to its arg u- notes differentiation of t li e fun ction wi Where the prime de aml G' ::/- 0. It has been F' ::/- 0 ass-II solu tious a rc cliaractcrized by nicut. Cl fl at in the vacuum off the wall. It has re wn in [11 5] that all Ja ss-I solu tions a ho thout loss of class-II solu tions a re unphysical. Wi 1 3) that also been shown in [1 = 0. In this case, the lu t ions fo r which G ' generali ty, we will c onsider class-I so Hon-van? J ? . . ? tn c components are is 11ng me l F ' (t - /z /) F ' (t + /z /) 3 2 (41rCT/ F 11~ (t - /z/) F 1 (t + /z /) F (t - /z /) , Where F > 0 and F ' > 0. whicli is ic form for t li e arb i trary fun ctiou F specif One must now find a n ction exists bl . term given in Eq. (5 . 12) . One such fu cornp ?:i, f 1 th e source < e with 3 orm [ll , ll 6, 117), and i s of the f (5.32) F (t - /z/) = e2,,.(t -/::/) 137 ... wli0re ,..,. - 2 is gi,?cs rise to tlie metri c - 1ra. Th r 2 + dJ/)] . (5.33) ds2 = e-2;,:/z/ [- dt2 + dz2 + c2"t (d . . Under th ?d ? ooi mate transio rmatwn e c I I- -t -1 (1 (5.34) - (' - 1, /.:I ) Z fi the met.? k le ta es the form I (5.35) d ? E (' . 19) ? t t h ? t ?? c 1?s o f' ? CJ . u te t ha me n the fo r m we ass ume 111 We no is r: 3-) ? 1 ? ? E qs. (-o. 33) a11 d (u. o . = st. surf?a ces of t 1c m etn cs 111 .. Now ' cons1?de r z co1 1 h lf of (2+ 1)-D de Sit ter spacetime wit tri c to ha The re. ulting space time is isome metric (5.36) spatial sectio11s, lik e A 1.5 etrics with fl at ell known , coordin ates resul ting in m w . . s d lly cover de Sitter spacet1me, an those 0 f E qs. (5 .33) and (5 .35 ) , only par tia . nate . ain wall spacetime. Under the coordi hence w?11 1 1on .Y par tia lly cover the dom ,. tra t? ns ormation [118] ~ h (,d) cos x + sinh (1,;t' )] t ln [cos f;; 1 cosl1 ( r,;t' ) sin x cos 0 X -;, cosh (K,t' ) cos;_+ s i11h (K.t' ) 1 cosh ( r,;t' ) sin x sin 0 y = fi:t') -;:, cos1J ( K,t') cos x + sinh ( th es e metric in Eq. (5.3 6) becom 7) ( 2+ sin 2 1 d0 2) (5.3 2 2 ("'t') dx x ds = -dt' + - coshK, 2 138 c wri tte!l in coor din atrs cet irn Wh icl . . 1)-D de Sitt.er spa1 IS Just the m rtri c for (2 + 5 2 x R. T he . . e topology of t his metric is t hat cove1 . t l , ie en t1re space t l!nc. Th T l ac1 . t? . 1t1s . at e 1 instan t o tu11e e topo logy. cl o111 a in \\":tll 1?s ted to have the sa m c c ? exp e 1 2. t' loiu a i11 wall has t he topology S ' t l e c ther w;:iv. Collside r tli e n in a llo of thr domain wall can !Jc? see The topology coord? . n mate transforma tio ~ l /2(t _ ;:;) = ~(';;(! -= ) T - Z _ p 1-,? 1-,; + ,y2) z p- 1;-2(t + z) 1,: F' l2(t - z) (:i-2 + T + -~K (.r2 + fl) -} e - 1e(I+:: ) + 1-,;e ' '( t -= ) :r F 112 (t - z) = :re"?(! -= ) X p1 /2 (t - z) = yc/\"( t -= ) } . V ? ? ? d. st ?o mate tran rrn at1 0n n wa ll ; a srnular coor Which cov i? th e e z > si ? 0 de of the dom ai tli ese coordinates , the < o side of tlie wall [1 13]. In or the z can be obtained f e rfinkowski form 5.35) take th Jnetri cs of Eq. (5 .33) a nd ( 2 (5.38) 2 2 2 '2 + d} ' + dZ . ds = -dT + d.\ e equation i wall previo11slv desc ribed by th th of the doma 11 Fur ermore, th e location e s li ret z :::: 0 is ? g iven by th e liyp erboloid of on now (5.39) ph id i. ? the de Sitte r metric. A gra c induced on th is hyperbolo I h. Where the metri F re 5? en in igu .1. n t 1s , 1. giv his fu t? wi? th the y-coord inatc suppressed of t nc 1011, the in teri or of the ed bv fi . of the domain wa ll is represen t e gure, the z > O si d h.vperboloicl , i. e. , 139 J t hC' y-coordinatr i s t uall.,? a 2-spherf' w11 eI hi ch is ac Consider the T = 0 circlf', w e lrnvc not s uppressed . At T = 0 w l (5.41) Z = --,,.{ '; . ( t -=l . t hat Z < 0. T hus . u11 d er '(t - z) > O, we 11otc From thr coordi11c:1 te restriction F IJ wall is bent in to a th i\1inkowski coordi11 ates. t l1 e domai o e t ransform ation t 5. 1. If one analytica lly ere, i.e, half of the T = O circle of Figure segment of a sph emainder of thr sph ere. p the r nd hi s solu tion to F'(t _ z ) < o, 0 11 c, pi cks u exte s t omain wall. Since i c d This sph mpletely en ?loses t hr ;; > O sid C' of t l ? ere now co . . = tli e :: < 0 side of t l1 e the spa. t? . ut .:: 0, e ime is assumed to be symrnctn c abo c . . the ed w1thm a sphere. T hus. to construc t domain JI 1? .. 1 so completely e11 closwa a perboloid th e interi or of the h y full J)a t ? r two copi rs ofS ? ce ime, one glu es toge th e th respec tive boundar ies. a long eir the domain wall i11 Minkowski observer 0 11 either side of t l if' wall, To an ec this most clearly by en .. 0 11 e can s rd be au accelerating sp h coo inates appear t o ? F. " 2 . H ll shown 111 1gurc o. ere, an r ? t? domai n wa consideri ig a con orni a l diagr am of the L }'2 z 2 T l ie . here R,2 =.,.\ " + + ? inertia] ob . .v e1. i? . nstaut R, w ser s denoted by a lrne of co y null line passing th ro ugh R = 1/ i;, wall is denoted by an asymptoticall domain accelerating toward ere = ote that for T < o, the obs erver sees a two-sph at T 0. . from her for T > 0. ay her 1 1 ? . , and then acceleratr n g aw ' ia tm g i ts collapse at T = o grt li er opies of thi s co11 forrn al di agram glued to of two c Thf' full spa,Ptirne c o11sists :et i11H' a ppears as in Figun~ 5.3 . Not<' a long? t l d . g spa c ie om arn wall. Th r resul ti11 1? k e . 1 fi m? t 11 .. v . that this spac t1? me 1ias no space ,e 140 T I F'>O F'< O Domain \ Wall I z X T=O circle sed . The 1 eet with the y-c: oordinate suppres Figure 0r.: id of one sh . ? : Hyp erbolo . . of a domarn wall s pacct1me. n in terior 0 f l Or egio 1e hyperboloid rep resents the z > t f this hyp erboloid , one glues toget l1 er two copies o irn e T'o obtain the full spacet alo l. ng th e domain w al 141 Domain Wall - -- - -- - - T=O R = 1/K I , ?? F igure 5 ? 2?? C on f'o rmal representa tion of the z > 0 cl ornaiu wall spacetime. T he heavy line reI) resen t s t l1 e ti. me evol ut1. on of. t he doma1.1 1 wa1 1 . f-A\. t eac1i 1 m. st ant o f' time ' t i1 e d om a m. wall is a two-sphere . For T < 0, t he wall accelerates toward an inert i~a I b s.e r.v er nnti.l 1. t reaches 1. t s m1.1 1.1 murn rad.w s of? 1 / r., at T = 0. It then O re-expa nd s away f.r om t he o bserver. I ?? Inertial Observer '-- --- - -- - --- / T= 0 Domain Wall ? : Conformal diagram representing the ent ire dom ain wall spacetime. Figure 0i:: 3 The regions to the left and ri ght of the domain wa ll are Minlw wski space with space?l ik e 1?1 1 fi m? ty cut away. 142 for a Do111ain Wall 5 on.4 .2 Junction Con diti I ? a 11 . F E "l 'OJJ1 CJ. ( 0r.: . 1 '.J.. ) , ? d ? 1?o n t?o r i'l C 01!1/'1 111 wl 1t HP re ' we will d ? t 1l e J?L S IIl Ct 10 11 CO ilSC llS ?u ? ? esses a deJta- f n ct1011 poss we see th h marn wall at t e s tres:-energy tensor for a do ? ? ? 1? ? to co11 ta 111 a Jum p c 1sc ont1- urr sing uJari t H insi c curn1 t ,v ? encr , we exp ect the extr ich e t he norllla1 vector with respect to wh ain wall. We tak nuity across the dom m the + side of the domain fro th iic curvature X ij is defi11 ed as poin t ing e <"xtrini: 11tit_,. o the - side. Next, w e introduce thr qua Wal] t (5.42) tensor for t l1 e doma in e form of the surface stress-energy \i\'e now wish to derive th the contracted form s by using E instein 's e quations (3. 