Noncanonical phase-space noncommutativity and the Kantowski ...

PHYSICAL REVIEW D 84, 024005 (2011)

Noncanonical phase-space noncommutativity and the Kantowski-Sachs singularity for black holes Catarina Bastos* Instituto de Plasmas e Fusaõ Nuclear, Instituto Superior Tećnico Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

Orfeu Bertolami† Departamento de Fı´sica e Astronomia, Faculdade de Cieˆncias da Universidade do Porto Rua do Campo Alegre, 687 4169-007 Porto, Portugal

Nuno Costa Dias‡,x and Joaõ Nuno Prata Departamento de Matema´tica, Universidade Luso´fona de Humanidades e Tecnologias Avenida Campo Grande, 376, 1749-024 Lisboa, Portugal (Received 10 April 2011; published 5 July 2011) We consider a cosmological model based upon a noncanonical noncommutative extension of the Heisenberg-Weyl algebra to address the thermodynamical stability and the singularity problem of black holes whose interior are described by the Kantowski-Sachs metric and modeled by a noncommutative extension of the Wheeler-DeWitt equation. We compute the temperature and entropy of these black holes and compare the results with the Hawking values. We observe that it is actually the noncommutativity in the momentum sector that allows for the existence of a minimum in the potential, which is the key to apply the Feynman-Hibbs procedure. It is shown that this noncommutative model generates a nonunitary dynamics that predicts a vanishing probability in the neighborhood of the singularity. This result effectively regularizes the Kantowski-Sachs singularity and generalizes a similar result, previously obtained for the case of Schwarzschild black holes. DOI: 10.1103/PhysRevD.84.024005

PACS numbers: 04.20.Dw, 11.10.Nx

I. INTRODUCTION Noncommutativity has been recurrently considered as a fundamental feature of space-time at the Planck scale [1]. Indeed, it is advocated that noncommutative geometry (NCG) captures the fuzzy nature of space-time at this scale, and that it should emerge as one of the main effects of quantum gravity [2,3]. In the context of string and M-theory, a configuration space noncommutativity arises as the low-energy effective theory of a D-brane in the background of a Neveu-Schwarz B field living in a space with spatial noncommutativity [1]. More recently, various aspects of noncommutative quantum field theory [4,5] and quantum mechanics [6–10] have been investigated. Noncommutativity has also been considered in the context of black holes (BH) to address the issues of thermodynamical stability and the removal/regularization of singularities. In this framework, we have argued in a series of papers that an additional momentum-momentum noncommutativity leads to interesting and relevant properties

*[email protected] † Also at Instituto de Plasmas e Fusaõ Nuclear, IST. [email protected] ‡ Also at Grupo de Fı´sica Matema´tica, UL, Avenida Professor Gama Pinto 2, 1649-003, Lisboa, Portugal. [email protected] x Also at Grupo de Fı´sica Matema´tica, UL, Avenida Professor Gama Pinto 2, 1649-003, Lisboa, Portugal. joao.prata@ mail.telepac.pt

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[11–13].1 In these works the BH is modeled with the Kantowski-Sachs (KS) metric [15] and, through the ADM procedure, the corresponding Wheeler-de Witt (WDW) equation is obtained and solved. In Refs. [11,12] we have considered the simplest case of a canonical phasespace noncommutativity for the two scale factors and the conjugate momenta of the KS geometry. In other words, we have considered the case where the fundamental algebraic structure of the theory is given by the extended Heisenberg algebra (EHA) which is the simplest possible generalization of the standard Heisenberg-Weyl algebra (HW). The EHA is canonical, globally isomorphic with the HW algebra and characterized by two additional noncommutative parameters, implementing the noncommutativity in the momentum and the configurational sectors. There are two noteworthy consequences of this choice of algebra: (1) To start with, the solution of the WDW equation factorizes into an oscillatory part of a ‘‘time’’ variable and a ‘‘spatial’’ part which obeys an ordinary Schro¨dinger-like differential equation. The associated potential of this equation exhibits a local minimum. This allows for an expansion around the minimum and the saddle point evaluation of the partition function in accordance with the 1

The impact of configuration space noncommutativity on BH has been thoroughly reviewed in Ref. [14]

