h ya . The tum ef- y quan acuum regions in haracterized b . y constructing the h can b e regarded M= tized b tize this dynamical system E h i . Both the v n op erator, h are called the rest mass y alue of cosh t dust shell is ignored, this matter shell y Berezin et al. [3,4]. If an = y the v y using Israel's junction condition [8], whic ariable is limited to the circumference radius B H haracterized b t of motion whic acuum regions should b e joined at the shell accord- of the shell and the global structure of spacetime can ov The original idea to quan orks as the equation of motion of the dust shell, and the as prop osed b estigated the Hamiltonia w ing to the Einsteincomplished b equations. Thisw program can b edynamical ac- v R b e classi ed b fect of incoheren can b e c the spacetime are also classicallyequation treated. of Then, motion onlyas the of the the energy shellHamiltonian equation op erator. whic is quan Althoughator the to Hamiltonian opb er-ev constructed is not unique, Berezin et al. in- t dust shell tells us the shell can b e c constan w an a-u.ac.jp a -u.ac.jp y z a-u.ac.jp y atsu ormhole ys .nago omim ucc.cc.nago [email protected] and Akira T ouc oshiro@alle gro.ph y k Hole and W View metadata,citationandsimilarpapersatcore.ac.uk . . t e y M wk- tion vit wski (June 10, 1995) o t step e con- y fully
GR-QC-9506032 oshimi Oshiro h corre- e-mail address: k e-mail address: c42615a@n e-mail address: y z y y mo dels x of infor- ,Y hild space- hanics is de- hto oted to con- ura y itself includes arzsc een classical and ormation of Blac vitational Collapse of a Dust Shell vitating dust shell w w am vit tum tensor of the y b e a Mink h y mo dels to shed a tum mec e fo cus our atten ables. Since an tum e ects of gra e function, the nature t dust shell at a nite y the shell collapse. It is v vitational collapse. One e b een dev t circumference radius, w a tum F acuum except the delta- hw v in Gra h quan tum gra e the problems concerning ap oration due to the Ha Kouji Nak vitational collapse of a spher- y lead to the parado vitational mass parameter ODUCTION artment of Physics, Nagoya University, Chikusa-ku, Nagoya 464-01, Jap et established, the presen Quan orks ha h the shell collapse form, w hisinv er, quan arious useful to Dep hma k hole ev ev yw tum gra w er at a constan h sp ecify the global structures of spheri- I. INTR hanical mo del of self-gra elop v o clarify the relation b et terpretation of w tum theories of gra , man e functions of the shell motion and the discrete hnical diculties but also conceptual ones v ations is to resolv ormhole states are forbidden when the rest mass a b er(s): 04.60.-m, 04.60.Ds, 04.70.Dy h has a nite gra t theory is not y t on some features of quan tum-mec ergence of the energy momen um arious time slicing on whic , the inner side of the shell ma 1 tum uctuations. tum spacetime whic Quan h as the in 7]. CS n In this mo del, one can consider the spherically sym- Historically = ould b e to dev A elop ed. By considering the static time slicing whic not only tec suc of time and the de nition of observ consisten mation loss [2]. Ho w new ligh In particular, quan ically symmetric shell, onin this whic pap er, has b een studied as one of suc The div ing radiation [1], whic function distribution of incoheren circumference radius. By themetry virtue of the spherical sym- time whic metric spacetime whic spacetime and the outer one is the Sc [3{ structing quan of the motiv the nal fates of blac obtain the w mass sp ectra whic cally symmetric spacetime formedfound that b w is comparable withquan Plank mass scale due toP the zero-p oin v sp onds to an observ sider v quan is considered. T ; 2 e ) n cek tro- (1.2) + aj adapt- 2 1, E 2 m er, H @ +4( (1.3) p ev 4 w m
(1.1) 8( 2 h y @R ximation where @R 1 tb alid when + (1.4) 2 wskian time of the inner side of : ; o 4 p k mass (Through- R m ) @ 2 e a de nite form , v h leads to the \Wheeler-DeWitt" equation, n rom their analysis, it is unclear that the sp ec- k constan B h i + 1 provided byCERNDocumentServer H hosen to b e the prop er estigating this equation, they obtain the v for the b ound states de ned b of 2( 4 In brought toyouby E is the Mink E m teger. F y using WKB appro er. Then, they obtained the = 1) and the time co ordinate is the Planc duce the sup er-Hamiltonianspace on whic an extended minisup er- b T in trum (1.2) is also v et al. [7] ga ing a simpler equation for the w c p ula of = alues m = v E G the shell. form where =1 and e denote the Planc CORE p E ving observ h that =m = m use units suc sp ectrum of the eigen describing the shell dynamicstime is of c a como where out this pap er w 1
equations and gives the equation of motion in a similar Nevertheless the two quantum treatments arriveatthe
2
manner to the Wheeler-DeWitt pro cedure. Following the same conclusion when m << n, (1.4) b ecomes meaning-
