Physics Letters B 686 (2010) 181–187

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Physics Letters B

www.elsevier.com/locate/physletb

Voros product, noncommutative Schwarzschild black hole and corrected area law

Rabin Banerjee a, Sunandan Gangopadhyay b,∗,1, Sujoy Kumar Modak a

a S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, Indiab West Bengal State University, Barasat, North 24 Paraganas, West Bengal, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 January 2010Received in revised form 9 February 2010Accepted 11 February 2010Available online 12 February 2010Editor: T. Yanagida

Keywords:Voros productNoncommutative Schwarzschild black hole

We show the importance of the Voros product in defining a noncommutative Schwarzschild black hole.The corrected entropy/area law is then computed in the tunneling formalism. Two types of correctionsare considered; one, due to the effects of noncommutativity and the other, due to the effects of goingbeyond the semiclassical approximation. The leading correction to the semiclassical entropy/area-law islogarithmic and its coefficient involves the noncommutative parameter.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The theoretical discovery of radiation from black holes byHawking [1,2] disclosed the first physically relevant window on themysteries of quantum gravity. The analysis is based on quantumfield theory in a curved background and reveals that black holesemit a spectrum that is similar to a thermal black body spectrum.This result made the first law of black hole mechanics [3] closelyanalogous to the first law of thermodynamics. This analogy ulti-mately led to the entropy for black holes and was consistent withthe proposal made by Bekenstein [4–7] that a black hole has an en-tropy proportional to its horizon area. All the above issues finallyled to the famous Bekenstein–Hawking area law for the entropyof black holes given by SBH = A/4. In a recent analysis [8] it wasproved that, taking the black hole entropy to be a state function,the standard Bekenstein–Hawking area law follows without usingthe first law of black hole mechanics.

However, most of these calculations were based on a semiclas-sical treatment and also on a commutative spacetime. The standardBekenstein–Hawking area law is known to get corrections due toquantum geometry or back reaction effects [9]. Recently, modifica-tions to the semiclassical area law due to noncommutative (NC)spacetime have also been obtained [10–13]. The motivation forthese investigations was that noncommutativity is expected to be

* Corresponding author.E-mail addresses: rabin@bose.res.in (R. Banerjee), sunandan@bose.res.in,

sunandan.gangopadhyay@gmail.com (S. Gangopadhyay), sujoy@bose.res.in(S.K. Modak).

1 Also, Visiting Associate at S.N. Bose National Centre for Basic Sciences, JD Block,Sector III, Salt Lake, Kolkata-700098, India.

0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2010.02.034

relevant at the Planck scale where it is known that usual semiclas-sical considerations break down. To obtain NC effects on the usualarea law, computation of the (NC) Hawking temperature was done.The Gibbs form of first law of thermodynamics was finally used toobtain the entropy. It was observed [10,12] that to the leading or-

der in the NC parameter θ , in the regimer2

h4θ

� 1, the (NC) arealaw was just an NC deformation of the usual semiclassical arealaw. Further, a graphical analysis revealed that when rh � 4.8

√θ ,

the NC version of the area law holds good to all orders in θ .In spite of this reasonable literature [10–13,35] on noncommu-

tative black holes, two outstanding issues remain. First, there isno clear cut connection of this type of noncommutativity with ourstandard notions of an NC spacetime where point-wise multipli-cations are replaced by appropriate star multiplications. Secondly,computations are confined strictly within the semiclassical regime.The present analysis precisely addresses these issues.

In this Letter, we have two objectives. First of all, we shall pointout that the Voros star product [14,15] plays an important role inthe obtention of the mass density of a static, spherically symmet-ric, smeared, particle-like gravitational source. This observation iscompletely new and has been missing in the existing literature[10–13,35]. To uncover the significance of the Voros product inwriting down the mass density, it is important to take a look atthe formulation and interpretational aspects of NC quantum me-chanics [16,17]. We observe that the inner product of the coherentstates (used in the construction of the wave-function of a “freepoint particle”), from which the mass density is written down, canbe computed by using a deformed completeness relation (involv-ing the Voros product) among the coherent states. Further, one cangive a consistent probabilistic interpretation of the wave-functionof the “free point particle” only when the Voros product is incor-

182 R. Banerjee et al. / Physics Letters B 686 (2010) 181–187

porated. Indeed the Gaussian distribution which follows from thisVoros type interpretation naturally includes the effect of the NCparameter θ and agrees with the structure given in the literature[10–13,35].