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Feynman-Hibbs procedure [16]. From the rules of statistical mechanics, we can then compute the temperature and entropy of the BH. The results are tantamount to the Hawking-Bekenstein quantities plus stringy and noncommutative corrections. One essential point of this construction is that the local minimum of the potential only appears if momenta noncommutativity is assumed. (2) The second consequence of the noncommutative algebra is that the spatial part of the wave function displays a pronounced damping behavior and, in fact, it was shown to vanish at infinity [11]. However, the obtained rate of decay is not sufficient to ensure square-integrability of the wave-function. But this is only marginally so. The latter result prompted the search for alternative noncommutative algebras which might lead to a fullfledged square-integrable wave function and the corresponding regularization of the BH singularity. The most obvious possibility would be to consider a one-parameter, noncanonical generalization of the EHA. This was done in Ref. [13], where we have proposed a noncanonical phasespace noncommutative algebra, which delivers a normalizable spatial wave function. This algebra is the simplest possible noncommutative, noncanonical extension of the HW algebra in the sense that: (i) it includes both position-position and momentummomentum noncommutativity; (ii) it only displays one extra parameter related with the noncanonical features; (iii) it is globally isomorphic with the Heisenberg-Weyl algebra. In Ref. [13] we have shown that this algebra does in fact regularize the singularity of the Schwarzshild BH at the quantum level. In the present work we will further investigate the implications of this algebra. We will compute the thermodynamical quantities of the BH, which will be shown to have extra noncanonical noncommutative corrections. Moreover, we will prove that this algebra also regularizes the singularity of the KS BH. Let us point out that some important issues related to our model remain unaddressed. Mostly relevant is the question of whether our noncanonical algebraic structure can be derived, in some appropriate finite dimensional limit, from a more fundamental theory like String theory or Loop quantum gravity. These theories are expected to produce some sort of low-energy noncommutative structure. Our algebra provides an interesting example of a simple noncommutative structure that solves longstanding conceptual issues like the thermodynamical stability or the singularity problem. The issue of whether or not a fundamental quantum gravity theory can be connected with this cosmological framework is a very interesting open problem. Furthermore, we know from previous works [11,12] that the dynamical features associated with our noncanonical

algebra (namely, the regularization of the BH singularity) are not shared by the canonical noncommutative Heisenberg algebra. A natural question that remains is then what other types of noncanonical algebras also lead to the regularization of the BH singularity. Several other issues concerning the properties of the new algebraic structure, like for instance the nature of the uncertainty relations associated to it and whether it yields a minimal length structure also deserve further investigation. We intend to return to some of these issues in a forthcoming work. This paper is organized as follows. In Sec. II, we review the main aspects of the noncanonical noncommutative algebra of Ref. [13] and derive the Schro¨dinger-type equation satisfied by the spatial part of the wave function. In Sec. III, we evaluate the partition function of the BH by the Feynman-Hibbs method and compute the temperature and entropy of the BH. In Sec. IV, we study the probability of the BH near the KS singularity. Finally, in Sec. V, we discuss some of the mathematical subtleties of the regularization result and present our conclusions. In this work we use units ℏ ¼ c ¼ G ¼ 1. II. PHASE-SPACE NONCANONICAL NONCOMMUTATIVE QUANTUM COSMOLOGY The Schwarzschild BH is described by the metric, 2M 2 2M 1 2 ds2 ¼ 1 dt þ 1 dr þ r2 d2 ; (1) r r where r is the radial coordinate and d2 ¼ d2 þ sin2 d’2 . For r < 2M the time and radial coordinates are interchanged (r $ t) so that space-time is described by the metric: 1 2M 2M 2 2 ds ¼ 1 dt þ 1 dr2 þ t2 d2 : (2) t t This is an anisotropic metric, thus for r < 2M, the interior of a Schwarzschild BH can be described as an anisotropic cosmological space-time. Indeed, the metric (2) can be mapped into the KS metric [17], which, in the Misner parameterization, can be written as pffiffi pffiffi pffiffi ds2 ¼ N 2 dt2 þ e2 3 dr2 þ e2 3 e2 3 d2 ; (3) where and are scale factors, and N is the lapse function. The following identification for r < 2M [12,13]: 1 pffiffi 2M 2M 1 ; 1 ; N2 ¼ e2 3 ¼ t t pffiffi pffiffi 2 3 2 3 2 e e ¼t : (4) is a surjective mapping that transforms the metric Eq. (3) into the metric Eq. (2). Moreover, we also have from (4) that pffiffi e2 3 ¼ tð2M tÞ (5) and so the Schwarzschild singularity (t ! 0) corresponds to

