Quantum-corrected finite entropy of noncommutative acoustic black holes

0.18 MB·English

by  M. A. Anacleto

Quantum-corrected finite entropy of noncommutative acoustic black holes
M. A. Anacleto, F. A. Brito, G. C. Luna, and E. Passos
∗ † ‡ §
Departamento de F´ısica, Universidade Federal de Campina Grande,
Caixa Postal 10071, 58109-970 Campina Grande, Para´ıba, Brazil
J. Spinelly
¶
Departamento de F´ısica-CCT, Universidade Estadual da Para´ıba
Juvˆencio Arruda S/N, Campina Grande, PB, Brazil
In this paper we consider the generalized uncertainty principle in the tunneling formalism via
Hamilton-Jacobi method to determine the quantum-corrected Hawking temperature and entropy
for2+1-dimensionalnoncommutativeacousticblackholes. Inourresultsweobtainanareaentropy,
5 acorrection logarithmic in leading order, a correction term in subleading order proportional to the
1 radiation temperature associated with thenoncommutative acoustic black holes and an extraterm
0 thatdependsonaconservedcharge. Thus,asinthegravitationalcase,thereisnoneedtointroduce
2 theultraviolet cut-off and divergences are eliminated.

n
a
J

1
I. INTRODUCTION
3

] The study of acoustic black holes was proposed in 1981 by Unruh [1] and has been extensively studied in the
h
literature [2–4]. Acoustic black holes was found to possess many of the fundamental properties of black holes in
t
- generalrelativityandhasbeendevelopedtoinvestigatetheHawkingradiationandotherphenomenaforunderstanding
p
quantum gravity. Thus, many fluid systems have been investigated on a variety of analog models of acoustic black
e
holes, including gravity wave [5], water [6], slow light [7], optical fiber [8] and electromagnetic waveguide [9]. The
h
[ models of superfluid helium II [10], atomic Bose-Einstein condensates [11, 12] and one-dimensional Fermi degenerate
noninteractinggas[13]havebeen proposedto createanacoustic black hole geometryinthe laboratory. A relativistic
1
version of acoustic black holes has been presented in [14, 15].
v
In Ref. [16]was investigated(1 + 1)-dimensionalacoustic black hole entropyby the brick-wallmethod. In orderto
9
7 obtain a finite result, they had to introduce the ultraviolet cut-off. So their calculation suggested that analog black
1 hole entropy has the cut-off problem similar to that of gravitational black hole entropy. More recently in [17] the
0 author uses transverse modes in order to cure the divergences.
0 The study onthe statisticaloriginofblackhole entropyhasbeen extensivelyexploredby severalauthors—see for
.
2 instance [18–20]. In Ref. [21], Kaul and Majumdar compute the lowestorder corrections to the Bekenstein-Hawking
0 entropy. They find that the leading correction is logarithmic, with
5
1 A 3 A
S ln +const.+
:
v ∼ 4G − 2 4G ···
(cid:18) (cid:19)
i
X
ontheotherhand,CarlipinRef.[22]computetheleadinglogarithmiccorrectionstothe Bekenstein-Hawkingentropy
r and shows that the logarithmic correction is identical to that of Kaul and Majumdar, plus corrections that depend
a
on conserved charges, as
A 3 A
S ln +ln[F(Q)]+const.+
∼ 4G − 2 4G ···
(cid:18) (cid:19)
where F(Q) is some function of angular momentum and other conserved charges.
The brick-wall method proposed by G. ’t Hooft has been used for calculations on the black hole, promoting the
understanding of the origin of black hole entropy. According to G. ’t Hooft, black hole entropy is just the entropy of
quantumfields outsidethe blackholehorizon. However,whenonecalculatestheblackholestatisticalentropybythis
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected]
¶Electronicaddress: [email protected]
2
method, to avoid the divergence of states density near black hole horizon, an ultraviolet cut-off must be introduced.