7) and Wall. One can do th i of the G?, uss C I ? - oc azz1 ? equ atw? ns: <.t (5.43) 3n + J,' ? b J, ?ab J, ?:.! - \ 11 a \ (5.44) I ?th J, ' = ? ? ? c w1 , .9a.b, \ ab D ; bn is the covariant cl en vat1ve associate Where = la vu where v'a a wall , k = l\?r, a , :in is the e domain = f.b i the extrinsic curvature of th l(bn D a , a nd eometry h, , G is th e E instein tensor re of the three g 00 00 Ri cci scalar curvatu 1 ,? 11 at ti s11r 1 ?a t 1r rr r,u ? ? normal to the wall. O ne can then s 10, is tl ' i c urnt spacelike J tress e takes the form [119 , 120 s - nergy tensor (5.45) f' equation yields e 'Y = 'Yi- Taking the trace of the a bov Wher (5.46) 143 1g Eq. (5.25), ,n, ll i?ffC' tlw IsrarJ it u t ing? t1 ?, 1. .. . bs t -15) aJJd us i1 Su 1, 1 s urtc 1\ into Eq . (3. n JU1J ction condi t io (5. -1 7) of a Domain Wal l 5 d4 3 Fiel ? ? Repulsive A cceleration aiu wall discussed t the accelera tion fi eld of th e dom tha We arc now ready to show . . . -gure o.2. the world . tha t rn Fi a bove is r l ? . s ily seen by notrng be ea epu s1ve . T his ca n . . repelled . To e dorn arn wall a nd 1s then line of ? . -. roaches t l1 a n iner tial o bser ver app y t l1 e Gauss- th? . athematical conte x t, we will emplo Prove mis re u 1t 111 a m or e Coda2 21? f orma lism , as done in [11 3} . s. (5 ..13) a11d (5.4 ..J. ) on es of Eq s t by taking the sum s and differenc \r\'r a r t . -1. 2) and (5 .45) , w e get the four l , and using E qs . (5 1vva l opposite sides of the equations (5.-1. 8) hr,cDbs cb 0 . . (S. -1.9) / D 1,1'"\red -. - Dn. A 0 , lar (5 .50) k s au 0 , ab 2 ( ,5 sau - 1 s 2) , (5 .51) 3 2+ ( - 16 -lr ab R f) is a potenti al fu re .,,, is a real scal ar fi ing proba biJit_v ra tes, tl1 e rntc for calculat a. According to our prescrip tion ninirn t? 1 c E 1? I r xp o11ent o t 1 uc 1c ran ?[ ' wall will be propor tiona l to the e of creat' i on a domain uations of motio n . t the solution to the Euclidean eq actiou evaluated a for the E uclidean all approximation , the expression -w In the context of th e thin ith the inclusion of 23) w given in Eq. (5 . action ? ? f the action IS Just the negat ive o a cosrnoJogicaJ te rm: 54) , r, :; .1, .. (5. -1 ;? r 3= - V fl d ( - - A ; ? V g G1 ,1, , f c1w (J 87f 2 e on the i11stanto n, 3 um e of tl1 e S rid g Wher ? egr ation of ,/Ti d 3 ol( gives the v e int 147 and intrgration of fa rl' 1x gin?s tl1 e rnl urn r of the ell tire lPns inst ant on. This rid ge is located at rad ius 1? = l /(27rO") from th<:' center of either fom-ba ll. T hus, t h<' rnlumr of the ridge aud the instanton its<'lf a. re given b:v YOl(S3) vo l(S'1) (S.SG) Hence, the act ion fo r nucleating a domain vvall has the va lue (5 .57) 5.5.2 Joining Instantons by Domain Walls A jump in extrinsic curva ture across a domain wall , like t hat given in Eq. (5.47) , can be used to join t hr two parts of a di sconnected iustanton by "surgery" . \Ne remove a small 4-ball of rad ius 11 frorn eacl1 i11s ta11ton. Their two three-surface boundari es have the same iutrinsic geometr~?, and their extrinsic curva tures are proport ional to the surface .metric. They can therefore be joined together in such a way as to satisf:v t he Israel matching condi tions, Eq. (5.47) , t hereby inserting a domain wall. However , th e surface energy density of t he dornaill wall used to join the i11- s tan to11 must be negative. To see t his, consider the instanton describing the nucleation of Schwarzschild-de Sitter black holes in A cosmological background . As de Sitter space is a nnihilating to nothing, successive three-spheres are shrink- ing. As Schwarz child-de Sitter black holes arc nucleated from nothing, successive three-spheres are expanding. However, for the domain wall instanton constrncted in , ec. 5.5. 1, as one approaches the domain wall , successive three sphere. a re expanding; as one rec des from the domain wall , successive three spheres arc sh rinking. C lea r ly, one' cannot use a doma in wall ,,?ith posiLin' e11C'rg_, - densit_,. to c:01rncct t h <" di sco11n0ctcd insta nt.on representi ng Lit e fo rmation of Schwarzschild- d e S itter black holes. However, if one uses a dornai 11 wa ll with negati,?c sm face energy d ensity, i. e., a virtual domain wall , t hen th is is possibl e. Note that a vir t ual d omain wall is not , ?ir t ual in thP sense t ]iat it corresponds to a E uclidean solu t ion of t he equations of Sec. 5.5. 1, for t'11 c ) + : ] Jrj 1 d' .-r . (5. 59) 149 I I Domain Wall ain wall . yo-yo" instanton for a virt ual dom F igure 5.5: The " . . . lidean actwn for the vir tual E (5 ) ave that t he Euc Using qs. .13 and (5 . 58), we h do . 1n ? am wall 1s given by 1 J\ (5.60) -- Id --- -- 487fJ(J"4 . v w - 87r0"2 d ? 1 te rnstan ton t 1en connec so a nd C l1amb11?1 1 c 1a im that the acti on for the Bous becomes (5 .61) 11 req ui res expla- tl b e twice IT. T lie fo rm of this ac tio and will b ie ounce action . . . gn between IN a11 d Ic1s- T he nation . F? t . e relative mmus sis t h Irs , we will addr es f I 1? ?? ? nng-e o 011en ta ? t10 n c r [9] 1 ? t thi? ? i?g n i?s t he resul t o a a uthors of c aim tha s mmus s Tl ? ? ? - n? 1- ses 1s on cntatwn cua nge a Pass? tl ugh the virt ual do main wall. 1 When mg u o ed fo ur- ua11 , tJ-1 e no rmals of which trad with each re mov te from first associa t ing a 1 to match at t 11 e do main ot! 1 iu ting. If t he tr.t rads a re are b 1 c JOsen to be ou twa rd po be reversed, !Jene e the relat ive rads must th ri en tation of one of the tet Wal] , e o ce of the vir tual presen e is a small corre ction due to the , t her minus sign . F ina lly dom ain wall. 150 T h C' general form of t hr boun ce act ion ca ll br written as (5.62) with I0 = 2(JN - Ids) and p = 1r. Bousso and Chamblin claim that there should be no volume terms in the actioll , provided the removed fo ur-balls have equal action ; hence, there is no 17" term in t he bounce actio11. T he reason for this stems from the ori entation reversal needed to match normals. T he resul ting subtraction of act ions cancels the vol ume terms. 'vVc arc now interested iu the consequcn ?es of summing the path integral over only connected paths; we disregard the disconnected instantoll and small p erturbations around it. As we have ment ioned before, the radius of the virtual domain wall is taken to at leas t as large as the P lanck leugth, 17 > l. The path integral is then (' - l o 1? d11 e - prJ2/2 11 > 1 ~ {?- [l - N( Jp)] e- Io . (5.63) Here, N(x) is given by (5.64) Note that for p = 1r , we havr. (5.G5) 5.6 Problems with the Patching Argun1ent In this section , we will address "vhat we feel am some furth er consequeuces and shortcomings of the patching argument proposed b:v Bousso and Chamblin . 151 e ori entation o f the chauge in th F irst 1. e 11 0 t li at t li ere sho uld b ? ,, WP uclicve me limi t . that th e . in ?.,va]l. Imagin e, in so tctrad when t domaraversrng the v irt ual v, or that it be hm?cs ected ? h at the conne cting boundar conn rnstan ton is sm oot fiJJi te interval [122) . Given a like a . r 3 re I isr with topology I x S , ,,?he cy m de ori entation. eith er witl1 a defini tr thi s m ? r e ctor can defin e a n ormal v ? aniiold , one . . e no-bo uJJ dary ? ro- n , t li ard or . . . ce tl11 s uorm al I S chose 111w g. Onou twaI d p orn tr n t normal. darv terms with respect to tha n PosaJ stiJ) 1 t 1 bou . ' u a es t 1at one calculates .. the cancellatwn of vol- . . 