133 Page 4 of 16 Eur. Phys. J. C (2017) 77 :133
does not exist). This highlights the effects of the nonlinear
electromagnetic field on the thermodynamical structure of
black holes. That being said, one can point out that, for vio-
lation of the mentioned condition, for spherically symmetric
AdS black holes, the temperature will be positive without
any root.
For small values of the horizon radius, the dominant term
is the charge term, while, for large values of the horizon
radius, the term will be dominant. For spherically sym-
metric AdS space-time, term is positive. Therefore, for
large values of the horizon radius, the temperature will be
positive. Therefore, there exists a root for the temperature,
r
0
, in which for r
+
< r
0
, the temperature is negative and
solutions are not physical.
On the contrary, due to the negative contribution of the
in spherical dS space-time, the temperature will be negative
for large values of the horizon radius. The effective term here
is the topological term. In other words, by suitable choices of
the different parameters, for a region of the horizon radius,
the topological term would be dominant, which results into
the formation of an extremum (maximum) and the existence
of two roots. Between these two roots, the temperature is
positive and solutions are physical (otherwise, the temper-
ature is negative). It is worthwhile to mention that, for the
hyperbolic horizon, the temperature will always be negative
in this case. Therefore, there is no physical solution at all in
this case.
Now, we focus on the effects of each parameter on nega-
tivity/positivity of the temperature. Evidently, by increasing
dimensions, the effect of the topological and charge terms
increase while the opposite takes place for .Asforthe
nonlinearity parameter, for r
+
> 1, if 0.5 < s < 1, then
the effects of the charge term is an increasing function of
s, while if 1 < s, the effects of the charge term will be a
decreasing function of s. It is worthwhile to mention that, for
r
+
< 1, for both cases of 0.5 < s < 1 and 1 < s, the charge
term is a decreasing function of the nonlinearity parameter.
We should point it out that we have excluded s = 1, since
it is the Maxwell case. In addition, the effect of the charge
term is an increasing function of the electric charge. Now,
by considering the mentioned effects, one is able to deter-
mine the thermodynamical behavior of these black holes in
the context of temperature and study different limits which
these black holes have.
3.1.2 CIM case
In the CIM case, the contribution of the charged term is
always toward negativity of the temperature. The dominant
term for small values of the horizon radius is the charge term,
which is negative. For medium and large values of the horizon
radius, the dominant terms, respectively, are the topological
and terms. Considering the dominance of different terms,
depending on the choices of topology and type of space-time,
the temperature could be one of the following cases:
(I) For dS space-time and k = 0, −1, all the terms in tem-
perature are negative. Therefore, the temperature will
be negative and the solutions are not physical.
(II) For dS space-time and k = 1, only the topological
term is positive whereas the charge and are negative.
Remembering that, for small and large black holes, the
dominant terms are q and terms, it is possible to find
two roots for the temperature and one maximum which
is located at the positive values of temperature. This
leads to presence of physical solutions only for medium
black holes while, for large and small black holes, phys-
ical black holes are absent.
(III) For AdS space-time, the term has positive contribu-
tion. Remembering that, for small and large black holes,
dominant terms are q and terms, respectively, irre-
spective of the topological structure of the black holes,
there exists a root for the temperature. For black holes
smaller than this root, the temperature is negative and
solutions are non-physical. For k = 0, −1, the tempera-
ture is only an increasing function of the horizon radius.
Whereas, by suitable choices of different parameters, in
the case of k = 1, the temperature may acquire one or
two extrema. The behavior of temperature in this case
is similar to T − r
+
diagrams in extended phase space
(similar ones in van der Waals like black holes), which
indicates that a second order phase transition takes place
for these black holes. It is worthwhile to mention that
extrema in temperature are matched with divergencies
in the heat capacity which results into the existence of
second order phase transition in the thermodynamical
structure of the black holes.
3.2 Heat capacity and stability
Next, we study the stability conditions of these black holes.
To do so, we employ the canonical ensemble which is based
on the heat capacity. The stability conditions are determined
by the behavior of heat capacity. In other words, the sign
of the heat capacity represents thermal stability/instability of
the system. The positivity indicates that system under consid-
eration is in thermally stable state. In addition, there are two
types of points which could be extracted by using the heat
capacity: bound and phase transition points. The bound point
is where the heat capacity (temperature) acquires a root. The
reason for calling it bound point comes from the fact that it
is where the sign of temperature is changed. Since the nega-
tive temperature is representing a non-physical solution, this
point marks a bound point. On the other hand, phase transi-
tion point is where a discontinuity exists for the heat capacity.
123
剩余15页未读，继续阅读
weixin_38628211
- 粉丝: 4
- 资源: 927
我的内容管理
展开
最新资源
- 构建Cadence PSpice仿真模型库教程
- VMware 10.0安装指南：步骤详解与网络、文件共享解决方案
- 中国互联网20周年必读：影响行业的100本经典书籍
- SQL Server 2000 Analysis Services的经典MDX查询示例
- VC6.0 MFC操作Excel教程：亲测Win7下的应用与保存技巧
- 使用Python NetworkX处理网络图
- 科技驱动：计算机控制技术的革新与应用
- MF-1型机器人硬件与robobasic编程详解
- ADC性能指标解析：超越位数、SNR和谐波
- 通用示波器改造为逻辑分析仪：0-1字符显示与电路设计
- C++实现TCP控制台客户端
- SOA架构下ESB在卷烟厂的信息整合与决策支持
- 三维人脸识别：技术进展与应用解析
- 单张人脸图像的眼镜边框自动去除方法
- C语言绘制图形：余弦曲线与正弦函数示例
- Matlab 文件操作入门：fopen、fclose、fprintf、fscanf 等函数使用详解
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