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NONCANONICAL PHASE-SPACE NONCOMMUTATIVITY AND . . .

t ! 0þ ) ;

! þ1:

(6)

On the other hand, the KS singularity is attained at (cf. Eq. (3)) ! þ1. In this paper we shall study the thermodynamics of the interior of the BH by considering the following noncanonical extension of the HW algebra [13]: 2 ^ ^þ ^ ¼ i 1 þ pffiffiffiffiffiffiffiffiffiffiffi P^ ½; 1þ 1 pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2 ^ þ ð1 þ 1 ÞP^ Þ ½P^ ; P^ ¼ ið þ ð1 þ 1 Þ pffiffiffiffiffiffiffiffiffiffiffi ^ þ 2 P^ Þ; ^ P^ ¼ ½; ^ P^ ¼ ið1 þ ð1 þ 1 Þ ½; (7) where , and are positive constants and ¼ < 1. Notice that this condition is experimentally satisfied in at least two distinct situations [8,18]. The remaining commutation relations vanish. For 0 it implies that the noncommutative commutation and uncertainty relations are themselves position and momentum dependent. Notice that ¼ 0 corresponds to the canonical phase-space noncommutativity [6,11,12]. We point out that the well-known Darboux’s Theorem ensures that we can always find a system of coordinates, where locally the algebra can be written as the HW algebra. For this algebra, this statement also holds globally, so that the algebra is isomorphic to the HW algebra. Actually, algebra Eqs. (7) is the simplest noncanonical extension that is globally isomorphic to the HW algebra. It is an open issue whether this algebraic structure can be physically motivated or derived from a fundamental theory. On the other hand, this noncanonical extension has a direct impact on the singularity problem as shown in Ref. [13]. The isomorphism between the algebra Eqs. (7) and the HW algebra is referred to as a Darboux (D) transformation. D transformations are not unique. Indeed, the composition of a D transformation with a unitary transformation yields another D transformation. However, the physical predictions are invariant under different choices of the D map. ^ c ; P^ ; ^ c ; P^ Þ obey the HW algebra: Suppose that ð c c ^ c ; P^ ¼ ½^ c ; P^ ¼ i: ½ c c

(8)

The remaining commutation relations vanish. Then a suitable transformation is: ^ ¼ ^ c P^ þ E ^ 2c ; 2 c ^ ^ ¼ ^ c þ P^ c P^ ¼ P^ c þ ; 2 2 c ^ ^ 2c : þ F P^ ¼ P^ c 2 c

(9)

Here, , are real parameters such that ðÞ2 þ pffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 , 2 ¼ 1 1 , and we choose the positive

4


solution (given the invariance of the physics under different choices of the D map [6]), and set E¼

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi F; F ¼ 1ð1þ 1Þ: 1þ 1

The inverse D map is easily computed: 1 ^ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ^ þ P^ 2 1 1 ^ P^ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P^ 2 1 1 ^ ^ ^ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P 2 1 1 ^ P^ c ¼ pffiffiffiffiffiffiffiffiffiffiffiffi P^ þ 2 1 2 F ^ þ pffiffiffiffiffiffiffiffiffiffiffiffi P^ : pffiffiffiffiffiffiffiffiffiffiffiffi 1 1þ 1

(10)

(11)

