Thermodynamics of an Evaporating Schwarzschild Black Hole in Noncommutative Space

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by  Kourosh Nozari

Thermodynamics of an Evaporating Schwarzschild
Black Hole in Noncommutative Space
Kourosh Nozari and Behnaz Fazlpour
7 Department of Physics, Faculty of Basic Science,
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University of Mazandaran,
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2 P. O. Box 47416-1467, Babolsar, IRAN

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e-mail: [email protected]
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J

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Abstract
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1
5 We investigate the effects of space noncommutativity and the generalized un-
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certainty principle on the thermodynamics of a radiating Schwarzschild black hole.
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0 We show that evaporation process is in such a way that black hole reaches to a
/
h
maximum temperature before its final stage of evolution and then cools down to a
t
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p nonsingular remnant with zero temperature and entropy. We compare our results
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with more reliable results of string theory. This comparison Shows that GUP and
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: space noncommutativity are similar concepts at least from the view point of black
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i hole thermodynamics.
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r PACS: 02.40.Gh, 04.70.-s, 04.70.Dy
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KeyWords: NoncommutativeGeometry,GeneralizedUncertaintyPrinciple,Black
Hole Thermodynamics
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1 Introduction
After thirty years of intensive research in the field of radiating black holes[1], various
aspects of the problem still remain under debate. For example the last stage of black
hole evaporation is not obvious in some respects. The string/black hole correspondence
principle [2] suggests that in this extreme regime stringy effects cannot be neglected. In
spite of the promising results that string theory has had in quantizing gravity, the actual
calculations of the Hawking radiation are currently obtained by means of quantum field
theory in curved space [3]. In fact the black hole evaporation occurs in a semiclassical
regime, namely when thedensity ofgravitonsis lower thanthat ofthematter field quanta.
Nevertheless, the divergent behavior of the black hole temperature in the final stage of
the evaporation remains rather obscure.
In addition to string theory itself, which provides an elegant framework for incorporation
of quantum gravity effects in black hole physics by direct state counting, several alter-
native approaches to incorporate quantum gravity effects in the calculation of black hole
thermodynamics have been proposed. These approaches can be classified as follows:
Generalized Uncertainty Principle(GUP)
•
Existence of a nonzero minimal length scale (which leads to finite resolution of
spacetime structure) can be addressed in GUP(see[4] and references therein). From
a heuristic argument, one can use GUP to find modification of Bekenstein-Hawking
formalism of black hole thermodynamics[5,6,7,8]. The main consequences of this
approach are summarized as follows:
Black hole evaporation ends up with a phase consisting a remnant with zero entropy
and there exists a finite temperature that black hole can reach in its final stage of
evaporation. This picture differs drastically with Bekenstein-Hawking prescription
which accepts the total evaporation of Black holes.
Modified Dispersion Relations(MDRs)
•
MDRs induced modification of black hole thermodynamics have their origin on loop
quantum gravity considerations(MDRs are signature of Lorentz invariance violation
at high energy sector of the field theory). Attempts to modify Bekenstein-Hawking
formalismbasedonMDRsshowmoreorlessthesamebehaviors asGUPframework,
butnowwefindsevereconstraintsonthefunctionalformofMDRswhenwecompare
our results with string theory more reliable results[9,10].
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Noncommutative Geometry
•
Noncommutativity eliminates point-like structures in favor of smeared objects in
flat spacetime. Based on this idea, several attempts have been performed to find
modificationofBekenstein-Hawking formalismofblackholethermodynamics within
noncommutative geometry[11,12]. The consequences of these attempts are as fol-
lows:
The end-point of black hole evaporation is a zero temperature extremal remnant
with no curvature singularity.
In this paper we are going to proceed one more step in the line of third alternative i.e.
Noncommutative Geometry. Our strategy differs with existing literatures in two main
respects: we don’t consider smeared picture of objects in noncommutative spacetime(as
has been considered in [11,12]), instead we deal with coordinate noncommutativity which
results modification of Schwarzschild radius. Also we consider possible generalization of
uncertainty principle within a string theory point of view. We calculate entropy-area rela-
tion and compare our results with more reliable results of string theory(calculated based
on direct state-counting). This comparison shows that GUP and space noncommutativity
are essentially similar concepts.
In which follows we suppose c = h¯ = G = 1.