Our second objective is to find quantum corrections to thesemiclassical Hawking temperature and entropy for noncommuta-tive Schwarzschild black hole. This is done by first computing thecorrection to the Hawking temperature by going beyond the semi-classical approximation in the tunneling method [25–31,8]. Usingthe corrected form of the Hawking temperature and the first law ofblack hole thermodynamics, the entropy is computed. In the litera-ture there are lot of papers which discuss the quantum correctionsto above entities for different black holes in many different ap-proaches [18–22,8,28]. However all of these works are confinedto the commutative case only and there is nothing concerningthe possible quantum corrections to the semiclassical Hawkingtemperature and entropy for noncommutative black holes. Herewe want to generalise our method [8,28] to the case of the NCSchwarzschild black hole and derive the appropriate corrections tothe black hole entropy. The result is seen to contain logarithmicand inverse horizon area corrections and holds to O(

√θ e−M2/θ ).

The coefficient of the logarithmic correction term is explicitly de-termined from the trace anomaly of the stress tensor [23,24]. Thiscoefficient is also found to have NC correction and holds to thesame order as the logarithmic correction to the entropy.

The Letter is organised as follows. In Section 2, we shall discussabout the role of the Voros product in defining a noncommuta-tive Schwarzschild black hole. In Section 3, the corrections to thearea law by the tunneling method are discussed. In Section 4, wepresent the computation of the coefficient of the logarithmic cor-rection to the area law. Finally, we conclude in Section 5.

2. Voros product and noncommutative Schwarzschild black hole

In this section, we present the role played by the Voros productin defining the mass density of the NC Schwarzschid black hole. Tobegin the discussion, we note that the NC effect in gravity can beincluded in several ways [32–39]. One of the ways in which thenoncommutative effect is incorporated is to take the spacetimeas noncommutative [xμ, xν ] = iθμν and use the Seiberg–Wittenmap to recast the gravitational theory in noncommutative spacein terms of the corresponding theory in usual commutative spaceleading to correction terms involving powers of θμν in the variousexpressions such as the metric, Riemann tensor, etc. [35–39]. Thismethod naturally implies the use of the Moyal star product [40,41].The twisted formulation of NC quantum field theory [42–46] is yetanother way of incorporating the effects of noncommutativity ingravity [33,34]. Alternatively, one can incorporate the effect of non-commutativity in the mass term of the gravitating object. This isdone by representing the mass density by a Gaussian distributioninstead of a Dirac delta distribution [47,13]. These ways of includ-ing noncommutative effects in gravity, are in general inequivalent.Also, the specific nature of the NC product, if any, that replaces thepoint-wise product, remains completely obscure. In this Letter, weshall employ the latter route to include the effect of noncommu-tativity in gravity. We shall show that there is a subtle connectionbetween the latter approach and the Voros star product just asthere is a direct connection between the former approaches andthe Moyal star product. To see this connection, we must first pay acareful attention to the formulation and interpretational aspects ofNC quantum mechanics. We present a brief review of these issuesfor the sake of completeness and also to pin down the impor-tance of the Voros product in defining the mass density of the NCSchwarzschild black hole.

Recently, in a couple of papers [16,17], it was suggested thatNC quantum mechanics should be formulated as a quantum sys-tem on the Hilbert space of Hilbert–Schmidt operators acting onclassical configuration space. In two dimensions, the coordinatesof NC configuration space satisfy the commutation relation

[x, y] = iθ, θ > 0, (1)

for a constant θ . The annihilation and creation operators definedby b = 1√

2θ(x + i y), b† = 1√

2θ(x − i y) satisfy the Fock algebra

[b,b†] = 1. The NC configuration space is then isomorphic to theboson Fock space

Hc = span

{|n〉 = 1√

n!(b†)n|0〉

}n=∞

n=0(2)

where the span is taken over the field of complex numbers. Thenext step is to introduce the Hilbert space of the NC quantum sys-tem. We consider the set of Hilbert–Schmidt operators acting onNC configuration space

Hq = {ψ(x, y): ψ(x, y) ∈ B(Hc), trc

(ψ†(x, y)ψ(x, y)

)< ∞}

.