The other related idea in order to cure the divergences is to consider models in which the Heisenberg uncertainty
relation is modified, for example in one dimensional space, as
~
∆x∆p 1+α2(∆p)2 ,
≥ 2
which shows that there exists a minimal length ∆x (cid:0)~α, where ∆(cid:1)x and ∆p are uncertainties for positon and
≥
momentum, respectively, and α is a positive constant which is independent of ∆x and ∆p. A commutation relation
for the generalized uncertainty principle (GUP) can be written as [x,p] = i~(1+α2p2), where x and p are the
GUP
positon and the momentum operators, respectively. Thus, using the modified Heisenberg uncertainty relation the
divergence in the brick-wall model are eliminated as discussed in [23]. The statistical entropy of various black holes
has also been calculated via corrected state density of the GUP [24]. Thus, the results show that near the horizon
quantum state density andits statisticalentropy are finite. In [25] a relationfor the correctedstates density by GUP
has been proposed
dn= d3xd3pe λp2,
(2π)3 −
where p2 =pip , and λ plays the role of the Planck scale and in a fluid at high energy regimes.
i
Recently, the authors in [26] using a new equation of state density due to GUP [27], the statistical entropy of a
2+1-dimensional rotating acoustic black hole has been analyzed. It was shown that using the quantum statistical
method the entropy of the rotating acoustic black hole was calculated, and the Bekenstein-Hawking area entropy
of acoustic black hole and its correction term was obtained. Therefore, considering the effect due to GUP on the
equation of state density, no cut-off is needed [35] and the divergence in the brick-wall model disappears.
There are several approaches to obtain the Hawking radiation and the entropy for black-holes. One of them is
the Hamilton-Jacobi method which is based on the work of Padmanabhanand collaborators [28] and also the effects
of the self-gravitation of the particle are discarded. In this way, the method uses the WKB approximation in the
tunneling formalismfor the computation of the imaginary partof the action. In [29] the authors Parikhand Wilczek
using the method of radial null geodesic determined the Hawking temperature and in [30] this method was used by
authors for calculating the Hawking temperature for different spacetimes. In Ref. [31] has been analyzed Hawking
radiation considering self-gravitation and back reaction effects in tunneling formalism. It has also been investigated
in [32] the back reaction effects for self-dual black hole using the tunneling formalism by Hamilton-Jacobi method.
In [34] has been studied the effects of the GUP in the tunneling formalism for Hawking radiation to evaluate the
quantum-corrected Hawking temperature and entropy for a Schwarzschild black hole. Moreover, the authors in [33]
have discussed the Hawking radiation for acoustic black hole using tunneling formalism.
In this paper, inspired by all of these previous work we apply the acoustic black hole metrics obtained from a
relativisticfluid ina noncommutativespacetime[36] viathe Seiberg-Witten maptostudy the entropyofthe acoustic
black hole. Whereas, on one hand our objective is to see if using the GUP in the tunneling formalism via Hamilton-
Jacobi method divergences are eliminated as in the gravitationalcase.
This is also motivated by the fact that in high energy physics both strong spacetime noncommutativity and quark
gluonplasma(QGP)may take placetogether. Thus,it seems to be naturalto look for acousticblack holesin a QGP
fluid with spacetime noncommutativity in this regime. Acoustic phenomena in QGP matter can be seen in Ref. [40]
and acoustic black holes in a plasma fluid can be found in Ref. [41].
Differentlyofthemostcasesstudied,weconsidertheacousticblackholemetricsobtainedfromarelativisticfluidin
anoncommutativespacetime. Theeffectsofthissetupissuchthatthefluctuationsofthefluidsarealsoaffected. The
sound waves inherit spacetime noncommutativity of the fluid and may lose the Lorentz invariance. As a consequence
the Hawking temperature is directly affected by the spacetime noncommutativity. Analogously to Lorentz-violating
gravitationalblackholes[42,43],theeffectiveHawkingtemperatureofthenoncommutativityacousticblackholesnow
isnotuniversalforallspeciesofparticles. Itdependsonthemaximalattainablevelocityofthisspecies. Furthermore,
the acoustic black hole metric can be identified with an acoustic Kerr-like black hole. It was found in [36] that the
spacetimenoncommutativityaffectstherateoflossofmassoftheblackhole. Thusforsuitablevaluesofthespacetime
noncommutativity parametera wider or narrowerspectrum of particle wavefunction can be scatteredwith increased
amplitudebytheacousticblackhole. Thisincreasesordecreasesthesuperressonancephenomenonpreviouslystudied
in [44, 45].
