PHYSICAL REVIEW D VOLUME 35, NUMBER 12 15 JUNE 1987

Fate of singularities in Bianchi type-I11 quantum cosmology

Jorma Louko Department of Applied Mathematics and Theoretical Physics, University of Cambridge,

Silver Street, Cambridge CB3 9EW, England (Received 2 February 1987)

We study the quantum cosmology of a locally rotationally symmetric vacuum Bianchi type-111 minisuperspace model with a non-negative cosmological constant. The quantum state chosen by the Hartle-Hawking proposal corresponds to a particular ensemble of classical Lorentzian spacetimes, most of which have negligible anisotropy. This quantum state ceases, however, to correspond to most of these spacetimes near their singularities. This means that in the Hartle-Hawking quantum state, classical spacetime ceases to be a valid concept near most of the classical singularities of the model.

I. INTRODUCTION

There is considerable evidence that the Hartle-Hawking (HH) proposal for the quantum state of the ~ n i v e r s e ' ~ ' may provide a conceptually simple explanation of the ob- served large-scale cosmological features of our Universe. Combining ideas of canonical and path-integral quantiza- tion of general relativity, this proposal defines the wave function of the Universe by a Euclidean path integral over compact four-geometries that are bounded by a prescribed three-geometry and regular matter fields that have prescribed values on the boundary. Although no precise definition of this integral is known, calculations in simple cosmological model^^-^ indicate that the classical solu- tions corresponding to the H H quantum state tend to have a long inflationary period, leading to a large, homogene- ous and isotropic universe with small-scale inhomo- geneities. Owing to its geometrical character, the propo- sal may also be extendible to more fundamental theories, of which general relativity is expected to emerge as a low-energy limit.

It is, on the other hand, far less clear what the H H pro- posal predicts about the most disturbing feature of classi- cal general relativity: spacetime singularities. As general relativity is not expected to be a fundamental theory of nature, it may not be very realistic to address the ultimate question of singularities in quantized models whose classi- cal dynamics is derived from Einstein's action. Neverthe- less, in simple cosmological models the predictions the H H proposal makes about singularities seem to be linked with the question as to whether the proposal also predicts a relation between the cosmological and thermodynamic arrows of time.'-'' Recent result^^"^ in a closed Friedmann-Robertson-Walker model and a Kantowski- Sachs model with a massive scalar field suggest that the solutions picked by the H H proposal do collapse into a fi- nal singularity after an initial inflationary phase and a matter-dominated phase, and that the thermodynamic ar- row of time points irreversibly toward the singularity. The problem, however, with even these simple models is that the saddle-point four-geometries that are assumed to give the dominant contribution to the path integral cannot

be found explicitly: in particular, these saddle-point geometries are neither Euclidean nor Lorentzian but com- plex, and it is not obvious how reliable the approximation of providing initial conditions for purely Lorentzian solu- tions by analytically continuing purely Euclidean ones remains in the matter-dominated and recollapsing phases of these solutions.

In this paper we study the relation of the H H proposal and spacetime singularities in a Bianchi type-I11 vacuum minisuperspace model with a non-negative cosmological constant. This model is not particularly relevant for our own Universe, since the classical solutions do not at any stage approximate a matter-dominated Friedmann- Robertson-Walker cosmology. What makes the model in- teresting, however, is that the singularities of the classical solutions are very similar to those in the Kantowski-Sachs model describing the interior of a black Further- more, finding the complex saddle-point four-geometries reduces to solving a simple algebraic equation, and the model gives therefore an arena for testing approximations used for finding the saddle points in more complicated models. We shall find that in the quantum state specified by the H H proposal, most of the classical singularities of our model are avoided in a genuinely quantum- mechanical manner: the wave function simply ceases to correspond to classical Lorentzian solutions near their singularities. This means that classical Lorentzian space- time ceases to be a valid concept when one approaches the region where the singularity classically should be, and the singularity disappears into quantum fluctuations. This is the prediction one would expect a proper quantum theory of gravity to give.

