Voros product, noncommutative inspired Reissner-Nordstr{\"o}m black hole and corrected area law

Abstract

We emphasize the importance of the Voros product in defining a noncommutative
inspired Reissner-Nordstr\"{o}m black hole. The entropy of this black hole is
then computed in the tunneling approach and is shown to obey the area law at
the next to leading order in the noncommutative parameter $\theta$.
Modifications to entropy/area law is then obtained by going beyond the
semi-classical approximation. The leading correction to the semiclassical
entropy/area law is found to be logarithmic and its coefficient involves the
noncommutative parameter $\theta$.

Full text

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arXiv:1012.4611v1 [hep-th] 21 Dec 2010
Voros product, noncommutative inspired Reissner-Nordstr¨ om
black hole and corrected area law
Sunandan Gangopadhyaya,c,∗, Dibakar Roychowdhuryb,†
aDepartment of Physics, West Bengal State University, Barasat, India
bS.N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata 700098, India
cVisiting Associate in S.N. Bose National Centre for Basic Sciences,
JD Block, Sector III, Salt Lake, Kolkata 700098, India
Abstract
We emphasize the importance of the Voros product in defining a noncommutative inspired
Reissner-Nordstr¨ om black hole. The entropy of this black hole is then computed in the
tunneling approach and is shown to obey the area law at the next to leading order in the
noncommutative parameter θ. Modifications to entropy/area law is then obtained by going
beyond the semi-classical approximation. The leading correction to the semiclassical en-
tropy/area law is found to be logarithmic and its coefficient involves the noncommutative
parameter θ.
Classical general relativity gives the concept of black hole from which nothing can escape. This
picture was changed dramatically when Hawking [1, 2] incorporated the quantum nature into
this classical problem. He showed by combining gravity and quantum mechanics that black
holes emit a spectrum that is similar to a thermal black body spectrum. This result made the
first law of black hole mechanics [3] closely analogous to the first law of thermodynamics. This
analogy ultimately led to the entropy for black holes and was consistent with the proposal made
∗sunandan.gangopadhyay@gmail.com, sunandan@bose.res.in
†dibakar@bose.res.in
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by Bekenstein [4, 5, 6, 7] that a black hole has an entropy proportional to its horizon area. All
the above issues finally led to the famous Bekenstein-Hawking area law for the entropy of black
holes given by SBH= A/4.
However, most of these calculations were based on a semiclassical treatment and also on a
commutative spacetime. The standard Bekenstein-Hawking area law is known to get corrections
due to quantum geometry or back reaction effects [8]. Recently, modifications to the semiclas-
sical area law due to noncommutative (NC) spacetime have also been obtained [9]-[14]. The
motivation for these investigations was that noncommutativity is expected to be relevant at the
Planck scale where it is known that usual semiclassical considerations break down.
In a recent paper [13], it has been pointed out that the Voros star product [15],[16] plays
a key role in the obtention of the mass density of a static, spherically symmetric, smeared,
particle-like gravitational source required in getting the NC inspired Schwarzschild black hole
[17],[18]1. Quantum corrections to the semiclassical Hawking temperature and entropy for NC
inspired Schwarzschild black hole have been obtained next. This is done by first computing
the correction to the Hawking temperature by going beyond the semiclassical approximation in
the tunneling method [32]-[38]. Using the corrected form of the Hawking temperature and the
first law of black hole thermodynamics, the entropy is computed. The result is seen to contain
logarithmic and inverse horizon area corrections and holds to O(√θe−M2/θ).
In this paper, we carry out the above analysis in the case of NC inspired Reissner-Nordstr¨ om
(RN) black hole. To obtain NC effects on the usual area law, computation of the (NC) Hawking
temperature is carried out. The first law of thermodynamics is then used to obtain the entropy. It
is observed that to the next to leading order in the NC parameter θ, in the regime r2
h/(4θ) >> 1,
the (NC) area law is just a NC deformation of the usual semiclassical area law as in case of the
NC inspired Schwarzschild black hole. Quantum corrections to the area law is then found by
following the procedure as mentioned above for the NC inspired Schwarzschild black hole. The
coefficient of the logarithmic correction term is explicitly determined from the trace anomaly of
