Physics Letters B 751 (2015) 278–283
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Stability of black holes based on horizon thermodynamics
Meng-Sen Ma
a,b,∗
, Ren Zhao
a,b
a
Department of Physics, Shanxi Datong University, Datong 037009, China
b
Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received
30 September 2015
Received
in revised form 19 October 2015
Accepted
23 October 2015
Available
online 27 October 2015
Editor:
M. Cveti
ˇ
c
On the basis of horizon thermodynamics we study the thermodynamic stability of black holes constructed
in general relativity and Gauss–Bonnet gravity. In the framework of horizon thermodynamics there are
only five thermodynamic variables E, P , V , T , S. It is not necessary to consider concrete matter fields,
which may contribute to the pressure of black hole thermodynamic system. In non-vacuum cases, we
can derive the equation of state, P = P (V , T ). According to the requirements of stable equilibrium
in conventional thermodynamics, we start from these thermodynamic variables to calculate the heat
capacity at constant pressure and Gibbs free energy and analyze the local and global thermodynamic
stability of black holes. It is shown that P > 0is the necessary condition for black holes in general
relativity to be thermodynamically stable, however this condition cannot be satisfied by many black holes
in general relativity. For black hole in Gauss–Bonnet gravity negative pressure can be feasible, but only
local stable black hole exists in this case.
© 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
Since the discovery of Hawking radiation we know that black
holes have temperature. Thus, the concept of entropy for black
holes proposed by Bekenstein is no longer an analogue. Not only
that, the works of Hawking made the entropy quantitative, namely
S = A/4. The laws of black hole mechanics [1] plus the generalized
second law of thermodynamics (GSL) [2] imply that black holes
are thermodynamic systems. The work of Jacobson [3] by deriv-
ing
the Einstein equation from a thermodynamic equation of state
and the work of Padmanabhan [4,5] by writing Einstein’s equa-
tions
for a spherically symmetric spacetime in the form of the first
law of thermodynamics make the connection between gravity and
thermodynamics very close. However, there are some differences
between black hole thermodynamics and conventional thermody-
namics,
such as the black hole entropy, which is proportional to
the horizon area but not volume, and the heat capacity of black
holes which may be negative.
Based
on the general first law of black hole thermodynamics:
dM = TdS + dJ+···, where “···” denote the possible additional
contributions from long range fields, many thermodynamic prop-
erties
of black holes have been studied. The couplings of different
*
Corresponding author at: Department of Physics, Shanxi Datong University, Da-
tong
037009, China.
E-mail
addresses: mengsenma@gmail.com, ms_ma@sxdtdx.edu.cn (M.-S. Ma).
matter fields to gravity will give different black hole solutions,
such as in general relativity (GR) Maxwell field leads to RN black
hole and Born–Infeld (BI) field gives the BI black hole. And these
black holes derived in the same gravitational theory with different
matter fields have very different dynamical and thermodynamical
properties.
In
fact, we can also treat the black hole thermodynamic system
from another perspective. According to the horizon thermodynam-
ics
proposed by Padmanabhan, one can obtain the thermodynamic
identity: dE = TdS − PdV from the field equations. One should
also analyze the thermodynamic properties of black holes on the
basis of this identity. In horizon thermodynamics, the pressure P
is
the (rr) component of energy–momentum tensor and the con-
crete
form of it is unnecessary. In other words, the thermodynamic
properties are relevant to the gravitational theories under consid-
eration
but insensitive to the concrete black hole solutions. In this
paper, we will analyze the thermodynamic stability of black holes
in the framework of horizon thermodynamics. In this case, only
two pairs of thermodynamic variables exist, which are the inten-
sive
quantities T , P and the extensive quantities S, V . One can
expect to obtain some different results from those obtained in
usual black hole thermodynamics.
The
paper is arranged as follows. In Section 2, we simply intro-
duce
the horizon thermodynamics and black hole thermodynamics
and demonstrate that they can be derived from the same field
equation. In Section 3 we will analyze the thermodynamic stability
of black holes in GR. The thermodynamic properties of Gauss–
http://dx.doi.org/10.1016/j.physletb.2015.10.061
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
.