(3)

Here trc denotes the trace over NC configuration space and B(Hc)

the set of bounded operators on Hc . To distinguish states in the NCconfiguration space from those in the quantum Hilbert space, wedenote states in the NC configuration space by |·〉 and states in thequantum Hilbert space by ψ(x, y) ≡ |ψ). Assuming commutativemomenta, a unitary representation of the NC Heisenberg algebrain terms of operators X , Y , P x and P y acting on the states of thequantum Hilbert space (3) is easily found to be2

Xψ(x, y) = xψ(x, y), Y ψ(x, y) = yψ(x, y),

P xψ(x, y) = h

θ

[y,ψ(x, y)

], P yψ(x, y) = − h

θ

[x,ψ(x, y)

].

(4)

The minimal uncertainty states on NC configuration space, whichis isomorphic to boson Fock space, are well known to be the nor-malized coherent states [48]

|z〉 = e−zz/2ezb† |0〉 (5)

where z = 1√2θ

(x + iy) is a dimensionless complex number. These

states provide an overcomplete basis on the NC configurationspace. Corresponding to these states we can construct a state (op-erator) in quantum Hilbert space as follows

|z, z) = 1√θ

|z〉〈z|. (6)

These states have the property

B|z, z) = z|z, z), B = 1√2θ

( X + iY ). (7)

We now introduce the momentum eigenstates normalised suchthat (p′|p) = δ(p′ − p)

|p) =√

θ

2π h2e

i√

θ

2h2 (pb+pb†), P i|p) = pi|p) (8)

satisfying the completeness relation

2 We use capital letters to distinguish operators acting on quantum Hilbert spacefrom those acting on NC configuration space.

R. Banerjee et al. / Physics Letters B 686 (2010) 181–187 183

∫d2 p|p)(p| = 1Q . (9)

With the above formalism in place, we observe that the wave-function of a “free particle” on the NC plane is given by [13,17]

ψ�p = (p|z, z) = 1√2π h2

e− θ

4h2 ppe

i√

θ

2h2 (pz+pz),

p = px + ip y . (10)

The position eigenstates |z, z), on the other hand, satisfy the fol-lowing completeness relation∫

θ dz dz

2π|z, z) � (z, z| = 1Q (11)

where the Voros star product between two functions f (z, z) andg(z, z) is defined as [14,15]

f (z, z) � g(z, z) = f (z, z)e←−∂z

−→∂z g(z, z). (12)

To prove this, we use (10) and compute∫θ dz dz

2π

(p′|z, z

)� (z, z|p)

= e− θ

4h2 (pp+p′ p′)e

θ

2h2 pp′δ(

p − p′) = (p′|p)

. (13)

The completeness relation for the position eigenstates in (11) im-plies that a consistent probabilistic interpretation of finding theparticle at position z can be given iff the point-wise multiplicationbetween the complex conjugated wave-function and the wave-function (10) is replaced by the Voros product [16]:

P (z) ∝ (p|z)e←−∂z

−→∂z (z|p). (14)

The computation of the above expression shows that the probabil-ity P (z) is independent of z and p as expected for a free particle.Once we have these observations and interpretations in place, wenow move on to write down the overlap of two coherent states|ξ, ξ ) and |w, w) using the completeness relation for the positioneigenstates in (11)

(w, w|ξ, ξ ) =∫

θ dz dz

2π(w, w|z, z) � (z, z|ξ, ξ ). (15)

A simple inspection shows that the following solution satisfies theabove equation

(w, w|z, z) = 1

θe−r2/(2θ); r = √

2θ |ω − z|. (16)

The Voros product, therefore, gives a specific representation of theDirac delta function since

limθ→0

1

θe−r2/(2θ) = 2πδ(2)(r). (17)

Correspondingly, in three space dimensions, a similar representa-tion for the delta function would be

limθ→0

1

(4πθ)3/2e−r2/(4θ) = δ(3)(r). (18)

This motivates one to write down the mass density of a static,spherically symmetric, smeared, particle-like gravitational sourcein three space dimensions as

ρθ (r) = M

(4πθ)3/2exp

(− r2

4θ

). (19)

The above arguments clearly point out the important role playedby the Voros product in defining the mass density of the NC

Schwarzschild black hole. Indeed our analysis provides a heuris-tic derivation and a possible justification for choosing (19) as themass density which is otherwise unclear in the original literature[13,11,49,10].

