On effective loop quantum geometry of Schwarzschild interior
Jerónimo Cortez, William Cuervo, Hugo A. Morales-Técotl, Juan C. Ruelas
aa r X i v : . [ g r- q c ] A p r On effective loop quantum geometry of Schwarzschild interior
Jer´onimo Cortez, ∗ William Cuervo,
1, 2, † Hugo A. Morales-T´ecotl, ‡ and Juan C. Ruelas § Departamento de F´ısica, Facultad de Ciencias,Universidad Nacional Aut´onoma de M´exico, Ciudad de M´exico 04510, M´exico Departamento de F´ısica, Universidad Aut´onoma Metropolitana Iztapalapa,San Rafael Atlixco 186, CP 09340, Ciudad de M´exico, M´exico. (Dated: September 26, 2018)The success of loop quantum cosmology to resolve classical singularities of homogeneousmodels has led to its application to classical Schwarszchild black hole interior which takesthe form of a homogeneous, Kantowski-Sachs, model. First steps were done in pure quantummechanical terms hinting at the traversable character of the would be classical singularityand then others were performed using effective heuristic models capturing quantum effectsthat allowed a geometrical description closer to the classical one but avoiding its singularity.However, the problem to establish the link between the quantum and effective descriptionswas left open. In this work we propose to fill in this gap by considering the path inte-gral approach to the loop quantization of the Kantowski-Sachs model corresponding to theSchwarzschild black hole interior. We show the transition amplitude can be expressed as apath integration over the imaginary exponential of an effective action which just coincides,under some simplifying assumptions, with the heuristic one. Additionally we further explorethe consequences of the effective dynamics. We prove first such dynamics imply some rathersimple bounds for phase space variables and in turn, remarkably, in an analytical way, theyimply various phase space functions that were singular in the classical model are now wellbehaved. In particular, the expansion rate, its time derivative, and shear become boundedand hence the Raychauduri equation is finite term by term thus resolving the singularitiesof classical geodesic congruences. Moreover, all effective scalar polynomial invariants turnout to be bounded.
PACS numbers: 04.60.Pp, 03.65.Sq, 04.70.Dy, 04.70.Bw, 98.80.Qc ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected]
I. INTRODUCTION
Two main questions of theoretical physics requiring the knowledge of the structure of spacetimeat a fundamental level are the nature of singularities appearing in classical general relativity andthe ultraviolet divergences of field theory. It is expected a quantum theory of gravity can providean answer for such questions as we have learned from simpler quantum physical systems whichimprove their behavior as compared to their classical analogues. Not only is compulsory to findhow these questions can be answered but in fact to be able to grasp what new concepts, if any, areneeded in the theoretical framework that replaces the origin of these issues.One candidate quantum gravity theory, loop quantum gravity (LQG) [1–3] which is a non-perturbative, background-independent approach to quantize general relativity is natural to considerin dealing with the nature of spacetime. In particular, the implementation of the loop quantumgravity program for cosmological models, which is known as loop quantum cosmology (LQC) [4–7],has led to the replacement of the big-bang singularity with a quantum bounce for homogeneousand isotropic models (see, for instance, the seminal works [8–12]). Also anisotropic [13–39], as wellas inhomogeneous models [40–53] have been studied.It has been argued in some cases [6] it is convenient to use effective models capturing theiressential quantum aspects mainly when the full quantum dynamics is unknown. The effectiveapproach has been tested by applying the effective dynamics to cases where quantum evolution isfully known, with the astonishing result that the effective dynamics matches quite well, even inthe deep quantum regime, with the full quantum dynamics of LQC [10, 11, 54, 55]. Clearly it iscrucial to determine whether and when an effective description is pertinent without relying on thefull quantum solution.Motivated by the success of LQC in the study of homogeneous cosmologies, the study of theSchwarzschild black hole interior by using LQC techniques was put forward in [26, 27] exploiting thefact that the interior Schwarzschild geometry is a particular homogneous Kantowski-Sachs model.Their results indicated that quantum Einstein equations were not singular. However, the answer tothe question what replaces the classical singularity was not answered. Further developments usingan effective approach were done in [28–39] (For a recent review see [56]) . Interestingly, [28] arguedthat there is a connection between the black hole and a Nariai Universe whereas [39] found thepresence of a white hole instead; the difference between these two works being whether a pair ofparameters in the quantization are scale factor dependent or constants, respectively. In particular[28] get quantum corrections both at singularity and horizon, as opposed to [39] which correctionsare limited to the would be singularity. Actually such difference appears already in cosmologicalmodels in regard to the inadequacy of the so called µ = constant prescription [6, 7]; in order tocorrectly describe the classical regime such parameter should be scale factor dependent. Yet [39]argue such analysis for cosmological models does not hold for Schwarzschild since it would alterthe notion of classical horizon. To us this is an unsettled issue which requires further study andmore information, for instance, the behavior of effective quantities, like geometric scalars in theeffective Raychaudhuri equation, or the effective Kretschmann and curvature scalars (see [35, 38]for first steps in this direction).The present work is aimed at filling the gap between