Energy distribution and thermodynamics of the quantum-corrected Schwarzschild black hole

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by  Mahamat Saleh

Energy distribution and thermodynamics of the quantum-corrected Schwarzschild
black hole
Mahamat Saleh,1, Bouetou Bouetou Thomas,2,3,4, and Timoleon Crepin Kofane5,3,4,
∗ † ‡
1Department of Physics, Higher Teachers’ Training College,
University of Maroua, P.O. Box 55, Maroua, Cameroon.
2Ecole Nationale Supe´rieure Polytechnique, University of Yaounde I, P.O. Box. 8390, Cameroon
3The African Center of Excellence in Information and Communication Technologies (CETIC)
4The Abdus Salam International Centre for Theoretical Physics,
P.O. Box 586, Strada Costiera, II-34014, Trieste, Italy
5Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box. 812, Cameroon
(Dated: January 25, 2017)
Inthiswork,energydistributionandthermodynamicsareinvestigatedintheSchwarzschildblack
7
hole spacetime when considering corrections due to quantum vacuum fluctuations. The Einstein
1
and Møller prescriptions were used to derive the expressions of the energy in the background.
0
The temperature and heat capacity were also derived. The results show that due to the quantum
2
fluctuationsinthebackgroundoftheSchwarzschildblackhole,alltheenergiesincreaseandEinstein
n energy differsfrom Møller’s one. Moreover, when increasing thequantumcorrection factor (a),the
a
differencebetweenEinsteinandMøllerenergies,theUnruh-Verlindetemperatureaswellastheheat
J
capacity of theblack hole increase while the Hawkingtemperature remains unchanged.
3
2 PACSnumbers: 04. 70. Dy,04. 20. Jb,97.60.Lf

]
c
The localization of energy and momentum in general black hole mechanics are, in fact, simply the ordinary
q
- relativity is one of the oldest problems in gravitation laws of thermodynamics applied to a system containing
r which still lacks a definite answer. The key point for a black hole[26]. All over the last decade, there have
g
[ the issue is whether the energy of spacetime may be lo- been several outstanding approaches toward a statisti-
calized or not. It was Sarracino and Cooperstock and cal mechanical computation of the Bekenstein-Hawking
1
other researchers that pointed the possibility that the (BH) entropy[27–30]. Black holes as thermodynamical
v
energy is localizable[1]. Through the years, many efforts systems are widely found in the literature[27–40].
9
weremadetodeveloptrulyinvariantgeneralprescription We know that the vacuum undergoes quantum fluctu-
2
9 for gravitationalfield energylocalization[2–8]. Using the ations. This phenomenon is the appearance of energetic
6 various prescriptions, many researchers derived energy particles out of nothing, as allowed by the uncertainty
0 and momentum expressions in various spacetimes[1, 9– principle. Vacuum fluctuations have observable conse-
1. 24]. The obtained results show that for general Kerr- quenceslikeCasimirforcebetweentwoplatesinvacuum.
0 Schild class of spacetimes, the expressions of the energy Due to quantum fluctuations, the evolution of slices in
7 coincideforalloftheprescriptionsexceptfortheMøller’s the black hole geometry will lead to the creation of par-
1 prescription. This divergence guides such investigation ticle pairs which facilitates black holes’ radiation[41].
:
v leading to the choice of an appropriate result for such Recently,Wontae andYongwan[42]investigatedphase
i spacetime[25]. Theoretical studies on black holes have transition of the quantum-corrected Schwarzschildblack
X
attracted substantial attention since the advent of gen- hole and concluded that there appear a type of Grass-
r eralrelativity. Theinvestigationofenergydistributionin Perry-Yaffe phase transition due to the quantum vac-
a
blackholesspacetimesis oneofthe hottopicsinmodern uum fluctuations and this held even for the very small
physics. size black hole. More recently, we investigate quasi-
normal modes of a quantum-corrected Schwarzschild
Black holes are thermal systems, radiating as black black hole and show that the scalar field damps more
bodies with characteristic temperatures and entropies. slowly and oscillates more slowly due to the quantum
During the past 40 years,researchin the theory of black fluctuations[43, 44].