111 herrn on cntatio11 sho uld not resul t Furt ore, a change o f . e element is defi n ed . tcgral, the volum ume terms W l rng a volum e rn1 ? en calcula t would involve t he coordinate syste m to be po ?t? mation to anoth er . . ? si Ive; a tra nsfor lrn gw e no well ? . Lastly, Bo 11sso and Chamb absolu te v l f the J acobian [122). a ue o . . . of the rnstanton should be re? . h . fo r the de Sit te r port10n defin ed ,ason w Y the a ct10n instanton. T he ir r the Nari ai b tracted in Eq. .61) , as opposed to the action fo U (5 . . . e two terms . . rnus sign betwe en th Proposal s to only s ti pula te a relative m seem 1), ne can add ac ti ons in Eq. (5.6 22) that o m blin do admi t [1 t Bousso and Ch a though they do JJ O . round , a l Provided lizes by an app ropriate backg one norma . . didate. namely two . v1011s can ggest . .fi . malization . We suggest an ob su or ' cl sp ecI c n . . ve ua1 domalll wal l. T ims, we ha , v1rt de Sitter 1 If 54 a Ia - s conn ecte d by (5.66) e action contain s i ar in that th reasona ble resu l t, bu t is pecul his gives a phys ically T ? a1 1 T l 11?s 1 ?s one i ?s sue rn w . 1 e vir tual dorna no con t1?I?b f' u t i ? on rom tl1 e pre sence of tl e a norm alizati on e nex t chapter w li en we defin We ?11 in th WI address m ore fully Procedure. ment involves co nnecting a ing argu onsequence of the patch One interesting c 152 cctecl iustauton de scribing t ring of ? t her. F irst. co11sid rr the com1 s ins a ntons toget ri ai . .--\s we liave a lf a Na 1 th e crcat ioJJ of h ihilat ion of a half de Sitrer S- and t li e a uu ng to t he Nariai . im agi11e connec ti seen the ''c?t ? . . 1s given b_v Eq. (5 .61) . Now ' c, , 10n e nnected instanton describes th side of th ? ? ? n . T his co is mstanton , a11 ot l1cr instanto tter S 1? Tl1e actio n creation of half a de Si and the annihilation of ha lf a Naria i for th? ? ton is given bv is mstan (5.67) that is created n for the spaceti rn e Here c n tion t hat tli e ac tio ' we iollow the c oJJ ve is given a for the uni verse t hat is destroyed 15 nd the act ion given a positive s ign a s via three bo 1111 c es. ariai black hole rninus ? n of N ign . To describe the pair creatio s action ? ? th is ins tantOIJ. T he resuJt iug . J we must co nnect t 1e on ? gmal rnstantou to for , n such bounces is (5.68) + h) = h + n fvdw ? I = h + n(h,, s pect multiple bo unces to be Jes that we ex This resul t d oes seem in t ui tiv e in bounces possess au action robable th ? sm? , three P an gle bounc es. In t l1i s case leading to a mo re , b ounce, th t ? reater thau a single a is a factor of nJ vdw > O g nucleation rate. ppressed exponentially su virtual do- s proposal is the size of the er problem t hat exists wi th thi Anoth al cl omaiJJ wall is ass um ed to the instantou . T he virtu ain wall used to connect ne can the11 ask m nects. O he radius of the uni verses it con t be llluch smaller than uni verse with b lack holes to e for the process of nucleating a Whether it is pos sibl e size of the co11 11 cc- e. In other words , th any "m em ory" o f its ini tial stat have ? . I d s? t er n contarne d m? t1i e 1? m?tia e i t ow the informat io tion do es not app ear to a lJ ome seuse then, i t cess. In s ihilation-recreat io11 pro spacetim e to surv ive t he ann 153 f us i11g qua nt um cos mology faced with t he origiu al problem o 0 11 ai11 SC(' lll s e is ag . . name]_,. lack bole pa ir c:reat10 u rates. and t he I e b no- Jo unda ry proposa l to calcula t r nclent a11d unrelate d th ed a nd wliat is destro vecl arc two inclcp at what is er at Processes. g a rn ed_v some of these p roblems h." clcti11in eek to re Iri the next chapter , we s ive c?oH11cction topolo gv. and a u alternat new 11 onnalizatio11 proced ure 154 Chapter 6 osal and Conne cting g Prop Alternative Pa tchin Topologies 6 ? 1 Introductio n ? ' t1 1 0 d t? s11r?g 1.c a 1 1 y con- sso a1Jd ClrnmbJ rn s me o In ChaJ)tc, r? 0,.. e c 1 i? s w cussed Bo u , ? ( TLJ) d esc r ?1? 1J r?n g? b l ack 11 0 1e n11- ? 1? ston es CI 1. structing co n t? e h muous 1 ? magm? ary tim h d pro bl ? 1 t h ei? r met o . i t I i eve a re some ems w cleation n ' J)om ? vve ted out what we bel ? eral formalism fo r surgically ent a more gen th ttemp t to implem In is chap ter, we a ing an identical re- w formalism inv olves first excis st ne con ructing CI THs. This he ifold describing t he i11sta11 ton. T section of the dis connected man gion from each IT FJ . t l n cre- g1? ve11 topology. T lw C 1 1s 1e ? ? n wi ll have a uncla O f t h1.s excised reg10 . r l J te t l bo ry d . .1? r 0ns. 1. 0 ca cu a w . f . he boundaries of L11 e remove 1eg ated by ' ng t? 1'd en t1 ymg alo more, at actions a re ad di tive. Further th f such a n instan ton , we assume ction o t ion without the a e connec sic curvat ure at t h accommodate a jump in ex trin We can action in to the t otal action . ditional matter addi t ion of an a d o normalize t he actions of efin e some proce d m e t e wi ll see tha t we m ust d t t 11 at ti . W ? t to no c , 11s me backgroun d. It is 1mporta11 the CIT H w?i t h respect to so 155 norma lization procPdurr is not implemented to make the act io11s tinit c. since t he instantons a rc compact a 11d haw finite act ion. Rather. we normalize Lo arr iw at ph:vsicall v sensible resul ts for nucleation rat es. Once \Vf' have la id out the general forma lism , we apply it to situations where t lw connect ing mauifold has topolog:v 5:J or 51 x 52 . Our rnoti \?ation for exploring r1lt<-'rnative conuccting topo logie. i. t" ?o-fold . Fi rst. we note t ha t if CITHs a re argued to be flu ctuations about discon11 cted instantons. we see no reasou wh:v t hese flu ctuations should uot include co1111ec:ti11g manifolds whosC' topology cau be of various types. One ca11 determine which co1111 C'cting to polog_v is prefe rred by calculating the ?orrespo11ding E uclidean ac:tiou for the CTTH and implement- ing th no-boundary formulation of quantum cosm ology. As we have seen, the prf'fcrrcd con11ecti 11 g topology will be the 0 11 <' corresponding to the least (most negat ive) act ion. As fo r our ecoud motivation, recall that within t he Bou. so and Chamblin for ma lism , whe11 a vir tual domain wa ll facil itates the con nection , t he conn ecting m anifold has topology 5:J , the radius of which is assumed to be much small er t han the ra lius of the instant.on. If one argues that the final state should li aw some "memory" of the origiual state, it is difficult to imagine how informa- tion conta ined in the initial state could be passed to the fin a l sta.te through a ma ni fo ld of a rbitrarily small dimensions. As argued in Chapter 5, thi s sit ua,t ion is not very different from a disconnected iustanton where it seems that an_v fi - na l state could be created from (c oHnected to) any ini t ial state. T h ' connecting top ology 5 1 x 5 2 is invo ked in an attempt to remedy this "information loss" problem, in that \Ve constrain the dimensions of t he 5 2 sectioll of the 01111cc tion to have a radius on the order of the instanton radius. Discussion of this second point leads naturaJly we believe, to a discussion of a 15G n process. tl1 ere . ogT? l ? l idea i. that in an a1 1nil1ilatio11-crertt io hol 11 clp c 1ypotlws1s. The ? ? 1 ? ? ? 1s ta t. t fi 1 ., t t 1c uu tia e o t1 JC mi s tate' . 1. c ld be a wav t? r r?n 1r at1 ?0 shou o orm n t o pass rorn J ? ? ? 1 t \ \ e 1 a ? 1 te. C trn , t ll'O llg1 l t1 l C th J orv O t? J? m tS lllltJa S a e fi 11 a l st ., te sl d 1 0U l Ja\?e a men . of S2 topology, wl1err the dimension that an S1 x use of a holographi c hypothesis, . 