It can be shown using Eqs. (9) and (11) that the algebra Eq. (7) implies the HW algebra, Eq. (8), and vice-versa. Of course, as already pointed out, we could consider other quadratic transformations relating the two algebras, which would, nevertheless, lead to the same physical predictions [6]. We consider now the Hamiltonian associated to the WDW equation for the KS metric and a particular factor ordering [11] pffiffi ^ H^ ¼ P^ 2 þ P^ 2 48e2 3 : (12) Upon substitution of Eqs. (9), we obtain: 2 2 ^ 2 H^ ¼ 2 P^ 2c 2 ^ 2c ^ c P^ c þ 2 P^ 2c þ 2 c 4 4 ^3 ^ 4c ^ c P^ þ 2F ^ 2c P^ F þ F2 c c c pffiffiffi pffiffiffi pffiffiffi ^ c þ 3 P^ 2 3E ^ 2c : 48exp 2 3 (13) c In the previous expression, the HW operators have the ^ c ¼ c , ^ c ¼ c , P^ ¼ i @ usual representation: @c c ^ which and P^ c ¼ i @@ c . Let us now define the operator A, corresponds to a constant of motion of the classical problem as discussed in Ref. [11] (which considers the same space-time setup, but assumes a canonical phase-space noncommutative algebra): ^ A^ ¼ P^ c þ : (14) 2 c A simple calculation reveals that for the noncommutative ^ algebra Eq. (7), A^ also commutes with the Hamiltonian H, ^ ¼ 0. This quantity corre^ A given by Eq. (12), i.e. ½H; sponds to the momentum P shifted by ð=ð22 ÞÞ, which can be seen as analogous to the canonical conjugate

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PHYSICAL REVIEW D 84, 024005 (2011) 2

momentum in the presence of a gauge field, where =2 corresponds to the electric charge and to the gauge field component. Thus, one seeks for solutions c ðc ; c Þ of the WDW equation, H^ c ¼ 0;

(15)

^ The most general which are simultaneous eigenstates of A. ^ solution of the eigenvalue equation A c ¼ a c with a real is [12,13] ic ; (16) a c a ðc ; c Þ ¼ Rðc Þ exp 2 c where Rðc Þ is an arbitrary C2 function of c , which is assumed to be real. From Eqs. (13), (15), and (16), one gets after some algebraic manipulation pffiffiffi pffiffiffi pffiffiffi 2 3 3a 2 00 2 R 48 exp c 2 3Ec þ R 2 ðc þ F3c ÞR þ F2 4c R þ a2 R 2 þ 2 þ 2aF 2c R ¼ 0: (17) The dependence on c has completely disappeared and one is effectively left with an ordinary differential equation d2 for Rðc Þ. Through the substitution, c ¼ z, d 2 ¼ c 1 d2 : and Rð ðzÞÞ ¼ ðzÞ one obtains a second order 2 2 c a dz linear differential equation, which is a Schro¨dinger-like equation 00a ðzÞ þ VðzÞa ðzÞ ¼ 0;

(18)

where the potential function, VðzÞ, reads: VðzÞ ¼ ðF2 z2 þ z aÞ2 pffiffiffi pffiffiffi pffiffiffi 3a 2 2 : þ 48 exp 2 3z 2 3 Ez þ

procedure, applied to the minisuperspace potential function, Eq. (19) depicted in Fig. 1. We present four different cases. The first case (a) describes the potential function for the commutative case ( ¼ ¼ ¼ 0), a strictly decreasing exponential function without a local minimum. The second case (b), the noncanonical noncommutativity in the configuration variables ( ¼ 0, ¼ 0:3) is considered, which reveals a potential function also without a local extremum. Finally, in the last two cases the noncommutativity is noncanonical and imposed (c) on the momentum sector ( ¼ 0), and (d) on the full phase-space. The last two cases admit a local minimum. Thus, as pointed out in Ref. [12] the potential has a local minimum only when the noncommutativity in the momentum sector is introduced. In the three last cases ‘‘the noncanonical’’ parameter was set to ¼ 0:3. This value is typical as are the values of the other noncommutative parameters. For some other values of these three parameters we encounter a qualitatively similar behavior for the potential function. When we set ¼ 0 in the potential, we recover the results of Ref. [12]. From Fig. 2 one concludes that the potential function has a minimum, which is located at small z values. Around the minimum the qualitative behavior of the two graphics [Fig. 2(a) and 2(b)] is quite similar. This allow us to safely neglect the term F2 z2 in the derivation of the noncommutative temperature and entropy of the BH. We stress that the Feynman-Hibbs method can be employed only when the noncommutative parameter associated with the momenta is nonvanishing. It is only under these circumstances that we have a local minimum of the minisuperspace potential. We can then expand the potential up to second order around the minimum. The potential function is, then, pffiffiffi a VðzÞ ¼ ðz aÞ2 þ 48exp 2 3 z þ 2 Ez2 : 2 (20)