2 Black Hole Thermodynamics in GUP Framework
The canonical commutation relations in a commutative spacetime manifold are given as
follows
[x ,x ] = 0, [x ,p ] = iδ , [p ,p ] = 0. (1)
i j i j ij i j
From a string theory point of view, existence of a minimal length scale can be addressed
in the following generalized uncertainty principle
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δxδp 1+β(δp)2 +γ , (2)
≥ 2
(cid:18) (cid:19)
where β is string theory parameter related to minimal length. Since we are dealing with
absolutely minimum position uncertainty we set γ = β p 2 and therefore the correspond-
h i
ing canonical commutation relation becomes
[x,p] = i(1+βp2). (3)
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The canonical commutation relations in commutative spacetime with GUP become
[x ,x ] = 0, [x ,p ] = iδ (1+βp2), [p ,p ] = 0. (4)
i j i j ij i j
Now consider the geometry of Schwarzschild spacetime with the following metric
dr2
ds2 = f(r)dt2 r2 dθ2 +sin2θdφ2 , (5)
− f(r) −
(cid:16) (cid:17)
where f(r) = 1 2M. There is a horizon at r = 2M with the following area
− r s
2π π
A = r2 dφ sinθdθ = 4πr2 = 16πM2. (6)
s s
Z0 Z0
Bekenstein-Hawking formalism of black hole thermodynamics gives the following relations
for temperature and entropy of black hole
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T = (7)
H
8πM
and
S = 4πM2 (8)
respectively. Within GUP framework, these equations should be modified to incorporate
quantum gravity effects. We use Bekenstein’s argument type considerations to find GUP
induced modification of black hole thermodynamics. For simplicity, consider the following
GUP
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δxδp 1+β(δp)2 . (9)
≥ 2
(cid:18) (cid:19)
A simple calculation gives,
δx β
δp 1 1 . (10)
≃ β ±s − (δx)2
h i
Here to achieve correct limiting result we should consider the minus sign in round bracket.
In original Bekenstein approach, from a heuristic argument based on Heisenberg uncer-
tainty relation, one deduces the following equation for Hawking temperature of black
hole,
δp
T . (11)
H
≈ 2π
Therefore, in the framework of generalized uncertainty principle, modified black hole
temperature is as follows
δx β
TGUP 1 1 . (12)
H ≈ 2πβ −s − (δx)2
h i
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Within black hole near horizon geometry, since δx r where r = 2M, one can write
s s
∼
this equation in such a way that can be comparable with equation (7):
M β
TGUP 1 1 . (13)
H ≈ πβ −s − 4M2
h i
which leads to the following relation
1 β β2
TGUP 1+ + , (14)
H ≈ 8πM 16M2 128M4
h i
up to second order in β. Obviously, when quantum gravitational effects are negligible,
that is when β 0, this relation gives (7) as a manifestation of correspondence principle.
→
Now consider a quantum particle that starts out in the vicinity of an event horizon and
then ultimately absorbed by black hole. For a black hole absorbing such a particle with
energy E and size R, the minimal increase in the horizon area can be expressed as
(∆A) 4(ln2)ER, (15)
min
≥
then one can write
(∆A) 8(ln2)δpδx, (16)
min
≥
where E cδp (with c = 1) and R 2δx. Using equation (10) for δp, we find
∼ ∼
2(ln2)A 4πβ
(∆A) 1 1 (17)
min ≃ βπ −s − A
h i
where we have defined A = 4π(δx)2. Now we should determine δx. Since our goal is to
compute microcanonical entropy of a large black hole, near-horizon geometry considera-
tions suggests the use of inverse surface gravity or simply the Schwarzschild radius for δx.
Therefore, δx r and defining 4πr2 = A and (∆S) = b = constant, then it is easy to
∼ s s min
show that,
dS (∆S) bβπ
min . (18)
dA ≃ (∆A)min ≃ 2(ln2)A 1 1 4πβ
− − A
h q i
Note that b can be considered as one bit of information since entropy is an extensive
quantity. Considering calibration factor of Bekenstein as ln2, the minimum increase
of entropy(i.e. b), should be ln2. Now we should perform integration. There are two
possible choices for lower limit of integration, A = 0 and A = A . Existence of a minimal
p
observable length leads to existence of a minimum event horizon area, A = 4π(δx )2.
p min
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So it is physically reasonable to set A as lower limit of integration. Based on these
p
arguments, we can write
A βπ
S dA. (19)
≃ ZAp 2A 1 1 4πβ
− − A
h q i
An integration gives
∞
A πβ A 4 n
S ln + c + , (20)
≃ 4 − 4 4 n A C
nX=1 (cid:16) (cid:17)
where is a constant. This is an interesting result which shows the logarithmic leading
C
order correction plus a power series expansion in terms of inverse of area. Up to third
order in 1, we find
A
A πβ A πβ 2 4 πβ 3 4 2 πβ 4 4 3
S ln + + 3 + , (21)
≃ 4 − 4 4 4 A 4 A − 4 A C
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where
= Ap + πβ ln Ap πβ 2 4 πβ 3 4 2 +3 πβ 4 4 3. (22)
C − 4 4 4 − 4 A − 4 A 4 A
p p p
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
It is obvious that when A = A , S 0 and therefore black hole remnant should have
p
→
zero entropy. A result which is physically acceptable since small classical fluctuations are
not allowed at remnant scales because of existence of minimal observable length.