earlier works, we solve classically the Einstein equations less when > m which means classical limit cannot b e
p
except the Hamiltonian constraint for the shell motion. obtained from this eigenvalue. Furthermore, it is shown
This means that the momentum constraint is classically that in the b ound states only the black hole formation
treated just like many minisup erspace mo dels. corresp onding to the range 1=2 In Sec.IV, we consider the quantum mechanics of col- the wormhole formation corresp onding to 0 lapsing shell, by using the one-parameter family of time is p ossible as classical solutions. slicings intro duced in the previous sections and we also Our purp ose in this pap er is to clarify the relation b e- discuss the global structure of spacetime in which the tween quantum and classical spacetime of this system. dust shell collapse forms. Our main result will b e ob- For this purp ose, wepay attention to the time slicing tained under the static time slicing. Furthermore, our on which the quantum mechanics is develop ed. Note consideration is restricted to the so-called b ound states that the previous treatments are essentially based on the 1 and wemust consider the cases E>1=2 and E<1=2 quantum mechanics on the other time slicing. Since separately in the static time slicing, these cases corre- the canonical formalism is based on the decomp osition sp ond to the black hole states and wormhole states re- of spacetime into space and time, a sp ecial attention sp ectively. Then, we discuss the relation b etween quan- must b e paid to the problem of time slicing which de- tum and classical solutions of this system and show that termines the foliation of spacetime. Furthermore it is there is no quantum states when E<1=2 and m . well-known in classical relativityhow a description of p We also consider the quantum mechanics on non-static black hole spacetime dep ends on the choice of time slic- time slicing to con rm that our arguments are natural ing: In the usual static chart the Schwarzschild horizon extension of the result obtained in the comoving frame. plays a role of the in nite redshift surface, while in the It can b e shown by the fact that the quantum version synchronous chart corresp onding to a freely falling ob- of the Hamiltonian constraint b ecomes identical with the server it is not any sp ecial surface [9]. This means that radial equation of (1.3) in the comoving limit. Finally, the horizon can b e regarded as a sort of b oundary of our consideration was summarized in Sec.V. Although the foliated spacetime only for a static observer. So the our simple mo del is a preliminary approach to quantum foliation of spacetime or the choice of observers in the gravity,itwould give a useful clue when one investigates spacetime is more imp ortant when one consider a black the problem of time slicing in a more complete theory. hole spacetime. To study the quantum mechanics of dust shell collapse, we use various time slicings in hop e that the physical I I. CLASSICAL EQUATION OF MOTION essence is indep endent of the time slicing. The black hole horizon is not a sp ecial surface for an observer who uses the prop er time along the shell history or the In this section, we derive the equation of motion for a Minkowskian time. However, a static observer outside collapsing dust shell, which is describ ed byvarious time the horizon will require a di erent b oundary condition slicing. Since the spacetime is spherically symmetric, one for the wave function due to the existence of the horizon can cho ose the metric of the form for him. By developing quantum mechanics for a static 2 a b 2 2 obsrever, we can obtain the mass sp ectrum which cor- ds = g dx dx + R d ; (a; b =0;1) (2.1) ab 2 resp onds to (1.2) and (1.4). Furthermore, we will show 2 where R is the circumference radius and d is the met- that wormhole states are also p ossible for a static ob- 2 ric of unit 2-sphere. For this metric matter elds de- server who stays inside the wormhole but when is same 0 1 p end on x and x which are time and radial co ordi- order of m no wormhole state is allowed owing to the p nates. Furthermore the spacetime is assumed to b e in zero-p oint uctuation of the shell motion. vacuum except the -function matter distribution at a This pap er is organized as follows. In Sec.I I, we derive nite circumference radius. We denote the world volume the classical equation of motion of the shell in terms of of this spherically symmetric shell as which is a (1+2)- various observers who de ne time slicings on the shell tra- dimensional hyp ersurface. In a neighb orho o d of , one jectory in spacetime. Although in nitely many observers can cho ose the Gaussian normal co ordinates [10], or time slicings can b e assumed in general, we restrict our consideration to a one-parameter family which con- 2 2 2 2 2 ; (2.2) ds = d + d + R d 2 tains twotypical time slicings corresp onding to the Gaus- sian normal co ordinates of a comoving