(19)

Equation (18) depends explicitly on the noncommutative parameters , , and the eigenvalue a. III. THERMODYNAMICS OF PHASE-SPACE NONCANONICAL NONCOMMUTATIVE BLACK HOLE In this section we compute de thermodynamical properties of the noncommutative BH using the previous phasespace noncanonical noncommutative quantum cosmological model. The procedure used here is the one developed in Ref. [12] for the canonical noncommutativity. We consider the NCWDW equation to study the quantum behavior of the interior of the Scharwzschild BH. To evaluate the noncommutative temperature and entropy of the BH, we compute the partition function through the Feynman-Hibbs

In order to use the Feynman-Hibbs procedure to evaluate the partition function, one expands the exponential term in the potential Eq. (20) to second order in powers of the z z0 variable. The minimum z0 is then obtained by solving the following equation pffiffiffi dV ¼ ðz0 aÞ þ 48 3ð1 þ 22 Ez0 Þ dz z0 pffiffiffi a þ 2 Ez20 ¼ 0: exp 2 3 z0 (21) 2 The minimum is thus defined by the relation pffiffiffi a þ 2 Ez20 exp 2 3 z0 2 ¼

ðz aÞ pffiffiffi 0 : 48 3ð1 þ 22 Ez0 Þ

(22)

Moreover, it should satisfy the second derivative criterion

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80 80 60

60

40 40

20

1

20

1

2

3

4

5

20 40 1

0

1

2

3

4

5

400 400 200

200

5

10

15

2

200

200

400

400

4

6

8

10

12

14

FIG. 1. Potential function for some typical values of , , and a: (a) commutative case, ¼ ¼ ¼ 0 and a ¼ 0; (b) noncommutative case 0 and ¼ 0; (c) momenta noncommutative case 0 and ¼ 0; and (d) full noncommutative case , 0. For cases (b), (c) and (d) the noncanonical parameter was set to ¼ 0:3.

400

400

200

200

2

4

6

8

10

12

14

2

200

200

400

400

4

6

8

10

12

14

FIG. 2. Approximation of the potential function. (a) the full potential function in the noncanonical noncommutative phase-space, ¼ 5, ¼ 0:1, ¼ 0:3 and (b) the potential function in the noncanonical noncommutative phase-space neglecting the term F2 z2 .

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pffiffiffi 3 2 2 2 E 6 ð1 þ 2 Ez0 Þ 3 pffiffiffi a 2 2 þ Ez0 l2 > 0; (23) exp 2 3 z0 2 pffiffiffi where l :¼ =4 3. Using these relations, the second order expansion of the potential function VðzÞ in powers of z z0 satisfies:

VðzÞ ¼ 48Bðz z0 Þ2 ðz0 aÞ2 þ 48k; (24) p ffiffiffi where B:¼ 6k½ð1 þ 22 Ez0 Þ2 ð 3=3Þ2 E l2 and pffiffiffi a k :¼ exp 2 3 z0 2 þ 2 Ez20 . Notice that the minimum is semiclassically stable as can be shown, following the arguments of Ref. [12]. So, we can rewrite the ordinary differential equation resulting from the NCWDW equation as 1 d2 1 2 2 2 þ 24½B l ðz z0 Þ ¼ ðz0 aÞ 24k : 2 dz2 2 (25) Comparing Eq. (25) with the Schro¨dinger equation of the harmonic oscillator, we can identify the noncommutative potential and the energy of the system, VNC ðyÞ ¼ 24ðB l2 Þy2 ;

(26)

in terms of the variable y z z0 . A quantum correction is introduced to the partition function through the potential, as given by the Feynman-Hibbs procedure [16]: BH 00 V ðyÞ ¼ 2BH ðB l2 Þ; (27) 24 NC where BH is the inverse of the BH temperature. The noncommutative potential with the corresponding quantum corrections then reads, V NC ðyÞ ¼ 24ðB l2 Þ y2 þ BH : (28) 12


Inverting this quantity, one obtains the BH temperature. Dropping the term proportional to M2 , as presumably M > >1, and considering the positive root, we obtain: TBH ¼

