PHYSICAL REVIEW D 71, 127502 (2005)

Absence of the Kasner singularity in the effective dynamics from loop quantum cosmology

Ghanashyam Date*The Institute of Mathematical Sciences, CIT Campus, Chennai-600 113, India

(Received 30 April 2005; published 21 June 2005)

*Electronic

1550-7998=20

In classical general relativity, the generic approach to the initial singularity is usually understood interms of the Belinskii-Khalatnikov-Lifschitz scenario. In this scenario, along with the Bianchi IX model,the exact, singular, Kasner solution of the vacuum Bianchi I model also plays a pivotal role. Using aneffective classical Hamiltonian obtained from loop quantization of the vacuum Bianchi I model, an exactsolution is obtained which is nonsingular due to a discreteness parameter. The solution is parametrized inexactly the same manner as the usual Kasner solution and reduces to the Kasner solution as thediscreteness parameter is taken to zero. At the effective Hamiltonian level, the avoidance of Kasnersingularity uses a mechanism distinct from the ‘‘inverse volume’’ modifications characteristic of loopquantum cosmology.

DOI: 10.1103/PhysRevD.71.127502 PACS numbers: 04.60.Pp, 98.80.Bp, 98.80.Jk

While the celebrated singularity theorems of classicalgeneral relativity imply that the backward evolution of anexpanding universe leads to a singular state, the nature ofthe singularity is elucidated in terms of the Belinskii-Khalatnikov-Lifschitz scenario [1,2]. The scenario viewsthe spatial slice close to the (spacelike) singularity as madeup of approximately homogeneous patches each of whichevolves according to the vacuum Bianchi IX model. As thesingularity is approached, the patches fragment indefi-nitely, asymptotically becoming infinitely small. The vac-uum Bianchi IX evolution in turn, can be viewed as asuccession of Bianchi I evolutions (Kasner epochs), inter-leaved by transitions among the Kasner epochs. This in-terleaved evolution continues indefinitely in a chaoticmanner. Apart from the indefinite number of Kasnerepochs and transitions among these, the monotonic de-crease of the volume and the singular nature of theKasner solution are responsible for the infinite fragmenta-tion of the homogeneous patches.

Singularities ‘‘evolving’’ from nonsingular physicalsituations are indicative of a breakdown of the extrapola-tion of the classical evolution i.e. dynamics of the Einsteinequation and call for an extension/modification of theclassical theory/framework. One natural avenue is to ap-peal to a corresponding quantum theory of gravity. Sincethe classical picture of the singular behavior involveshighly dynamical geometries with arbitrarily large curva-tures, a quantum theory which does not depend on anypreselected background geometry is likely to be mostsuitable for obtaining the required extensions.

Loop quantum gravity (LQG) is precisely such a back-ground independent approach [3]. While the LQG for thegeneral inhomogeneous situations is quite complicated, itsmethods can be implemented and tested in simpler con-texts of spatially homogeneous geometries. Indeed, loopquantization of the so-called diagonalized, Bianchi class A

address: shyam@imsc.res.in

05=71(12)=127502(4)$23.00 127502

models has been carried out [4,5] and shown to be non-singular within the quantum framework. In the quantumframework, nonsingularity means nonbreakdown of thefundamental dynamical equations (which are partial dif-ference equations) and boundedness of relevant operatorssuch as the inverse triad operator which enter the quanti-zation of curvature components, matter densities, etc.

It is of course more intuitive and convenient to obtain themodifications implied by the quantum theory in the famil-iar geometrical setting of classical general relativity i.e.obtaining the modifications to the Einstein dynamics keep-ing the kinematical framework of Riemannian geometry intact. This has been done systematically for the isotropicmodels [6–8] in terms of an effective Hamiltonian. Thederivation of the effective Hamiltonian is based on theobservation that if the fundamental dynamical equationadmits a solution which is WKB approximable (i.e. theamplitude and the phase are slowly varying in a suitablesense) in some domain, then to the order �h0 one obtains aHamilton-Jacobi equation from which an effective classi-cal Hamiltonian can be read off, also valid within the samedomain. The largest possible domain of validity of such anapproximation is constrained by the classical ‘‘turningpoints’’ dictated by the effective Hamiltonian. The formof the effective Hamiltonian so obtained does not dependon details of the presumed solution. Being o� �h0�, theeffective Hamiltonian is insensitive to factor orderingissues.