the description using loop quantum modeland that using an effective dynamics for the Kantowski-Sachs model representing the Schwarzschildblack hole interior. We will derive the effective Hamiltonian constraint via the path integralapproach starting from the quantum Hamiltonian in the so called improved dynamics scheme withthe quantization parameters depending on the scale factors and consider the transition amplitudebetween two basis states labelled with different values of a time parameter. After performing theusual partition of the time interval we get the effective action, S eff , as the argument of an imaginaryexponential that is to be integrated upon according to Feynman’s prescription. It is from S eff thatthe effective Hamiltonian H eff will be extracted. Thereafter we will analyze in an analytical mannerthe effective Hamiltonian theory associated to H eff and its impact on the behavior of relevantscalars. More precisely, we will prove that the effective expansion scalar, its time derivative andshear are bounded. Moreover, it is demonstrated that every scalar polynomial invariant, so, inparticular, the Ricci and Kretschmann scalars, are bounded in the effective approach.This paper is organized as follows. Section II is devoted to the classical setting of the theory. Westart by recalling that the Schwarzschild interior geometry can be described by a Kantowski-Sachsmodel. Thereafter, we recast the model in connection variables and perform a qualitative canonicalanalysis of the classical dynamics identifying the singular behavior of curvature invariants. Next,in Section III, within the framework of the improved dynamics prescription, we get the effectiveHamiltonian constraint by using the path integral approach. This along the lines of [22] for BianchiI. The effective loop quantum black hole interior geometry is analyzed in Section IV, where it isshown that classically divergent quantities are actually bounded in the effective approach. InSection V we discuss and summarize our main results.Throughout this work we will denote by µ to the so-called improved dynamics for homogeneousmodels, which has previously been denoted in literature by ¯ µ ′ (see, for instance, [32]). II. CLASSICAL THEORYA. The interior geometry in connection variables
As it is well known, a Schwarzschild black hole of mass M (i.e., a spherically symmetric vacuumsolution to general relativity) is described by the metric ds = − (cid:18) − GMr (cid:19) dT + (cid:18) − GMr (cid:19) − dr + r (cid:0) dθ + sin θdφ (cid:1) , (1)in Schwarzschild coordinates T ∈ R , r ∈ R + , 0 ≤ θ ≤ π and 0 ≤ φ ≤ π . At r = 0 there is a truesingularity (the Kretschmann scalar blows up as r →
0) which is wrapped by an event horizonlocated at the so-called Schwarzschild radius, r s = 2 GM (where the Schwarzschild coordinatesbecome singular). The exterior (i.e., r > r s ) spacelike ( ∂/∂r ) a and timelike ( ∂/∂t ) a vector fieldsswitch into, respectively, timelike and spacelike vector fields at the black hole interior (i.e., 0 The determinant of the fiducial metric (4) is given by ˚ q = sin θ , so that the densitized triad˚ E ai = √ ˚ q ˚ e ai reads ˚ E ai = sin θ ˚ e ai . The compatible densitized triad ˚ E a ∂ a = ˚ E ai τ i ∂ a , which takesvalues in the dual of su (2), and its corresponding su (2)-valued co-triad ˚ ω a dy a = ˚ ω ia τ i dy a , areexplicitly given by˚ ω a dy a = τ dx + τ dθ − τ sin θdφ, ˚ E a ∂ a = τ sin θ∂ x + τ sin θ∂ θ − τ ∂ φ , (5)where τ i are the standard generators of SU (2), satisfying [ τ i , τ j ] = ǫ kij τ k .Note that integrals over R × S involving spatially homogeneous quantities will generally diverge,given the non-compact character of the x -direction. To circumvent this feature, which, for instance,is an obstacle to properly calculate the Poisson brackets, one restricts x to an interval of finite length L , w.r.t. the fiducial metric, and then perform all integrations over a finite-sized cell V = [0 , L ] × S of fiducial volume V = 4 πL .Now, by imposing the Kantowski-Sachs symmetry group R × SO (3) in the full theory, one getsthat the symmetric connection A = A ia τ i dy a and triad E = E ai τ i ∂ a can be written, after gaugefixing of the Gauss constraint, as follows A = L − c (˚ ω x dx ) + b (˚ ω θ dθ + ˚ ω φ dφ ) + Γ , E = p c (cid:16) ˚ E x ∂ x (cid:17) + L − p b (cid:16) ˚ E θ ∂ θ − ˚ E φ ∂ φ (cid:17) . (6)Here, Γ = Γ ia τ i dy a = cos θτ dφ is the spin-connection compatible with the triad density E . Coeffi-cients b , c , p b and p c , which are all only functions of time, capture the non-trivial information aboutthe symmetry reduced model. From Eqs.(5)-(6), it follows that the Kantowsi-Sachs connection andtriad are explicitly given by A = L − cτ dx + bτ dθ − bτ sin θdφ + τ cos θdφ, (7) E = p c τ sin θ ∂ x + L − p b τ sin θ ∂ θ − L − p b τ ∂ φ . (8)The phase space resulting from the symmetry reduction and gauge fixing processes is the sym-plectic space Γ = [( b, p b , c, p c ) , Ω], with symplectic form [27, 31]Ω = 18 πGγ Z V d y (cid:0) dA ia ∧ dE ai (cid:1) = 12 Gγ ( dc ∧ dp c + 2 db ∧ dp b ) , (9)where γ is the so-called Barbero-Immirzi parameter. The only non vanishing Poisson bracketsdefined by the reduced symplectic form (9) are { b, p b } = Gγ, { c, p c } = 2 Gγ. (10)Let us remark that, in fact, we will not consider the whole of the phase space Γ . Indeed, as apart of the gauge-fixing procedure, p b can be chosen to be a strictly positive function [32], p b > p c correspond to regions with triads of opposite orientations [27, 32],then p c can be chosen to be strictly positive as well.Recall that a 3-metric q ab is related with its compatible densitized triad E ai by qq ab = E ai E bi .Thus, from Eqs. (3) and (8) it follows that X = p b / ( L p c ) and Y = p c ; i.e., in terms of the triadvariables, the Kantowski-Sachs metric reads ds = − N ( t ) dt + p b L p c dx + p c (cid:0) dθ + sin θdφ (cid:1) . (11)Thus, with respect to the metric (11), the length of the interval [0 , L ] in the x -direction, the areaof S and the volume of the cell V = [0 , L ] × S , are respectively given by l = p b / √ p c , A S = 4 πp c , V = 4 πp b √ p c (12)Now, in terms of the reduced canonical variables, ( b, p b ) and ( c, p c ), the Hamiltonian constrainttakes the form [27] C Ham = 16 πG H class = − πNγ (cid:20) bc √ p c + ( b + γ ) p b √ p c (cid:21) , (13)which defines H class . By choosing the lapse function equal to one, from Eqs.