Solving Einstein’s equations with the above mass density incor-porated in the energy–momentum tensor leads to the following NCSchwarzschild metric [11,13]

ds2 = − fθ (r)dt2 + f −1θ (r)dr2 + r2(dθ2 + sin2 θ dφ2) (20)

where

gtt(r) = grr(r) = fθ (r) =(

1 − 4M

r√

πγ

(3

2,

r2

4θ

)). (21)

The event horizon of the black hole can be found by settinggtt(rh) = 0 in (20), which yields

rh = 4M√π

γ

(3

2,

r2h

4θ

). (22)

Since this equation cannot be solved in a closed form we take the

large radius regime ( rh2

4θ� 1) where we can expand the incomplete

gamma function to solve rh by iteration. Keeping up to the leadingorder 1√

θe−M2/θ , we find

rh � 2M

(1 − 2M√

πθe−M2/θ

). (23)

3. Corrected area law from quantum tunneling

Now for a general static and spherically symmetric spacetimethe Hawking temperature (T H ) is related to the surface gravity (κ )by the following relation [49]

T H = hκ

2π(24)

where the surface gravity of the black hole is given by

κ = 1

2

(dfθdr

)r=rh

. (25)

Therefore the Hawking temperature for the noncommutativeSchwarzschild black hole is found to be

T H = h

4π

[1

rh− r2

h

4θ3/2

e− rh2

4θ

γ( 3

2 ,r2

h4θ

)]. (26)

To write the Hawking temperature in the regimer2

h4θ

� 1 as a func-tion of M we use (23). Keeping up to the leading order in θ , weget

T H � h

8π M

[1 − 4M3

√πθ3/2

e−M2/θ

]. (27)

We shall now use the first law of black hole thermodynamics tocalculate the Bekenstein–Hawking entropy. The first law of blackhole thermodynamics is given by

dS = dM

T H. (28)

Hence the Bekenstein–Hawking entropy up to leading order in θ isfound to be

S =∫

dM

T H= 1

h

(4π M2 − 16M3

√π

θe− M2

θ

)

+ O(√

θ e− M2θ

). (29)

184 R. Banerjee et al. / Physics Letters B 686 (2010) 181–187

The same expression of Bekenstein–Hawking entropy was foundearlier in [49] by the tunneling method. In order to express theentropy in terms of the noncommutative horizon area (Aθ ), weuse (23) to obtain

Aθ = 4πr2h = 16π M2 − 64

√π

θM3e− M2

θ . (30)

Comparing Eqs. (29) and (30), we find that at the leading orderin θ , the noncommutative black hole entropy satisfies the area law

S = SBH = Aθ

4h. (31)

This is functionally identical to the Bekenstein–Hawking area lawin the commutative space.

Hence we have analytically observed that in the regimer2

h4θ

� 1,the noncommutative version of the semiclassical Bekenstein–Hawking area law holds up to leading order in θ . This motivatesus to investigate the corrections to the semiclassical area law upto leading order in θ .

To do so, we first compute the corrected Hawking temper-ature T H . For that we use the tunneling method by going be-yond the semiclassical approximation [25]. Considering the mass-less scalar particle tunneling, the Klein–Gordon equation under thebackground metric (20) is given by

− h2

√−g∂μ

[gμν√−g ∂ν

]Φ = 0. (32)

It is worthwhile to point out that since noncommutativity is com-ing here through the matter sector (Φ), the form of the Klein–Gordon equation does not change with respect to the spacetimecoordinates. For simplicity we restrict ourselves to the radial tra-jectory so that only the r − t sector of the metric (20) is necessary.Since Eq. (32) cannot be solved exactly, we proceed by choosing astandard WKB ansatz for Φ as

Φ(r, t) = exp

[− i

hS(r, t)

], (33)

where,

S(r, t) = S0(r, t) +∞∑

i=1

hi Si(r, t). (34)

Now substituting (34) in (32) and equating coefficients of differentpowers in h to zero one obtains a set of partial differential equa-tions [25–31,8]. They can be simplified to find the solution for n-thorder in h, given by the first order partial differential equation

∂ Sn

∂t= ± fθ (r)

∂ Sn

∂r, (35)

where (n = 0, i; i = 1,2, . . .). For the lowest order in h (n = 0) weare left with a semiclassical Hamilton–Jacobi type equation

∂ S0

∂t= ± fθ (r)

∂ S0

∂r. (36)

Now we can choose the functional form of the semiclassical actionS0(r, t) by looking at the symmetry of the background metric (20)as

S0(r, t) = ωt + S0(r). (37)

