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Black Holes from Stars to Galaxies – Across the Range of Masses
Proceedings IAU Symposium No. 238, 2006
V. Karas & G. Matt, eds.
c
2007 International Astronomical Union
doi:10.1017/S1743921307005546
Uncertainty principle for the entropy of
black holes, de Sitter and Rindler spaces
Tetsuya Hara, Keita Sakai, Shuhei Kunitomo and Daigo Kajiura
Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan
email: hara@cc.kyoto-su.ac.japan
Abstract. By a simple physical consideration and uncertain principle, we derive that tempera-
ture is proportional to the surface gravity and entropy is proportional to the surface area of the
black hole. We apply the same consideration to de Sitter space and estimate the temperature
and entropy of the space, then we deduce that the entropy is proportional to the boundary sur-
face area. By the same consideration, we estimate the temperature and entropy in the uniformly
accelerated system (Rindler space). The cases in higher dimensions are considered.
Keywords. Uncertainty principle – black hole entropy – de Sitter space – Rindler space
1. Introduction
Although it has passed almost 30 years since the discussion of the thermodynamics of
black hole and event horizon began, there are still many problems about the fundamental
concepts. One of them is that the entropy of the black hole is proportional to its surface.
The other is that the temperature of the black hole and event horizon is related to the
acceleration strength. These results could be understood if heuristic assumptions are
adopted. One of the key concept is the uncertainty principle. We apply it to the black
hole, de Sitter space and Rindler space to derive the characteristic features about the
entropy in these spaces.
2. Black hole
As the black hole of mass M has the gravitational radius r
g
=2GM/c
2
, there is the
corresponding momentum ∆p = /(2c
1
r
g
) from the uncertainty principle ∆p∆x /2,
putting ∆x as ∆x c
1
r
g
. If the black hole is formed by photons corresponding to this
momentum, it could be speculated that the temperature of the black hole is proportional
to the surface gravity kT ∆pc 1/(4c
1
GM) M/r
2
g
and it could be explained that
the entropy of the black hole is proportional to the surface of the black hole.
If the black hole is formed by the black body radiation of temperature T , the volume
V , total photon number N and total entropy S are given by
V =
Mc
2
=
15
π
2
(8πG)
4
M
5
c
7
,N= nV =
30ζ(3)
π
4
8πG
c
M
2
240
π
3
G
c
M
2
,
S/k
B
= sV /k
B
=
32π
3
GM
2
/c
, where the following radiation density , number
density n, and entropy density s of the radiation temperature T are used ( =˜aT
4
,
n =
2ζ(3)
π
2
k
B
T
c
3
=0.244
k
B
T
c
3
,s=
4
3
˜aT
3
,
n
s
k
B
=
2
π
2
ζ(3)
π
2
15
×
4
3
=
45ζ(3)
2π
4
0.2776, being
˜a = π
2
k
4
B
/(15
3
c
3
)andζ(3) = 1.202 the radiation constant and zeta function).
Using the surface of the black hole A =4πr
2
g
=16πG
2
M
2
/c
4
, and taking c
1
=
3π
4
,the
following relation is derived, S/k
B
c
3
A/(4G) A/(4
2
p
), where
p
=
G/c
3
is the
Planck length.
379
380 T. Hara et al.
3. De Sitter space
The SNe Ia and WMAP observations confirm that our universe is now accelerating.
For simplicity, we consider the universe with Λ term as de Sitter space with metric
ds
2
= −
1 −
Λ
3
r
2
c
2
dt
2
+
1 −
Λ
3
r
2
−1
dr
2
+ r
2
dθ
2
+sin
2
θdφ
2
.
The characteristic point of this space is that there is the horizon with the radius
of
Λ
=
3/Λ (de Sitter horizon). Applying the uncertainty principle for this length,
the energy ∆E for the photon or particle is estimated as ∆E ∆pc
c
2∆x
k
B
T.
Taking ∆x = c
3
Λ
, the temperature T is given by k
B
T =
c
2c
3
Λ
c
2c
3
√
3
√
Λ. When
we put c
3
= π and the acceleration of the space as a = c
2
Λ/3, it becomes the one
k
B
T = c
√
Λ/(2π
√
3) = a/(2πc) what Gibbons and Hawking have derived.
Because the cosmological constant is related to the vacuum energy density ρ
Λ
as ρ
Λ
=
Λc
2
8πG
the number density of the particle is given by n
Λ
∆E
c
2
= ρ
Λ
. Assuming that particle
energy is given by ∆E = c/2c
3
Λ
, the number density n
Λ
becomes as n
Λ
=
ρ
Λ
c
2
∆E
=
Λc
4
8πG
2c
3
Λ
c
=
3c
3
4πG
c
3
Λ
. Taking the volume of the universe V as V =
4
3
π
3
Λ
, the total
number is given by N
Λ
= n
Λ
V =
3c
3
4πG
c
3
Λ
4
3
π
3
Λ
= c
3
c
3
G
2
Λ
= c
3
2
Λ
2
p
.
If we assume the particle as Bose particle such as photon, the total entropy is propor-
tional to the total number as ( N/(S/k
B
)=0.2776 )
S
k
B
∼
N
0.2776
∼ 4c
3
2
Λ
2
p
. Using the
area of the de Sitter horizon A =4π
2
Λ
, it is expressed as
S
k
B
∼
c
3
π
A
2
p
, where the entropy
is proportional to the horizon area A. If we take c
3
= π/4, it becomes S/k
B
= A/(4
2
p
)
which is derived by Gibbons and Hawking.
4. Rindler space
Unruh Effect has been said that in the uniformly accelerating coordinate (Rindler
coordinate) he or she (observer) seems to be in a bath of blackbody radiation at the
temperature T which is related to the acceleration κ(= a)ask
B
T = κ/(2πc).
There is a characteristic length
κ
= c
2
/κ due to the acceleration κ. Applying the
uncertainty principle to this length, the energy ∆E of the particle is given by ∆E
∆pc
c
2∆x
c
2c
4
κ
κ
2c
4
c
k
B
T, where we put ∆x = c
4
κ
. The relation between the
temperature and the acceleration κ is given by k
B
T
κ
2c
4
c
. If we take c
4
= π,theresult
is the same derived by Unruh.
In the following we consider the relation of this temperature to the entropy as k
B
S =
A
4
2
p
. One way of the derivation is to assume that the acceleration κ is the gravitational
acceleration by the mass M, which is composed of the blackbody radiation of temperature
T . If we take the volume of the considering region as V , the mass is given as M =
˜aT
4
c
2
V.
The acceleration κ due to this mass is given by κ =
GM
V
2
3
=
G˜aT
4
c
2
V
V
2
3
=
Gπ
2
15
k
4
B
T
4
3
c
3
V
1
3
c
2
.
Here we use the relations k
B
T = c/(2c
4
κ
), the above equation becomes κ =
c
2
κ
=
π
2
15
1
16c
4
4
Gc
c
2
V
1
3
4
κ
. Then the region size is given by V
1
3
=
15×16c
4
4
π
2
3
κ
2
p
.
As the total entropy is S =
4
3
˜aT
3
V , the entropy per surface S/V
2
3
is given by
S/k
B
V
2
3
=
4
3
π
2
15
k
3
B
T
3
3
c
3
V
1
3
=
8
3
c
4
2
p
.
If we take c
4
=
3
32
and A = V
2
3
, the relation
S
k
B
A
=
1
4
2
p
is derived, which means that the
entropy per surface is 1/(4
2
p
). Finally, we derived the relation S/(k
B
)=A/(4
2
p
), which
is common for the black hole and de Sitter space.