Surface gravity and Hawking temperature from entropic force viewpoint

Abstract

We consider a freely falling holographic screen for the Schwarzschild and Reissner-Nordstr\"om black holes and evaluate the entropic force \`a la Verlinde. When the screen crosses the event horizon, the temperature of the screen agrees to the Hawking temperature and the entropic force gives rise to the surface gravity for both of the black holes. Comment: 7 pages, 1 figure; RevTeX4

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arXiv:1003.2049v3 [gr-qc] 2 Apr 2010
Surface gravity and Hawking temperature
from entropic force viewpoint
Ee Chang-Young,∗Myungseok Eune,†and Kyoungtae Kimm‡
Department of Physics and Institute of Fundamental Physics,
Sejong University, Seoul 143-747, Korea
Daeho Lee§
Basic Science Research Institute, Sogang University, Seoul 121-742, Korea
Abstract
We consider a freely falling holographic screen for the Schwarzschild and Reissner-Nordstr¨ om
black holes and evaluate the entropic force ` a la Verlinde. When the screen crosses the event horizon,
the temperature of the screen agrees to the Hawking temperature and the entropic force gives rise
to the surface gravity for both of the black holes.
Keywords: entropic force, Hawking temperature, surface gravity
∗cylee@sejong.ac.kr
†younms@sejong.ac.kr
‡helloktk@naver.com
§dhleep@sogang.ac.kr
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Hawking radiation results from the quantum effect of fields in a classical geometry with an
event horizon [1]. Thus the study of Hawking radiation may provide a clue in constructing
the theory of quantum gravity. Considering its importance, it may be useful to have various
interpretations of the effect from different angles, which may lead to a further understanding
of the nature of black holes.
There have been several ways of providing the interpretation of Hawking radiation. One
of them is the proposal suggested by Robinson and Wilczek [2], which is that Hawking
radiation plays the role of preserving general covariance at the quantum level by canceling
the diffeomorphism anomaly at the event horizon(see also [3, 4]). The flux of Hawking
radiation can be also obtained through the scattering analysis and there have been the
studies of the grey body factor for various black holes to calculate the Hawking temperature
[5–14].
Recently, it has been suggested by Verlinde that gravitational interaction can be inter-
preted as a kind of entropic force through the holographic principle and the equipartition
rule [15]. This entropic formulation of inertia and gravity has been used to study thermody-
namics at the apparent horizon of the Friedmann-Robertson-Walker universe [16], Friedmann
equations [17], Newtonian gravity in loop quantum gravity [18], holographic dark energy [19–
21], an extension to Coulomb force [22], entropic corrections to Newton’s law [23], the Plank
scale effect [24], and the gauge/gravity duality [25]. There have been many works for the
entropic force in the cosmological models [26–31] and the black hole backgrounds [32–37].
In this paper, we study the Hawking temperature and surface gravity by calculating the
entropic force on the holographic screen for spherically symmetric and static black holes.
We consider a test particle with mass m which approaches the holographic screen from a
distance ∆x (see FIG. 1). In this case the change of entropy associated with the information
on the boundary is assumed to be [15]
∆S = 2πm∆x, (1)
where we have taken the natural units of c = ¯ h = kB= 1. From the thermodynamic laws,
the entropic force F acting on the test particle is given by [15, 38]
F∆x = T∆S, (2)
where T is the temperature on the screen. Substituting Eq. (1) into Eq. (2), the entropic
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FIG. 1: A test particle with mass m approaches the holographic screen, which is the boundary ∂V
of the sphere with volume V with radius r. ∆x is the distance that the particle moves.
force can be written as
F = 2πmT. (3)
It is assumed that the total number N of the fundamental bits is proportional to the area
A of the screen, i.e.,
N =A
G,
(4)