InourstudyweshallfocusontheHamilton-Jacobimethodtodeterminetheentropyofanacousticblackholeusing
the GUP and considering the WKB approximationin the tunneling formalism to calculate the imaginary part of the
action in order to determine the Hawking temperature and entropy for noncommutative acoustic black holes. We
anticipatethatwehaveobtainedtheBekenstein-Hawkingentropyofacousticblackholesanditsquantumcorrections.
There is no need to introduce the ultraviolet cut-off and divergences are eliminated.
3
The paper is organizedas follows. In Sec. II we briefly review the steps to find the noncommutative acoustic black
hole metrics. In Sec. III we consider the WKB approximation in the tunneling formalism and apply the Hamilton-
Jacobi method to compute the temperature and entropy. In Sec. IV we compute the statistical entropy. We develop
explicitly computations for the magnetic and electric sectors. Finally in Sec. V we present our final considerations.
II. NONCOMMUTATIVE ACOUSTIC BLACK HOLE
In this section let us briefly review the steps to find the noncommutative acoustic black hole metric in (3+1)
dimensions fromquantumfield theory. The noncommutativity is introducedby modifying its scalarandgaugesector
by replacingthe usual product of fields by the Moyalproduct [46–49] — see also[50, 51] for relatedissues. Thus, the
Lagrangianof the noncommutative Abelian Higgs model in flat space is
1
ˆ = Fˆ Fˆµν +(D φˆ) Dµφˆ+m2φˆ φˆ bφˆ φˆ φˆ φˆ, (1)
µν µ † † † †
L −4 ∗ ∗ ∗ − ∗ ∗ ∗
wherethehatindicatesthatthevariableisnoncommutativeandthe -productistheso-calledMoyal-Weylproductor
star product which is defined in terms of a real antisymmetric matri∗x θµν that parameterizes the noncommutativity
of Minkowski spacetime
[xµ,xν]=iθµν, µ,ν =0,1, ,D 1. (2)
··· −
The -product for two fields f(x) and g(x) is given by
∗
i
f(x)∗g(x)=exp 2θµν∂µx∂νy f(x)g(y)|x=y. (3)
(cid:18) (cid:19)
As one knows the parameterθαβ is a constant,real-valuedantisymmetric D D- matrixin D-dimensionalspacetime
×
with dimensions of length squared. For a review see [49].
Now using the Seiberg-Witten map [46], up to the lowest order in the spacetime noncommutative parameter θµν,
we find the corresponding theory in a commutative spacetime in (3+1) dimensions [47]
1 1 1
ˆ = F Fµν 1+ θαβF + 1 θαβF D φ2+m2 φ2 bφ4
µν αβ αβ µ
L −4 2 − 4 | | | | − | |
(cid:18) (cid:19) (cid:18) (cid:19)
1 (cid:0) (cid:1)
+ θαβFαµ (Dβφ)†Dµφ+(Dµφ)†Dβφ , (4)
2
where F =∂ A ∂ A and D(cid:2)φ=∂ φ ieA φ. (cid:3)
µν µ ν ν µ µ µ µ
− −
Now, if we decompose the scalar field as φ= ρ(x,t)exp(iS(x,t)) into the original Lagrangian,we have
= 1F Fµν 1 2~θ B~ +ρ(θ˜gµνp+Θµν) S S+θ˜m2ρ θ˜bρ2+ ρ (θ˜gµν +Θµν)∂ ∂ √ρ, (5)
µν µ ν µ ν
L −4 − · D D − √ρ
(cid:16) (cid:17)
where = ∂ eA /S, θ˜= (1+θ~ B~), B~ = A~ and Θµν = θαµF ν. In our calculations we consider the case
µ µ µ α
where Dthere is n−o noncommutativity·between sp∇ac×e and time, that is θ0i = 0 and use θij = εijkθk, Fi0 = Ei and
Fij =εijkBk.