(10) and (13) we obtainthat the dynamics is dictated by˙ b = { b, H class } = − γ √ p c (cid:0) b + γ (cid:1) , (14)˙ c = { c, H class } = 12 γ p / c (cid:0) b p b − bcp c + γ p b (cid:1) , (15)˙ p b = { p b , H class } = 1 γ √ p c ( bp b + cp c ) , (16)˙ p c = { p c , H class } = 1 γ (2 b √ p c ) . (17)A direct calculation shows that the Ricci and Kretschmann scalars of the metric (11) are,respectively, R = 2 p c ¨ p b + p b (2 + ¨ p c ) p b p c , (18) K = R abcd R abcd = 12 p b p c (cid:2) p c (cid:0) p b ˙ p c − p c ˙ p b ˙ p c ¨ p b + 2 p c ¨ p b (cid:1) (19)+ 4 p b p c (cid:0) p c ¨ p b (cid:0) p c − p c ¨ p c (cid:1) + ˙ p b (cid:0) − p c + 2 p c ˙ p c ¨ p c (cid:1)(cid:1) + p b (cid:0) p c + 2 p c ˙ p c (2 − p c ) + p c (cid:0) p c (cid:1)(cid:1)(cid:3) . It is not difficult to see, by using Eqs.(16)-(17), that ¨ p b = bc/γ and ¨ p c = ( b/γ ) − 1. Substitutingthe latter expressions into Eqs.(18)-(19), as well as by imposing the constraint (13) and employingthe dynamical equations (14)-(17), we get that on the constraint surface R = 0 , K = 12 γ (cid:18) b + γ p c (cid:19) . (20)Solving the equations (14) and (17), (see Eq. (26) below) one gets ( b + γ ) = a /p c , with a being a constant depending on initial conditions and γ . Hence the Kretschmann scalar goes as1 /p c . Explicitly, K = 12 a γ p c . (21)Thus, the Kretschmann scalar blows up as p c tends to zero, corresponding to the classical singu-larity.Let us now examine the solutions to the system (13)-(17), and let us inspect the behavior ofthe expansion scalar and shear. B. Solutions, expansion scalar and shear To start, notice that cp c is a constant on the constraint surface. Indeed, from Eqs.(15) and (17)it follows that { cp c , H class } = − γ π C Ham . (22)Let us denote the constant cp c by γK c ; i.e., cp c = γK c . (23)Since the sign flipping K c → − K c is associated to the time reversal t → − t [32], the two regions, K c > K c < 0, are causally disconnected. Let us consider the region K c > 0, as in [32] (forthe sake of completeness, we will also discuss the opposite choice, K c < 0, at the end of the presentsection). Provided that p c > 0, we have that c must be a strictly positive function of time t . Since C Ham = 0 implies that b and c must have opposite signs (see Eq. (13)), we then conclude that b < 0. On the other hand, viewed as a quadratic equation in b , the constraint (13) has discriminant D = γ ( K c − p b ) , where we have used (23). Thus, to keep b real, D must be non-negative, whichimplies that p b is bounded from above p b ≤ K c . (24)Now, note that Eqs.(14) and (17) are actually decoupled equations from the rest of Hamilton’sequations. Thus, we have that dp c db = − bp c ( b + γ ) , (25)which solution is given by p c = p c (cid:18) b + γ b + γ (cid:19) . (26)Here, p c and b stand for initial conditions at t = t . Since ˙ b < b isa monotonically decreasing function of time t and, by virtue of Eq.(26), so is p c . Now, substitutingEq.(26) in Eq.(14), we get that˙ b = − α (cid:0) b + γ (cid:1) , α = (cid:2) γ √ p c ( b + γ ) (cid:3) − . (27)So that, g ( b ) = − γ α ( t − t ) + g ( b ) , g ( s ) = γs ( s + γ ) + arctan (cid:18) sγ (cid:19) . (28)Clearly, g < b < 0. It is easy to see that g decreases monotonically as b evolves in time,which in turn implies that g is a monotonous decreasing function of t . Indeed, a straightforwardcalculation shows that dg/dt = − γ α ; i.e., g is a monotonically decreasing function of time.Note, in addition, that − π/ < g . Thus, the relationship (28) makes sense (i.e., it is a well-definedrelationship providing b ( t ) at each given t value) only if 2 γ α ∆ t < π/ g ( b ), where ∆ t = ( t − t ).Hence, as ∆ t approaches to the maximal value ∆ t f ,∆ t f = 12 γ α h π g ( b ) i , (29)the solution b ( t ) will tend to b → −∞ . Then, from Eq.(26) it follows that the solution p c ( t ) willtend to zero as ∆ t approaches ∆ t f . By substituting Eq.(26) into c = γK c /p c [see Eq.(23)], we getthat c = γK c p c (cid:18) b + γ b + γ (cid:19) . (30)Hence, the solution c must tend to infinity as ∆ t → ∆ t f . By using Eq.(23) and Eq.(26) into theconstraint equation C Ham = 0 [see Eq.(13)], we obtain that p b = − γK c b ( b + γ ) . (31)Thus, in the limit when ∆ t tends to ∆ t f , the solution p b goes to zero as 1 / | b | . (Note that p b /p c diverges as | b | when ∆ t → ∆ t f ).Relations (26), (28), (30) and (31) provide the solution to the system (13)-(17). Once thesolution b ( t ) is obtained from Eq.(28), the rest of solution functions, namely p c ( t ), c ( t ) and p b ( t ), aredetermined by substituting b ( t ) into equations (26), (30) and (31), respectively. The time domainof the solution functions is t ∈ [ t , t + ∆ t ], with ∆ t ≤ ∆ t f ; given an initial data ( b , p b , c , p c ) at t = t , with b ∈ R − , p b ∈ (0 , K c ), and c , p c ∈ R + , the solution will tend to the ‘endpoint’( b → −∞ , p b → , c → ∞ , p c → , (32)as t approaches t f = t + ∆ t f . From (12), (26) and (31), it follows that the length l , the area A S and the cell volume V will behave as l ∼ | b | , A S ∼ /b and V ∼ / | b | as t → t f .Let us now consider the congruence of timelike geodesics defined by comoving observers inKantowski-Sachs spacetime (11), with N = 1; that is, the associated vector field to the congruenceis ξ a = ( ∂/∂t ) a . Thus, the expansion scalar θ corresponds to ˙ V /V , where V = 4 πp b √ p c is thecongruence’s cross-sectional volume. A simple calculation shows that θ = ˙ p b p b + ˙ p c p c . (33)By using Eqs.