The paper is organized as follows. In Sec. I1 we intro- duce the model and examine the local and global proper- ties of the classical solutions. The H H wave function and the corresponding Lorentzian solutions are found in Sec. 111. The results are summarized and discussed in Sec. IV.

11. CLASSICAL DYNAMICS

We start from the spatially homogeneous minisuper- space metric

3760 1987 The American Physical Society

35 - FATE OF SINGULARITIES IN BIANCHI TYPE-111 . . . 3761

Here d f i 2 is the metric on a compact two-dimensional manifold V of constant negative Ricci curvature -2. Such manifolds are locally isometric to the unit pseudo- sphere: they are constructed by factorizing the pseudo- sphere by a discrete group in a way analogous to imposing periodic boundary conditions in flat Euclidean space." The coordinate r is taken to be periodic with period 2 ~ . The overall prefactor is p 2 = 2 ~ / v , where G is Newton's constant and v is the volume of V. In the classification of the local symmetries of spatially homogeneous space- times, the metric (1) is known as the locally rotationally symmetric case of Bianchi type I11 (Refs. 13 and 14). It may also be viewed as an analytic co2tinuation of the Kantowski-Sachs metric," in which d n is replaced by the metric on the unit two-sphere.

Inserting the metric (1) into the Einstein action15

and integrating over the spatial dimensions gives rise to the minisuperspace action

where the overdot denotes d / d t and A=p212. We shall consider only the case A 2 0. Although the Einstein equa- tions for a general Bianchi type-I11 metric are not deriv- able from the spatially homogeneous form of the action (2), the locally rotationally symmetric case makes an ex- ception, and the Einstein equations for the metric (1) are obtained by varying (3) with respect to a, 6 , and N (Ref. 14). The general solution in the gauge b = t is

where D and H are constants. We may take t > 0. For N to be positive we must require ( h / 3 )t + t + D > 0. El- iminating t and N from (4) gives the trajectory

where we have defined c = a '6. Some of these trajectories are shown in Fig. 1.

The solution (4) is reminiscent of the Schwarzschild-de Sitter solution,16 with the constant D being the analogue of the Schwarzschild mass. In the case h=O, in particu- lar, the terms containing d t2 and dr2 are similar to those in the exterior Schwarzschild solution except in their overall signs. We shall now look at the global properties of the solution (4) for the various values of A and D.

(i) A=O, D>O. The solution is analogous to the negative-mass Schwarzschild solution, and the singularity at t =O is a real curvature singularity. Suppressing the pseudosphere, the Penrose diagram is shown in Fig. 2. The pseudosphere will be suppressed in all Penrose dia- grams of this section. -

(ii) A > 0 , D > 0. Near t =0, the solution behaves as in case (i), and the singularity at t =O is a real curvature singularity. As t + cc , however, the solution approaches de Sitter space, and the future timelike infinity is space- like. The Penrose diagram is shown in Fig. 3.

(iii) h=O, D <O. The solution is analogous to the positive-mass exterior Schwarzschild solution. If r were not periodic, the surface t = -D would be similar to the Schwarzschild horizon, and the spacetime could be ex- tended past this horizon just like the Schwarzschild solu- tion. As r is periodic, however, the surface t = -D is in

c A fact analogous to the surface that separates the Taub part from the Newman-Unti-Tamburino (NUT) part in Taub- NUT space." Although local curvature is regular, this surface is topologically singular, and the spacetime cannot be continued in a way preserving geodesic completeness and manifold structure" (the resulting extension would be

- 3 b

\ x t = O

FIG. 1 . Some general Lorentzian trajectories ( 5 ) in the case h > 0. The trajectories end either on the b axis, on the c axis, or FIG. 2. The Penrose diagram of the solution (4) in the case at the origin, depending on the sign of D. In the case h = O , the h = O , D > 0. Solid (dashed) lines indicate curves of constant t trajectories would be straight lines. ( r ) . Points whose r coordinates differ by 2n-n are identified.