Thus,linearizingthe equationsofmotionaroundthe background(ρ0,S0), withρ=ρ0+ρ1 andS =S0+ψ we find
the equation of motion for a linear acoustic disturbance ψ given by a Klein-Gordon equation in a curved space
1
∂ (√ ggµν∂ )ψ =0, (6)
µ ν
√ g −
−
where g = bρ0 g˜ just represents the acoustic metrics in (3+1) dimensions, being the metric components, g˜
µν 2cs√f µν µν
given in the form
g˜ = [(1 3~θ B~)c2 (1+3~θ B~)v2+2(~θ ~v)(B~ ~v) (~θ E~) ~v],
tt − − · s− · · · − × ·
1 vj Bj θj
g˜ = (~θ E~)j(c2+1) 2(1+2~θ B~) (~θ E~) ~v + (~θ ~v)+ (B~ ~v),
tj −2 × s − · − × · 2 2 · 2 ·
1 h ivi Bi θi
g˜ = (~θ E~)i(c2+1) 2(1+2~θ B~) (~θ E~) ~v + (~θ ~v)+ (B~ ~v),
it −2 × s − · − × · 2 2 · 2 ·
g˜ = [(1+θ~ B~)(1+c2) (h1+θ~ B~)v2 (~θ E~) ~v]δiij +(1+~θ B~)vivj.
ij · s − · − × · ·
f = [(1 2~θ B~)(1+c2) (1+4~θ B~)v2] 3(~θ E~) ~v+2(B~ ~v)(~θ ~v). (7)
− · s − · − × · · ·
4
We should comment that in our previous computation we assumed linear perturbations just in the scalar sector,
whereas the vector field A remain unchanged.
µ
In the following we shall focus on the planar noncommutative acoustic black hole metrics in (2+1) dimensions [36]
to address the issues of Hawking temperature and entropy of three-dimensional acoustic black hole in the tunneling
formalism via Hamilton-Jacobi method considering the generalized uncertainty principle. For the sake of simplicity,
we shall consider two types of a noncommutative spacetime medium by choosing first pure magnetic sector and then
we shall focus on the pure electric sector.
III. TUNNELING FORMALISM VIA HAMILTON-JACOBI METHOD
In this section we consider the WKB approximation in the tunneling formalism and apply the Hamilton-Jacobi
method to calculate the imaginary part of the action in order to determine the Hawking temperature for a
noncommutativeacousticblackhole. Inourcalculationsweassumethatthe classicalaction satisfiesthe relativistic
I
Hamilton-Jacobi equation to leading order in the energy
gµν∂ ∂ +m2 =0, (8)
µ ν
I I
where m is the mass of the scalar particle. Thus, we can neglect the effects of the self-interactionof the particle. We
begin our analysis with the pure magnetic sector (the case B~ =0 and E~ =0), as we shall show below.
6
A. Pure Magnetic Sector - the case B~ 6=0 and E~ =0
Theacousticlineelementinpolarcoordinatesonthenoncommutativeplanein(2+1)dimensions,uptoanirrelevant
position-independent factor, in the nonrelativistic limit was obtained in [36] and is given by
ds2 = (1+θ B ) [(1 3θ B )c2 (1+3θ B )(v2+v2)]dt2 2(1+2θ B )(v dr+v rdφ)dt
− z z − z z s− z z r φ − z z r φ
+ (1+θzBz)(d(cid:8)r2+r2dφ2) . (9)
(cid:9)
whereB isthemagnitudeofthemagneticfieldinthez direction,θ isthenoncommutativeparameter,c = dh/dρ
z z s
is the sound velocity in the fluid and v is the fluid velocity. We consider the flow with the velocity potential
p
ψ(r,φ)=Alnr+Bφ whose velocity profile in polar coordinates on the plane is given by
A B
~v = rˆ+ φˆ, (10)
r r
where B and A are the constants of circulation and draining rates of the fluid flow.