observer and the where is the prop er time whichwould b e measured by static Schwarzschild co ordinates of an observer who stays an observer comoving with the shell. The co ordinate is at a nite circumference radius, and derive the p ossible the prop er distance from along the geo desics which are spacetimes as the solutions of the Einstein equations. In orthogonal to , and is assigned to the world volume Sec.I I I, weintro duce the Hamiltonian constraint which of = 0. Imp osing the spacetime to b e orientable on , corresp onds to the time-time comp onent of the Einstein 2 dx = dR + d; (2.7) wemay regard one side of as b eing the \outer side" (>0) and the other side as b eing \inner side" (<0). where and are constant parameters at the outside The whole spacetime is constructed by connecting the shell. This form is useful b ecause one can recover the the Schwarzschild spacetimes of the outer side and the static and comoving time slicings bycho osing the param- Minkowski spacetime of the inner side. The Einstein eters to b e =0,and = 0, resp ectively. Except the equations give the \junction condition" for the neighb or- sp ecial cases the time slicing corresp onds to an observer ho o d of , whichiswell-known as Israel's formula [8,10]. who is not comoving with the shell but infalling toward In the case of the Minkowski-Schwarzschild junction, the the origin R = 0. In the following we will study the pa- nontrivial conditions are given by [10], rameter dep endence of the equation of the shell motion in b oth classical and quantum levels. [@ R]= ; [@ A]= 0; (2.3) R The rst task is to rewrite the junction condition (2.3) using the co ordinates x and t. The general co ordinate where @ denotes the partial derivative with resp ect to its transformations from (2.2) into (2.6) should b e subscripted co ordinate, and A represents all the metric functions on the spacetime. The bracket [A] means the d = N cosh dt + U sinh dx; (2.8) di erence of A between the outer and inner sides, d = N sinh dt + U cosh dx: (2.9) [A] = lim A lim A: (2.4) The b o ost angle is related to the velo city of the shell !+0 ! 0 in the (t; x) frame as follows The rst equation of (2.3) contains the total energy of 2 U dx dust shell de ned by = R where is the surface tanh = : (2.10) energy density of the dust shell. From the divergence of N dt the energy-momentum tensor of dust shell, one can easily where the subscript means the derivative along a line see that is a constant of motion. of constant . In this frame the junction conditions (2.3) For our convention of calculation, let us intro duce are given by quasi-lo cal mass de ned by N R [(@ R) ]= U cosh ; [(@ R) ]= sinh ; x t t x M = (1 g @ R@ R); (2.5) R R 2 (2.11) which is conserved in eachvacuum region of the spher- ically symmetric spacetime. Of course, M = 0 in the where t and x are treated as indep endentvariables in the Minkowski side, and M (6= 0) in the Schwarzschild side calculation of the partial derivatives. represents the gravitational mass of the shell. The for- On the other hand, b ecause the quasi-lo cal mass M mula (2.5) is useful to estimate the derivatives of R in de ned by (2.5) invariant under the co ordinate transfor- b oth sides, which are involved in (2.3). mation (2.8) and (2.9), it must satisfy with the conditions Nowwe discuss the time slicings to describ e the shell 2 2 motion. \Time slicings" usually mean foliations of a (@ R) (@ R) 2M t x x+ t+ 1= ; (2.12) whole spacetime. Foliations are spaces in a spacetime 2 2 R N U in which one can set observers. In our mo del, however, in the Schwarzschild side, and the equation of motion for the collapsing shell can b e re- duced to a lo cal equation on . Hence, the necessary 2 2 (@ R) (@ R) t x x t pro cedure is to set a radial direction of co ordinate sys- 1= ; (2.13) 2 2 N U tem near , which is called \time slicing" in this pap er. Let us denote the radial co ordinate by x and rewrite the in the Minkowski side. By the virtue of the time slicing metric into the form (2.7) and the co ordinate transformations (2.8) and (2.9), we obtain the relations 2 2 2 2 2 2 2 : (2.6) ds = N dt + U dx + R d 2 dx dR = ; (2.14) We can refer to a lo cal observer near whose world line is dt dt along a constant x. There are twotypical observers. One dR d is a comoving observer corresp onding to the time slicing = = N sinh ; (2.15) dt dt x = . The world line of this observer is emb edded in x x . Another is a static observer who stays at a constant dR 1 d 1 = = U cosh : (2.16) circumference radius x = R (dx = dR in (2.6)). We call dx dx t t the ab ove time slicings \comoving slicing" and \static 2 2 slicing," resp ectively. Our idea is to give a more general If the metric comp onents N and U in the co ordinate form of the time slicing as follows, system (t; x) are assumed to change continuously at the 3 2M shell, substitution of (2.15) and (2.16) into (2.11) leads 2 2 N = 1 f (t) ; (2.24) to