ZNC

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ¼ exp½22BH ðB l2 Þ: (29) 2 48ðB l Þ BH

The thermodynamic quantities of the Schwarzschild BH can now be computed. Starting from the noncommutative internal energy of the BH, UNC ¼ @@BH lnZNC , UNC ¼

1 þ 4ðB l2 ÞBH ; BH

(30)

the equality UNC ¼ M, allows for obtaining an expression for the BH temperature: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 16 BH ¼ 1 1 2 ðB l2 Þ : (31) 2 8ðB l Þ M

(32)

We can clearly see that this quantity is positive (cf. Equation (23)). If E ¼ 0, we recover the results of Ref. [12]. That is, this noncanonical noncommutativity introduces new corrections to the noncommutative temperature. Furthermore, the mass dependence of the noncommutative temperature remains the same as the 1 Hawking temperature for a BH, TBH ¼ 8 M . As in Ref. [12], we can recover the Hawking temperature for a 0 value. Thus, equating Eq. (32) with the Hawking temperature and using the stationary condition Eq. (22), we obtain for ¼ 0:1, ¼ 0:3 and a ¼ 18:89: z0 ¼ 2:26369

0 ¼ 0:487:

(33)

And thus, since cannot be exactly equal to zero, we can regard 0 ¼ 0:487 as a reference value which yields the Hawking temperature, and as increases we get a gradual noncommutative deformation of the Hawking temperature. Notice that this value differs from the one obtained using a canonical phase-space noncommutativity [12]. This occurs as the noncanonical model has more parameters, , E and F, it is obvious that they will affect the value of 0 . The entropy is calculated using the expression, SNC ¼ lnZNC þ BH UNC . Thus, the entropy for the phase-space noncommutative BH is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 M2 M2 16B þ 1 2 SBH ¼ þ 2 16B 16B M sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 3 M2 16 8B 1 þ 1 2 B 2 : (34) ln 2 2 B M M Neglecting as before terms proportional to M2 as M 1, we finally obtain

Finally, the noncommutative partition function is given by

4 ðB l2 Þ: M

SBH ’

M2

pffiffi 8ð6k½ð1 þ 22 Ez0 Þ2 33 2 E l2 Þ 1 3M2 pffiffi : ln 2 6k½ð1 þ 22 Ez0 Þ2 33 2 E l2

(35)

For the reference value ¼ 0 , we recover the Hawking entropy, plus some ‘‘stringy’’ corrections: sffiffiffiffiffiffiffi 2 lnð8 MÞ: (36) SBH ¼ 4 M2 þ ln 3 In summary, as in the case of the temperature, we obtain that the noncommutative entropy of the BH is the Hawking entropy plus additional contributions and some noncommutative corrections.

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IV. THE SINGULARITY PROBLEM Let us now turn to the singularity problem and further elaborate on the issues discussed in Ref. [13]. The wave function Eq. (16) solution of the NCWDW equation is oscillatory in c and the remaining c -sector is squareintegrable and, hence, in the following, it is natural to consider hypersurfaces of constant c . This suggests that a measure d ¼ ð c Þddc should be assumed for the definition of the inner product in L2 ðIR2 Þ and the evaluation of probabilities. Starting from this premise, we were able to prove in Ref. [13] that the probability vanishes in the vicinity of the Schwarzschild singularity (corresponding to c , c ! þ1, cf. Eq. (6)). Here, we wish to investigate under which circumstances the entire KS singularity (c ! þ1) could be regularized by our noncanonical noncommutative model. It is worthwhile reviewing our arguments. A generic solution of the NCWDW equation H^ c ¼ 0 can be written as Z c ðc ; c Þ ¼ CðaÞ c a ðc ; c Þda; (37)

jj c jjL2 ðIR2 ;dÞ ¼

R

c ðc ; c Þ c ðc ; c Þd

1=2

¼

R


where CðaÞ are (for the time being) arbitrary complex constants and c a ðc ; c Þ is a simultaneous solution of A^ c a ¼ a c a and H^ c a ¼ 0, c ic ; (38) exp a c a ðc ; c Þ ¼ a 2 c Here a ðc Þ is a solution of Eq. (18). This equation is asymptotically of the form (cf. Figure 2) 00a ðzÞ Kz4 a ðzÞ ¼ 0;

z ! þ1;