1 ? t? iat1.0 . vive w t ' rn orn n ot ? a spacetune sur th 2 ar(' all d b c 1e S owe to arge, w ould allo JC olographic hypothe ses as the? an 111? 1 11? 1? ? . e review severa l h at10n-creat10n pro cess. W de h two-dimensional '?screens" can enco d to cosmology and determine whic applie in r whether the si ze of hell deterrn the infor t? ? . Wr t ma 1011 of the bulk space time com1 ect io11 topolog y. 1 2 th allows them to '"tr averse" a11 S x S ese screens CITHs 6 -2 Alternative F ormalism for 1? t? a1 c u 1a t?m ? ? t? 1srn or c g t,l 1e pair ? n alterna tive orma Iii 11 a this s ect?1 0 11 , we \.VJ ? consider ? will defin e a 11 orrn ali za- d black lwles usin g CTTHs. W e Pro uction rates o f neutral ? J ? TJ 1e two different conn f'ct1011 topo ogies. schc o re t'ion ' me t? or t h e act ion and e xpl d S e Sit ter to e itter transi- uatio . ?11 r are the follo wing: (1) a d sit ns we w1 conside d (2) a d e SI' tt er logy S ?1 ri n d 5 1 x S2 , an topo tion vica a connectm? g manifo ld of S 1 x S2 logie s S 3 and . to iai t ransition via th e connecting topo ar 6 2 1 alism in General ? ? Form tial state annihilat es to st g a nucleation pro cess wl1 crC' an ini We art by assum in g. The instanton created from nothi ll th lowed by a fin al s tate llo ing and is fol ifold. Let I, ed by a disconnecte d man ting thi nucleation process is describ media ectcd inst,rnton. Si milarly. let clisconll the action of the in itial state of the denote rn cn- the disconnected iJJstanton. As 11 e final state o f denote the actiou for th 157 urn c from each ? o rtio11 of the au identical four- rn l ill excise tionccl above. we w s will be deno ted ese excised region ed iustanton. The boundari es of th disconnect . nd T+, respectively uclidean times L a T_ and LT+' and are located at the E by L he con- ari es. The h,vpers urfacc a t which t a long these bound We tli en identify = t d er1n Lime T T x . f\.?ote th a be designated by ~ rx at E ucli nection is made w iJl rtherm ore , let us a ss ume u re t is sorn f' small Euclidean time. F r == 7 x ? f, whe t the geodesic surfa ce L; at time T; and tha th begins at a totally at the ini tial state . We have constru cted tally geodesic surf ace I; at tim e Tl1 fin al state ends at a to < Furthermore, we Tl. undaries ill the rcg iou T; < T a manifold which has 11 0 bo ifold. The action in der m t ? . h' 1 are regular every where 0 11 this man consi e n cs w 1c 1 . . the reg?? _ l can be written as [123] 1011 ,; ::; T ::; T (6. 1) T < L , integral in to thr ee in tervals, T; < the a bove vVe can now break up < < . Hence, we h ave T r_ < T+ T T < T+ , and 1 (6.2) I = i, + I x + f1, where l r_ ,J (G .3) 1 (R- 21\ )Jgd .x, I ; --167r, T; 1 l T+( (G.4) - R-2/\)/gd 4 :E, I x -167r T_ 4 (G.5) --1 l TJ (R - 2J\)/gd .x. 11 1G1r r+ e the manifold is t otally y terms at I:; and '?1 si11 c We have not writt en boundar uuclary terms are zero. hese boundaries, a nd hence, sucli bo geodesic at t 158 /'ii as + 1 formalism. 01H' can wri tr R , tion f x . In tl1 r 3 Lc't us co11 sid -?r thC' ac 11"2 3- R) - 2(../h A ').0 + R,Jij = -N../h (X JJ\ ?iJ - 2 (6.6) 2( ../hJ< Ni - ../h h 11 N,J),;. der metri cs which a re reg- mp tions and cons i an now consider modifv our assu We c face I:rx. T he in tr insic lar anifold except at perhaps the sur u everywhere 0 11 th e m 1 I 1 ? ? re, )ut t 1e extrrn s1c cur- ontinuous cveryw 1e th cree-metric h ?. 1?s ass umed to b e lJ e can tl, en evalua te the surface I:rx. W y have a fiuite d iscontinuity a t vature ma 1 o x in the limi t "--1---., ... ' i?. e. ' - 3R ) - [-l- r x+r[N ../h,(J\.i jf( ij - 1(2 I X - Jim l Q l67r }Tx -( d3 .x] (6.7) N 1 jN),;J rir - ../h,h': 2(../h,K) .o + 2(../h, X . . 11. Tlrns , we J, avc In the lin11?t t h fi ? ird terms varns ?, e rst and th (6.8) ? f' o f' I Tl 1 . 1 ere a re t 1ree ? ? ? nal pri-nci ?p le to t he rcsultrng orm O I! e can ap P1 Y a van at10 ? ? r?e ? ar b1. trary rn t 11 e ? rn g w a t ?d Fi? m e the va.n atwns 1 scenarios 0 1 er. rst , w e ass u ? cons ations of h ;j at T = Tx . ri + < < TJ , a.s well as a.re the ,?a 7 r_ and T 7 regions i < r < s, then each term in the uch condi tion If the a t? unde r s c 10 ? to be extremize d 11 1s n vari ation is tlie usual nden tly. Tlie res ul t of such h indep e 11 variat ion must v anis as we . . s < r < T x an d T x < T < TJ , G , he regwn T; Einstein e t? 1 O rn t qua ions " = cenario involves the re at 7 = Tx . A second s ntinuity of the ex trinsic curvatu s co s an a e tliat this matte r distribution i e assum ter action to J. W addi t ion of a ma t variation descri bed he ell at T = T x . T dense, infinitely thin matter sh tl infinitely 11 e d . E, .r ns tcm? ' as 1 nary s e q ua t??i ons as we ? the or r iu the first enari ?o would yield sc is second scenari o is ap ter. Th Israel ? . ? last. ch tion condi tion d escribed in tlir Junc 159 nblin fonnalisrn. Oue ca l l accommo- Bousso a nd Clrnr precisely tha t us cl in the roduci ng a n add il iona l out int discont inui t ies iu t he e xtrinsic curva t ure with date ns in /IJ,,, to be ,1rbitrarv urce. In this scena rio, one allows tl1 c , ?ariat io lllatter so 1 ese < T < TJ , but fixes hu c1t T = T x . Und r r t.l r_ a nd T+ in t he regions Ti < T < a1Jd instein 's equa t ions in t l1 e regions T, < T < r_ E conditions , we gC't t he 11sua l T = T x . is no constrain t 011 the extri11sic curvat" urc at 7 + < 7 < TJ , but there d 11 . \Ve will cribe how one can calc ulate the act ions Ji an I will now d es art of the di . co1rnc'cted g a small , iden tical fom -vo]urne from each p be excis in region . T hen , one can four- volume of a11 excised m a nifold . Let V denote the write (6.9) I z (6 .10) ac11 part of racted because we a re e xcising a region from e where t he J, . are s ubt the dis ?onnected manifold . have de Sitter space as their ini t ia l consider All nu cleation proce se w e will ction, t iug attention to CITI-ls , wr " ill normalize all a ? tatc. Since we a rc rest ric H . We will call e action of a connected d e Sit te r to de Sit ter IT wit h resp ect to t h d IT H H. T lie connecting manifo ld for the backgro un t his the background CIT euting the manifold connecting the CITH reprC's will have the sam e topolo gy as i11 general, a llow the s ize of t he con- th owever , we can~ nucleation process. H ITH ground CITH to be liffer ent frorn that of t he C ng ma nifold in the backne t i rocess. representing the nuclea tion p 160 First ormalization Sch eme ? ? ke One posc 1? n 1s to ta ? ,.,? 1?b 1e nonna 1zat1o (G.11) lized action will tl i en be gire11 by The norma (G. 12) eat ion rate will th en be given by The 1Jucl r A exp [-2(1 - h g)] , (6.13) 1:) + 4(/ ? - 1;.)} exp [- 2(11 - l ;)]. A exp [-2(Ix - 1 cleatio11 = /{ -, then wr will recover the nu me 1 x = 1~ a nd [ 1. Notice that if w ass u turn cosmology. q11au the no-bo1mdary fornrnlati on of d from rate as determine Second Normaliz ation Scheme ction for the bac kgrotmd me, when evaluat ing the a th In is normaliza tion sche g mall climeusious of the conncctiu imit where initiall y s ke the l CITH, we will ta constaut . We den ote imcusions rernainrge d ifold tend to zero , and initi ally la man -1 . s of the connectin g manifold by i ally smalJ dimen sion th it e collective set o f in is given by The normalizing act ion (G.1-1) Nucleation Rate e Action and Interpretation o f th value of 1 - h g will h a, e the t has the most 11 cgativr In general, the p rocess tha rn1 cleation proce ss. T hcrC' a rc rate, i.e. , is the m ost ?'preferred" largest nucleation 161 i ke . s. Firs t . 