R where f (t) is an arbitrary function of t. The factor f (t) [ ]= ; [ ]= 0: (2.17) may b e eliminated by using the co ordinate transforma- R tion dT = f (t)dt.However, one cannot determine this Note that the parameter = at the outer side must + function by the Einstein equations. Since the junction b e di erent from = at the inner side. Since we are conditions are lo cal conditions, all the metric comp onents interested in an observer lo cated at the outer side of the on the shell dep end only on the time co ordinate t, and shell, only is treated as a parameter for sp ecifying + the lapse function N (t) is treated as an arbitrary func- the time slicing. Then b ecomes a function dep ending tion by the virtue of the gauge freedom f (t) ofachoice on R.Furthermore, (2.12) and (2.13) for the quasi-lo cal of t by a lo cal observer. In the next section, we develop mass in b oth sides are rewritten into the forms the canonical quantum theory using this arbitrariness of N . 2M 2 cosh 1 + 2 2 1 = ; (2.18) + Before discussing the quantization pro cedure, it is b et- 2 R U U ter to give the brief derivation of the classical solution of this system, since the vacuum spacetimes in the outer and and inner regions of the shell are classically treated. In 2 cosh 1 2 2 the comoving frame, the di erence of the quasi-lo cal mass = ; (2.19) 2 U U between the outer and inner sides of leads to the for- mula resp ectively. If and are omitted from Eq.(2.17), 0 0 (2.18) and (2.19), we obtain the metric comp onent M = (R + R ); (2.25) + 2 1 2M 2M 2 2 where prime denotes the derivative with resp ect to . = + ; (2.20) + 1 + + 2 U R R Then the junction conditions (2.3) allow us to write ex- 0 0 plicitly R and R as follows, + which shows that g = 1 in the comoving time slicing xx ( =0; = 1), while g =1=(1 2M=R) in the static + xx 1 1 0 0 2E + ; R 2E : (2.26) = R = + time slicing ( =1; = 0). The velo city of the shell + 2 R 2 R motion measured by the time co ordinate t should b e de- Because the classical motion for b ound states is limited ned by(dR=dt) . Then, by using (2.10), (2.14), (2.18) in the range, and (2.20), we arrive at the nal form of the equation of motion 0 R ; (2.27) 2(1 E ) 2 dR 1 V (M ; ; ; R)= 0; (2.21) + 0 0 2 the derivatives R and R must satisfy the condition 2N dt 2 + 0 0 R 2E 1; R 1: (2.28) + where the p otential V = V (M ; ; ; R) is given by 0 1 The equalities hold just at the turning p oint R = 0 =(2(1 E )) of the shell motion. Recall that R are B C V prop ortional to the comp onents of the extrinsic curva- B C 1 1+ V = V ; (2.22) B C 1 2 tures K of of the outer and inner sides. Hence, @ A 1 2 + + 2E K is always p ositive, while K b ecomes negative when 2 R E<1=2. The signs of K are essential to the global 2 1 structure of spacetime constructed by the junction of the 2E+ ; (2.23) V =1 1 4 R outer and inner spacetimes. In the range of E given by 0 and and E are constants de ned by = = and + shows a wormhole formation and a black hole formation, E = M= resp ectively. In the limit !1, the p oten- resp ectively (see Fig.1) [10]. The gravitational mass M tial V coincide with V in the comoving system. This 1 can b e also negative, even if the lo cal energy condition represents the motion of the shell describ ed by an ob- 2 = R > 0 is imp osed, and it do es not contradict to server corresp onding to the time slicing parameter \" the p ositive mass theorem [12] b ecause it has no asymp- who is accelerated against gravity pro duced by the shell. 2 totically at region and has a timelike singularityatthe It is worth noting that the metric comp onent N = g tt Schwarzschild side. Thus the relation b etween the value cannot b e determined by the junction conditions. The of E and the global structure of spacetime is very clear, reason can b e seen in the pro of of Birkho 's theorem and we use this relation also in quantum mechanics of [11]. When one cho oses R to b e the radial co ordinate, the the shell, in which the motion is sp eci ed by a discrete vacuum Einstein equations for the spherically symmetric eigenvalue of E . vacuum spacetime give 4 (a) secondary constraint coincides with (2.21). The Euler- Lagrange equation which can b e derived by the variation with resp ect to R is the rst derivative of (2.21). Be- ehave the Lagrangian (3.1), the usual step of the + cause w canonical formalism leads to the canonical momentum conjugate to R dR P = ; (3.2) N dt (b) and the Hamiltonian N 2 H = (P + V ); (3.3) 2 + We wish to emphasize that (2.21) is a result of the Hamil- tonian constraint H = 0 which is generated by the varia- tion with resp ect to N . Our mo del can keep the prop erty of a constrained system in a similar manner to the full canonical theory of general relativity. The next step is to quantize the Hamiltonian con- t given by (c) strain 2 P + V =0: (3.4) e use the usual commutation relation [R; P ]= ih + If w and the simplest factor ordering, (3.4) is