There are various types of corrections that arise. Themost dramatic one is the correction implied by the non-trivial quantization of inverses of various classical quanti-ties such as scale factors, triad components, volume, etc.[9,10]. In the isotropic context, this has led to a variety ofimplications [11–14]. For the anisotropic context, thisleads to the suppression of chaotic behavior of theBianchi IX model [15] with the further result that asymp-totically for vanishing volumes, a Bianchi IX solutionapproaches a Kasner solution in a stable manner [16].

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BRIEF REPORTS PHYSICAL REVIEW D 71, 127502 (2005)

For the vacuum Bianchi I model, there is no potentialterm and no scope for a modification of dynamics due toquantizations of inverses of triad components. Although atthe quantum level, there is no singularity [4], the effectiveclassical Hamiltonian derived from a continuum approxi-mation is identical to the Einstein Hamiltonian leading tothe same singular Kasner solution. Recently however analternative method of deriving the effective classicalHamiltonian has been devised, in the context of isotropicmodels [8], which uses the WKB ansatz directly at thedifference equation level bypassing the step of first deriv-ing the Wheeler–De Witt differential equation to be fol-lowed by WKB approximation. The same method can alsobe applied in the anisotropic context which leads to aneffective Hamiltonian different from the EinsteinHamiltonian. An exact, nonsingular solution of thisHamiltonian for the vacuum Bianchi I model is the resultpresented here.

A Bianchi I space-time is specified in terms of the metricof the form,

ds2 � dt2 �XI

a2I �t��dxI�2; (1)

where t is the synchronous time. The vacuum Einsteinequations then lead to the well-known Kasner solution:aI�t� � t2�I where �I are constants satisfying the condi-tions

PI�

2I � 1 �

PI�I. For subsequent comparison with

the new solution, it is convenient to describe the timeevolution in terms of a new time coordinate � correspond-ing to the lapse N :� a1a2a3, defined by N d� � dt. Thescale factors then evolve as aI � e�I�.

Loop quantization of all diagonalized, Bianchi class Amodels has been given in [5]. Briefly, it may be summa-rized as follows. The kinematical Hilbert space is spannedby orthonormalized vectors labeled as j�1; �2; �3i, �I 2R. These are properly normalized eigenvectors of the triadoperators pI with eigenvalues 1

2�‘2P�I, where � is the

Barbero-Immirzi parameter and ‘2P :� 8�G �h :� � �h. Thevolume operator is also diagonal in these labels witheigenvalues V� ~�� given by �12�‘

2P�

3=2���������������������j�1�2�3j

p. Here

we have used the vector notation to denote the triple��1; �2; �3�. Imposing the Hamiltonian constraint opera-tor on general vectors of the form jsi �

P~�s� ~��j ~�i leads

to the fundamental difference equation for the coefficientss� ~��. Here the sum is over countable subsets of R3. Thereare further gauge invariance conditions [4] which do notconcern us here.

In the present context of the vacuum Bianchi I model,the fundamental difference equation takes the form [5]:

X~�12

A12� ~�; ~�12�s� ~�; ~�12� cyclic � 0; (2)

where ~�12 � ��1; �2; �01; �02� with each of the �� being �1;

s� ~�; ~�12� � s��1��0�1��0�01;�2��0�2��0�02;�2�;

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�0 is an order 1 parameter, and

A12� ~�; ~�12� � V� ~�; ~�12�d��3���1�2 �01�02�; (3)

d��� :�� ���������������������������

j1�0��1jp

����������������������������j1��0��1j

p� � 0;

0 � � 0:

In the summary above, we have used the nonseparablekinematical Hilbert space and also made the parameter�0 explicit [17].