Here ω is the conserved quantity corresponding to the time trans-lation Killing vector field in the background metric (20) and isrepresented by the Komar energy integral

ω = 1

4π

∫∂Σ

d2x√

p(2) nμσν∇μKν . (38)

This is defined on the boundary (∂Σ ) of a spacelike hypersurfaceΣ and pij is the induced metric on ∂Σ , while p(2) = det pij . Unitnormal vectors nμ and σν are associated with Σ and ∂Σ respec-tively, whereas, Kν is the timelike Killing vector. When observedat infinity this is the energy of the spacetime and matches withthe commutative mass (M) of the black hole. But unlike the stan-dard Schwarzschild black hole, in this case the effective energyexperienced by a particle at finite distance is not the same as ex-perienced at infinity. Because of the noncommutative effects it ismodified at finite distances. It will be discussed more elaboratelylater on (in Section 4) in order to find the leading correction to thesemiclassical entropy. For now we proceed with the calculation ofcorrected temperature and entropy of the black hole (20). Putting(37) in (36) we have

S0(r) = ±ω

∫C

dr

fθ (r), (39)

where the +(−) sign stands for the ingoing (outgoing) particle.The contour C is chosen such that it starts from just behind theevent horizon to the outer region, left to right, in the lower half ofthe complex plane, avoiding the singularity at the event horizon.Using (37) and (39) we finally find the semiclassical action as

S0(r, t) = ω

(t ±

∫C

dr

fθ (r)

). (40)

Since for all n, Sn(r, t)-s satisfy the similar type of differentialequations, the solutions for any Si(r, t) can differ from S0(r, t)only by a proportionality constant. The most general solution forthe scalar particle action is then given by

S(r, t) =(

1 +∞∑

i=1

γi hi

)S0(r, t), (41)

where γi -s are the proportionality constants and have the dimen-sions of h−i . Since in (3+1) dimensions in the unit c = G = κB = 1,Planck length (lp) and Planck mass (mp)3 are proportional to

√h,

we can readily express (41) as

S(r, t) =(

1 +∞∑

i=1

βi hi

(Mrh)i

)S0(r, t)

= ω

(1 +

∞∑i=1

βi hi

(Mrh)i

)(t ±

∫C

dr

fθ (r)

), (42)

where βi -s are dimensionless constants which can be related tothe trace anomaly of the stress tensor of the scalar field. The ingo-ing (Φin) and outgoing (Φout ) scalar field modes can now be foundby substituting S(r, t) from (42) into (33) with proper choice ofsign. As a result the ingoing and outgoing probabilities are givenby

Pin = |Φin|2

= exp

[2

h

(1 +

∑i

βihi

(Mrh)i

)(ω Im t + ω Im

∫C

dr

fθ (r)

)]

(43)

3 l2p = hG3 , m2

p = hcG .

c

R. Banerjee et al. / Physics Letters B 686 (2010) 181–187 185

Pout = |Φout|2

= exp

[2

h

(1 +

∑i

βihi

(Mrh)i

)(ω Im t − ω Im

∫C

dr

fθ (r)

)].

(44)

respectively. Since the incoming mode can always get inside theevent horizon, one has Pin = 1, which gives Im t = − Im

∫C

drfθ (r) .

Substituting this in (44) we get

Pout = exp

[−4ω

h

(1 +

∑i

βihi

(Mrh)i

)Im

∫C

dr

fθ (r)

]. (45)

Now the corrected Hawking temperature for the noncommutativeSchwarzschild black hole can be identified by using the principle ofdetailed balance [50] which relates the ingoing and outgoing prob-abilities as Pout = exp(− ω

T H)Pin . Taking Pin to be unity and Pout

from (45) one can calculate the corrected Hawking temperature as

T H = h

4

[1 +

∑i

βi hi

(Mrh)i

]−1(Im

∫C

dr

fθ (r)

)−1

. (46)

Since the tunneling phenomena takes place in the close vicinity ofthe event horizon, we can express fθ (r) = fθ (rh)+ (r − rh) f ′

θ (rh)+O(r − rh)2 = (r − rh) f ′

θ (rh) + O(r − rh)2. Substituting this in (46)and performing the contour integral we finally get the correctedHawking temperature as

T H = T H

[1 +

∑i

βi hi

(Mrh)i

]−1

. (47)

Hence, once again applying the first law of black hole thermody-namics with this corrected Hawking temperature, we obtain thefollowing expression for the corrected entropy/area law:

Sbh = Aθ

4h+ 2πβ1 ln Aθ − 16π2β2h2

Aθ

+ O(√

θ e− M2θ

)

= SBH + 2πβ1 ln SBH − 4π2β2h

SBH+ O

(√θ e− M2

θ), (48)

where Aθ and SBH are defined in (30) and (31) respectively. Thisexpression is functionally identical to the corrected entropy/arealaw for the standard Schwarzschild black hole [28,8]. Howeverthere is an important difference. This expression of correctedentropy has both noncommutative and quantum corrections. Al-though here we have restricted ourselves only to the leading ordercorrection due to the NC parameter (θ ), one can try to include allorder θ corrections. This is technically more involved and we shallnot address this issue in this Letter. Now we move on to the nextsection to compute the coefficient β1 in the above expression.

4. Calculation of the coefficient β1

By making an infinitesimal scale transformations to the metriccoefficients in (20), the coefficient β1 can be related to the traceanomaly in the following way [8]:

β1 = − M

4πωIm

∫d4x

√−g⟨T μ

μ

⟩(1)

= − M

4πωIm

∞∫ iβ∫ π∫ 2π∫r2 sin θ

⟨T μ

μ

⟩(1)dr dt dθ dφ. (49)

rh 0 0 0

Here, 〈T μμ〉(1) is the trace anomaly calculated for the first loop

expansion and ω is given by the Komar energy integral (38) eval-uated near the event horizon. The one loop trace anomaly of thestress tensor for the scalar fields moving in the background of a(3 + 1)-dimensional curved manifold is given by [23,24]

⟨T μ

μ

⟩(1) = 1

2880π2

(Rabcd Rabcd − Rab Rab + ∇a∇a R

). (50)

For the metric (20), the invariant scalars are explicitly found as

Rabcd Rabcd = 48M2

r6+ M2e−(r2/2θ)

4πr6θ5

×[

r10 + 16α1 + 32θ3e(r2/4θ)Γ

(3

2,

r2

4θ

)α2

+ 768e(r2/4θ)Γ

(3

2,

r2

4θ

)](51)

where

α1 =(

r6θ2 −√

π

θr5θ3e(r2/4θ) − 4

√π

θr3θ4e(r2/4θ)

),

α2 =(

r5

√θ

− 24√

πθ2e(r2/4θ) + 4θ2(

r2

θ

)3/2),

Rab Rab = M2e−(r2/2θ)(r4 − 8r2θ + 32θ2)

8πθ5, (52)

R = − Me−(r2/4θ)(r2 − 8α)

2√

πθ5/2. (53)

Note that in the commutative limit (θ → 0) the above re-sults match with the known results of the standard vacuumSchwarzschild spacetime metric, for which Rabcd Rabcd = 48M2

r6 ,

Rab Rab = 0, R = 0. To find the trace anomaly (50), we now evalu-ate

∇a∇a R = − Me−r2/2θ

8πθ5(r2/θ)1/2

[2M

(r2 − 12θ

)(r2/θ

)3/2θ

+ √πer2/4θα3 + 4Me(r2/4θ)Γ

(3/2, r2/4θ

)α4

](54)

where

α3 = (r5 − 22r3θ + 72rθ2 − 2M

(r4 − 20r2θ + 48θ2)),

α4 = (r4 − 20r2θ + 48θ2).

Exploiting all these results, the trace anomaly is calculated from

(50), up to the leading order in O(e− r24θ ), as

⟨T μ

μ

⟩(1) = 1

2880π2

[(48M2

r6− 4M2

√πθ5/2

e−r2/(4θ)

r

)

− Me−r2/(4θ)

8√

πθ9/2

[r4 − 2Mr3 − 22r2θ + 40Mrθ

]]

+ O(e−r2/(2θ)

). (55)

Substituting this in (49) and performing the integral yields

β1 = M

180πω

[1 + 2M√

πθ

(1 − 2M2

θ

)e−M2/(θ)

]

+ O(√

θ e−M2/(θ)). (56)

To compute ω, we calculate the Komar energy integral (38). For thespacetime metric (20) one has the following expressions for the

186 R. Banerjee et al. / Physics Letters B 686 (2010) 181–187

Killing vectors (K μ), its inverse (Kν ) and the unit normal vectors(nμ , σν )

Kμ = (1,0,0,0), Kν = − fθ (1,0,0,0), (57)

nμ = f −1/2θ (1,0,0,0), (58)