reduced to the Schrodinger equation, 2 d 2 h +V =0; (3.5) 2 FIG. 1. Penrose diagrams of p ossible spacetimes con- dR structed by the Minkowski-Schwarzschil d junction with where p otential V is given by (2.22) and has the form E<1. The tra jectories of the dust shell in the Minkowski and Schwarzschild spacetimes are drawn. The two spacetimes 2 2 2 (R R )(R R )(R + R ) [ +1+ 2E ](1 E ) 0 1 2 are glued at the shell b oundary. These gures show (a) a ; V = 2 2 ( + E ) R(R R ) black hole spacetime for 1=2 3 0 0 condition requires that b oth R and R are p ositive), (b) a + (3.6) 0 wormhole spacetime for 0 + 0 0 R > 0), and (c) a negative mass spacetime (in which R < 0 + and the constants R , R , R and R are given by 0 1 2 3 0 and R ) without any asymptotic at region. 1=2 1=2 m(2E + )h mh R = ; R = ; (3.7) 0 1 2 + 1+2E 2(1 E ) I I I. CANONICAL QUANTIZATION 1=2 1=2 mh mh R = ; R = : (3.8) 2 3 2(1 + E ) 2( + E ) Nowwe give the Hamiltonian which generates the equation of motion given by (2.21). The pro cedure is to Let us explain some implications of these radii. Because use the gauge freedom previously mentioned. Although of the condition jE j < 1 of the b ound state, the classical the Hamiltonian is not uniquely determined, we seek the motion of the shell has the maximum radius R which 1 simplest one here. For this purp ose, we consider the La- corresp onds to a turning p oint. In the time slicing pa- grangian for (2.21) rameterized by , the classical motion has also the min- 2 imum radius R = R where the in nite redshift o ccurs 0 dR N for the corresp onding observer. Note that R is equal to V: (3.1) L = 0 2N dt 2 2M in the static limit = 0, which coincides with the true horizon radius. From the p otential (3.6), R = R 0 _ Because (3.1) do es not include N , there is a primary is a turning p oint of the shell. Since the WKB feature, constraint that is the canonical momentum conjugate in general, breaks down at turning p oints, this means to N must weakly vanish. Furthermore, there is a sec- that the semiclassical description b ecome meaningless at ondary constraint which can b e obtained by the variation R = R [13]. The p otential V diverges at R = R where 0 3 of (3.1) with resp ect to N . It is easy to con rm that the we obtain the regular singular p oint of the di erential 5 equation (3.5). This radius R b ecomes smaller than R , of the wave function is p ossible in the region R 3 0 0 if the inequality E>1=2 =2 holds. Then, we can must stop at R = R owing to the in nite p otential bar- 3 consider the region R < R 3 0 sically forbidden and disapp ears only in the comoving here is that the wave function vanishes at R = R and 3 limit !1(i.e., R ! 0). Only when < 3 there R !1. Although it is dicult to solve (4.1) exactly,an 0 exist the additional b ound states with E in the range approximate calculation of the eigenvalue E is p ossible. 1 2 allowed region R 0 1 R R E 2 2 regions 0 0 1 3 1 < 1 < 1 ; (4.3) =1+ R R 1+E similar to the quantum eld theory in a Rindler space- 3 time, in which b oth sets of mo de functions in the left- since the wave function is de ned in the region R 3 and right-handed wedges together are complete on all of 1. As will b e seen later, the eigenvalue E is in the range Minkowski space [14]. 1=2 The Schrodinger equation (3.5) means that wehave the Hilb ert spaces H of which dep ends on the time R 3 2 slicing parameter , in particular, as a consequence of 1 < 1+ < (4.4) R 2 the existence of the classically forbidden region R < 3 R< R (or R 0 0 3 remains nearly constant in (4.1). Then, intro ducing the < 3). Hence, for each ,we can give a discrete set of non-dimensional variable z , the eigenvalues of E , which is the unique observable in p 2 this quantum system. Because the vacuum spacetimes (1 E ) 2m R R 3 ; (4.5) z = outside and inside the shell are classically treated, the 1=2 E h global structure of the whole spacetime is sp eci ed only by E . Then the Hilb ert space can b e regarded as a set The approximate form of (4.1) can b e written as follows, of spherically symmetric spacetimes (such as black holes, 2 2 d 1 k p 1=4 wormholes) which the collapsing shell forms. + + =0 (4.6) 2 2 dz 4 z z IV. MASS EIGENVALUES where is treated as a constant, and 4 2 m (1 2E ) (1 + 2E )(1 + E ) 1 2 In this section, we study the quantum mechanics ; (4.7) p = + 4 4E 4 on various time slicings using the Schrodinger equation r 2 3 (3.5). We mainly consider the static time slicing =0 m (2E 1)(2 + E 2E ) (1 + E ) k = (4.8) 2 and then the other time slicing is considered to con rm 4E 1 E that our argument is the natural extension of that in [7]. The general solution is a sup erp osition of the two Wit- taker's function M (z ) and M (z ), one of whichis k;p k; p de ned by A. Static Time Slicing 1 n X (2p + 1) (p k + n +1=2) z p+1=2 z First, we consider the typical time slicing which corre- M (z )=z e k;p (2p + n + 1) (p k +1=2) n! sp onds to a static observer. When = 0, the Schrodinger n=0 equation (3.5) can b e written in the form (4.9) 2 2 2 d (1 E ) (R R )(R R ) R + R 0 1 2 2 h + =0 The b oundary condition at z !1selects the unique 2 2 2 dR E (R R ) R 3 solution W (z ) which exp onentially decrease as z in- k;p creases and is written by the sup erp osition (4.1) (2p) ( 2p) where M (z )+ M (z ): W (z )= k;p k; p k;p (1=2 p k ) (1=2+ p k) 1=2 mh 1=2 R =2M =2mE h ; R = ; 0 1 (4.10) 2(1 E ) 1=2 1=2 mh mh Now another b oundary condition at the regular singular R = ; R = : (4.2) 2 3 p oint z =0 (R=R ) determines the eigenvalue E . The 3 2(1 + E ) 2E p+1=2 b ehavior M (z ) ! z in the limit z ! 