holes in general relativity has brought to light strong In this paper, the energy distribution and thermody-
hints of a very deep and fundamental relationship be- namics of a quantum-correctedSchwarzschild black hole
tween gravitation, thermodynamics, and quantum the- are investigated in order to highlight the energetic and
ory. The cornerstone of this relationship is black hole thermodynamical behaviors of the black hole when the
thermodynamics, where it appears that certain laws of vacuum fluctuations are taken into account.
According to the work of Kazakov and Solodukhin on
quantum deformation of the Schwarzschildsolution [45],
the backgroundmetric ofthe Schwarzschildblackhole is
∗Electronicaddress: [email protected] defined by
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] ds2 =f(r)dt2 f(r)−1dr2 r2(dθ2+sin2θdϕ2), (1)
− −
2
where where
2M 1 r hµl = hlµ = gνn [ g(gµnglm glngµm)] . (11)
f(r)= + U(ρ)dρ. (2) ν − ν √ g − − ,m
− r r Z −
Since Einstein’s energy-momentum complex is re-
For an empty space, U(ρ) = 1. Thus we obtain the
stricted to quasi-cartesian coordinates, we can express
Schwarzschildmetric
the above metric( 5) in quasi-cartesian coordinates de-
ds2 = 1 2M dt2 1 2M −1dr2 r2(dθ2+sin2θdϕ2), fined as:
(cid:0) − r (cid:1) −(cid:0) − r (cid:1) − (3) T = t+r− f(r)−1dr
where M is the black hole mass. The event horizon of  x = rsinθcoRsϕ (12)
the black hole is localized at r =2M.  y = rsinθsinϕ
EH
z = rcosθ
Taking into account the quantum fluctuation of the
vacuum, the quantity U(ρ) transforms to [42] 
The metric can then be expressed as:
e−ρ ds2 = dT2+dx2+dy2+dz2
U(ρ)= , (4) − 2
qe−2ρ− π4GR +(1−f(r))(cid:20)dT − xdx+yrdy+zdz(cid:21) . (13)
where G = G ln(µ/µ ), G is the Newton constant
R N 0 N Theenergyofthephysicalsystemis,onceagain,given
and µ is a scale parameter.
by the formula
The background metric of the quantum-corrected
Schwarzschildblack hole can then be read as
E = θ0dx1dx2dx3. (14)
E 0
Z Z Z
ds2 =f(r)dt2 f(r)−1dr2 r2(dθ2+sin2θdϕ2), (5) θ0 is evaluated using equations ( 10), ( 11) and ( 13)
− − 0
and substituted in ( 14) to get the energy
with f(r) = 2M + √r2 a2 and a2 = 4G /π. The
− r r− R
event horizon(cid:0)of such black hol(cid:1)e is located at the radius
1 r
rEH =√4M2+a2. It is clear that the area of the event EE =M r2 a2+ . (15)
− 2 − 2
horizon increases with the quantum-correction parame- p
ter, a. The behavior of these energies is represented on Figs.
The energy-momentum complex from Møller’s pre- 1 and 2.
scription can be derived as[11, 14]:
1 1.003
τµ = εµλ, (6) EE
ν 8π ν,λ EM
1.0025
where the superpotentials are
1.002
∂g ∂g
εµνλ =√−g( ∂xνkσ − ∂xνσk)gµkgλσ. (7) Energy1.0015
The energy of the physical system in a four- 1.001
dimensional background is given by
1.0005
E = τ0dx1dx2dx3. (8)
M 0
Z Z Z 1
2 3 4 5 6 7 8 9 10
r
Substituting equation( 5) in ( 7), the Møller’s energy
distribution for the quantum-corrected Schwarzschild
FIG. 1: Behavior of Einstein and Møller energies versus the
black hole can be written as: radial position r for M =1.
a2
E =M + . (9)
M
2√r2 a2
These figures show that the energies decrease when
−
increasing r and for fixed r, Einstein energy is greater
The energy distribution from Einsten’s energy-
than Møller energy. At the event horizon, r = r =
momentum complex is[2, 14]: EH
√4M2+a2, the two energies coincide when a = 0 and,
1 for a = 0, the gap between the two energies increases
θνµ = 16πhµν,ll, (10) when i6ncreasing a.