(14), (17), (26) and (31), as well as calculating ˙ p b by employing Eq.(31), it is notdifficult to see that θ = α (cid:18) b + γ b (cid:19) (cid:0) b − γ (cid:1) . (34)Clearly, the expansion scalar is a monotonically decreasing function of time (recall that b < t ). What is more, irrespective of the initialcondition b , θ → −∞ as t tends to the maximal value t f (i.e., the volume shrinks to zero as t → t f , invariably). At finite proper time ∆ t f , the congruence of timelike geodesics develop acaustic, and the cell volume becomes zero; in fact, the geodesics of the congruence turn out to beinextendible (i.e., incomplete). Note, however, that depending upon the initial condition b , therewould be a stage where the volume, in fact, will enlarge. Indeed, observe that θ is strictly positivefor − γ/ √ < b < 0, it is zero at b = − γ/ √ 3, and it is strictly negative for b < − γ/ √ 3. Thus, if theinitial condition b is in ( − γ/ √ , V max , at b = − γ/ √ 3, and afterwards it will monotonically decrease up to zero volume, at t = t f (time at which p c vanishes and the Kretschmann scalar blows up).A direct calculation shows that the time derivative of the expansion scalar (34) is given by˙ θ = − α (cid:18) b + γ b (cid:19) (cid:0) b + 2 γ b + γ (cid:1) , (35)0where we have used (27). Since ˙ θ is strictly negative, the expansion scalar θ is a monotonicallydecreasing function of t (as we have already pointed out).The shear, which is given by (see for instance [38]) σ = 12 σ ab σ ab = 13 (cid:18) ˙ p b p b − ˙ p c p c (cid:19) , (36)reads explicitly as follows σ = α (cid:18) b + γ b (cid:19) (cid:0) b + γ (cid:1) . (37)From Eqs.(34), (35) and (37), it is a simple exercise to see that ˙ θ = − (1 / θ − σ , which isnothing but Raychaudhuri’s equation. (Recall that the congruence is hypersurface orthogonal, sothat there is no rotational term. In addition, the term R ab ξ a ξ b = R is identically zero on shell).Let us remark that by considering K c < 0, one gets that c < p c is strictly positive) andthat b > b and c must have opposite signs). Exactly as above, it is shown that p b ≤ | K c | .The expressions for p c , c and p b [respectively, Eqs. (26), (30) and (31)] will be the same ones,though now with K c < b > 0. Of course, equation (27) governing the dynamics of b is thesame one, so is its solution (28); but now with g > 0, provided that b > 0. Since ˙ b < 0, then b is a monotonically decreasing function of t , which implies that g decreases in time. Rather thantechnical, the important difference between conventions K c > K c < K c is a time reversal, so it is natural to write the solution to Eq.(27)as g ( b ) = 2 γ α ( t − t ) + g ( b ) , g ( s ) = γs ( s + γ ) + arctan (cid:18) sγ (cid:19) , (38)with t > t (for instance, t would denote the ‘present time’, whereas t stands for an earlier time).Clearly, 0 < g < π/ b ∈ R + , and it is a monotonically decreasing function of t . Equation(38) implies that b ( t ) will tend to b → ∞ as t → 0; so that, p c → c → −∞ and p b → t approaches zero. In particular, we have that as t → 0, the Kretschmann scalar will diverge as b ,whereas the cell volume will collapse to zero as V ∼ /b .The explicit expression for the expansion scalar of a congruence of timelike geodesics constructedfrom comoving observers (i.e., with associated vector field ξ a = ( ∂/∂t ) a ) is also given by Eq.(34).Note that θ > < t < t ∗ , where t ∗ is ‘the cosmological time’ at which b ( t ∗ ) = γ/ √ θ iszero at t ∗ , and it is strictly negative for t > t ∗ . The expansion scalar is, in fact, a monotonicallydecreasing function in time. Note, in addition, that θ → ∞ at t = 0. The shear and the timederivative of θ have, of course, exactly the same expressions as above.1Now, by considering the congruence of ‘past-directed comoving world lines’, which associatedvector field is ξ a = − ( ∂/∂t ) a , one gets that the backward in time (BT) expansion scalar is givenby θ BT ( τ ′ ) = − α (cid:18) b + γ b (cid:19) (cid:0) b − γ (cid:1) , (39)where b is evaluated at ( t − τ ′ ) and the parameter τ ′ is from zero to t (so that t = 0 correspondsto the limit τ ′ → t ). Thus, θ BT ( τ ′ ) → −∞ within a finite ‘proper time’ t ; that is to say, thevolume shrinks to zero as we approach the ‘initial singularity’. III. QUANTUM SCHWARZSCHILD INTERIOR In this section we implement a path integral quantization of the Schwarzschild black hole interior.To do so we make use of its Kantowski-Sachs form as well as the similarity of the latter with BianchiI model, both being anisotropic homogeneous models. In particular, rather than starting fromscratch with the LQC techniques (see e.g. [6, 17, 27]) we perform a sequence of transformationsin phase space that ultimately allow us to identify adequate holonomy type variables for the KSmodel at the hamiltonian level, first, and, second, to introduce its path integral quantization. Wefollow closely the analyisis for Bianchi I in [22]. In this way an effective action, and hence aneffective hamiltonian, can be identified from the transition amplitude of the quantum KS model.This effective hamiltonian will be used in the next section to analyze the effective geometry forSchwarzschild interior.Let us notice that an effective Hamiltonian was proposed in [28, 31] motivated by severalprevious results. Essentially it was defined by the heuristic replacements b → sin( µ b b ) /µ b and c → sin( µ c c ) /µ c in (13), where the use of µ b , µ c follows from their appearence in a form of thehamiltonian constraint in which curvature terms are expressed by holonomies along elementarysquares of length related to them [6]. Our approach, that was described above, is different fromthis simple replacement, however, we will regain the effective hamiltonian of [31]. Now variouscriteria turn out to be necessary in the holonomy version of the construction [6]. They include (i)the area of the elementary squares used in the holonomies should not be less than the minimumarea