the stress tensor [39],[40]. This coefficient is also found to have NC correction.
To begin the discussion, we follow the analysis in [13] based on the formulational and in-
terpretational aspects of NC quantum mechanics to highlight the important part played by the
1Note that there are ways in which the Moyal star product [19] is used to incorporate the NC effect in gravity
[20]-[24]. The twisted formulation of NC quantum field theory [25]-[31] is yet another way of incorporating the
effects of noncommutativity in gravity.
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Voros star product in writing down the mass and charge densities of a static, spherically sym-
metric, smeared, particle-like charged gravitational source required in getting the NC inspired
RN black hole
ρ(M)
θ
(r)=
M
(4πθ)3/2exp
Q
(4πθ)3/2exp
?
?
−r2
4θ
−r2
4θ
?
?
ρ(Q)
θ
(r)=
.(1)
Solving Einstein’s equations with the above mass and charge densities incorporated in the energy-
momentum tensor leads to the following NC inspired RN metric [10],[12]
ds2= −fθ(r)dt2+ f−1
θ(r)dr2+ r2(d˜θ2+ sin2˜θdφ2) (2)
where
gtt(r)=grr(r) = fθ(r) = 1 −4M
r√πγ
?3
?
2,r2
4θ
?
+Q2
πr2
?
F(r) +
?
2
θrγ
?3
2,r2
4θ
??
(3)
F(r)=γ2
?1
2,r2
4θ
?
−
r
√2θγ
?1
2,r2
2θ
.
The event horizon of the black hole can be found by setting gtt(rh) = 0 in (2). Since this
equation cannot be solved in a closed form, we take the large radius regime (r2
h
4θ>> 1) where we
can expand the incomplete gamma function to solve rhby iteration. Keeping upto the next to
√θe−r2
leading order
1
0
4θ, we find
rh≃ r0
?
1 −
r0
2√πθ(r0− M)
?
2M −
Q2
√2πθ
?
e−r2
0/(4θ)
?
(4)
where
r0= M +
?
M2− Q2
(5)
is the horizon radius of the commutative RN black hole.
Now for a general static and spherically symmetric spacetime, the Hawking temperature
(TH) is related to the surface gravity (κ) by the following relation [41]
TH=?κ
2π
(6)
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where the surface gravity of the black hole is given by
κ =1
2
?dfθ
dr
?
r=rh
. (7)
Therefore in the regime (r2
h
4θ>> 1), the Hawking temperature for the NC inspired RN black hole
is found to be (upto order
1
√θe−r2
0/(4θ))
TH
≃
?
2πr3
0
?
Mr0− Q2+
Mr2
0
√πθ(r0− M)
0/(4θ)
?
3M − r0−
Q2
√2πθ
−(r0− M)
2θ
r2
0
?
e−r2
0/(4θ)
+
Q2r4
√24πθ2e−r2
0
?
. (8)
We shall now use the first law of black hole thermodynamics to calculate the Bekenstein-Hawking
entropy. The first law of black hole thermodynamics is given by [42]
S =
?
dM
TH
+
?
Y dQ(9)
where
Y=−ΦH
TH
Q
rh
−
∂
∂Q
?
dM
TH
(10)
ΦH
=
.
Using eq.(8), it can be easily shown that Y = 0 for the NC inspired RN black hole. Hence the
Bekenstein-Hawking entropy upto next to leading order in θ is found to be
S =
?
dM
TH
= πr2
0
?
1 −
r0
√πθ(r0− M)
?
2M −
Q2
√2πθ
?
e−r2
0/(4θ)
?
.(11)
In order to express the entropy in terms of the NC horizon area (Aθ), we use eq.(4) to obtain
Aθ= 4πr2
h= 4πr2
0
?
1 −
r0
√πθ(r0− M)
?
2M −
Q2
√2πθ
?
e−r2
0/(4θ)
?
. (12)
Comparing eq(s)(11, 12), we find that at the leading order in θ, the NC black hole entropy
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satisfies the area law
S = SBH=Aθ
4?. (13)
This is functionally identical to the Bekenstein-Hawking area law in the commutative space.
Hence we have analytically observed that in the regime
r2
h
4θ>> 1, the NC version of the semiclas-
sical Bekenstein-Hawking area law holds upto leading order in θ. This motivates us to investigate
the corrections to the semiclassical area law upto leading order in θ.
To do so, we first compute the corrected Hawking temperature˜TH. Following the tunelling
method in [32] which goes beyond the semiclassical approximation, one obtains the corrected
Hawking temperature as
˜TH= TH
?
1 +
?
i
˜βi?i
(Mrh− Q2/2)i
?−1
.(14)
Now applying the first law of black hole thermodynamics once again with this corrected Hawking
temperature, we obtain the following expression for the corrected entropy/area law :
Sbh
=
Aθ
4?+ 2π˜β1lnAθ+ O(
SBH+ 2π˜β1lnSBH+ O(
√
θe−r2
√
0/(4θ))
=
θe−r2
0/(4θ)) (15)
where Aθand SBH are defined in (12) and (13) respectively. This expression is functionally
identical to the corrected entropy/area law for the standard Schwarzschild black hole [35, 42].
However there is an important difference. This expression of corrected entropy has both non-
commutative and quantum corrections. Although here we have restricted ourselves only to
the leading order correction due to the NC parameter (θ), one can try to include all order θ
corrections. This is technically more involved and we shall not address this issue in this paper.
Now we move on to compute the coefficient˜β1 in the above expression. By making an
infinitesimal scale transformation to the metric coefficients in (2), the coefficient˜β1 can be
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