3 Black Hole Thermodynamics in Noncommutative
Geometry
A noncommutative space can be realized by the coordinate operators satisfying
[xˆ ,xˆ ] = iθ , i,j = 1,2,3, (23)
i j ij
where xˆ’s are the coordinate operators and θ is the noncommutativity parameter with
ij
dimension of (length)2. Canonical commutation relations in noncommutative spaces read
[xˆ ,xˆ ] = iθ , [xˆ ,pˆ ] = iδ , [pˆ,pˆ ] = 0, (24)
i j ij i j ij i j
Now, we note that there is a new coordinate system
1
x = xˆ + θ pˆ , p = pˆ (25)
i i ij j i i
2
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with these new variables, x ’s satisfy the usual(commutative) commutation relations
i
[x ,x ] = 0, [x ,p ] = iδ , [p ,p ] = 0. (26)
i j i j ij i j
Note that noncommutativity is an intrinsic characteristic of underlying manifold.
For a noncommutative Schwarzschild black hole, we have[13,14]
2M
f(r) = 1 , (27)
− √rˆrˆ
(cid:16) (cid:17)
where rˆsatisfies (25). The horizon of the noncommutative Schwarzschild metric as usual
satisfies the condition gˆ = 0 which leads to
00
2M
1 = 0. (28)
− √rˆrˆ
If in this relation we change the variables xˆ to x , and then using(25), the horizon of the
i i
noncommutative Schwarzschild black hole satisfies the following condition
2M
1 = 0. (29)
− (x θijpj)(x θikpk)
i − 2 i − 2
q
This leads us to the following relation
2M x θ p θ θ p p
1 1+ i ij j ij ik j k + (θ3)+... = 0, (30)
− r 2r2 − 8r2 ! O
where θ = 1ǫ θ . Using the identity ǫ ǫ = δ δ δ δ , one can rewrite (30) as
ij 2 ijk k ijr iks jk rs − js rk
follows
2M M 1
1 L~.~θ p2θ2 (p~.~θ)2 + (θ3)+... = 0, (31)
− r − 2r3 − 8 − O
(cid:20) (cid:16) (cid:17)(cid:21)
where L = ǫ x p , p2 = ~p.p~ and θ2 = ~θ.~θ . If we set θ = θ and assuming that
k ijk i j 3
remaining components of θ all vanish (which can be done by a rotation or a re-definition
of the coordinates), then L~.~θ = L θ and p~.~θ = p θ. In this situation equation (31) can be
z z
written as
M 1
r3 2Mr2 L θ p2 p2 θ2 + (θ3)+... = 0. (32)
− − 2 z − 8 − z O
(cid:20) (cid:16) (cid:17) (cid:21)
Since p2 = p2 +p2+p2, one can write (p2 p2)θ2 = (p2 +p2)θ2 and therefore (32) can be
x y z − z x y
written as follows
ML θ M
r3 2Mr2 z + p2 +p2 θ2 + (θ3)+... = 0. (33)
− − 2 16 x y O
(cid:16) (cid:17)
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Since Schwarzschild black hole is non-rotating, we set L~ = 0 and therefore L = 0( this
z
means that space noncommutativity has no effect on the Schwarzschild geometry up to
first order of space noncommutativity parameter). So we find
M
r3 2Mr2 + p2 +p2 θ2 + (θ3)+... = 0. (34)
− 16 x y O
(cid:16) (cid:17)
With the following definitions
M
a 2M = r , η p2 +p2 θ2, (35)
≡ − − s ≡ 16 x y
(cid:16) (cid:17)
and considering only terms up to second order of θ, the radius of event horizon for non-
commutative Schwarzschild black hole becomes
a 2a3 27η +√108a3η +729η2 1/3 a2 2a3 27η+√108a3η +729η2 −1/3
rˆ − + − − + − − .
s
≡ 3 54 ! 9 54 !