(39)

(K is a positive constant) and so its solutions satisfy, cf. [pag. 75 [19]] (C 2 IR) C K1=2 3 a ðzÞ ’ exp i z ; z ! þ1; a 2 IR; (40) z 3 and are thus square-integrable. Hence we may impose the normalization: 2 Z Z j c a ðc ; c Þj2 dc ¼ dc ¼ 1: (41) a c With the assumed measure, we then have: 1=2 R CðaÞCða0 Þ c a ðc ; c Þ c a0 ðc ; c Þdc dada0 :

(42)

Notice that, in general, the functions c a ðc ; c Þ are not orthogonal as they are solutions of a hyperbolic-type equation (see comment in the discussion below). We may nevertheless apply the Cauchy-Schwartz inequality to obtain: Z Z jj c jjL2 ðIR2 ;dÞ jCðaÞjjj c a jjL2 ðIR2 ;dÞ da ¼ jCðaÞjda: (43) We conclude that for CðaÞ 2 L1 ðIRÞ the wave function Eq. (37) belongs to L2 ðIR2 ; dÞ, i.e. it is square-integrable on constant c hyper-surfaces. Hence the probability of system reaching the Schwarzschild singularity is: Rþ1 Rþ1 Rþ1 2 2 Pðr ¼ 0; t ¼ 0Þ ¼ lim ~c ~ c j c ðc ; c Þj dc ¼ 0 1 j c ðc ; Þj d ¼ ~ lim (44) ~ ; !þ1 ; !þ1 c

c

c

c

which effectively regularizes the singularity as point out in Ref. [13]. We next consider the more general KS singularity, corresponding to c ! þ1, (c.f. Eq. (3)). We thus wish to compute probabilities of the type: Z Z Z Pðc 2 I; c ! þ1Þ ¼ lim j c ðc ; Þj2 d ¼ lim j c ðc ; c Þj2 dc ; (45) c !þ1

I

c !þ1

IR

I

where I is some compact subset of IR. A simple calculation shows that: j c ðc ; c Þj2 ¼ And thus:

R R IR

0 IR CðaÞCða Þa

2 R c c ic ic 0 c ic 0 0 a a ¼ CðaÞ a da : (46) exp dada exp a a IR

Z Z c ic 2 exp a da CðaÞ dc : a c !þ1

Pðc 2 I; c ! þ1Þ ¼ lim

I

IR

(47)

Notice that a 2 C2 ðIRÞ \ L2 ðIRÞ and thus a 2 L1 ðIRÞ, that is: MðaÞ jja jjL1 ðIRÞ ¼ supx2IR ja ðxÞj < 1;

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for each a 2 IR:

(48)

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PHYSICAL REVIEW D 84, 024005 (2011) 1

We have already assumed that CðaÞ 2 L ðIRÞ. Given the fact that the potential Eq. (19) varies smoothly with respect to a, it is reasonable to expect that for a suitable choice of constants CðaÞ, we may admit the following regularity conditions: (1) CðaÞ 2 L1 ðIRÞ (2) CðaÞMðaÞ 2 L1 ðIRÞ We then have: Z Z ic a daj2 dc j CðaÞa c exp I IR Z 2 jIj jCðaÞjMðaÞda < 1; (49)

In the second part of the paper we have showed that the KS singularity (just as the Schwarzschild singularity, studied in a previous paper) is regularized by the noncanonical, noncommutative regime. This is the main feature of our model and a direct consequence of the fact that the eigenstates a are square-integrable, in spite of being associated to a continuous set of eigenvalues a. This property is shared by all Hamiltonians of the form H ¼ p2 þ VðxÞ with a potential asymptotically like VðzÞ ’ Kz2þ (for some K, > 0) as z ! þ1 [19]. The reason for the apparent paradox of having square-integrable eigenstates and a continuous spectrum is that the Hamiltonians H defined on their maximal domain

whereRwe have used the regularity condition 2), and where jIj ¼ I dc < 1, since I is compact. We thus have a uniform convergence on every compact interval I and all c . This means that we can safely interchange the integral and the limit to obtain: Z c Z Pðc 2 I; c ! þ1Þ ¼ lim CðaÞ a I c !þ1 IR 2 ic a da exp dc : (50)