where G is a given constant for the moment and will be identified as the Newton’s constant
later on. When a total energy E is divided uniformly over the bits N, the temperature is
determined by the equipartition rule [15]
E =1
2NT,(5)
as the average energy per bit.
Now, we consider the Schwarzschild-like black holes which include both Schwarzschild
and Reissner-Nordstr¨ om (RN) black holes with the line element given by
ds2= gµνdxµdxν= −f(r)dt2+dr2
f(r)+ r2dΩ2
2, (6)
where f(r) vanishes at the event horizon rH.
In order to consider a freely falling holographic screen, we introduce the Painlev´ e-
Gullstrand (PG) line element [39] of the Schwarzschild-like geometry (6). The ingoing (+)
and outgoing (−) PG coordinates can be written as
ds2= −dτ2+
?
dr ±
?
1 − fdτ
?2
+ r2dΩ2
2, (7)
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or, equivalently,
ds2= −f dτ2± 2
?
1 − fdτdr + dr2+ r2dΩ2
2. (8)
Note that constant-time slices are completely flat in this coordinates system.The PG
coordinates are related to the Schwarzschild-like coordinates by
dτ = dt ±
√1 − f
f
dr.(9)
The conserved energy of the gravitational field over a spacelike hypersurface V at a certain
time is given by the Komar integral [40]
E(V ) =
1
4πG
?
∂V∇µξνnµσνdA, (10)
where nµand σµare the unit normal vectors perpendicular to the hypersurface V and to its
boundary ∂V with constant radius r, respectively. The normal vectors satisfy nµnµ= −1
and σµσµ= 1. And ξαis a timelike Killing vector field which satisfies the Killing equation
of ∇µξν+ ∇νξµ= 0.
For r > rH, the Killing vector and the unit normal vectors are given by ξµ= δµ
τ, nµ= −δτ
µ,
and σµ= δµ
rin the PG coordinates. Then, from Eq. (10), the energy is given by
E =r2
2Gf′(r), (11)
where′denotes a differentiation with respect to r. The entropy defined on the freely falling
holographic screen located outside the horizon is given by
S =
A
4G.
(12)
The above relation can be restated as
N = 4S (13)
via Eq. (4). Using the equipartition rule for the energy E in Eq. (5) and the relation (13),
we get the following relation [41, 42]
T =E
2S.
(14)
Then, from Eqs. (11) and (12) the temperature on the holographic screen with the constant
radius r is given by
T(r) =
1
4πf′(r). (15)
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At the horizon, the surface gravity κ is defined with the Killing vector ξµby the following
relation
ξµ∇µξα= κξα.
2f′(rH). Thus, the temperature (15) evaluated at
(16)
In the Schwarzschild-like geometry, κ =1
the horizon (r = rH) is related to the surface gravity as follows:
T(rH) =
1
4πf′(rH) =
κ
2π.
(17)
Since the Hawking temperature TH is related to the surface gravity by TH =
κ
2π[43], the
temperature at the horizon T(rH) is the same as the Hawking temperature.
Finally, the entropic force is obtained from Eqs. (3) and (15) as
F = 2πmT(r) =1
2mf′(r), (18)
where m is the mass of the test particle. For the Schwarzschild case f(r) = 1−2GM/r, the
entropic force becomes the Newton force
F =GmM
r2
, (19)
where M is the mass of the black hole and G is identified as the Newton’s gravitational
constant. For the RN case f(r) = 1 − 2GM/r + GQ2/r2, the entropic force becomes
F =GmM
r2
−GmQ2
r3
, (20)
where M and Q are the mass and charge of the black hole, respectively. Note that the force
in Eq. (20) contains the gravitational effect only, since the test particle’s charge does not
play any role in this consideration. Via the metric, only the geometry which is determined
by the energy of the system plays a role here. At the event horizon(r = r+), the entropic
force in the RN black hole becomes the usual expression
F =m(r+− r−)
2r2
+
= mκ,
where r±≡ GM ±√G2M2− GQ2.
In Ref. [22], the RN black hole was considered. There the motivation came from a mis-
match between the Hawking temperature and the energy in the entropic force formulation.
However, the energy used there contains only the gravitational mass not including the elec-
tromagnetic energy. On the other hand, we obtain the energy using the Komar integral
with the RN metric, which contains both mass and charge contributions to the energy. This
yields the correct relation in our case.
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