3
corrected Schwarzschild black hole is given by
1.25
E
E
E
M r = 4M2+a2. (17)
H
1.2
p
Using this expression,the Hawking temperature reads
1.15
Energy ~
1.1 TH =T|r=rH = 8πM, (18)
which corresponds to the Hawking temperature of the
1.05 Schwarzschildblackholefreefromanykindofcorrection.
The entropy of the black hole is given by
1
0 0.2 0.4 0.6 0.8 1 A
a S = =πr2 =4πM2+πa2. (19)
4 H
FIG. 2: Behavior of Einstein and Møller energies versus the
quantumcorrection parameter a for M =1 and r=rEH. Substituting Eq. (19) into Eq. (18) yields
1
T = ~ S a2 −2 . (20)
H
The temperature of the black hole is given by the 4π (cid:18)π − (cid:19)
Unruh-Verlinde matching[46–49]
The heat capacity of the black hole is
~ a2
T = (2M + ) (16)
4πr2 √r2 a2 ∂S
− C =TH = 2S+2πa2. (21)
(cid:18)∂T (cid:19) −
Its behavior is represented on Fig. 3. H
Wecanclearlyseethatthequantumvacuumfluctuations
0.05 contribute to the heat capacity of the black hole by a
a=0.1
0.045 a=0.9 positive quantity Ca =2πa2.
In summary, we have studied energy distribution and
0.04
thermodynamics for the Schwarzschild black hole when
0.035 consideringquantumcorrectionsduetothequantumvac-
0.03 uumfluctuations. Concerningenergydistribution,wede-
T rived the energy from Einstein and Møller prescriptions.
0.025
Their behaviors show that when a = 0, the two ener-
0.02
gies coincide (E = E = M). When introducing the
E M
0.015 quantum-correction to the black hole, the Møller energy
0.01 becomesgreaterthantheEinsteinenergy,andthediffer-
ence increases when increasing the quantum-correction
0.005
2 2.5 3 3.5 4 4.5 5 parametera(seeFigs. 1and2). Theeventhorizonradius
r
of the black hole as well as its Unruh-Verlinde temper-
ature increase with the quantum-correction parameter.
FIG. 3: Variation of the temperature versus the radial posi-
tion r for differentvalues of thequantum-correctionparame- TheHawkingtemperaturestaysunchangedalthoughthe
ter a. horizon changes. The heat capacity increases with the
quantum-correction parameter. Thus we can conclude
that quantum vacuum fluctuations generate positive en-
Through this figure, we can see that the temperature ergyincreasingtheenergyoftheblackholeandhelpsta-
decreases when increasing r. Moreover, when increasing bilizing thermodynamically the black hole. At the event
thequantum-correctionparametera,thetemperaturein- horizon,quantumvacuumfluctuationsincreasetheener-
creases. gies (see Fig. 2) without changing the Hawking temper-
The Hawking temperature is the temperature at the ature which remains proportional to the inverse of the
event horizon. The horizon radius r of the quantum- black hole mass.
H
[1] Cooperstock F I and Sarracino R S 1978 J. Phys. A: [2] Einstein A 1915 Preuss. Akad. Wiss. Berlin 47 778.
Math. Gen. 11 877.
4
[3] Papapetrou A 1948 Proceedings of the Royal Irish [28] Carlip S 1997 Phys. Rev. D 55 878.
Academy A 52 11. [29] Strominger and Vafa 1996 Phys. Lett. B 379 99.