(36)
Two other roots of (34) are not real. In the case of commutative space, η = 0, and
therefore we recover usual Schwarzschild radius, r = 2M. Since a η, we can expand
s
≫
equation (36) to find the following relation for Schwarzschild radius in noncommutative
space
η 27η2
rˆ = a . (37)
s − − a2 − 2 a5
Since a = r we have considered only the real parts of our equations. One can write
s
−
η = Mα, where
1
α = p2 +p2 θ2, (38)
16 x y
(cid:16) (cid:17)
and therefor η = rs α. In this manner, we can write equation (37) as follows
2
α 27α2
rˆ = r + . (39)
s s − 2r 8 r3
s s
After calculation of Schwarzschild radius of black hole in noncommutative space, we have
all prerequisites to calculate thermodynamics of black hole in noncommutative spacetime.
First we consider black hole temperature. The Hawking temperature of Schwarzschild
black hole in noncommutative space can be given by the following relation
M
Tˆ = , (40)
H 2πrˆ rˆ
s s
where substitution of rˆ leads to the following generalized statement
s
Tˆ = M r α + 27α2 −2 (41)
H 2π s − 2r 8 r3
(cid:16) s s (cid:17)
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which leads to the following relation
1 α 3α2
Tˆ 1+ . (42)
H ≈ 8πM 4M2 − 8M4
h i
Figure 1 shows the plot of black hole temperature versus its horizon radius in three
candidate models. As this figure shows, within GUP and Noncommutative geometry,
black hole before its terminal stage of evaporation reaches to a maximum temperature
and then cools down to a zero temperature remnant.
Now we calculate entropy of black hole in a noncommutative spacetime. In the standard
Bekenstein argument, the relation between energy and position uncertainty of a given
particle is given by(see [10] and references therein)
1
E . (43)
≥ δx
Within a noncommutative framework, we suppose δx = rˆ . Therefore, we find the follow-
s
ing generalization
1
E , (44)
≥ rˆ
s
which substitution of rˆ from (39) leads to
s
1
E . (45)
≥
r α + 27α2
s − 2rs 8 rs3!
Since r = 2M, this relation implicitly shows the modification of standard dispersion
s
relations which has strong support on loop quantum gravity[9]. In this manner, the
increase of event horizon area is given by
1
∆Aˆ 4(ln2) . (46)
≥
1 α + 27α2
− 2rs2 8 rs4!
which leads to the following relation
dS ∆S min) ln2
( . (47)
dAˆ ≈ ∆Aˆ min) ≃ 4(ln2) 1
(
1−2αrs2+287αrs42!
Therefore we can write
dS 1 α 27α2
1 + . (48)
dAˆ ≃ 4" − 2r2 8 r4#
s s
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Now we should calculate dAˆ. Since
Aˆ = 4πrˆ rˆ , (49)
s s
we find
dAˆ = 1+γ 4πα 2 +γ 4πα 3 +γ 4πα 4 dA, (50)
1 A 2 A 3 A
" #
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
where γ ’s are some constant γ = 7, γ = 27, γ = 3(3)6 and A = 4πr2. We can
i 1 − 2 4 3 − 2 s
integrate (48) to find (cid:16) (cid:17)
A πα A πα 4 πα 4 2
S ln +κ ( )2 +κ ( )3 +
≃ 4 − 2 4 1 2 A 2 2 A
(cid:16) (cid:17) (cid:16) (cid:17)
πα 4 3 πα 4 4 πα 4 5
κ ( )4 +κ ( )5 +κ ( )6 , (51)
3 2 A 4 2 A 5 2 A
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
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where κ ’s are some constant κ = 29, κ = 41, κ = 1305, κ = 63 3 , κ =
i 1 2 2 − 3 4 4 − × 2 5
310 . Generally, this relation(cid:16)can be written as (cid:16) (cid:17)
40
(cid:17)
∞
A πα A 4 n
S ln + c + (52)
≃ 4 − 2 4 n A C
nX=1 (cid:16) (cid:17)
Where
∞
Ap + πα ln Ap c 4 n. (53)
C ≃ − 4 2 4 − n A
nX=1 (cid:16) p(cid:17)
This is an interesting result which shows the modified entropy of black hole within non-
commutative geometry. In the case of commutative spaces α = 0 and this equation yields
the standard Bekenstein entropy,
A
S . (54)
≃ 4
Equation (52) is very similar to (20). As a result we see that GUP and Noncommutative
geometry give the same area dependence to the modified entropy of black hole. This
feature may inherently reflect the fact that GUP and spacetime noncommutativity are
not different in essence. Figure 2 shows the entropy-area relation for an evaporating
black hole in bekenstein-Hawking and the noncommutative geometry view points. Within
noncommutative geometry approach black hole in its final stage of evaporation reaches to
a zero entropy remnant.
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