Let us now consider the transformations of the time and the azimuthal angle coordinates as follows
(1+2θ B )Ardr
z z
dτ =dt+ ,
[(1 3θ B )c2r2 (1+3θ B )A2]
− z z s − z z
ABdr
dϕ=dφ+ . (11)
r[c2r2 A2]
s −
In these new coordinates the metric becomes
ds2 = (1 2Θ) 1 (1+6Θ)(A2+B2) dτ2+θ˜ 1 (1+6Θ)A2 −1dr2 2θ˜2Bdϕdτ +θ˜r2dϕ2, (12)
− − − c2r2 − c2r2 − c
(cid:20) s (cid:21) (cid:20) s (cid:21) s
where Θ=θ B and θ˜=1+2Θ.
z z
The radius of the ergosphere is given by g00(re) = 0, whereas the horizon is given by the coordinate singularity
g (r )=0, that is
rr h
B2
r˜ =(1+6Θ)1/2r , r = r2 + , (13)
e e e s h c2s
A
r˜ =(1+6Θ)1/2r , r = | |. (14)
h h h
c
s
5
Now for B =0 (no rotation), we have the metric of stationary acoustic black hole given by
ds2 = f(r)dτ2+f(r) 1dr2+θ˜r2dϕ2, (15)
−
−
where f(r) = (1 2Θ) 1 r˜h2 , with c = 1. Thus, we obtain the Hawking temperature of the noncommutative
− − r2 s
acoustic black hole in te(cid:16)rms of r˜(cid:17)h as
f (r˜ ) (1 2Θ)
T˜ = ′ h = − +O(Θ2), (16)
h
4π 2πr˜
h
or in terms of r , we have
h
T˜ =(1 5Θ)T +O(Θ2), (17)
h h
−
where T =(2πr ) 1 is the Hawking temperature of the acoustic black hole.
h h −
Near the horizon, when r r˜ , we have f(r) 2a(r r˜ ), where a is the surface gravity of acoustic black hole
h h
→ ≈ −
horizon.
We shall now consider the Klein-Gordon equation for a scalar field Φ and apply the tunneling method via the
Hamitlon-Jacobi ansatz
1 m2
∂ √ ggµν∂ Φ=0 (18)
√ g µ − ν − ~2
(cid:20) − (cid:21)
(cid:0) (cid:1)
where m is the mass of the scalar particle. In this way, using the WKB approximation,we can write
i
Φ=exp (t,r,ϕ) . (19)
~I
(cid:20) (cid:21)
Then, to the lowest order in ~, we have
gµν∂ ∂ +m2 =0. (20)
µ ν
I I
Considering the metric (15) the equation (20) becomes
1 1
(∂ )2+f(r)(∂ )2+ (∂ )2+m2 =0. (21)
− f(r) tI rI θ˜r2 ϕI
Now due to the symmetries of the metric, we can suppose a solution of the form
= Et+W(r)+J ϕ. (22)
ϕ
I −
As a consequence
dW(r)
∂ = E, ∂ = ∂ =J . (23)
t r ϕ ϕ
I − I dr I
where J is a constant and the classical action is given by
ϕ
E2 f(r)(m2 J2 )
− − θ˜r2
= Et+ dr +J ϕ. (24)
I − q f(r) ϕ
Z
In this way, by taking the near-horizon approximation, f(r) 2a(r r˜ ), the spatial part of the action function,
h
≈ −
reads
E2 2a(r r˜ )(m2 J2 )
1 − − h − θ˜r2 2πiE
W(r)= dr = . (25)
2a q (r r˜ ) 2a
Z − h
Therefore, the tunneling probability for a particle with energy E is given by
2πiE
Γ exp[ 2Im ]= . (26)
≃ − I a
6
On the other hand, we have Γ exp( βE), where β is the inverse temperature β =1/T˜ . Thus, acoustic black hole
h
≃ −
temperature is given by
a
T˜ = . (27)
h
2π
Since a=f (r˜ )/2, we can get the temperature in terms of the horizon radius, given by
′ h
1 2Θ
T˜ = − =(1 5Θ)T . (28)
h h
2πr˜ −
h
Note that the temperature is modified due to noncommutativity of spacetime, which precisely agrees with the result
(17).