0 means k;p As previously mentioned, for the b ound states in the that can satisfy the b oundary condition only when range 1=2 0 (1=2+p k)=1; (4.11) R is classically allowed. Though a quantum p enetration 1 6 b ecause the -function do es not vanish on the real axis. Although the sp ectrum of E obtained here is consis- Since the function has p oles at non-p ositiveintegers, tent with the assumption 1=2 this condition is reduced to b e also checked bynumerical calculation of (4.1), which is based on the standard sho oting metho d of solving two- n = k p 1=2; (4.12) p oint b oundary-value problems [15]. In Fig.2 the eigen- values of E are plotted for m = 1 and m = 100, which where n is a non-negativeinteger. From (4.12) we can corresp onds to the quantum numb ers of n which runs give the limiting b ehavior of the mass eigenvalue: In the 2 from 0 to 10. We note that the approximate sp ectrum limit n m , (4.12) coincides with the numerical results within the ac- 4 m curacy of our numerical co de if cho osing the constant E 1 : (4.13) 2 8(n +1) = 1 in (4.13) and =4=3 in the case of (4.14) [16]. Hence we can claim that (4.12) is useful to discuss the If = 1, (4.13) corresp onds to the sp ectrum (1.2) ob- qualitative b ehavior of the sp ectrum E = E (n). tained by Berezin [4] and (1.4) Hajcek et al. [7] in the The b ehaviors of the sp ectrum of E written by (4.13) limit n>> 1 of highly excited states. On the other hand and (4.14) are plausible as a quantum version of gravi- 2 in the opp osite limit n< tational collapse. Recall that n is the quantum number comes more imp ortant, and wehave of the shell motion, and 1 E can b e regarded as the 1 3 gravitational binding energy p er unit rest mass energy. 1 2n+1 p E + : (4.14) We can nd that the gravitational binding energy re- 2 2 3 m mains small when the kinetic energy of the shell motion 2 dominates the inertia of the shell (n m ), while the (a) binding energy b ecomes large when the inertia dominates .For the ground state (n =0)we ob- 1 the kinetic energy tain E ! 1=2 in the limit m 1, which corresp onds to k hole formation. The quantum 0.99 the classical limit of blac 2 1=3 zero-p oint uctuations generate the term (2=m ) in 0.98 (4.14), where is taken to b e equal to 4=3. The contri- E bution of this zero-p oint uctuations can seriously a ect 0.97 alpha = 1 m is not so large. numerical results the motion of the shell, if m<1 which is required in 0.96 Note that the restriction the comoving time slicing disapp ears in this static time 0.95 slicing. This is a consequence of the inner b oundary con- dition. In this static time slicing, the central singularity 0.94 y the in nite redshift surface, and the inner 0 2 4 6 8 10 is hidden b n b oundary condition is set up at a nite R. Hence we can construct quantum mechanics of the collapsing dust shell without taking account of \information loss" at the (b) tral singularity. 0.6 cen The eigenvalues (4.12) (or (4.13), (4.14)) shows that h corresp onds to 0.59 the global structure of spacetime whic these eigenvalues is limited to only a black hole for- 0.58 mation, i.e. the wave functions which has a supp ort only in R 0.57 3 states(Fig1.(a)). This do es not means that there is no E ormhole state. When E<1=2, we can also consider the 0.56 numerical results w alpha = 4/3 wave function whose supp ort exists only in 0 0.55 In this case, the classically allowed region also exists in R 1 0 1 3 0.54 0 holds due to the condition E<1=2. In this case, one ximation that de ned by (4.4) 0.53 cannot use the appro 0 2 4 6 8 10 t. So, wemust numerically solve the n is nearly constan Schrodinger equation (4.1) and the result is shown in FIG. 2. The eigenvalues of E for the wave functions of Fig.3: For xed m, the eigenvalue of E monotonically black hole formation in a static time slicing. The validity decreases in the range 0 of the approximate formula (4.12) checked bynumerical cal- eigenvalues plotted in Fig.3 corresp onds to the global culations. The value of m is chosen to b e (a) m =1:0000 and structure of a wormhole formation i.e. the wave function (b) m = 100:00. The sp ectrum given by (4.12) is drawn by whose supp ort exists only in 0 solid lines which corresp onds to (a) = 1 and (b) =4=3 3 to the wormhole states(Fig.1(b)). When m>>1, there resp ectively. 