gap ∆ found in the spectrum of the area operator of the full theory, (ii) physical quantitiesmust be independent of a fiducial metric introduced along the analysis, as well as (iii) avoidanceof large quantum gravity effects in classical regimes [9]. Such criteria led to propose the following2form of the µ ′ s [17, 28, 31] : µ b := s ∆ p c and µ c := √ ∆ p c p b . (40)Let us begin with the hamiltonian description and consider the classical constraint (13). It isconvenient to define first the following set of canonical variables [22] λ b := p b √ G ~ , λ c := r p c G ~ , ϕ b := b √ G ~ , ϕ c := c r p c G ~ , (41)which Poisson brackets take the form ~ { ϕ l , λ j } = γδ l,j l, j = b, c. (42)Let us observe that the following variables k b := ϕ b λ c = bµ b √ ∆ , k c := ϕ c λ b = cµ c √ ∆ , (43)have exponential forms U b := e i √ ∆ k b = e i bµ b , U c := e i √ ∆ k c = e i cµ c , (44)that are amenable for the application of LQC techniques. In the sequel the following identity willbe of much help sin( √ ∆ k l ) √ ∆ = U l − ( U l ) ∗ √ ∆ , l = b, c. (45)Using the set of variables λ b , λ c , k b , k c , given in eqs. (40), (41) and (43), in the classical constraint(13) yields the following form H µ = − ~ γ (cid:20) λ b k b )( λ c k c ) + λ c λ b ( λ b k b ) + γ G ~ λ b λ c (cid:21) . (46)Next we consider a small argument approximation for the ( λ b k b ) and ( λ c k c ) factors in (46) throughthe following relations ( λ b k b ) ≈ √ ∆ h λ b sin( √ ∆ k b ) i := Φ b , (47)( λ c k c ) ≈ √ ∆ h λ c sin( √ ∆ k c ) i := Φ c , (48) The two other possibilities that were explored, do not satisfy all these criteria, yielding inconsistent physics. Seefor example [38] and references therein. √ ∆ k b ) and sin( √ ∆ k c ) will be understood according to (45) so that our elementaryvariables will be λ l , U l , l = b, c . Their Poisson brackets can be obtained upon combination of (42),(43) and (44). The result is { λ b , U b } = ℓ i ~ U b λ c , { λ c , U c } = ℓ i ~ U c λ b , (49)with ℓ := γ √ ∆.Let us now proceed to quantization. Our elementary quantum observables will be b λ b , b λ c , ˆ U b andˆ U c . The Hilbert space of this system will be H (2)P oly = H P oly ⊗ H P oly with H P oly = L ( R Bohr , d µ H )and R Bohr is the Bohr compactification of the real line and d µ H is its Haar’s measure [59]. We usea basis of eigenkets | ~λ i := | λ b , λ c i of the operators ˆ λ b and ˆ λ c . These basis satisfy h ~λ ′ | ~λ i = δ ~λ ′ ,~λ , (50)where δ λ ′ ,λ is a Kronecker delta. To represent ˆ U ′ s we make use of the commutation relations h ˆ λ b , ˆ U b i = ℓ ˆ U b ˆ λ c , (51) h ˆ λ c , ˆ U c i = ℓ ˆ U c ˆ λ b , (52)which follow from the application of Dirac’s prescription to the Poisson brackets (49). They leadto ˆ U b | ~λ i = | λ b + ℓ /λ c , λ c i , ˆ U c | ~λ i = | λ b , λ c + ℓ /λ b i . (53)Now we proceed to implement the quantum version of (47) and (48) as [60]ˆΦ b := 1 √ ∆ (cid:20)q ˆ λ b \ sin √ ∆ k b q ˆ λ b (cid:21) and ˆΦ c := 1 √ ∆ (cid:20)q ˆ λ c \ sin √ ∆ k c q ˆ λ c (cid:21) , (54)whose action on the basis areˆΦ b | ~λ i = − i √ λ b √ ∆ hp λ b + ℓ /λ c | λ b + ℓ /λ c , λ c i − p λ b − ℓ /λ c | λ b − ℓ /λ c , λ c i i , (55)ˆΦ c | ~λ i = − i √ λ c √ ∆ hp λ c + ℓ /λ b | λ b , λ c + ℓ /λ b i − p λ c − ℓ /λ b | λ b , λ c − ℓ /λ b i i . (56)Hence the quantum version of the hamiltonian constraint (46), using (47) and (48) first at theclassical level and then their quantum version (54), becomesˆ H µ = − ~ γ " ˆΦ b ˆΦ c + ˆΦ c ˆΦ b + ( ˆΦ b ) ˆ λ c λ b + ˆ λ c λ b ˆ(Φ b ) + γ G ~ ˆ λ b ˆ λ c , (57)4for which a symmetric ordering has been introduced. Its building blocks are defined to act uponthe eigenbasis | ~λ i , according to (55) and (56) while the ˆ λ factors act diagonally.This hamiltonian will be used now to obtain Feynman’s formula for the propagator to go fromstate | ~λ i ; τ i i at proper time τ i to | ~λ f ; τ f i at time τ f > τ i . It takes the form h ~λ f ; τ f | ~λ i ; τ i i = h ~λ f | e − i∆ τ ˆ H µ / ~ | ~λ i i , ∆ τ = τ f − τ i . (58)To calculate explicitly such propagator we consider as usual a partition of the time interval ∆ τ and split, accordingly, the time evolution operator ase − i∆ τ ˆ H µ / ~ = N − Y n =0 e − i ǫ ˆ H µ / ~ , where N ǫ = ∆ τ. (59)Then using ˆ I = X ~λ n | ~λ n ih ~λ n | , (60)together with (59) allow us to rewrite (58) as h ~λ f ; τ f | ~λ i ; τ i i = X ~λ N − ,...,~λ N − Y n =0 h ~λ n +1 | e − i ǫ ˆ H µ / ~ | ~λ n i , (61)where ~λ f = ~λ N y ~λ i = ~λ . Next we consider that for small ǫ h ~λ n +1 | e − i ǫ ˆ H µ / ~ | ~λ n i = δ ~λ n +1 ,~λ n − i ǫ ~ h ~λ n +1 | ˆ H µ | ~λ n i + O ( ǫ ) . (62)The matrix elements h ~λ n +1 | ˆ H µ | ~λ n i can be calculated using (55) and (57): h ~λ n +1 | ˆ H µ | ~λ n i = ~ γ ∆ (cid:26)p λ b,n +1 λ b,n p λ c,n λ c,n +1 ( P n + Q n ) + 4 γ ∆ G ~ λ b,n λ c,n δ ~λ n ~λ n +1 + p λ b,n λ b,n +1 λ b,n + λ b,n +1 (cid:18) λ c,n λ b,n + λ c,n +1 λ b,n +1 (cid:19) R n (cid:27) . (63)where P n = (cid:16) δ λ b,n +1 ,λ b,n + ℓ /λ c,n +1 − δ λ b,n +1 ,λ b,n − ℓ /λ c,n +1 (cid:17) (cid:16) δ λ c,n +1 ,λ c,n + ℓ /λ b,n − δ λ c,n +1 ,λ c,n − ℓ /λ b,n (cid:17) ,Q n = (cid:16) δ λ b,n +1 ,λ b,n + ℓ /λ c,n − δ λ b,n +1 ,λ b,n − ℓ /λ c,n (cid:17) (cid:16) δ λ c,n +1 ,λ c,n + ℓ /λ b,n +1 − δ λ c,n +1 ,λ c,n − ℓ /λ b,n +1 (cid:17) ,R n = (cid:16) δ λ b,n +1 ,λ b,n +2 ℓ /λ c,n +1 − δ λ b,n +1 ,λ b,n + δ λ b,n +1 ,λ b,n − ℓ /λ c,n +1 (cid:17) δ λ c,n +1 ,λ c,n . (64)At this point we can see from (55)-(56) and hence in the matrix elements of ˆ H µ given by Eq.