Dmax ðHÞ ¼ f c 2 L2 ðIRÞ:H c 2 L2 ðIRÞg

IR

From the regularity condition 2) the functions CðaÞa ðc Þ regarded as functions of a (for each fixed c ) belong to L1 ðIRÞ. Indeed, jCðaÞa ðc Þj jCðaÞMðaÞj, for all c 2 IR. On the other hand, the Riemann-Lebesgue Lemma [20] states that every L1 function has a continuous Fourier transform, which vanishes at infinity, i.e.: Z lim fðaÞeðic =Þa da ¼ 0 (51) jc j!1

IR

for any f 2 L1 ðIRÞ. We conclude that: Pðc 2 I; c ! þ1Þ ¼ 0

(52)

for any compact set I. This effectively regularizes the KS singularity as well. V. DISCUSSION AND CONCLUSIONS In this work we obtain the noncommutative temperature and entropy of the Schwarzschild BH using a noncanonical noncommutative extension of a KS cosmological model. This formulation properly generalizes the full commutative, as well as the configurational and the momentum (canonical) noncommutative, models. We showed that the use of the Feynman-Hibbs method is only meaningful if one admits a nonvanishing noncommutativity in the momentum sector (i.e. 0) and we find that the noncanonical noncommutativity introduces further corrections to the noncommutative temperature and entropy obtained in Ref. [12]. Just as in Ref. [12], the Hawking quantities for the BH are recovered for a nonvanishing value of 0 , which in the present setup corresponds to 0 ¼ 0:487.

(53)

are not self-adjoint operators. In fact, they are the adjoint operators of the symmetric Hamiltonians Hð0Þ : SðIRÞ ! L2 ðIRÞ;

! H ð0Þ ¼ ðp2 þ VÞ

(54)

(where SðIRÞ is the set of Schwartz functions on IR) and act on domains larger than the domains of the self-adjoint extensions of Hð0Þ . Since H is not self-adjoint, it is not surprising it generates a nonunitary time evolution. It is worthwhile to compare these results with the full commutative case ( ¼ ¼ ¼ 0) and the momentum noncommutative case ( ¼ ¼ 0, 0). In the full commutative case (as also for configurational noncommutativity, i.e. ¼ ¼ 0 but 0) the potential Eq. (19) behaves asymptotically as VðzÞ ’ a2 . Consequently, the Hamiltonian Eq. (54) has deficiency indices (0, 0), i.e. it is essentially self-adjoint, and its (unique) self-adjoint realization is defined on the maximal domain Eq. (53). Hence, its spectrum is continuous, its eigenstates are not normalizable, and the time evolution is unitary. Most of these properties are also shared by the momentum (canonical) noncommutative case, where the potential behaves asymptotically like VðzÞ ’ z2 . The operator H, defined on the domain Eq. (53), is still self-adjoint. Its eigenstates display a pronounced damping behavior which, nevertheless, is not sufficient to make them normalizable. The spectrum is continuous and the evolution unitary [12]. Once we give up the self-adjointness of the Hamiltonian, we may become apprehensive that the corresponding eigenvalues may not be real. We stress that a self-adjoint operator is a sufficient condition for the reality of the eigenvalues. However, it is by no means necessary. In fact, a wide class of non self-adjoint Hamiltonian operators has recently been studied which exhibit real eigenvalues [21]. These Hamiltonians are PT-symmetric, that is they display an invariance under parity and time reversal transformations. An unbroken PT-symmetry can be shown to be equally sufficient for a real spectrum. Our Hamiltonian in Eqs. (18) and (19) looks asymptotically like p2 Kx4 (with K > 0). This Hamiltonian has been analyzed in [21] and shown to be PT-symmetric.

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ACKNOWLEDGMENTS The work of C. B. is supported by Fundaçaõ para a Cieˆncia e a Tecnologia (FCT) under the grant SFRH/ BPD/62861/2009. The work of O. B. is partially supported

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by the FCT grant PTDC/FIS/111362/2009. N. C. D. and J. N. P. were partially supported by the grants PTDC/MAT/69635/2006 and PTDC/MAT/099880/2008 of FCT.

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