IV. THE STATISTICAL ENTROPY
In this section we consider the generalized uncertainty principle (GUP) in the tunneling formalism and
apply the Hamilton-Jacobi method to determine the quantum-corrected Hawking temperature and entropy for a
noncommutative acoustic black hole. Let us start with the GUP [37, 38], which is an extension of [39]
αl α2l2
∆x∆p ~ 1 p∆p+ p(∆p)2 , (29)
≥ − ~ ~2 !
whereαis adimensionlesspositiveparameter,l = ~G/c3 =M G/c2 10 35mis the Planklength, M = ~c/G
p p − p
≈
is the Plank mass and c is the velocity of light. Since G is the Newtonian coupling constant, the correction terms in
p p
the uncertainty relation (29) are due to the effects of gravity.
Now the equation (29) can be written as follows.
~(∆x+αl ) 4α2l2
p p
∆p 1 1 , (30)
≥ 2α2lp2 −s − (∆x+αlp)2!
wherewehavechosenthenegativesign. Sincel /∆xisrelativelysmallcomparedtounitywecanexpandtheequation
p
above in Taylor series
1 α α2
∆p 1 + + . (31)
≥ ∆x − 2∆x 2(∆x)2 ···
(cid:20) (cid:21)
As we have chosen G = c = k = 1, so we also choose ~ = 1, and we have l = 1. In these units the uncertainty
B p
principle becomes
∆x∆p 1. (32)
≥
Now using the saturated form of the uncertainty principle we have
E∆x 1, (33)
≥
which follows from the saturated form of the Heisenberg uncertainty principle, ∆x∆p 1, where E is the energy of
≥
a quantum particle. Therefore, we can rewrite equation (31) in the form
α α2
E E 1 + + , (34)
G ≥ − 2(∆x) 2(∆x)2 ···
(cid:20) (cid:21)
So by using Hamilton-Jacobi method the tunneling probability of a particle with corrected energy E becomes
G
2πiE
G
Γ exp[ 2Im ]= . (35)
≃ − I a
comparing with the Boltzmann factor (e E/T), we obtain the acoustic black hole temperature
−
T =T˜ 1 α + α2 + −1. (36)
h − 2(∆x) 2(∆x)2 ···
(cid:20) (cid:21)
7
Here we will choose ∆x=2r˜ . Thus, we have the corrected temperature due to the GUP
h
(1 2Θ) α α2 −1
T = − 1 + + . (37)
2πr˜ − 4r˜ 8r˜2 ···
h (cid:20) h h (cid:21)
Heretheentropyisassumedtosatisfythearealaw,sinceitdependsonlyonthegeometryofthehorizon. Thus,using
the laws of black hole thermodynamics we get the entropy in terms of horizon area of the acoustic black hole as
dE κdA dA˜ dA˜ πα π2α2
S = = = =(1+2Θ) 1 + + ,
Z T Z 8πT Z 8πr˜hT Z 4 (cid:20) − 2A˜ 2A˜2 ···(cid:21)
A˜ πα A˜ π2α2
= (1+2Θ) ln + . (38)
"4 − 8 4 − 32A˜/4 ···#
We can write this expression for entropy in terms of horizon radius as given by
2πr˜ πα 2πr˜ π2α2
S =(1+2Θ) h ln h T˜ + , (39)
h
4 − 8 4 − 8 ···
(cid:20) (cid:21)
where A˜ = 2πr˜ = (1+3Θ)A is the horizon area of the noncommutative acoustic black hole and A = 2πr is the
h h
horizon area of the acoustic black hole. The correction terms are due to quantum effects. The second term is a
correction logarithmic that appears in the leading order and is similar to the existing results for gravitational black
hole in 4 dimensions and that are also derived by other methods. The third term is a correction term to the area
entropyinsubleadingorderandisproportionaltotheradiationtemperature,T˜ =(1 5Θ)T ,ofthenoncommutative
h h
−
acoustic black hole. The equation (39) can be written in terms of r as
h
S 2πr πα 2πr π2α2 πα
S˜= =(1+3Θ) h ln h (1 5Θ)T ln(1+3Θ)+ , (40)
h
(1+2Θ) 4 − 8 4 − 8 − − 8 ···
Note that the logarithmic term is obtained due to the contribution α(∆p) in the GUP, while the gravitational case,
for example, for a Schwarzschild black hole the logarithmic correction is obtained from the quadratic term α2(∆p)2
in the GUP.