σν = f 1/2θ (0,1,0,0), (59)√

p(2) = r2 sin θ. (60)

Using these, Eq. (38) is simplified as

ω = 1

8π

π∫θ=0

2π∫φ=0

r2 sin θ (∂r fθ )dθ dφ. (61)

Finally, integrating over the angular variables, we get

ω = M

[1 − r√

πθ

(1 + r2

2θ

)e−r2/(4θ)

]+ O

(√θ e−r2/(4θ)

). (62)

Note that at spatial infinity (r → ∞), the Komar energy (ω) isnothing but the commutative mass (M) of the spacetime as ex-pected. However for finite distance the effective energy involvesNC corrections. Also, for the commutative limit (θ → 0), ω = Mholds at any radial distance outside the event horizon which is awell-known fact for the standard Schwarzschild black hole. Nearthe event horizon (23), the above expression for ω simplifies to

ω = M

[1 − 2M√

πθ

(1 + 2M2

θ

)e−M2/θ

]+ O

(√θe−M2/(θ)

). (63)

Substituting this in the expression for β1 in (56), we obtain:

β1 = 1

180π

[1 + 4M√

πθe−M2/θ

]+ O

(√θe−M2/θ

). (64)

Exploiting (48) and (64), we find the cherished result for the cor-rected entropy/area law (up to leading order correction) for the NCSchwarzschild black hole

Sbh = Aθ

4h+ 1

90

(1 + 4M√

πθe−M2/θ

)ln

Aθ

h+ O

(√θ e− M2

θ)

= SBH + 1

90

(1 + 4M√

πθe−M2/θ

)ln SBH + O

(√θ e− M2

θ). (65)

This is the general expression for the entropy of NC Schwarzschildblack hole where both the NC and quantum effects have beentaken into account. The first term in this expression is the semi-classical entropy and the next term is the leading correction. It islogarithmic in nature. The coefficient of the logarithmic correctionis different from the standard Schwarzschild black hole [28,8] dueto the presence of noncommutative parameter (θ ). In the commu-tative limit θ → 0, the expression for the corrected entropy exactlymatches with the standard Schwarzschild case where the coeffi-cient of the leading correction is 1

90 , obtained in the path integral[23], euclidean [51] and tunneling [28,8] formalisms.

5. Conclusions

We now conclude by making the following comments. In thisLetter we have shown the importance of the Voros star prod-uct in writing down the mass density of a noncommutativeSchwarzschild black hole. To point out the role played by the Vorosproduct, we have first taken recourse to a rigorous formulation ofNC quantum mechanics [16,17]. It has been observed that the in-ner product of the coherent states (used in the construction of the

wave-function of a “free point particle”), can be computed by usinga deformed completeness relation (involving the Voros product)among the coherent states. This inner product is then used to writedown the mass density by making a dimensional lift from two tothree space dimensions. A consistent probabilistic interpretation ofthe wave-function of the “free point particle” can also be givenonly when the Voros product is incorporated. The Gaussian distri-bution of the mass density indeed follows from this Voros type in-terpretation and naturally includes the effect of the NC parameterθ and agrees with the structure given in the literature [10–13,35].

Another part of the Letter dealt with the entropy/area law cor-rections of the NC Schwarzschild black hole. A general result forNC Schwarzschild black hole entropy/area law was found, takingboth quantum and NC effects into account. For this we first usedthe tunneling method by going beyond the semiclassical approx-imation and calculated the corrected Hawking temperature. Thisresult involved the (NC) semiclassical Hawking temperature at thelowest order and corrections at higher orders. Using this modi-fied temperature and the first law of black hole thermodynamicswe then calculated the corrected entropy. The (NC) semiclassicalBekenstein–Hawking value was reproduced at the lowest order andhigher order corrections contained logarithmic and inverse powersof horizon area. The coefficient of the leading (logarithmic) termwas fixed by using the trace anomaly of the scalar field stresstensor. The trace anomaly and the Komar energy integral for NCSchwarzschild metric were explicitly calculated to determine thiscoefficient. The value of the coefficient was found to have NC cor-rection. We also show that the commutative limit of the correctedentropy/area law of the NC Schwarzschild black hole matches withthe standard result for the Schwarzschild black hole [23,51,28,8].

Acknowledgements

One of the authors (SKM) thanks the Council of Scientific andIndustrial Research (CSIR), Government of India, for financial sup-port.

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