Thermodynamics of black holes and the symmetric generalized uncertainty principle Abhijit Duttaa∗, Sunandan Gangopadhyayb,c†
arXiv:1408.3928v1 [gr-qc] 18 Aug 2014
a
Department of Physics, Adamas Institute of Technology, Barasat, Kolkata 700126, India b
Department of Physics, West Bengal State University, Barasat, Kolkata 700126, India c
Visiting Associate in Inter University Centre for Astronomy & Astrophysics, Pune, India
Abstract In this paper, we have investigated the thermodynamics of Schwarzschild black holes using the symmetric generalized uncertainty principle which contains correction terms involving momentum and position uncertainty. We obtain the masstemperature relation and the heat capacity of the black hole using which we compute the critical and remnant masses. The entropy is found to satisfy the area law upto leading order corrections from the symmetric generalized uncertainty principle.
The understanding of the thermodynamic properties of black holes has been one of the most remarkable achievements in theoretical physics. Recently, the idea of a minimal length equal to the Planck length in various theories of quantum gravity [1],[2] have led to a serious study of black hole thermodynamics [3]-[5] and its quantum corrected entropy [6]-[8]. In this paper we will study the thermodynamics properties of Schwarzschild black holes using the symmetric generalized uncertainty principle (SGUP) [9] β 2 l2 γ2 h ¯ 1 + 2 (δx)2 + 2p (δp)2 δxδp ≥ 2 L h ¯ (
)
(1)
where lp is the Planck length (∼ 10−35 m), γ and β are dimensionless constants and L is another uncertainty constant. The L → ∞ corresponds to the GUP case. We obtain the mass-temperature relationship from which we compute the heat capacity of the black hole. From this, we calculate the critical and remnant masses in terms of the Planck mass and the constants γ and β appearing in the SGUP (1). We then compute the entropy keeping leading order corrections from the SGUP. The well known area law is recovered with corrections from the SGUP. ∗ †
[email protected] [email protected],
[email protected]
1
To start with we consider a Schwarzschild black hole of mass M . Near the event horizon of the black hole, the momentum uncertainty and the temperature for a massless elementary particle are related as [6] T =
(δp)c kB
(2)
where c is the speed of light and kB is the Boltzmann constant. At thermodynamic equilibrium, the temperature of the black hole will be equal to that of the particle. Also, near the horizon of the Schwarzschild black hole, the position uncertainty of a particle can be expressed in terms of the Schwarchild radius [6],[10] δx = rs ; rs =
2GM c2
(3)
where is a calibration factor, rs is the radius of Schwarzschild black hole and G is the Newton’s universal gravitational constant. To relate the temperature with the mass of the black hole, the GUP (1) has to be saturated β 2 l2 γ2 h ¯ 1 + 2 (δx)2 + 2p (δp)2 . δxδp = 2 L h ¯ (
)
(4)
Using eqs.(2, 3), the above relation can be put in the following form
Mp2 c2 1 1 γ2 + M= 4 kB T kB T L2
2M Mp
!2
h ¯ Mp c
!2
β2 + k T B (Mp c2 )2
(5)
2
h = Mp c2 and Mp = c Glp (Mp being the Planck mass) has been used. where the relations c¯ lp To fix , we set the constants γ and β to zero which gives
M=
Mp2 c2 . 4kB T
(6)
Comparing this with the semi-classical Hawking temperature T = the value of = 2π. Hence eq.(5) can be written as
Mp2 c2 1 1 γ2 M = + 8π kB T kB T L2
8πM Mp
!2
h ¯ Mp c
!2
Mp2 c2 8πM kB
[3], [4], yields
β2 + k T . B (Mp c2 )2
(7)
Introducing the following relations M0 =
8πM kB Mp c L ; L0 = ; T0 = L= 2 Mp Mp c h ¯ lp
(8)
eq.(7) can be expressed as M0 =
1 γ 2 M 02 + + β 2T 0. T 0 L02 T 0
(9)
Now by definition, the heat capacity of the black hole is given by C = c2
dM dT 2
(10)
which by using eq.(9) gives
kB 2β 2 T 0 − M 0 C= . 2 8π T 0 − 2γ M0 L02
(11)
To get the remnant mass (where the radiation process stops), we set C = 0 and this leads to v
Mrem
u 1 β u Mp t . = 2 2 2 4β 4π 1 − 2 γ 2¯h4
(12)
L Mp c
The condition that remnant mass is real and does not diverge leads to the following inequality 4β 2 γ 2 h ¯2 < 1. L2 Mp2 c4
(13)
Now from eq.(9), we can express the temperature in terms of the mass as r
0
0
M −
M 02 1 −
T =
4β 2 γ 2 L02
− 4β 2 (14)
2β 2
where the negative sign before the square root has been taken to reproduce eq.(6) in the γ, β → 0 limit. The above relation readily leads to the existence of a critical mass below which the temperature becomes a complex quantity v
u β 1 u Mcr = Mp t . 2 2 2 4β 4π 1 − 2 γ 2¯h4
(15)
L Mp c
Eqs.(12) and (15) imply that the remnant and critical masses are equal and for both the masses, the condition in eq.(13) applies. Also, in the limit γ → 0, both the results reduce to those found in [11]. We now move to calculate the entropy which from the first law of black hole thermodynamics reads S=
Z
c2
dM kB Z dM 0 = T 8π T0
(16)
where we have used eq.(8) to write the second equality. Substituting eq.(14) in eq.(16) and carrying out the integration expansion keeping terms up to leading order in γ 2 and β 2 yields S kB
8πM 4πM 2 β2 = − ln 2 Mp 8π Mp SBH SBH β2 = − ln kB 16π kB
!
γ2 − 24πL02
8πM Mp
!3
3 √ γ 2 SBH 2 β2 − ln(16π) − 8 π 02 16π 3L kB
3
(17)
where
SBH kB
=
4πM 2 Mp2
is the semi-classical Bekenstein-Hawking entropy for the Schwarzschild 2
2
= 4lp2 SkBH black hole. In terms of the area of the horizon A = 4πrs2 = 16π G cM , eq.(17) 4 B can be recast in the following form S A A β2 = 2− ln kB 4lp 16π 4lp2
!
√ 8 πγ 2 lp2 β2 − ln(16π) − 16π 3L2
A 4lp2
!3 2
(18)
which is the famous area law with corrections from the SGUP. We conclude by summarizing our findings. In this paper, we study the effect of the SGUP in the thermodynamics of Schwarzschild black holes. We obtain the masstemperature relation and the heat capacity of the black hole using which we compute the critical and remnant masses which are found to be equal and are consistent with our earlier findings [11], [12]. From the expression for the critical mass, we also obtain an inequality involving the constants γ and β. Finally, we compute the entropy and recover the area theorem with the SGUP corrections. We observe that the SGUP leads to a 3 correction term of the form A 2 .
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