To derive the effective Hamiltonian, one assumes thatthere exist solution(s) of the partial difference equationwhich have a slowly varying amplitude and phase at leastin some region of large volume. Explicitly, defining~p��I� :�

12�‘

2P ~�, one introduces an interpolating func-

tion, �p�, such that s� ~�� � � ~p��I�� � C� ~p� expf i�h�� ~p�g and assumes that the amplitude and phaseare slowly varying functions of ~p in the sense that higherorder terms in the Taylor series about any ~p in the relevantregion are smaller than the lower order terms when com-pared over the quantum geometry scale q :� 1

2��0‘2P.Taylor expanding the interpolating wave function � ~p; ~�IJ�, �IJ� � �12; 23; 31�, it is straightforward to checkthat the leading terms (in powers of �h) in the real part of theequation are o� �h0� while those in the imaginary part areo� �h�. The o� �h0� terms involve only the first order partialderivatives of the WKB phase. Identifying K1 :���1;0;0� ~p� etc., one infers the Hamiltonian system fromthe Hamilton-Jacobi equation as fpI; KJg � � IJ, with theHamiltonian (� :� �0�),

NHeff� ~p; ~K� � �2

�

�p1p2 sin�K1

�sin�K2

� cyclic

�; (4)

where � will be referred to as the discreteness parameter.The effective Hamiltonian is periodic in �KI and we

may restrict our attention to ��< �KI < �. The imagi-nary part of the equation however requires the domain ofvalidity to be restricted further in order to be self-consistentwith the assumption of slow variation of the interpolatingwave function. The restriction is pI � q and small neigh-borhoods of � �

2 are to be excluded. Thus along the �KIaxes, the effective Hamiltonian is a good approximation inthe intervals: ���;� �

2 � �, �� �2 ; �2 � �, ��2 ; ��

for some small positive . The values �KI � � �2 will turn

out to be the turning points of the trajectories, ~p���, of theeffective dynamics.

For �KI � 0, one can use ��1 sin�KI � KI. Then theeffective Hamiltonian goes over to the EinsteinHamiltonian and the dependence on � drops out. Keepingonly the �h0 terms, we have effectively taken jpIj largerthan the quantum geometry scale set by q. Thus it is clearthat Einstein dynamics is reproduced for large values oftriads and small values of their conjugates KI. In thisregime, the Hamilton’s equations for pI, combined with

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BRIEF REPORTS PHYSICAL REVIEW D 71, 127502 (2005)

the relation of the triad components to scale factors, jpIj �aJaK (and cyclic permutations), identifies the KI’s ascomponents of the extrinsic curvature of the constant tslices: KI � � 1

2daIdt . The effective Hamiltonian thus de-

viates from the Einstein Hamiltonian mainly in the regionof the phase space with not too small values of �KI.

The small value of �KI can be achieved by taking thelimit �! 0 (KI fixed) which then removes any restrictionon KI’s. From a purely classical perspective, such a viewmay be welcome especially since the parameters �0, �,absent in the Einstein dynamics, disappear. However, froma quantum perspective, the discreteness parameter cannotbe zero [17]. For a nonzero value of the discretenessparameter, the effective Hamiltonian represents an exten-sion of Einstein dynamics for phase space regions beyondsmall extrinsic curvatures and large triad components.With nonzero �, the effective dynamics is to be viewedas a way of extending the Einstein dynamics by incorpo-rating the discrete feature of quantum geometry.

We use geometrized units (� � 1 � c) so that all quan-tities have dimensions of powers of length. The triadvariables pI and �h have dimensions of �length�2; the effec-tive Hamiltonian, scale factors, the synchronous time allhave dimensions of length while the � has dimensions of�length��2. Using the naturally available quantum geome-try length scale of

���q

p, all dimensionful quantities below

are rendered dimensionless. We will continue to use thesame symbols though.

The Hamilton’s equations from the effectiveHamiltonian are

dpI

d�� �2pI cos�KI

�pJ sin�KJ

�pK sin�KK

�

; (5)

dKId�

� 2sin�KI�

�pJ sin�KJ

�pK sin�KK

�

; (6)

0 �

�p1 sin�K1

�

�p2 sin�K2

�

cyclic: (7)

It follows immediately that pI sin�KI=� :� ��I=2 areconstants of motion, with ��1�2 cyclic� � 0 �

