Physics Letters B 742 (2015) 225–230
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Entropy relations and the application of black holes
with
the cosmological constant and Gauss–Bonnet term
Wei Xu
a,∗
, Jia Wang
b
, Xin-he Meng
b,c
a
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
b
School of Physics, Nankai University, Tianjin 300071, China
c
State Key Laboratory of ITP, ITP-CAS, Beijing 100190, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received
22 October 2014
Accepted
14 January 2015
Available
online 19 January 2015
Editor: M.
Cveti
ˇ
c
Keywords:
(A)dS
black hole
First
law of thermodynamics
Smarr
relation
Thermodynamic
bound
Thermodynamic
relation
Based on entropy relations, we derive the thermodynamic bound for entropy and the area of horizons for
a Schwarzschild–dS black hole, including the event horizon, Cauchy horizon, and negative horizon (i.e.,
the horizon with negative value), which are all geometrically bound and comprised by the cosmological
radius. We consider the first derivative of the entropy relations to obtain the first law of thermodynamics
for all horizons. We also obtain the Smarr relation for the horizons using the scaling discussion. For the
thermodynamics of all horizons, the cosmological constant is treated as a thermodynamic variable. In
particular, the thermodynamics of the negative horizon are defined well in the r < 0side of space–time.
This formula appears to be valid for three-horizon black holes. We also generalize the discussion to
thermodynamics for the event horizon and Cauchy horizon of Gauss–Bonnet charged flat black holes
because the Gauss–Bonnet coupling constant is also considered to be thermodynamic variable. These
results provide further insights into the crucial role played by the entropy relations of multi-horizons in
black hole thermodynamics as well as improving our understanding of entropy at the microscopic level.
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
In order to understand the entropy of black holes at the mi-
croscopic
level, the entropy product of multi-horizon black holes
has been investigated widely in many previous studies [1–24].
The entropy product is always independent of the mass of black
holes, which is universal for many charged and rotating black holes
[1–13], black rings, and black strings [14]. In fact, the entropy
product, when combined with Cauchy horizon thermodynamics,
can be used to determine whether the corresponding Bekenstein–
Hawking
entropy can be written as a Cardy formula, thereby pro-
viding
some evidence for a CFT description of the corresponding
microstates [14,15]. Therefore, it is important to study the thermo-
dynamics
of the Cauchy horizon.
However,
the mass-independence of the entropy product fails
for some multi-horizon black holes [15–19]. Hence, the entropy
sum [12,13,16,20,23] and other thermodynamic relations [16,17,
20–22,24] are
introduced, which are also mass-independent in
*
Corresponding author.
E-mail
addresses: xuweifuture@gmail.com (W. Xu),
wangjia2010@mail.nankai.edu.cn (J. Wang), xhm@nankai.edu.cn (X.-h. Meng).
some cases and they appear to be universal. In particular, this
applies to the relation T
+
S
+
= T
−
S
−
, which is closely associated
with the mass-independence of the entropy product. This can also
be understood well in physical terms by the holographic descrip-
tion,
i.e., the thermodynamic method of black hole/CFT (BH/CFT)
correspondence [7,25–30]. The thermodynamic relations T
+
S
+
=
T
−
S
−
may used as criteria to determine whether there is a two-
dimensional
CFT dual for black holes in the Einstein gravity the-
ory
and other diffeomorphism-invariant gravity theories [7,25–30].
In addition, it has been shown that the thermodynamic relation
T
+
S
+
= T
−
S
−
is equivalent to the central charge being the same
(i.e. c
R
= c
L
) for some two-horizon black holes. However, the in-
terpretations
of other thermodynamic relations are not clear. Thus,
the present study focuses on entropy relations and their applica-
tions
to black hole thermodynamics.
Based
on entropy relations, we derive the thermodynamic
bound for entropy and the area of horizons for a Schwarzschild–dS
black hole, including the event horizon, Cauchy horizon, and nega-
tive
horizon, which are all geometrically bound and they comprise
the cosmological radius. We consider the first derivative of entropy
relations to obtain the first law of thermodynamics for all horizons.
We also obtain the Smarr relation for horizons using the scaling
discussion. For the thermodynamics of all horizons, the cosmolog-
http://dx.doi.org/10.1016/j.physletb.2015.01.018
0370-2693/
© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.