3762 JORMA LOUKO

FIG. 3. The Penrose diagram of the solution (4) in the case h > 0, D > 0. Curves of constant t ( r ) are horizontal (vertical) lines. The boundaries r =O and r = 2 ~ are identified.

an orbifoldt8j. We shall call this kind of behavior a Taub-NUT (TN) singularity. The Penrose diagram is shown in Fig. 4.

(iv) h > 0 , D <O. The surface ( h / 3 ) t 3 + t +D=O is a T N singularity similar to that in case (iii), and the only qualitative difference from case (iii) is that the future timelike infinity is now spacelike. The Penrose diagram is shown in Fig. 5.

(v) h=O, D=0. The solution is locally flat. If the pseu_dosphere were not compactified, the metric - d t 2 +dR would cover the interior of the future light cone in three-dimensional Minkowski space, and t =O would be a

FIG. 4. The Penrose diagram of the solution (4) in the case h=O, D <O. The solution (4) covers region I, and the solid (dashed) lines indicate curves of constant t ( r ) . Points whose r coordinates differ by 2 ~ - n are identified. Regions 11, 111, and IV give the orbifold extension past the Taub-NUT singularity t = - D .

FIG. 5. The Penrose diagram of the solution (4) in the case h > 0, D < 0. The only difference from Fig. 4 is that the time- like infinities are now spacelike.

coordinate singularity on the light cone. With the pseudo- sphere compactified, the surface t =0 becomes a T N singularity.

(vi) h > 0 , D =O. The solution is locally de Sitter space. When H ~ = 3/h, we have at t =O a T N singularity similar to that in case (v). If the pseudosphere were not compactified and we regarded p just as an arbitrary con- stant of dimension length, the solution with this value of H would cover the region T > (x2+ y2)"*, of the de Sitter hyperboloid

with the metric

When ~ ~ # 3 / h , there is a conical singularity at t =0, corresponding to a 6 function in local curvature.

111. QUANTUM DYNAMICS

We quantize the model in the standard way by going to the constrained Hamiltonian form of the action (3) and applying canonical quantization.'9 The classical super- Hamiltonian constraint turns into the quantum Wheeler- DeWitt equation

where we have adopted the factor ordering proposed by Hawking and page6 and used the null minisuperspace coordinate pair (b,c) . Following the H H proposal,'22 we choose the particular solution of (8) that is defined as

where the path integral is over all compact positive- definite four-geometries of the form

FATE OF SINGULARITIES IN BIANCHI TYPE-111 . . . 3763

that have the three-geometry specified by the arguments of YHH as their single boundary. IE is the Euclidean Ein- stein action15

which for the metric (10) takes the form

The lower limit of the t integration in (12) corresponds to the boundary of the four-manifold, where a and b have the values that appear as the arguments of YHH in (9). The upper limit, denoted by t = t , , corresponds to the point where the four-manifold closes in a smooth way. We have adopted the convention that t < t , because this will allow us to write the classical Euclidean solutions in a convenient coordinate system. The equality of (11) and (12) follows by integrating the first term in (1 1) by parts and noticing that the second term in (1 1) cancels the sub- stitution term from the boundary of the four-manifold but not the one from t = t , .

The definition of the path measure in (9) is an as yet open question. A well-known problem is that IE is not positive definite, and it is now known whether there exists a complex contour that would make the integral conver- gent. Another problem is how to formulate in the path measure the smoothness of the closing of the four- geometry at t = t , . We are not addressing these questions in detail here. However, as it is a general principle that the fixed quantities in a path measure should reflect the fixed quantities in the corresponding classical variational principle, we may gain some insight into the latter prob- lem by working out the fixed quantities of the classical variational principle based on our IE . This is what we shall do next.