PI�

2I �

�PI�I�

2 following from (7).If all the �I are zero, then all the pI, KI are also

constants. Since the lapse is nonzero, the pI are nonzeroconstants and the solution represents the usual Minkowskispace-time. The effective dynamics thus retains theMinkowski space-time as a solution indicating a goodclassical limit of the quantum dynamics. For a nontrivialsolution, then,

PI�

2I � 0 must hold and as usual, by a

constant scaling of the N (or of �), one can arrange the�I to satisfy

PI�I � 1. Thus the one dimensional parame-

ter space of these solutions is exactly the same as that of theusual Kasner solutions. (The special cases of the form�1 � 1, �2 � �3 � 0 are independent of � in their behav-ior and are not considered here. Thus all �I are assumed to

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be nonzero.) It follows that exactly one of the �I’s must bestrictly negative and the remaining two strictly positive and�J �K � 1� �I > 0.

Since �I are nonzero constants, neither pI nor sin�KIcan vanish and thus cannot change sign. For definiteness,let us restrict our attention to ‘‘positively oriented’’ triadand, in particular, take pI > 0, 8 I. Then, sgn�sin�KI� ��sgn��I�, which fixes the two quadrants to which the‘‘angles’’ %I :� �KI must be confined along a solutioni.e. either 0< �KI < � or ��< �KI < 0. Clearly, pI

approach 1 as �KI approach the end points of the intervalsand take the minimum value �j�I j

2 for �KI � � �2 . The

traversal of %I with � is also fixed by (6): %I must decreaseif sin%I is positive and increase if sin%I is negative i.e. wemust have %I ! 0� as �! 1 (%I ! �� as �! �1).

Eliminating KI in favor of �I, pI, leaves us with anequation for pI, namely,

dpI

d�� ��1� �I�

�������������������������������pI�2 � �

��I2

�2r

: (8)

The � is determined by the quadrant to which the angle�KI belongs. The solution is easily obtained as

pI��� � �j�Ij2

coshf�1� �I���� ��I�0�g: (9)

Notice that a triad component attains its smallest value,�j�I j2 , at � � ��I�0 while for large j�j it behaves as pI �

��j�Ij=4� exp��1� �I�j�j�.In terms of the scale factors, aI �

volumepI , the solution is

given by

a2I ��� � �1� �I

2��coshf�1� �J���� ��J�0�g�

�coshf�1� �K���� ��K�0�g�

�coshf�1� �I���� ��I�0�g��1�: (10)

For comparison, the triad and the scale factor for theKasner solution are

pI��� � pI0e�1��I��; a2I ��� � �aI�20e

2�I�: (11)

One can recover the Kasner solution from the modified oneby taking the limit ��I�0 ! �1, �! 0 such that� j�I j

4 e��1��I���I�0 � pI0.

For the Kasner solution, two scale factors vanish and athird one diverges such that the volume vanishes exponen-tially with �, as �! �1. This translates into a finitesynchronous time t in the past, making the Kasner solutionsingular.

By contrast, for the modified solution, none of the triadvariables can become zero at any � and the volume nevervanishes. Furthermore, due to the hyperbolic cosine func-tion, for both asymptotic times, the modified solutionapproaches the large volume behavior of the Kasnersolution. In particular, the scale factor behaves as

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BRIEF REPORTS PHYSICAL REVIEW D 71, 127502 (2005)

a2I ! ���1� �I�=4� exp�2�Ij�j� as j�j ! 1. Conse-quently, exactly one scale factor vanishes while the re-maining two diverge as j�j ! 1. Thus if one begins with acubical cell at some finite �, then the cell will becomeplanar after both forward and backward evolutions (underthe Kasner evolution the cell will become planar in the‘‘future’’ and one dimensional in the ‘‘past’’). Since �! 1correspond to t! 1, the vanishing/diverging behavior ofscale factors never occurs for finite t and the modifiedsolution is nonsingular.