[4] Tolman R C 1930 Phys. Rev. 35 875 [30] AshtekarA,BaezJ,CorichiAandKrasnovK1998Phys.
[5] Landau L D and Lifschitz E M 1951 The Classical The- Rev. Lett. 80 904.
ory of Fields (Addison-Wesley Press, Reading, Mas- [31] Hawking S W 1975 Commun. Math. Phys. 43 199.
sachusetts., New York) p.317. [32] Hawking S W 1974 Nature 30 248.
[6] BergmannPGandThomsonR1953Phys. Rev.89400. [33] Bekenstein J D 1973 Phys. Rev. D 7 2333.
[7] WeinbergS1972 Gravitation and Cosmology: Principles [34] Pete M, arXiv:physics.pop-ph/0906.4849v1.
and Applications of General Theory of Relativity (John [35] FarmanyaA,AbbasiSandNaghipourA2008ActaPhys.
Wiley and Sons, Inc., NewYork) p.165. Pol. A 114 651.
[8] Goldberg J N 1958 Phys. Rev. 111 315. [36] ZhaoR,ZhaoH-XandHuS-Q2007 Mod. Phys. Lett. A
[9] Xulu S S 2007 Int. J. Theor. Phys. 46 2915. 22 1737.
[10] RadinschiIandGrammenosT2008Int.J. Theor. Phys. [37] Gibbons G W and Hawking S W 1977 Phys. Rev. D 15
47 1363. 2738.
[11] Vagenas E C 2006 Mod. Phys. Lett. A 21 1947. [38] ParikhMKandWilczekF1998Phys.Rev.D58064011.
[12] Vagenas E C 2004 Mod. Phys. Lett. A 19 213. [39] Bardeen J M, Carter B and Hawking S W 1973 Comm.
[13] Xulu S S,arXiv:gr-qc/0304081v1. Math. Phys. 31 161.
[14] Vagenas E C 2003 Int. J. Mod. Phys. A 18 5949. [40] Mahamat S, Bouetou B T and Kofane T C 2012 Gen.
[15] VirbhadraK S 1990 Phys. Rev. D 41 1086. Relativ. Gravit. 44 2181.
[16] VirbhadraK S 1990 Phys. Rev. D 42 2919. [41] Mathur S D 2009 Class. Quantum Grav. 26 224001.
[17] VirbhadraK S 1990 Phys. Rev. D 42 1066. [42] Wontae K and Yongwan K 2012 Phys. Lett. B 718 687.
[18] AguirregabiriaJM,ChamorroAandVirbhadraKS1996 [43] Mahamat S, Bouetou B T and Kofane T C 2014 Astro-
Gen. Rel. Grav. 28 1393. phys. Space Sci. 350 721.
[19] Xulu S S 1998 Int. J. Mod. Phys. D 7 773. [44] Mahamat S.,BouetouB.T. andKofaneTC2016Astro-
[20] RadinschiI 1999 Acta Physica Slovaca 49 789. phys. Space Sci. 361 137.
[21] RadinschiI 2000 FIZIKAB 9 43. [45] KazakovDIandSolodukhinSN1994Nucl.Phys.B429
[22] RadinschiI 2000 Mod. Phys. Lett. 15 803. 153.
[23] Xulu S S 2007 Int. J. Theor. Phys. 46 2915. [46] Verlinde E2011 JHEP 04 029.
[24] Mahamat S, Bouetou B T and Kofane T C 2011 Com- [47] Unruh W G 1976 Phys. Rev. D 14 870.
mun. Theor. Phys. 55 291. [48] Konoplya R A 2010 Eur. Phys. J. C 69 555.
[25] WeiY-H 2008 Chin. Phys. Lett. 25 2782. [49] LiuY-X,WangY-QandWeiS-W2010 Class. Quantum
[26] Wald R M 2001 Living Rev. Rel. 4 6. Grav. 27 185002.
[27] Carlip S 1995 Phys. Rev. D 51 632.