01H' ca n \ ' !(' " ' a11 action l three, pc) . - ssi1) 1 c Ways to 111 terp ret om resul t cesses T H. ompariJJg this a ct ion fo r n 1rio11s p ro I Eq. (G.12) as t lie act ion fo r a i \'C' rnluc is one' \\'rl,V t.o determ ine and de t ? ? which has the rn ost n egat crmmmg t int crprc tatio11 based l iffcrC'11 which J)ro - mu late a c cess 1s rnost likely. 0 11 e can fo r can be tl10 11ght of as th 11 rate in Eq. (G .1 3). T lie first cxpo11en tia l on (' nucleatio d A. In this sensC', Lile ns far calle buti ng to the prefact or , which we haw t lr con t ri ut 1 - h can be th ought of as f l uctuations a bo in t action orrections to sadd le p o 1 ions to t !H' prefactor w ill t ribu tt h l poin t. Typi cally, these addi t ional con e sacld I ? l . topo l ? ns to cl r tcrnrnw \\' 11 c 1 c o1mcct 111g ogy be small l s a m ea ? a? ncI t icy can serve a interpretation invoh ?e s attr rnp t ing to s 11rn 15 red . A thi rd m ethod o f most p refe r 11d Cli a 111blin did fo r . r1 s Bousso a possible s izes of t he connecting rn aui fo lcl over 1? ? ?1 ? o a 11a 1_ vs1 ?s a re s11rn a r _v pe the rad ? ? t? a vu? ? ,Jrn, t s o t ua l clorn arn wall. T he res11l ts of r ? ru t l1 e last cha p ter, ther e is tha t as was shown in to t li e second in terpr etation, in reduction can be used to he prefac tor. Th e ex t ent of this only a sm a ll cha nge i n t . etation . will d iscus? eaciJ in te rpr determ ? l - J me w 11c 1 topology 1s m ost prefcn cd. \\ e in t h , e example that fo llo w 6 e Sitter ?2 ?2 de Sitter to d e Sit ter to de Sitter C ITH, consis ting of e will now consider t he background d W e i tter and the recreation of a d t h spacetime e near-a nnihilat ion o f a de Sitter cons ists of two se?a ra te e disconnected instan ton for tJ1is process spa.cetirne. T h 1ratcly t lie sit uat ions w h ere t li c ini tial and sphcres. We wilJ consid er sepc li a lf-fo ur- I 2 logy 5:i and x S . by a 111anifoJd of topo fin aJ !:>tates are conn e cted I G2 Connection via an :i Manifold \Ye, start ' 1)} ' remov 1?1 1 g a s111 ;:d] fo ur-hall from eacl1 part of the iusta nton. The bou11d ary of the removed region will have topolog~? 5:i. T he CITH will be con- st ructed si? mp 1Y 1) )' 1. d cnti.f ~r.r n g the bouudan. cs of. the exci.s ed rl'giollS. Since the radi u?s of t he? connect,? on w1? 1I be assumed to he s1rn1 II reI at,., ?e to LI 1c coswo I og1.c a I radius of the de Sitter instanton. we can work in th e approximation that the removed four-ball is embedded in fo ur-dimensional flat space. The coordinates for the embedding are .T = x z =1 =; ,,s ,i sn i n xs ein 0 o s 0co , s q>, Y = 17 sin X sin 0 sin q> , w == '1/ cos X , wh ere O ~ x ~ 1r. O ~ f) ~ 1r, and o ~ cp ~ 2n. T he res ulting metri c is given by (6. 15) 2 4 T he volu me occup1?e 1 d by t lu?s f?o ur-ball 1?s gi? ven by Vo l(D ' ) == 1r '1/ /?~ - I n t I1 1?s region, R = 2A. Thus, t he act ion of th is removed vol ume becomes --1 Ii' cl'1 .r Jrj(R - 2/\), 167f . D'1 2 1 - - ~: Vol(D' ) , 1 - 1r/l. (6. lG) 4 --'// 16 Hence, by Eq. (6 .9), we have ( G. 17 ) We must now calculate the action I , . In the region where 7 < 7 " successive Lbree-spheres are shrinking a.s one approaches t he surface '.:,,. Hence, the unit 163 In the rrgiou T > T x . success i,?e d normal in this rC'gion i s given by -8/fJ,7- outwar e, the a,,?a_, . frolll the surface ~r ,. . Henc tlirce spheres arc expandi ug as one rnovrs ? in these 11 by 8/or;. The trac<' of the extrin sic rnrvat11n outward normal is givc regions is the11 calcu lated as - 3 (G.18) A _= -- , 17 and I >- has the forrn -2.- j (I<+ - /\."_) J;; r/3.1;, 81r. 3 - -Vol(S 3 ), 41r17 .19) -311r r ') . (6 2 nce, the total t 1 = f fo r the proccs ? vn! arc cousiJer ing. He Finally, we note tha 1 action for the CIT H is given by 37r 37r 2 7r J \ 4 .20) I = - - + -11 + -11 (G J\ 2 8 cheme, ormalizing act ion . In t h e first normalizatio11 s Now, we turn to the n the 11ormalizing action can be wri tten as (G.21) T he normali zed act ion is (6.22) f the cheme, sin ce we lrnvc assu m ed that t he radiu o Iu u s the second normalizatio di us, 3 uclidean de Sitter cosm ological ra S connection is small co mpared to the E lG-1 limi t r;-+ 0. Tillis. w e lu-n-c the we will apply Eq. (6.14) in 2 (- 31r) 2:\ 31r (6.23) i\ rmalized act ion is T he no f 2 7f _1\ -1 (6.24) 37 . I - h g = -.r; + -172 8 nd Chamblin is similar to the res ult of Bousso a ote that for r; = - l /(21ro-) , t his [9], to second order in r;. ld Connection via a n 5 1 x 5 2 Manifo wheJJ we connect the r CITH onstruct the de S itter to de Sitte \Ve will now c this , we will excise a 2 anifold of topology 5 1 x 5 . To do st in a nton with an m tanton ; of the half of the disconnected ins 2 x 5 2 from each region of topology D ised two-ball will be of a two-ball. The radius of the exc 2 here , D is the to pology oted by c. Further more, 2 will be den by a and the radi us of the excised 5 denoted l 011ce again es of computing t he actiou , we wil or purpos we ass ume a ? c. F sp ace. 1 We note is embeddeackground i11.stanto n may be different from the d ing mani nstanton , we arri ve at a variety of of the connecting manifold in t he nucleation i nly seems reasonable to assume that the cli111e11 sion. are the re ul ts. However. it o o11 process. and this is what we wil1 lcati same ill l>ot11 the b<1ckgro11nd an d nu ting 111aJ1ifolds cefor th assume. A furth er assu mption will be tliat for connec hen 1 5 2 th e rad ius of the 5 '2 portion w ill be assumed to be on the of topology 5 x , zon, (o r eq ui valently, the cosmologic al) hori order of the radius of the black hole ~ he radius of ~ 1/ ../A. Fmthermore, a r1, wh ere we recaJJ th at 17 is t i. e., c see that the action of Eq . (G.24) 3 e the 5 conn ect ion. With these a ssumptions , w nccted wi th a ws that the action of a de Sitt er to de Sitter CITH whell cou sho t likely. TJiis is followed by a de Sitter to de Sitter manifold of topology 5 3 is mos nifold is 5 1 x S'2, as the action of Eq. (G .35) sl1ows. CITH when the connecting ma that a de Sitter to a ri ai Consideration of the actions (6 .40) and (G.46) show topology 53 of is 11 ext rnost pre ferred, transition when connected by a manifold f topology 5 1 x 5 2 . followed by the same process w hen connected by a manifold o abov<' by considerillg tl1 c pair c re- as One can come to the same con clusions the i11i tial state is ation rates. Since we can onl y speak of such a rate when 172 . . 1 1 s? iv . . trans1t1011 , ?ia an differrnt from t l fi a 1 n state. we? seP t 1at n cc 1ttcr to !'