Let Sa, 6b, and SN be variations of a, b, and N such that 6a and Sb vanish on the boundary of the four- manifold, and let L denote the integrand in (12). The variation of the action (12) is, after integrating by parts the terms involving 6 i and 6b,

where La, Lb, and L,v are the usual variational deriva- tives given by

The Euler-Lagrange equations obtained by setting La, Lb , and L,v to zero are the Euclidean Einstein equations for

the metric (101, and the general solution in the gauge b =t is

where b and H are constants. We may take t > 0. By the definition of t , , either a or b must vanish at t =t, . From (15) we see that the only possibility is

and the constant 5 may be rewritten as

To avoid a conical singularity at t = t , , we must impose

which implies

The geometry is then regular at t = t , , the coordinate singularity being similar to the one in the Euclidean Schwarzschild metric.15 The conditions (16) and (1 8) are therefore necessary and sufficient to make the classical solutions close in a regular way: they are also sufficient to make the substitution term in (13) vanish. Hence, the ap- propriate boundary conditions for the classical variational principle are (16) and (18), supplemented with the vanish- ing of 6a and 6b on the boundary of the four-manifold. In particular, 66 need not vanish at t , . This suggests that the path measure in (9) cannot be thought of as an ordi- nary Lagrangian or Hamiltonian path measure that would just fix a and b at both ends and not take into account the substitution term in (12) (Ref. 20).

We shall assume that the path integral (9) may be ap- proximated by contributions from the extrema of I E , the classical solutions given by (151, (17), and (19). For prescribed a and b on the boundary of the four-manifold, t , is determined from

and the classical action is

We see that IE (21) may take both positive and negative values, and the model is therefore a counterexample to Horowitz's conjecture that the classical value of IE be al- ways negative." Other counterexamples are provided by the Kantowski-Sachs and Bianchi type-I models discussed in Ref. 9.

We shall now study the cases h=O and h > 0 separately.

JORMA LOUKO - 3 5

When h=O, (20) has one real positive solution, corre- sponding to a real positive-definite four-geometry, and a pair of complex-conjugate solutions, corresponding to complex four-geometries. We adopt the view that at the semiclassical level, the components of VIHH coming from the separate extrema may be treated separately.22 As the component coming from the real extremum is not oscilla- tory and does therefore not correspond to Lorentzian spacetimes, we shall ignore this component.

When a << b the complex extrema are

and the wave function is

As the prefactor is slowly varying compared with the os- cillating part, YHH corresponds through the WKB ap- proximation to those Lorentzian trajectories for which S=ab acts as the Hamilton-Jacobi (HJ) f ~ n c t i o n . " ~ We shall call Lorentzian trajectories obtained from the HH wave function in this fashion Lorentzian HH trajectories. The HJ equations of motion reduce to

da -- db -O,

(24)

which implies D = O but leaves H arbitrary. As (24) must be considered approximate, however, we may allow D to be nonzero if it is so small that 1 da /db 1 << 1.

When a >>b we have

tion is not valid, and no Lorentzian HH trajectories exist. In Fig. 6 we show the Lorentzian HH trajectories in the

(b,c) coordinate system. When c <<b3, the trajectories are straight lines that would pass near the origin. Before reaching the singularity, however, these trajectories either penetrate the region c >>b or acquire a large da /db, and VIHH ceases to correspond to them. This means that clas- sical Lorentzian spacetime ceases to be a valid concept when one approaches the region where the singularity classically should be, and we may say that the Lorentzian HH trajectories fade out of existence before reaching the singularity. Notice that the purely real second-order term in (22) breaks the WKB approximation before imaginary higher-order terms could modify the HJ function.

When h > 0, (20) has one positive solution and two pairs of complex conjugate solutions. The real solution will be ignored as above, but the complex solutions now give rise to two distinct oscillating components in THH. As the cosmological constant has a characteristic associated length scale p(3/h)"2, we may expect this length scale to divide the configuration space into domains where the behavior of VIHH is qualitatively different. The corre- sponding characteristic value of the dimensionless vari- ables a and b is ~ ~ ( 3 / h ) ' / ~ , and that of c is x3.

When b >>X or c << b 3 or both, one component of VIHH is

where

The prefactor is not slowly varying, the WKB approxima-

FIG. 6. The Lorentzian Hartle-Hawking trajectories in the case h = 0. The trajectories fade out near the dotted line c = b '.