Consider the behavior of the volume, V2 � p1p2p3. Forj�j ! 1, V2 ! �3

64 j�1�2�3je2j�j, and therefore it musthave a minimum, V�, for some ��. It is clear that for �larger (smaller) than all ��I�0, V2 is monotonic. Thus ��must lie in the interval of the minimum and maximumvalues of the parameters ��I�0. The minimum volume, V�,reached by any particular solution depends on ~� as well ason ~�0. For any given ~�, the smallest possible minimumvolume is attained for solutions for which all ��I�0 areequal and is given by

�V��min �

��������������������������3j�1�2�3j

8

s(12)

which can be arbitrarily small though strictly positive.So far we have focused on the features of the exact

solution. Recall that for the validity of the effectiveHamiltonian we also need pI * q, or in the dimensionlessvariables used above, pI * 1. This puts a restriction on thetime � for which any specific solution can be trusted as anapproximation. Specifically, coshf�1� �I���� ��I�0�g *2

�j�I j. Had we used the effective Hamiltonian obtained

from the continuum approximation [5,16] which is thesame as the Einstein Hamiltonian, we would get the usual

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Kasner solution and obtain the restriction on � as expf�1��I���� ��I�0�g * 4

�j�I j. For ease of comparison, we have

just written pI0 :��j�I j4 expf��1� �I���I�0g so that for

large �, both solutions match. It follows that the � regimeof validity for the effective solution is larger than that forthe Kasner solution. However, since 2

�j�I j> 2

� > 2, the pa-rameters ��I�0 are outside the domain of validity and so isthe smallest (nonzero) value of triad components. The‘‘bounce’’ in the triad components is thus in the quantumregime. This is different from the isotropic case [12] wherethe bounce implied by effective dynamics can occur in thedomain of validity of the effective Hamiltonian, dependingupon the details of the matter Hamiltonian. Thus, in thesimplest of the anisotropic models, although the effectivedynamics inferred is nonsingular due to bouncing triadcomponents, these bounces lie in the quantum domainand for a ‘‘reliable’’ removal of singularity, one still needsto appeal to the quantum theory.

In summary, we make three points: (1) the method ofeffective Hamiltonian can be extended to homogeneous,anisotropic models and leads to nonsingular effective dy-namics with the exact solution parametrized by the sameKasner parameters; (2) unlike (at least some of) the iso-tropic models, for reliable singularity removal, one has toappeal to quantum theory; and (3) the quantum theorystipulates modifications of the classical Einstein dynamics,not only for small triads (pI) but also for larger values oftheir conjugates (KI) and is responsible for thenonsingularity.

I would like to thank Martin Bojowald and GolamHossain for useful remarks. This work was initiated duringmy visit to AEI, Golm during December 2004. The warmhospitality is gratefully acknowledged.

[1] V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifschitz,Adv. Phys. 13, 639 (1982).

[2] A. D. Rendall, gr-qc/0503112.[3] C. Rovelli, Living Rev. Relativity 1, 1 (1998); T.

Thiemann, Lect. Notes Phys. 631, 41 (2003); A.Ashtekar and J. Lewandowski, Classical QuantumGravity 21, R53 (2004).

[4] M. Bojowald, Classical Quantum Gravity 20, 2595 (2003).[5] M. Bojowald, G. Date, and K. Vandersloot, Classical

Quantum Gravity 21, 1253 (2004).[6] M. Bojowald, Classical Quantum Gravity 18, L109

(2001).[7] G. Date and G. M. Hossain, Classical Quantum Gravity

21, 4941 (2004).[8] K. Banerjee and G. Date, Classical Quantum Gravity 22,

2017 (2005).

[9] M. Bojowald, Phys. Rev. D 64, 084018 (2001).[10] M. Bojowald, Pramana 63, 765 (2004).[11] M. Bojowald, Phys. Rev. Lett. 89, 261301 (2002); G. Date

and G. M. Hossain, Phys. Rev. Lett. 94, 011301 (2005).[12] G. Date and G. M. Hossain, Phys. Rev. Lett. 94, 011302

(2005).[13] G. M. Hossain, Classical Quantum Gravity 22, 2511

(2005).[14] S. Tsujikawa, P. Singh, and R. Maartens, Classical

Quantum Gravity 21, 5767 (2004).[15] M. Bojowald and G. Date, Phys. Rev. Lett. 92, 071302

(2004).[16] M. Bojowald, G. Date, and G. M. Hossain, Classical

Quantum Gravity 21, 3541 (2004).[17] A. Ashtekar, M. Bojowald, and J. Lewandowski, Adv.

Theor. Math. Phys. 7, 233 (2003).

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