(57), that states supported on a regular (equally spaced) ~λ − lattice do not fit into our quantum KSmodel. This is a difficulty that also appears in the Bianchi I models and thus, to proceed further,5we can use the approximation proposed in [22] for that case. It consists of exploiting the fact thatwe are looking for a continuous yet quantum effective approximation [61, 62]. Hence, effectively,one can replace at leading order the Kronecker deltas by Dirac’s in (64). This implies that one isapproximating at leading order a description from H (2)P oly to H (2)S ch = H S ch ⊗ H S ch , H S ch = L ( R , dx ),so that ~λ is now a continuous variable. Within this approximation it is useful to adopt the followingintegral form of Dirac’s delta δ ( λ n +1 − λ n ) = 12 πγ Z R d ϕ n +1 e − i ϕ n +1 ( λ n +1 − λ n ) /γ . (65)Then, eq. (62) can be expressed as h ~λ n +1 | e − i ǫ ˆ H µ | ~λ n i = (cid:18) πγ (cid:19) Z d~ϕ n +1 e − i ~ϕ n +1 ( ~λ n +1 − ~λ n ) /γ × (cid:26) ǫ γ ∆ (cid:20) M n + N n + L n + γ ∆ G ~ λ b,n λ c,n (cid:21)(cid:27) + O ( ǫ ) , (66)where M n = p λ b,n +1 λ b,n p λ c,n λ c,n +1 sin( √ ∆ ϕ b,n +1 /λ c,n +1 ) sin( √ ∆ ϕ c,n +1 /λ b,n ) , (67) N n = p λ b,n +1 λ b,n p λ c,n λ c,n +1 sin( √ ∆ ϕ b,n +1 /λ c,n ) sin( √ ∆ ϕ c,n +1 /λ b,n +1 ) , (68) L n = p λ b,n λ b,n +1 λ b,n + λ b,n +1 (cid:18) λ c,n λ b,n + λ c,n +1 λ b,n +1 (cid:19) sin( √ ∆ ϕ b,n +1 /λ c,n +1 ) . (69)here ~ϕ = ( ϕ b , ϕ c ). This last expression allow us to rewrite the propagator in the form h ~λ f ; τ f | ~λ i ; τ i i = (cid:18) πγ (cid:19) N Z d~λ N − ...d~λ Z d ~ϕ N ... d ~ϕ e i / ~ S Nµ + O ( ǫ ) , (70)where S Nµ = ǫ N − X n =0 − ~ γ ~ϕ n +1 ~λ n +1 − ~λ n ǫ + ~ γ ∆ hp λ b,n +1 λ b,n p λ c,n λ c,n +1 × (cid:16) sin( √ ∆ ϕ b,n +1 /λ c,n +1 ) sin( √ ∆ ϕ c,n +1 /λ b,n ) + sin( √ ∆ ϕ b,n +1 /λ c,n ) sin( √ ∆ ϕ c,n +1 /λ b,n +1 ) (cid:17) + p λ b,n λ b,n +1 λ b,n + λ b,n +1 (cid:18) λ c,n λ b,n + λ c,n +1 λ b,n +1 (cid:19) sin( √ ∆ ϕ b,n +1 /λ c,n +1 ) + γ ∆ G ~ λ b,n λ c,n (cid:21) . (71)Now we take the limit N → ∞ and Eq. (71) takes the form S µ = lim N →∞ S Nµ = Z τ f τ i d τ (cid:26) − ~ γ ~ϕ · ˙ ~λ + ~ γ ∆ (cid:20) λ b λ c sin( √ ∆ ϕ b /λ c ) (cid:16) √ ∆ ϕ c /λ b ) + sin( √ ∆ ϕ b /λ c ) (cid:17) + γ ∆ G ~ λ b λ c (cid:21)(cid:27) . (72)Therefore we can see that the effective hamiltonian is H eff µ = − ~ γ ∆ (cid:20) λ b λ c sin( √ ∆ ϕ b /λ c ) (cid:16) √ ∆ ϕ c /λ b ) + sin( √ ∆ ϕ b /λ c ) (cid:17) + γ ∆ G ~ λ b λ c (cid:21) . (73)6Using (41) to return to the original variables ( b, c, p b , p c ) we get finally H eff µ = − Gγ " √ p c sin µ b bµ b sin µ c cµ c + p b √ p c (cid:18) sin µ b bµ b (cid:19) + γ p b √ p c . (74)Let us emphasize that the classical hamiltonian (13) is recovered by taking the small argumentlimit | µ l l | << l = b, c ) in the effective hamiltonian (74); i.e., the classical model, namely, theclassical hamiltonian and equations of motion, are recovered from the effective one in the regime | µ l l | << b, c, p b , p c ) lie in the constraint surface H eff µ = 0, “evolving” along the gaugeintegral curves of the hamiltonian vector field generated by H eff µ . In the next section we will focuson analyze how geometrical quantities behave in the effective quantum scenario provided by H eff µ . IV. EFFECTIVE LOOP QUANTUM DYNAMICS To investigate the effective geometry, let us begin by considering the Hamilton equations as-sociated to the hamiltonian (74), ˙ ζ = { ζ, H eff µ } , with the Poisson brackets (10), and the effectivescalar constraint H eff µ = 0,˙ b = − γ µ b − sin ( bµ b ) [sin ( bµ b ) − cµ c cos ( cµ c ) + 2 sin ( cµ c )]2 γ √ ∆ µ b , (75)˙ c = γ µ b + 2 bµ b cos ( bµ b ) [sin ( bµ b ) + sin ( cµ c )] − sin ( bµ b ) [sin ( bµ b ) + 2 cµ c cos ( cµ c ) + 2 sin ( cµ c )]2 γ √ ∆ µ c , (76)˙ p b = √ ∆ cos ( bµ b ) [sin ( bµ b ) + sin ( cµ c )] γµ b µ c , (77)˙ p c = 2 √ ∆ cos ( cµ c ) sin ( bµ b ) γµ b . (78)At this point two remarks are in order. First, provided that both p b and p c are strictly positivequantities, the first term in (74) must be strictly negative in order to satisfy the constraint. Second,since γµ b µ c = γ ∆ /p b and γµ b = γ ∆ /p c [cf. Eq. (40)], Eq.(77) and Eq.(78) can be written in theform d (ln p i ) /dt = f i ( bµ b , cµ c ), i = a, b .7A solution to the effective model is a sufficiently smooth , real solution to Eqs. (75)-(78)which, in addition, satisfies the scalar constraint (74). Let us refer to solutions of the effectivemodel as effective solutions. Since the dynamics is pure gauge, each point in the constraint surfaceis an appropriate initial condition for effective solutions. Now, to fix notation, let χ be theinitial condition ( b , c , p b , p c ) at t = t (the reference initial time) to the effective solution χ =( b, c, p b , p c ).From Eq. (74), it follows that effective solutions χ must satisfy, in particular, thatsin( µ b b ) = − sin( µ c c ) ± s sin ( µ c c ) − ∆ γ p c . (79)Since effective solutions χ are real ones, the discriminant must necessarily be nonnegative, so thatsin ( µ c c ) ≥ ∆ γ /p c . Thus, in particular, we have that p c is bounded from below by ∆ γ ; i.e., p c ≥ ∆ γ . (80)This expression implies that the area of S [cf. Eq. (12)] cannot be less than 4 π ∆ γ in the effectivegeometry. By using inequality (80) into the relations defining µ b and µ c [cf. Eq. (40)], we get that∆ γp c ≤ µ b ≤ γ , ∆ γp b ≤ µ c ≤ p c γp b . (81)Thus, in particular, γµ b ≤ . (82)Using again that the first term in (74) must be strictly negative it must be the case that sin( µ b b )and sin( µ c c ) must have opposite constant signs,sin( µ l l ) > , sin( µ l ′ l ′ ) < , (83)with l being equal to b or c , and l ′ being the complementary of l ; that is, for l = b ( l = c ), l ′ = c ( l ′ = b ). Strict inequalities (83), and the continuity of the functions µ b b and µ c c , imply that2 n π < µ l l < (2 n + 1) π and (2 m − π < µ l ′ l ′ < m π , for some n , m ∈ Z fixed and determinedby the effective solution χ ; in fact, by its corresponding initial condition χ . Indeed, given aninitial condition χ , we will have that sin( µ l l ) > µ l ′ l ′ ) < 0, so that n is thegreatest integer n satisfying that n < ( µ l l ) / π , whereas m is the least integer m satisfying that Let f = ( l, p l ), l = a, b , be a solution to Eqs. (75)-(78), and let us suppose that f is, at least, of class C . Thus, itfollows from Eqs. (75)-(78) that f is, in fact, a C ∞ function. This bound is consistent with that found in Ref. [32], where