FIG. 7. The Lorentzian trajectories corresponding to in the case h > O . The trajectories fade out before reaching the singularity.

FATE O F SINGULARITIES IN BIANCHI TYPE-I11 . . . 3765 35 -

and

The prefactor varies slowly, and treating S =ab[ 1 + ( h / 3 )b 2 ] ' / 2 as the HJ function reveals that the Lorentzian H H trajectories have negligible D. At b <<X, where the cosmological constant is much smaller than the spatial Ricci scalar, (26) reduces to (23) . At b >>X, where the spatial curvature is small compared with the cosmo- logical constant, the oscillating part of (26) reduces to that in a Bianchi type-I minisuperspace model.9 This is the limit behavior one would expect.

When b <<X and c << b 3 , the formula (26) is not valid, and one can verify that T ~ A is not rapidly oscillating. The Lorentzian H H trajectories corresponding to are therefore as shown in Fig. 7. The trajectories have negli- gible D, and all of them fade out before reaching the singularity.

When b >>X, and c >>x2b, the second component of \VHH is

- \ v ~ A = exp( - M cos [ [$] '" .b2] ,

where

and m is given by (28) . The prefactor varies slowly, and the corresponding Lorentzian trajectories are again those for which D is negligible. When b >>X and c <<x2b, $ 2 ) . HH 1s not rapidly oscillating, and this is the case also when b <<X and c >>x3. When b <<X and c <<x3, how- ever, we get

FIG. 8. The Lorentzian trajectories corresponding to in the case h > 0. The singularity is reached only by a narrow pen- cil peaked around the trajectory c = b 3 + ( 3 /h )b.

Solving the HJ equations with the argument of the cosine as the HJ function gives H 2 = 3 / A but leaves D arbitrary, and the conditions b <<X, c <<x3 imply 1 D 1 <<X. The formula (31) remains valid everywhere near the trajectory on which H Z = 3/h and D =0, reducing to (29) when b >>X. The Lorentzian H H trajectories corresponding to T E A are therefore as shown in Fig. 8. The trajectories fade out before reaching the singularity except for a nar- row pencil peaked around the trajectory on which H 2 = 3 / h and D =O.

When A goes to zero with fixed a and 6 , goes to the wave function of the case h=O, and \ v ~ A goes to zero.

IV. DISCUSSION

The anisotropy of the classical Lorentzian trajectories can be characterized by the shear of the world lines that are comoving with the metric (1). The magnitude of this shear averaged over the spatial directions is given byI3

I 1 2

1 r i b u2= -

p 2 ~ ' lo-% 1 '

a quantity which has the dimension of inverse length squared. In the case h=O, where there are no dimension- ful parameters in the model, a dimensionless quantity a may be obtained by multiplying u2 by p2 (ab2 )2 /3 , which is the square of the characteristic spatial length. Although this may seem rather ad hoc, one finds that the Lorentzi- an H H trajectories are such that a is small, and these tra- jectories fade out when a grows to the order of unity. In the case A > 0 , a second dimensionless quantity /3 may be obtained by multiplying u2 by ( 3 / h ) p 2 , which is the square of the characteristic length associated with the cosmological constant. For *;A, the Lorentzian H H tra- jectories are such that at least one of a and /3 is small, and these trajectories fade out when the smaller of a and /3 grows to the order of unity. For *EL, on the other hand, the Lorentzian H H trajectories peaked around the critical one on which H 2 = 3 / h and D =0 do reach the singulari- ty, where both a and /3 diverge.

In the case h> 0 , it might be argued that the H H pro- posal does not see the singularity of the critical trajectory on which ~ ~ = 3 / h and D =0, since this T N singularity results from compactifying the pseudosphere and the con- ditions (16) and (18) do not depend on this compactifica- tion. The singularities of the neighboring trajectories could then be understood through the auantum uncertain- ., ty principle. This would, nevertheless, not explain why similar T N singularities are avoided in the case A=O. If O< h << 1 , one can verify that the prefactor in *& (31) is much smaller than the corresponding prefactor in *$A. This might be argued to imply a relatively low probability for the singular Lorentzian H H trajectories.