7 are many b ound states whose eigenvalues of E satisfy the is not so much di erent from the case = 0, and we also inequality E<1=2. Together with the case E>1=2, we imp ose the b oundary condition; the wave function must 1=2 can take the limit to the p ositive mass classical solutions vanish at the regular singular p oints R = mh =(2( + of b ound state in which E takes an arbitrary value in E )) and R = 1. Then, we can derive the eigenvalues 0 E in a similar manner. The approximation that = eld theory in Rindler spacetime. Note that there is no 1 R =R is constant in (3.5) leads to 2 b ound state in < m 2:4m , and wormhole space- p 4 2 m (1 + ) times do not exist in the case < m due to the zero : (4.16) E 1 2 p oint uctuations of the shell motion. 8(n +1) 0.3 2 in the limit n>>m . This shows that the -dep endence constraint for bound states n=0 2 E is not so sensitive in the limit n>>m .However, n=1 of 0.25 n=2 ximated form (4.16) will not remain valid as n=3 the appro b ecome in nitely large. To clarify the b ehavior of E for 0.2 the ground states in the limit >>1, wemust solvenu- merically the Schrodinger equation (3.5). By varying the E 0.15 parameter ,we can consider the extrap olation from the static time slicing to the comoving one. In particular, for 0.1 m 1, we can compare the sp ectrum of E with (1.4) in the comoving limit. The numerical results for m = 1 are 0.05 plotted in Fig.4, which con rm that the mass sp ectrum converges to (1.4) as increases and E remains larger 0 =2. On the other hand, as an example of the sp ec- 0 1 2 3 4 5 6 than 1 m trum for m>1, the eigenvalues E for m = 10 are plotted in Fig.5. We nd the common tendency that the energy FIG. 3. The eigenvalues of E for the wave functions level at the ground state decreases as increases. The of wormhole formation in a static time slicing. The remarkable p oint for m>1(i.e., >m ) is that the p m-dep endence of E is shown for the quantum numb ers n =0, p mass sp ectrum do es not keep the condition of E>1=2 V means a 1, 2 and 3. The solid line given by R >= min 3 nevertheless the wave function has its supp ort only in rough upp er b oundary of the allowed range of E . R 3 E<0 are allowed, if is suciently large. (This might Wemust also note that the state of E =1=2 is for- means that H in (1.1) cannot b e p ositive self-adjoint B bidden like the case E<1=2. It can b e also seen from op erator in general.) Then observable states exist in the the necessary condition for the existence of b ound states. range 1 p Let us denote the minimum value of the p otential V by and these corresp ond to the global structure of black hole V (E ) < 0 which dep ends on E . Since we imp ose the min and wormhole formations (see Fig.1). b oundary condition j = j = 0, all eigenvalues R=0 R=R 3 E of b ound states must satisfy the condition 1 p R (E ) > : (4.15) 3 0.95 V (E ) min E =1=2 ;(<<1), we obtain V (E ) When 0.9 min 2 2 2 m (1 + ). Then the inequality is reduced to m > = , which cannot b e satis ed when ! 0. Furthermore E 0.85 lambda = 0 lambda =1 e can giveaphysical interpretation of E 6=1=2. The w lambda =5 lambda = 10 lambda = 50 classically allowed region is R 1 1 0 0 0.8 lambda = 100 alpha = 1 E ! 1=2, so the shell is con ned in this narrow 0as spectrum given by Hajicek et.al wever, it is imp ossible due to the uncertainty region. Ho 0.75 relation P R h,thus E =1=2 is forbidden due to tum e ect of the shell motion. the quan 0.7 0 2 4 6 8 10 n FIG. 4. The -dep endence of the sp ectrum E (n) for B. Non-static Time Slicing m =1:0000. The time slicing parameter is chosen to b e =0:0000, 1:0000, 5:0000, 10:000, 50:000 and 100:00. As Based on the result obtained in the static time slicing, increases, the sp ectrum of E approaches to (1.4) drawn by we consider the Schrodinger equation (3.5) in the case the dashed line. 6= 0 except that there is no b ound states in the region 0 3 8 0.9 solutions which describ e wormhole or black hole forma- 0.8 tion. One may regard the regions 0 0.7 R 0.6 spacetime resp ectively in a similar way to the quan- 0.5 tum eld theory in Rindler spacetime [14]: The set of e eigenvalues of E in our 0.4 classical solution with p ositiv E 0.3 mo del corresp onds to the complete set of mo de func- lambda=0 owski spacetime, and black hole and 0.2 lambda=1 tions in whole Mink lambda=2 wormhole states corresp ond to the mo de functions in 0.1 left- and right-handed wedge resp ectively.Furthermore, 0 we showed that the wormhole states cannot exist when -0.1 < m 2:4m as a result of the zero p oint uctuations p -0.2 or a comoving time slicing also, we 0 2 4 6 8 10 of the shell motion. F n can see the same quantum e ect. Although it is not unclear whether E<0 is p ermissible or not, observable FIG. 5. The -dep endence of the sp ectrum E (n) for states exist in the range 1 p m =10:000. The time slicing parameter is chosen to b e only black hole states are p ossible when < m .Thus, p =0:0000, 1:0000 and 2:0000. It is shown in this gure our conclusion is that the wormhole formation with small that the sp ectrum do es not remain in the range 1 >E >1=2 mass is eciently suppressed in the quantum collapse of and even the negative mass states are allowed. dust shell. Then, in our mo del, it is dicult to create small wormhole due to the quantum uctuations of the Finally,we consider the wave function given by the shell motion. Schrodinger equation (3.5) in the comoving frame.