Thus, for α=0 and Θ=0 we just reproduce the usual semiclassicalarea law for the Bekenstein-Hawkingentropy,
S =A/4=2πr /4. Moreover,choosing α=12/π, we have
h
2πr 3 2πr 9
S˜=(1+3Θ) h ln h 18(1 5Θ)T ln[1+2Θ+O(Θ2)]+ , (41)
h
4 − 2 4 − − − 4 ···
and the resulting logarithmic correction to the entropy becomes, 3/2ln(A/4) and which has the same correction
−
for gravitational black holes in four dimensions. The last term in (41) is independent of the horizon radius and
corresponds to an extra term that depends on a conserved charge c = e(1+2Θ). Notice that both the logarithmic
gravitational-like term and the appearance of the conserved charge corrections are enforced by the existence of the
linear correctionin the GUP we areconsideringin Ref. (29). This conservedchargeis obtainedfromthe equationsof
motion for the field A (pure magnetic sector) of Lagrangian(5), ie
µ
( B~)j =2eρ(1+2Θ)uj, j =1,2. (42)
∇×
Therefore, in our model considering a linear GUP given by (29) (the linear correction term in the momentum
uncertainty is required), we obtain logarithmic corrections similar to that of Kaul and Majumdar [21] and one extra
term that depends on a conserved charge in a manner analogous to that obtained in [22].
It is also interesting to note that the equation (39) has a form similar to the result presented in Ref. [20] and
references therein, for black holes in two dimensions that have the following correction to the entanglement entropy
c c Λ
S = σ + ln , (43)
h
6 6 ǫ
(cid:18) (cid:19)
where c is a central charge, Λ is an IR cut-of, ǫ is a UV cut-of and σ =ψ /2 is given by the value of the field ψ on
h h
the horizon. ψ is the scalar field of the Polyakovaction [20].
8
A. Pure Electric Sector - the case B~ =0 and E~ 6=0
In the present subsection we repeat the previous analysis for the pure electric sector (B~ =0 and E~ =0). As in the
6
earlier case we take the acoustic line element obtained in [36], in polar coordinates on the noncommutative plane, up
to first order in θ, in the ‘non-relativistic’ limit, given by
3θ (A2+B2+θr( A+ B)
ds2 = 1 ( A+ B) 1 Er Eφ dτ2
− 2r Er Eφ − − c2r2
(cid:20) (cid:21)(cid:26) (cid:20) s (cid:21)
+ 1 θ ( A+ B) 1 A2+θErAr −1dr2+r2dϕ2 2 B + θEφ rdϕdτ , (44)
(cid:20) − r Er Eφ (cid:21)"(cid:18) − c2sr2 (cid:19) #− (cid:18)csr 2cs(cid:19) )
where θ r =θ(~n E~)r , θ φ =θ(~n E~)φ. The radius of the ergosphere is given by g00(r˜e)=0, whereas the horizon
E × E ×
is given by the coordinate singularity g (r˜ )=0, that is
rr h
θ A+θ B 1 (θ A+θ B)2 θ (θ )2
r˜ = Er Eφ Er Eφ +4r2, r˜ =r Er 1+ Er (45)
e 2c2s ± 2s c4s e h± h"2cs ±s 4c2s #
where r = (A2+B2)/c2 and r = A/c are the radii of the ergosphere and the horizon in the usual case. For
e s h | | s
θ =0, we have r˜ =r and r˜ =r .