!an ai ic n ,?ia a n S 1 x S2 co1rn Pc--an thC' s,WH' (n-rnsitio s :J conucrtion is more preferred th tio JJ . e. it seems thri t tJ1 c corn i ce ting topo logy rg um ents considered ab ov Bv th e a topology S1 x S2 S"l the conn ectillg ? is a lways preferred . However. we fee l that sum in the pach in tegral . This res ul ts from e histories over which to Posse. ses m or 1 "bumps'? that can be pla ced 0 1 th con t ribution of Planck sized per t urbations or e I describing a dC' Si cter t o th 2 anifold. The action ' S portion of the conne cting m h 1 2 conn ec ting topology is gi ven i11 Eq. (6.43) . Suc a ria i transi tion via an S x S he co11 trib11 tio11 from th e "bumps" ion depends on a and c. In general, t an act 11 as ral can be formally wri t te dep ends on c. Hence, t he pat l1 integ (6.51) / da de D[bumps] r,- t . on to the path integral f rom tlic ever, we will show tha t t he largest cont rihu ti How hole the order of the cosmo logical or bla ck bumps occurs when c i s la rge, say on radius. ust specifj, the geometri es over which ry out such a path integ ral, one m To car 2 me g the ?011necting topolo gy S 1 x S , we ass u to s um . Since we are con siderin ls a11d two-spheres give the -bal th metri cs which are dir ect products of two at q. (6 .25) is an tribution to the path int egral. Tl1e metric of E most important con considered the case of c ? a. try. In this example, we example of such a geome poses he Ricci scalar curvatur e decom For such a m etric ' it is e asy that t. to show nd th e Ricci scalar for th e two- sum of the Ricci scala r for the two ball a into a ecting calculate the action I * for this conn sphere, i. e. , R = RD2 + R 2 . Let us now 173 mT 111 11 (note t li at t lii s is llOt the fu ll actio11 for tlie CITH). \re h a ifoJd l 1 I * - lfor j(R - 2A)fijrt .r , 1 1 2.i\.fijct1.1:) (G .52) ~ - ; ( / RIJ2 ,jgd".1: + j R sz fijd :1; - j1 71 al Ga 11ss-Bo11uct theorem to the integrals over Application of the two-dime nsion tl1c sc?aJ?,r? c urva t ? ? c. urc t l1 en gi.v es. 1 I*~ - - - (XD 2l 5-2 + \ s2 l 0 2 - 2.'\ 1 02 xs2) , (G.53) 1G7i The e Euler characteristic of the m anifold that is subscripted. wli crc X denotes th total ac t? 10n o f' the CITH is given by (6.54) I -- --571+ J* . 2/\. liicl1 is what we will use in 1.3) , w This is of thr fo rm of the action in Eq . (G.- ur-volume t r of this calcula tion. Under tll<' ass um ptio11 that the fo li c I"Pmainde the action depends only on the t of t11e conn ection v'iJ2x s 2 is small , vvc llote tha ependent e, we l1 ave shown that when r is large, the action is ind topology. Henc of the bum ps on the two-sph ere. from the bump ?. A reasona ble way to now consider the contribu t ion Let us he 5 2 ing the 5 2 in to many Planck si zed a.rc'as is by di viding t consider par tition t, im agine placing Plauck size d perturbations or 2 in to Ac = exLJ7fc such cell s. N the sum over bumps is to co11 sider how "b to calculateumps" in these cells. A way th ere are e can place these bumps on t he two-sphere. For example, many ways on ce the second bump, etc. How ever, Ac - l to pla Ac ways to place the first bum p, f the bumps, to place n bump s on the two-sphere, due to the indistinguisha bili ty o tr coun ting. Hence, one can wri oue must di vide by a factor of n' to avoid over (6.55) 174 f' ne11 t1?a i ? 1 ? to a n expo l occurs 011ly f?o r large' c. Hence, where? t, 1e eva 11at1on o t 11 e surn 111 over r? in t l1 e path intC'gral reduces t aO we call reasonably ass ume that t l1 c s11 aussiaJJ . wit h Tlie resul tiHg ill tegration is on"! r o. wl1i ch is _ju st a G single term. the resu l t l rx., r - aJ(P.1r/:Hrr/1rl) da, ./1 (6.56) at extendiJJg the upper limi t of wh re N(.x) is as given ill Eq. (5.64). vVe note t h all clrnngr iu the in tegral, siu c e m ost of ts in a sm in tegration over a. to oo resul t few standa rd deviations. T he resul t for the e firs t li e contribu tion com es frorn th path integral is t hen (6.57) path in tegral, ant to note that. t lH~ contribu tion to tlir prefactor of the I t is import entia l sion in sq uare bracke ts , is no w on the ord er of the expon given by t lw expres ow, was not the case. It is also of the saddle poin t act ion , s omething that un t il n is wou ld not happen in the case when the connec ting 1111Porta n t to note that th 17 tains a two-sphere of radius 17- Since topology was 5 3 . This three-sphere con lan ck leHgths, the !lumber of Planck of a few P IS assumed to be of the or d er the be placed on this two-splie re is mu ch smaller than in sized bumps t hat can nential 1 5 2 connec ting topology i. e .. t here would not be an expo case of the 5 x cont ribution from this sum . refactor of the path in tegra l p However , this exponenti a l c ontributioll to the cleat ion rate. Vle calculate tli e nucleation rate with would not change the nu hich wou ld have tl1 e same con necting topology as respect to a background CIT H, w r pa t l1 in tegral for the backgr n1111d th e CITH d escribing th e nucle ation process. T h 175 oneutial contribution to t he prefactor. Hcuce. would have a n identical ex p CITH ssion for the rate . t he two exponentials wou ld cancel iu the C'xprc ographic 6.5 Connecting Topolog ies and the Hol Hypothesis a holographi c principle ca11 shed on the pos- ection, we di scuss what ligh t In t hi s s olo- rent patching topologies. In ge neral , the h sible physical interpretatio n of diffe e geometry and the numbe r ween spacetim graphic principl e is a relat ionship bet acetime. It has been motivated of degrees of freedom , N , contain ed in the bulk sp ing argument [124] . iderations of black hole en tropy. Consider t he follow b_v cons of as composed of poses a P lanck scale cutoff, s pace can be thought If one im , for ch of which contains a few degrees of freedom. Hence Planck sized cubes, ea , since thermodynamic entr opy S . Furthermore a region of vol lime V , N(l ?) ~ V lly disord ered system. t exceed N , we expect that S (V) ~ V for a maxima canno bly rnaller. kenstein sho.wed (c.f. [125] ) , the entropy is considera However, as Be e cntrop_v of a black hol e is ation . First, th Let 11s start with two piec s of inform a of t he bl ack hole eve11t ho ri zo11 . Second , n by i = A/4 , where A is tl1e areg ive Sb1 amics states that the entrop y t he generalized second law of black hole tli crmod_vn he black hole can never de crease, i.e., e t of a black hole plus the ent ropy outsid black hole that swallO\.VS a stat ionary osbli + a oSout 2: 0. Now, let HS consider s of thi ? body d self-gravity with radius R and eucrg_v E. The radiu body of limite 2. ility of the body implies tha t E ~ R / is defin ed by 41r R 2 = A ; gravitational stab arna must increase by at le ast 81r RE ack ho! swallows the body , its When the bl iu zed second law, the change 2 nerali (cYA = 41rR 2: 81rRE). To not violate the ge 176 hole must not exceed 27i R E. Heuce. th op_v of the region outs id r the black e entr /-1. Hmn,,?cr , ccupy a volume l ?, S ~ .4 for systc?ms of limited se]f -grm?it,r that o clf.-gravit,r and black holes tems of limi ted s Bekrnstri 11 poi nts ou t tha t both sys culty li rs in defining R for obr,? th ? ? I d H a lJ s,?stems sho11Jcl . The di ffi ?' is >oun . ence, a particular system. 't resul t was proposed by b oth A more rad ical in te rprrtat io n of Bekenstein 's of freedom in ide the [126] and Susskind [127]. Th ey claim that all degrees Hooft e, Ar / p1 ,v ing t 11 at under b]ac], h 1 1 ? an be ignored. Henc J v ( l ' ) ~ A 4, irn 0 \ e 10n zou c a phvsical system can be e ncoded 1formatio11 a bout certai 11 condi tion ?, all of the i1 at a densit.v of one bit of urface such th on its boundary, i. e., a tw o-d imensional s ta per Planck area i not exceeded. da e to more genera] ts have 1 een made to app ly a lw lographic principl Attemp n defining the bounding t ions, such as cosmology. H owever , diffic ul ties arise i si tua ary. , a nd open or fl at uni vers es may possess no bound regiou , since both closed ees of freedom of the bulk e(s) a re the degr The question then is, on wliat surfac spacetirne encoded ? : smcc xplore the a bove qu estion text111 the fo lJowing con \iVe wou ld like to e nisonal, th iple implies that the world i. , in essence, two-d ime e holographic princ ting topologies that all ow consider whether conn ec it is an interesting questio n to ferred over other connectin g topologies , e large are pre at least two dimensions to b at ion conta iH ecl iH the iHit iaJ spacetime can sun-ive rm in the sense that t l1 e info two-dimensioJ1 a! surfaces can rmine on what a tunneling process. One must dete 11 attempt to answer the encoded . In athe bulk information of a sp acetime be uss various hologrnpliic J ro posals and see if they ca11 bove quest ions, we will dis c a topology. fur ther justifjr a S 1 x S 2 connec ting 177 The Fischler-Susskind Proposal 6. 