( ! Although the -dep endence of E is not so essential to 1), which has the form the set of observalb e states in our argument, to set a 2 2 time slicing, we can easily see the relation b etween ob- d 1 2 2 h + 2E + = 0: (4.17) 1 2 servalb e states and geometries (wormhole states or black dR 4 R hole states, etc.). This means that the time slicing or Note that this Schrodinger equation corresp onds to the observers in a spacetime plays an imp ortant role when Wheeler-DeWitt equation (1.3), and our Schrodinger one classify the observable states in quantum theory of equation may b e regarded as a natural extension of (1.3). gravity. Since quantum theories are based on the Cauchy In terms of the requirement of unitarity and of p ositivity problem and, usually, the con guration space of quantum of energy,Hajcek et al [7] gave the b oundary condition gravity is three-space geometries, wemust sp ecify a set that for the b ound state E<1 the wave function must of observers or initial time slicing at rst to clarify the set vanish at the origin and at in nity. Then the discrete of observable states. In this sence, the notion of the time sp ectrum of E was shown to b e (1.4) which b ecomes slicing or observer will b ecome to b e more imp ortantin meaningless when m>1. As discussed byHajcek et a fully consistent quantum gravity. al., this means that when m>1 one cannot imp ose the p ositivityofE or the regularity of the wave function at ACKNOWLEDGMENTS R = 0. In our approach the regularityofwave function at R = R are set up in any time slicing parameter , 3 even if m> 1. Therefore, we can discuss the comov- Wewould like to thank H.Ishihara, J.So da, T.Tanaka ing limit through the extrap olation of the results to the and M.Hotta for valuable discussions. range 1. In this sense, as previously mentioned, ob- servable states exist in the range 1 is suciently larger than m , there is no essential di er- p ences in a set of observable states. On the other hand, there are no wormhole states 0 p These states are suppressed by the zero p oint uctuations [1] S.W. Hawking, Commun. Math. Phys. 43, 199, (1975), of the shell motion. Phys. Rev. D14, 2460 (1976). [2] D.N.Page, \BLACK HOLE INFORMATION" in Pro- ceedings of the 5th Canadian Conference on General V. SUMMARY AND DISCUSSION Relativity and Relativistic Astrophysics, University of Waterloo,13-15 May, 1993, edited by R.B.Mann and In summary,by studying this mo del of dust shell col- R.G.McLenaghan (World Scienti c, Singap ore, 1994) lapse, we obtained the wave function for the b ound states [3] V.A. Berezin, N.G. Kozimirov, B.A. Kuzmin and I.I. with p ositive eigenvalues of E under the static time slic- Tkachev, Phys. Lett. B 212, 4, 415 (1988). ing. When >>m, these can recover the classical p [4] V.A. Berezin, Phys. Lett. B 241, 2, 194 (1990). 9 [5] H.Ishihara, S.Morita and H.Sato, Mo d. Phys. Lett. 6, 10,855, (1991). H.Ishiraha, S.Morita, KUNS-1012 April 11, 1990, (preprint). M. Visser, Phys. Rev. D43,2,402 (1991). V.A. Berezin, Bures-sur-Yvette rep ort, 1991 (un- published ). I.H. Redmount, W.-M. Suen, Phys. Rev. D 49, 10, 5199 (1994). [6] P.Hajcek, Commun. Math. Phys. 150, 545 (1992). [7] P.Hajcek, B.S. Kay and K.V. Kuchar, Phys. Rev. D46, 12, 5439 (1992). [8] W. Israel, Nuovo Cimento 44B, 463 (1967). [9] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th edition (Pergamon Press, Oxford, 1989). [10] S.K. Blau, E.I. Guendelman and A.H. Guth, Phys. Rev. D35, 1747 (1987). V.A. Berezin, B.A. Kuzmin and I.I. Tkachev, Phys. Rev. D36, 10, 2919 (1987). [11] S.W. Hawking and G.F. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973). [12] R.Scho en and S.T.Yau, Phys. Rev. Lett 43, 1457 (1979); Commun. Math. Phys. 79, 47 (1981); 231 (1981). E.Witten, Commun. Math. Phys. 80, 381 (1981). [13] P.Hajcek, Phys. Rev. D31, 785 (1985), K.Nakamura, Y.Oshiro, A.Tomimatsu, Prog. Theor. Phys. 90,4,861 (1993). [14] N.D.Birrel and P.C.W.Davies Quantum elds in curved space (Cambridge Univ. Press. Cambridge, 1982). [15] W.H.Press et al. Numerical Recipes: the Art of Scienti c Computing (Cambridge Univ. Press. Cambridge, 1985). [16] =4=3 can b e easily obtained by the estimation as follows. When E ! 1=2, the classically allowed region b ecome to b e narrow due to the R R ! 0. Since 1 0 the eigenvalues of E is determined by the b ehavior of the wave function in the classicall y allowed region, 1+R =R 1+ R =R 4=3 when E ! 1=2. 2 0 2 1 10