p c ≥ ∆ γ / . µ l ′ l ′ ) / π < m . By continuity, µ l l and µ l ′ l ′ must remain, respectively, in (cid:0) n π, (2 n + 1) π (cid:1) andin (cid:0) (2 m − π, m π (cid:1) ; otherwise, the constraint will be violated. We then have disjoint sectors,and they are as many as the distinct pairs ( n , m ) that the initial conditions define. Althoughwe will perform our analysis by considering a generic sector, it is worth remarking that it is onlywithin the (0 , µ d | d | << N l = (2 n + 1) , if n ≥ | n | , if n < , N l ′ = m , if m ≥ | m | + 1) , if m ≤ N d (with d being b or c ) we have that µ d | d | is confined to be in (cid:0) π ( N d − , πN d (cid:1) .Explicitly, given an effective solution χ , the quantities µ b | b | and µ c | c | are bounded by π ( N b − <µ b | b | < πN b and by π ( N c − < µ c | c | < πN c , where N b and N c are (strictly) positive fixed integersdetermined by the initial condition χ .Now, inequalities π ( N d − < µ d | d | and µ d | d | < πN d imply that 0 < | sin( µ d d ) | ≤ ≤ | cos( µ d d ) | < 1. Thus, for any given phase-space function g , we will have the strict inequality | g cos( µ d d ) | < | g | . (84)In particular, we have that the strict inequality | sin( µ d d ) cos( µ d ′ d ′ ) | < | ˙ p b | and | ˙ p c | . From Eqs. (77)-(78) -written in terms of the explicit expres-sions for γµ b µ c and γµ b - it follows by using the triangle inequality, the boundedness of the sinefunction and Eq.(84) that | ˙ p b | < (cid:18) γ √ ∆ (cid:19) p b , | ˙ p c | < (cid:18) γ √ ∆ (cid:19) p c . (85)To get the first inequality we have used, in addition, the relationship sin(2 bµ b ) = 2 cos( bµ b ) sin( bµ b )in Eq. (77). Similar calculations employing relations (80)-(81) in Eqs. (75) and (76) shows that | ˙ b | and | ˙ c | are bounded from above by | ˙ b | < √ ∆ + (cid:18) µ c | c | γ ∆ / (cid:19) p c ≤ (cid:18) µ c | c | γ ∆ / (cid:19) p c , | ˙ c | < (cid:18) µ c | c | + 3 µ b | b | γ ∆ / (cid:19) p b . (86)Since µ d | d | < πN d , we obtain that | ˙ b | < (cid:18) πN c γ ∆ / (cid:19) p c , | ˙ c | < (cid:18) πN c + 3 πN b γ ∆ / (cid:19) p b . (87)Inequalities (85) and (87) imply that | ˙ χ | is bounded from above by F ( N b ,N c ) | χ | , where F N b ,N c ) = max (cid:26) γ ∆ + (4 + 2 πN c + 3 πN b ) γ ∆ , γ ∆ + (2 + πN c ) γ ∆ (cid:27) . N b , N c ) turn out defined for t ∈ R . In addition,let us remark that effective solutions are bounded by the exponential function. Indeed, recall thatEq.(77) and Eq.(78) can be written in the form d (ln p d ) /dt = f d ( bµ b , cµ c ). Thus, combining Eqs.(78) and (80), employing the boundedness of the sine and Eq. (84), it is not difficult to see that∆ γ ≤ p c < p c e | t − t | /γ √ ∆ . (88)Similarly, from Eq.(77) it follows that p b e − | t − t | /γ √ < p b < p b e | t − t | /γ √ . (89)Since π ( N d − < µ d | d | < πN d , using inequalities (80) and (88)-(89), we get that γπ ( N b − < | b | < πN b ( µ b ) e | t − t | /γ √ ∆ , (90) π ( N c − µ c ) e − | t − t | /γ √ < | c | < πN c p b ∆ γ e | t − t | /γ √ (91)Thus, in contrast to the classical model, b and c are finite quantities at every time: there is nota finite proper time limit, t f , at which b and c will become infinite. Besides, for all t ∈ R , p c isbounded from below by a positive number, namely ∆ γ , and p b is a strictly positive quantity aswell. Note, in addition, that Eqs. (88)-(89) prevent the metric (11) to have a coordinate singularityin the effective approach.Let us now focus on the behavior of geometrical and invariant quantities. From Eqs. (77)-(78),using the explicit expressions for γµ b µ c and γµ b , it immediately follows that | θ eff | = (cid:12)(cid:12)(cid:12)(cid:12) ˙ p b p b + ˙ p c p c (cid:12)(cid:12)(cid:12)(cid:12) = 1 γ √ ∆ (cid:12)(cid:12)(cid:12)(cid:12) sin( bµ b + cµ c ) + 12 sin(2 bµ b ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ √ , (92) | σ eff | = 1 √ (cid:12)(cid:12)(cid:12)(cid:12) ˙ p b p b − ˙ p c p c (cid:12)(cid:12)(cid:12)(cid:12) = 1 γ √ (cid:12)(cid:12)(cid:12)(cid:12) sin( cµ c − bµ b ) − cos( cµ c ) sin( bµ b ) + 12 sin(2 bµ b ) (cid:12)(cid:12)(cid:12)(cid:12) < γ √ , (93)where we have used Eq.(84) to get the strict inequality in the last term of Eq. (93). The bound-edness of the expansion scalar ensures, in particular, that the volume of a cell will remain differentfrom zero at any finite proper time. Indeed, let V r = (4 πp b √ p c ) | t r be the volume of the cell V = [0 , L ] × S at an arbitrary reference proper time t r of comoving observers in the effectiveKantowski-Sachs geometry (with N = 1), and let t be any other finite proper time. It is a simplematter to see that | θ eff | ≤ /γ √ 4∆ implies that V r e − | t − t r | /γ √ ≤ V ≤ V r e | t − t r | /γ √ , (94)0where V is the volume of the cell V at time t . The volume V is a well-defined, strictly positivequantity at any finite proper time t ∈ R and, consequently, the congruence of timelike geodesicsdefined by comoving observers [i.e., the integral curves of the vector field ξ a = ( ∂/∂t ) a ] will notdevelop a caustic (at finite proper times).Let us now demonstrate that the effective Ricci and Kretschmann scalars, R eff and K eff , are infact well-behaved, finite quantities. Provided that R eff and K eff have second order terms in thetime derivatives of p b and p c , we shall first calculate { ˙ p d , H eff µ } . By using Eqs. (77)-(78), as wellas Eqs. (75)-(76), a straightforward calculation shows that¨ p b = p b γ ∆ γ µ b sin(2 bµ b ) (cid:20) sin( bµ b ) cos( cµ c ) + 14 sin(2 cµ c ) (cid:21) + sin(2 cµ c ) (cid:20) sin( cµ c ) cos( bµ b ) + 14 sin(2 bµ b ) sin ( bµ b ) (cid:21) + cos( cµ c ) (cid:2) sin(2 cµ c ) cos( bµ b ) + cos( cµ c ) sin ( bµ b ) sin(2 bµ b ) (cid:3) ( µ b b − µ c c ) ! , (95)¨ p c = − cos( bµ b − cµ c ) + p c γ ∆ ( bµ b ) (cid:2) ( cµ c ) (cid:3) − 12 sin(2 bµ b ) sin(2 cµ c ) − sin ( bµ b ) cos( bµ b + cµ c ) + sin(2 bµ b ) [1 + sin( bµ b ) sin( cµ c )] ( µ c c − µ b b ) ! . (96)Employing the triangle