The case h=O is very similar to the vacuum Bianchi type-I model with a positive cosmological constant, for

3766 JORMA LOUKO 35 -

which the H H wave function was found in Ref. 9. By an analysis similar to ours, one can verify that the Lorentzi- an H H trajectories also in this Bianchi type-I model have negligible anisotropy and that all of them fade out before reaching the singularity.

In conclusion, the mechanism by which the H H propo- sal in our model avoids most of the classical singularities is that the wave function simply ceases to correspond to the classical Lorentzian trajectories before these trajec- tories would reach the singularity. This warns us that in general, the approximate knowledge of YHH and the Lorentzian H H trajectories in a limited region of the con- figuration space need not imply that VIHH continues to correspond to these Lorentzian trajectories in the remain- ing regions. For example, in the region where c << b and c <<b, our PEA (26) coincides up to a preexponential fac- tor with the function

. . , . which is an exact solution of the Wheeler-DeWitt equa- tion (8). The lack of the preexponential factor in (26) is irrelevant, since we have not defined the path measure in (9) beyond the classical approximation. Continuing Po along the D =O Lorentzian trajectories by the WKB ap- proximation would lead us to conclude that (33) remains the correct formula for PEA all the way to the singularity and that the Lorentzian H H trajectories therefore do reach the singularity, as this WKB approximation for V10 is in fact exact. From the analytic result, however, we know this not to be the case.

In more complicated minisuperspace models, the com- plex saddle-point four-geometries that are assumed to give the dominant contribution to the H H wave function can- not in general be found by analytic methods. In models with a massive scalar field, the usual way of finding the Lorentzian H H trajectories is first to find those purely Euclidean extrema for which the value of the scalar field is very large and approximately constant, and then to use analytic continuations of these extrema to provide initial conditions for the Lorentzian H H t r a j e c t ~ r i e s . ~ ~ , ~ " ~ When this method is applied in a closed Friedmann- Robertson-Walker model and a Kantowski-Sachs model with a massive scalar field, the Lorentzian H H trajectories

are found to start with the value of the scalar field large and approximately constant in time, giving rise to a large effective cosmological constant and a period of exponen- tial e ~ ~ a n s i o n . ~ ~ ~ " ~ When followed further in time, these Lorentzian trajectories enter a phase where the scalar field oscillates and the trajectories behave as a matter- dominated universe. Finally there follows a recollapse, which reaches a singularity for all these trajectories in the Kantowski-Sachs model and almost all in the Friedmann-Robertson-Walker model. It is likely that this method is reliable in finding the exponentially expanding phase of the Lorentzian H H trajectories, since during this phase the value of the scalar field does not change appre- ciablv and the method is known to be exact in a Dure de Sitter minisuperspace m ~ d e l . ~ , ~ It is not obvious, how- ever. whether the H H wave function continues to corre- spond to these trajectories in the matter-dominated and recollapsing phases. It was recently suggestedi0 that this question could be studied by continuing the H H wave function along these purely Lorentzian trajectories from the exponential phase to the matter-dominated and recol- lapsing phases by using the WKB approximation. Nu- merical analysis showed that the WKB approximation remains self-consistent all the way to the recollapse singu- larity, and this was argued to imply that the Lorentzian H H trajectories do reach the singularity.10 In light of the results in our model and the discussion in the previous paragraph, there seems to exist the possibility that this ar- gument might be hazardous. We would therefore feel that the possible recollapse of the Lorentzian H H trajectories in these models would require further study, especially since this recollapse may have far-reaching consequences for the cosmological and thermodynamic arrows of time and the final fate of our universe.'-lo

ACKNOWLEDGMENTS

I wish to thank J. Halliwell, S. Hawking, R. Laflamme, J. Sidenius, and S. Wada for helpful discussions and com- ments, and the relativity group of DAMTP for their hos- pitality. This work was initiated under a research grant at NORDITA, Copenhagen, and its completion was sup- ported by a research grant from the University of Helsinki and by the British Council.

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