5.1 lation of a holographic princip le w e will disc uss is that proposed by The first formu wiJl hence fo r th refer to as the FS proposal. Fischler and Susskiud [128], " ?hich we 111i 11 g a Fri edmm1-Ro ber tson-Wal kcr They form ulate thei r proposal by fi rst assu fi at spatia l sect ions, t he Lorentz ian rnct ri c of wlti cli is (FR\V) cosmology wit l1 giV(' l l l)y (6.58) rica1 spat ia l re t he num ber of spatial dim ens ions is d. NC'xt , co11 sidcr a spl1e wh e r ord inate size R with boundarv B. Now, co nsider the li gl1 t- lik region of co from B to t!1r center of r. Let R 11 be surface L fo rmed by past- ingoing l igh t rays ri zon, a11d consider t li e fo ll owing ical ho t he coord inate dista11ce to the c osmolog F igure 6. 1). If R < Ru (corresponding to the boundary B ), 1 t li ree cases (see its t ip to t l1 e fu ture of the singul ari ty th t hen the surface L fo rms a lighLco ne wi 1 = = R (boundary B ), the11 the tip of t he Jightcone is located at at t 0. If R 11 2 u (boundary B ) , then the lightcon e is 3 t he singulari ty at t = 0. Lastly, if R > R tli rough L. In t l1 c first case, thi s t runcated. ow, consider t l1e en tropy passillg ntropy con tain ed in r at ti ni e t. T his is also true for t J1 e second is eq ual to t he e ssing thro ugl1 L is less than that cast' - However , fo r the last case , the cntropv pa ss ing in r at time t. T he FS proposal, is thell , that the entropy pa contained a rea of the bouJJ d ing sm face B. Quant itatively, th rough L never exceeds the (6.59) s constant , corn oving entropy de11s i ty. Sin ce R11 and a are full ction where a is t l1 e alid at all t imes. of t ime, t he a bove condi tion mus t be v 178 Cosmological Horizon , =const. r cl mal diagram for a fl at F RW 1111 iverse . The smfaces Bi , B2, a u F igure 6. 1: Confor < , R = RH , and R > R11 , re pectively. R B3 correspond to situations wher e R 11 ys are designated by L, , L2, a11d L:i , The corresponding past-ingoi ng light ra respectively Sit- ly this proposal to a de Sitte r n11iver.se. Consider t he de \iVe now app e curvat ure and a(t) ter metri c of Eq. (3. 102) with sp atial sectioll .s of posit iv dinate size of the hori zon is giv e n by (l / Ii ) cash Ht. T he coor ?/. dt Ill .60) XH = ! - = 2 arc tan ( r ) , (6 . - oo a(t) he boundary a rea of the horiz on is given b_v T (6.61) The volume inside this sphere is given b_,? (G.62) Heuce, t he entropy /area ratio i given by S 2Xll - Sill (2,\". ll ) (G .63) = 2(x1-1) sin 2 A C7 2a ( YJJ ) ' 179 ing nt! uc of . (G.G7) 0 A , ea and )., is the affine pa rameter of t he null where .4 is t he cross sectio nal ar 0. One stops following the ligh t- he light-ray for 0 ~ congruence, one follows t e. For such a light likc hen a caustic is reached , i .e., 0 becomes positiv ray w N ~ eA/4. persurface L bounded by B , we have S ~ A/4, and hy <:.r Bekenstein 's bound in the appropri ate limi t. Bousso would like to rec ov w11 ereas to entropy in a spatia l region , However , Bekenstein 's pr oposal refers wo prop osals in the rop osal refers to mill hyp ersurfaces . To link the t Bousso 's p T li e- o proves what he calls th e "Spacelike P roj ection appropria te limit , Bouss ssessing a fu ture-directed of a closed surface B po orem " : Let A be the a rea region V be con- L with no IJoundary other t han B . Let the spatia l ligh t-sheet f L wi t l1 any spacelike hyper surface in the in tersec tion of t he causal past o tained n V. grees of freedom presen t o containing B. Let N v be t he to tal number of de T hen , N; ., ~ A/ 4. 182 To use the B0 11 sso proposal for our purposes. ,,?c 111 11 st si mph ? reverse the a hove prescription. Recall t hat we ,Yish to drt.cnnine the t\\?o-cl i111cnsiom1 I surface B that encodes the informatioll co11ta.i nP.d iu the bu lk spacct irn c. \ 1\ic start by drawing a Penrose d iagram of tl1r spacct ime of in tr rcst. We Lhc11 sli ce the spacctime in to a fami ly of null- rays. Note that t here will exist two in<'quiva le11 t null sli cings; si mpl _v choos' 0 11 c. Next, ident ify t he normal, trapped , and ant i -trapped regions of t he spacetime. Iu each region, draw a wedge. the legs of which point in t he direction of negative expansion of the light- rays. 0 11 a given li ght rav, proj ec t each point towards t l1 e tip of the local we lge , onto the nearest point B;. T l1i s nea rest poin t B will occur where the direct io11 of the tip cha ll gC's. or vd1e11 a spacet ime boundarv 2 is reached. This should be done for each light-ray in the family. The surfaces Bi wi ll for rn a hypers urface , or screen , on whi cl1 the in forrnatio11 contained in t he spacetime sli ced by the null-rays can be projPctecl. Vie will now apply this proposal to d, Sitter spacetime. The conformal di a- gram is given in Figure 6.3. Consider a 111111 congruence of past-outgoing rays. O ne can start at the surface x = 0, and projec t points back until one reaches 1- . This is a screen-hypersurface of infini te size , onto which t he degrees of freedom of ha lf of de Sitter ?pacetime can be proj ected. The other half can be projected onto 1+. V./e can now apply the spacelike projectiou theorem to t he screens that form the event horizon E of an observer at x = 0. These screens are all of size 41r / H 2 = const ., and eucode half of de Sitter spacet ime (see Figure 6.2b). vVe now turn to the ariai spacetirne. A conformal diagram of thi spacetirn e is shown in Figure 6.4. To find a screeu-hypersurface that encodes Lh e informat ion of the bu lk spacetim e, we start by slicing the spacetirn c in to past-outgoing null- rays. It then becomes clear that 1- is tlw screen-hyper ?urface onto which the 183 1+ / / ' / ' ' 'A ' // / ' 0 >' ,, / II < t:! II / ' X / ' X / ' / / ' ' / V ' / / ' / ' / r (a) (c) (b) Figure 6.3: Part (a ) is a conformal diagram for de Sitter space time. The dashed lines are null surfaces indicating cosmological horizons. The wedges are drawn in accordance with the prescription described in the text. Part (b) shows how the spacetime can be proj ected onto two screens of infinite area, namely 1+ and 1- . In part ( c) , the spacelike projection theorem is used to project half of de Sitter space onto E, the event horizon of an observer at X = 0. 