e e h h
p
Now for B =0 (no rotation)and =0 with c =1, we have the metric of stationary acoustic black hole givenby
φ s
E
4θ r 5θ r
ds2 = f(r)dτ2+ 1 Er h f(r)−1dr2+ 1 Er h r2dϕ2, (46)
− − r − 2r
(cid:18) (cid:19) (cid:18) (cid:19)
where
3θ r r2 θ r
f(r)= 1 Er h 1 h Er h . (47)
− 2r − r2 − r
(cid:18) (cid:19)(cid:18) (cid:19)
Now we obtain the Hawking temperature of the acoustic black hole as
T = f′(r˜h+) =(1 2θ ) 1 + (θ2). (48)
h r
2π − E 2πr˜h+ O
Thus, considering the tunneling formalism via Hamilton-Jacobi method with the GUP (29) we obtain the corrected
temperature as
(1 2θεr) α α2 −1
T = − 1 + + . (49)
2πr˜h+ (cid:20) − 4r˜h+ 8r˜h2+ ···(cid:21)
In this way, the entropy of the noncommutative acoustic black hole, near the black hole horizon, is
S =(1+2θε ) 2πr˜h+ παln2πr˜h+ π2α2T˜ + , (50)
r h
4 − 8 4 − 8 ···
(cid:20) (cid:21)
where 2πr˜h+ is the horizon area of the noncommutative acoustic black hole, ie, A˜=(1+θεr/2)A and A=2πrh the
horizon area of the acoustic black hole. Again, the correctionterms are due to quantum effects. Thus the entropy in
terms of r is
h
S θε 2πr πα 2πr 5θε π2α2 πα θε
S˜= = 1+ r h ln h 1 r T ln 1+ r + . (51)
h
(1+2θε ) 2 4 − 8 4 − − 2 8 − 8 2 ···
r (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Note that the second term is a logarithmic correction that appears in the leading order as in the gravitational case
in 3+1 dimensions and the third correction term is proportional to the radiation temperature of the acoustic black
hole. Moreover,choosing α=12/π, we have
θε 2πr 3 2πr 5θε 3 θε
S˜= 1+ r h ln h 18 1 r T ln 1+ r + . (52)
h
2 4 − 2 4 − − 2 − 2 2 ···
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
9
The last term in (52) is independent of the horizon radius and corresponds to an extra term that depends on a
conserved charge c = e(1+θε /2). This conserved charge is obtained from the equations of motion for the field A
r µ
(pure electric sector) of Lagrangian(5), i.e.,
θε
E~ =2eρw 1+ r , (53)
∇· 2
(cid:18) (cid:19)
where we have considered the normalization v u /w =1, since in the near horizon regime v c =1.
r r r s
≡ ∼
V. CONCLUSIONS
In summary, by considering the GUP, we derive the noncommutative acoustic black hole temperature and entropy
using the Hamilton-Jacobi version of the tunneling formalism. Moreover, in our calculations the Hamilton-Jacobi
method was applied to calculate the imaginary part of the action and the GUP was introduced by the correction to
the energy of a particle due to gravity near the horizon. The GUP allows us to find quantum corrections to the area
law. The noncommutative nature of the spacetime also suggests the existence of corrections in terms of conserved
electric charge.
Acknowledgments
We would like to thank CNPq, CAPES and PNPD/PROCAD - CAPES for partial financial support.
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