inequality, condition (84), and the boundedness of µ b [cf. (82)] as wellas of the sine function, we get that | ¨ p b | < γ ∆ (cid:18) 54 + (cid:2) µ b | b | + µ c | c | (cid:3)(cid:19) p b , | ¨ p c | < γ ∆ (cid:18) 112 + 2 (cid:2) µ b | b | + µ c | c | (cid:3)(cid:19) p c . (97)Using that µ d | d | < πN d , we arrive to | ¨ p b | < (cid:18) π [ N b + N c ]4 γ ∆ (cid:19) p b , | ¨ p c | < (cid:18) 11 + 4 π [ N b + N c ]2 γ ∆ (cid:19) p c ≤ (cid:18) 13 + 4 π [ N b + N c ]2 γ ∆ (cid:19) p c , (98)where the last inequality in the second expression follows from 1 ≤ p c / ( γ ∆).In order to simplify notation, let us introduce the quotients x := ˙ p b /p b , y := ˙ p c /p c , v := ¨ p b /p b and w := ¨ p c /p c . So, inequalities (85) and (98) read as follows | x | < γ √ ∆ , | y | < γ √ ∆ , | v | < (cid:18) π [ N b + N c ]4 γ ∆ (cid:19) , | w | < (cid:18) 13 + 4 π [ N b + N c ]2 γ ∆ (cid:19) . (99)The Ricci scalar, which is given by R eff = 2 v + w + (2 /p c ) [cf. Eq. (18)], is thus bounded by | R eff | ≤ | v | + | w | + 2 p c . (100)1By using Eq. (80) and Eq. (99) we have that the Ricci scalar [in the sector labelled by ( N b , N c )]is bounded by | R eff | < (cid:18) 11 + 4 π [ N b + N c ] γ ∆ (cid:19) . (101)Let us now focus on the Kretschmann scalar, K eff . From Eq. (19), it is easy to see that interms of the quotients x , y , v and w , K eff is given by K eff = 4 v + 3 w − vw − vxy + 4 wyx + 6 vy − wy + 6 x y − xy + 72 y + 2 p c y + 4 p c . (102)Clearly, the effective Kretschmann scalar turns out to be a bounded quantity. The explicit bound isobtained by using the inequalities (80) and (99), as well as the triangle inequality. A straightforwardcalculation shows that | K eff | < ξγ ∆ , ξ = 4 (cid:0) N b + N c ] π + 59[ N b + N c ] π + 160 (cid:1) + 232 . (103)In addition, since ˙ θ eff = v − x + ( w − y ) / 2, we get for effective solutions that (cid:12)(cid:12)(cid:12) ˙ θ eff (cid:12)(cid:12)(cid:12) ≤ | v | + x + 12 | w | + 12 y < γ ∆ (cid:18) 354 + 2( N b + N c ) π (cid:19) (104)This, together with Eqs. (92)-(93), proves that ( R ) eff is a bounded quantity as well.In general, we have that any quantity of the formΛ := N X j =1 C j (¨ p b ) n j (¨ p c ) m j ( ˙ p b ) r j ( ˙ p c ) s j ( p b ) α j ( p c ) β j , (105)where n j , m j , r j and s j are nonnegative integers, and α j and β j are any two real numbers, is abounded quantity on shell. Indeed, from inequalities (85) and (98) it follows that | Λ eff | < N X j =1 | C j | (cid:18) A bc γ ∆ (cid:19) n j (cid:18) B bc γ ∆ (cid:19) m j (cid:18) γ √ ∆ (cid:19) r j (cid:18) γ √ ∆ (cid:19) s j ( p b ) n j + r j + α j ( p c ) m j + s j + β j , (106)where A bc := 5 + 4 π [ N b + N c ] and B bc := 13 + 4 π [ N b + N c ]. Then, we have that in the effectiveapproach of the KS model, any effective quantity Λ eff of the form (105) will be bounded by (106).Provided that any scalar polynomial invariant P associated to the metric (11) -with the lapsefunction being set to the unit constant function- will take the form (105), as it is actually thecase for the Ricci and Kretschmann scalars, we can assert that in the effective geometry of the KSmodel P eff will be bounded everywhere, even though its classical counterpart is not (i.e., even if P class diverges at some regime).2 V. DISCUSSION The quest for the fundamental nature of spacetime may shed light on long standing problemsas the singularities appearing in classical general relativity and the ultraviolet divergences of fieldtheories. Hence quantum gravity theories that endow with quantum character to spacetime acquireparticular interest. Loop quantum gravity, in particular, has yielded homogenous cosmologicalmodels in which the classical singularity is replaced by a quantum bounce and thus Schwarzschildinterior which classically amounts to a homogeneous, Kantowski-Sachs, model is amenable for asimilar treatment. Indeed the loop quantization of Schwarschild interior showed that the would beclassical singularity is actually traversable and later on some heuristic effective models confirmedthe same result but also added possible replacements for the singularities like another black hole,a Nariai universe or a white hole. However, connecting the quantum treatment with the effectivemodel was left open. In this paper we have advanced a proposal that links the loop quantumdescription of the Schwarzschild interior with an effective model that is based on a path integralscheme. Specifically we have built a transition amplitude between two loop quantum states of theKantowski-Sacks model as a path integral, Eq. (61), consisting of an imaginary exponential of anaction in phase space from which the effective heuristic hamiltonian constraint descends, Eqs. (70),(72) and (74). Although this strategy was originally used for homogeneous isotropic, as well assome anisotropic, models the particular case of Kantowski-Sachs had not been dealt with before.Armed with the effective constraint we embarked in the study of the ensuing dynamics thathappened to lead to rather simple analytic bounds for the basic phase space variables, Eqs. (88)-(91) and their time derivatives. In particular expansion and shear turn out to be bounded too asit is the volume, Eqs. (92)-(94). Similarly, by considering the second order time derivatives of thebasic phase space variables, according to the effective dynamics, we get that both the effective Ricciand Kretschmann scalars, Eqs. (101) and (103), are bounded. This bounded character actuallyholds for any product of the form (105) containing second order and first order time derivatives aswell as powers of the variables p b , p c . It is a remarkable fact that analytic results were obtained fromthe effective dynamics which, although simple in appearance, could only be treated numerically inprevious works.There are several interesting points which can be further explored along the lines we havefollowed in the present work. 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