184 1+ \ \ \ V I I\ \ \ \ I \ I ) V V .. < I \ I \ \ \ I\ \ V \ '? : r (a) (b) \ \ I \ \ I V I \ I \ I \ I \ I I ' ?: (c) T he formal di agram for th e Nari ai spacetime. .4: Part (a) shows a con F igure 6 s the black hole black hole singul ari ty , the dot-dashed line e jagged line deuotes th awn i11 l horizons. The wedge s arc dr izons. and the dotted liues t he cosmologica hor tion iven i11 t he text. Part (b) sliows the projec rescription g accordance witli the p projection of the spa cetime onto he space time onto 1- . Part ( c) shows the of t izou each of finit? the spacelike project iou theorem to t he region inside the black hole. Hence, as shown in F igure 6.4b , a screen on which one can encode the in format ion of the spacetirne is 1- , whi ch is infini te in size. However , as shown in F igure 6.4c, one ca11 encode t he in terior of the black hole on t he bl ack hole horizo n. One can t br n apply the Spacc likC' Projection Theorrrn to t he region ou ts ide the black hole. and encode this region onto the cosmol ogica l horizon . 6.5.4 Application to Nucleation Processes We have surveyed a numbPr of holographic hypothesis. What , if anything can t hey tell us abou t t he viabili t>' of vari o11 s co11 nect i11g topologies? For a nucleat ion procrss to have "memorv?' of it. ini tial state. informa tion from this initia l state must, in some vague sense, "survive" the near-annihilat ion-recreat ion process . I t seems that for t his to have any possibility of occurring, t he cormecting manifold must have at leas t some dimensions that are large enough to fac ilita te the passage of information . Co11 sider the de Sitter initi a l state. Some of t he holographic proposals that we have considered indicate that the cosmological horizon is a scrcen-hypersurfac<' onto which t he bulk information of th is spacet ime can be encoded. Each two- dimensional screen on this hypersurface has a radius r = j3Ti.. and a rea A = 121r / Lambda. Next , consider the aria i spacet imc as t he fin a l state. Here, we have seen t hat one or bot li hori zons (cosmologica l and black hole) a re necrssar:v to eIJ code the in formation of t hi s spacetirne. Eacl1 hori zon has a, radius r = /!iA a nd a rea .4 = 41r / A. Hence, it seems t hat a connecting manifold t hat has two of 186 essary to a llow information to pass from m ensions of at least th is siz e is nec its di (' 5 1 x 5 2 connecting manifold , thP h the initial state to the final s tate. Indeed. for t ii.., 2 t ion can be of this size. For ex arnple, we ca11 let c = /I radius c of the 5 sec lack hole li orizon . Also, we could ize of the b whi ch would correspond to the s nd the cosmological = /i!A, which would encompass both the black hole alet c me. horizon of the Nariai spacet i 6.6 Conclusion ced an alternative fo rmalism to construct con- In this chapter , we have in trod u cleation of black holes in a presenting the nu tinuous imaginary time his tories re ng this formalism, we studi ed two different connecting de Sitter spacetime. Usi of various connecting topol ogies 3 1 2 . The study topologies, namely 5 and 5 x 5 sum over all pos- tivated by the path integral formalism, i.e., one shou ld was mo tories cting initial and final states . We believP that such his sible histories conne determined that a connect ion ould contain various connec ting topologies. We w ion which was more tated by a manifold of topo logy 5 3 resulted in an act facili manifold of pared to the action of a c onnection facilitated by a negative com uantum 1 5 2 In the context of the no-b oundary formulation of q topology 5 x . 53 connection would be more probable than cosmology, this indicates t hat an 3 n ection where tli c 5' x 5 2 connection. However, we believe that a11 5 con an small is problematic in tha t it is not much difforeut radius is constrained to be the nton. Furthermore, it is not clear how than the case of the disconn ected insta state could survive the ormatiou possessed by the b ulk spacetirne of the initi al inf If one takes the claims of hol ography seri- near-annihilat ion , recreatio n process . o dirnenions of the connecti on remain ously, then a connection wh ere at least tw 187 large, but \\'h r re t lw , ?o l11rn e rem ai ns s11 rn ll. co11lcl a ll o\\' in fo rrn at ion to s11n?iw the nucleation process. \,Vit h this in mi11 d . we' co11 sicl crccl a coun rct iug ma ui fo ld of topo logy 5 1 x 5 2 , where the dirncusions oft he 5 2 port ion of the ma ni fo ld could be large , say ou the order of the cosmologica l (or black hole) l1 ori zon . For this par t icul a r connecting topolog_v. the pat l1 in tegra l possrssPs many mor C' historiPs over whi cl1 to sum than the 5 3 c-0111H'c ti ug manifold . T hese histories resul t from 2 1 Pl anck s ized p er t urba tions that can be pl aced on the 5' por tion of the 5 x 5'2 connecting ma ni fo ld. We showed that the contribu t ion to the path integral from t hese per t urbations was au exponentia l, and ,vas of the same ord er as t he saddle point con t ribution. However, we noted t ha t such a cont ribut.ion would uot be present iu an expression for the nucleation rate, since it would be cancelled when one norma lizes wi th resp ec t to an appro pria te background ins tantou . In some sense then , there seems to be 110 observational distinction between connecting top ologies in thi s context. Attempting to determin e a possible distinct ion wo uld const itu te interesting future work . 188 Appendix A Tunneling Calculation of Transmission Coefficient via the WKB Approximation In t11is app endix we derive the t ransmission coeffi cient T for t nnn eling thro ugh a potential ba rri er. Our d eri vati on uses tll e techniques of ordina ry quan t 11m m echa nics a nd is done in tlie context of thr vVKB approxima tion , or serniclassical limi t. T he form a lism discussed here is prese11 tcd in mos t in trodu -tory text books on qua n t um m echanics, c. f [131). We consider a particle incident on a po tenti al U(x ). In the semiclassical limit , tli e typical wavelength of th e p article is sm all compared to the distance over which th e potential vari es significantly. vVe star t "vi th the time-independen t Schrodinger equation (A.I) We the11 assume th e tunnelin g behavior of a particle is well approximated by a WKB wavefun c ti on of tl1 e form 1Pw1. = 21r /l,/ p , we get dd>x. / ? l. I is justifi ed . This is the condition underwhi ch the WKB appr oximation pli cation . We arc T his condition presents an immediate problem for our ap r , at these interested in the behavior of the particle at the tu rning points. Howeve 191 ' = cL? / cL:r =/ 0, oln-iousl.r , ?iolati ug t]1 (' aboYr co ndit iou . Sin er wr poiuts , JJ 0 and eacl1 side of :i: 1 a nd .T 2 , wr wi ll ass um e that the WKB a pproximatio n is valid 011 y to relate t he WKB vvavefun ctious in the se regio11s. must find a wa rclatio11ship between the WKB ,rnvcfunctio ns in regions far from F inding the l1rodiugcr eq uation cx;:ictlv near ;:i the t 11rni11g point is done h~- first solving t lie Sc nt and arriving at a wavefun ction . T hi s wa.vcfunction is t hen c-Iassical tmniug poi urning point . T he fo rm of t hi s wavefn 11 c- extrapolated to regions far from the t these t ion far from the t urning point wi ll rese mble t he vVKB wavefunctions in ns . Knowing the relationsliip between f orms of thr, extrapolated wavefun c- regio se n in the regions far from the turning point will give us equations we can u tio ns are known as conn ection formulae. uatio to conn r,ct these two regions ; these eq weel! the \1/KB wavefunctions in t li e bet \iVe then ass ume that t l1 c rela tionship of t he same form as t he extrapolated regions fa r from the turning poin t will b e ions. solu tio 11. T hus, we usr the connectio n formu lar to relate t he \t\TKB solut he connec tion fo rmulae are given by [13 1 J T Classical Region Forb idden Region 1 n1 n [- l_ ; ?x1JJ