#### Some results on a black hole with a global monopole in Poincaré gravity

Eur. Phys. J. C
Some results on a black hole with a global monopole in Poincaré gravity
Valdir B. Bezerra 1
Cristine N. Ferreira 0
Elton P. J. Alvarenga 0
0 Núcleo de Estudos em Física, Instituto Federal de Educação, Ciência e Tecnologia Fluminense , Rua Dr Siqueira 273, Campos dos Goytacazes, RJ 28030-130 , Brazil
1 Departamento de Física, Universidade Federal da Paraíba , Caixa Postal 5008, João Pessoa, PB 58059-970 , Brazil
The aim of this work is to study the thermodynamics and spin current of a system corresponding to a black hole containing a global monopole in the context of Poincaré gravity theory which is an extension of general relativity, in the sense that the intrinsic angular momentum of matter is also a source of gravitational interaction. Thus, in this work we find the solution corresponding to the spacetime under consideration by taking into account that the action which describes this system contains terms corresponding to the curvature and torsion. The metric obtained is a function of mass, solid angle deficit and the coupling constants of the quadratic terms of the curvature and torsion. In this model, the stability of the system is studied through the analysis of the Hawking temperature and the specific heat. In this context it was also studied the critical temperatures of the system considering positive or negative cosmological constant. In the vicinity of the black hole with a global monopole, where there is a logarithmic correction due to the relationship between the torsion and curvature fields, some analysis were done. We also study the AdS/dS limit where the black hole is analyzed from the topological point of view. Although the effect of spin current density at low energies is negligible, in the vicinity of strong gravitational fields it can generate an appreciable effect due to spin gravity coupling.
1 Introduction
One of the most intriguing phenomena that can be observed in
the universe concerns to the fact that it experiences an
accelerated expansion, a phenomenon confirmed recently from the
observations of supernova explosions. Due to these
observations with supernovae it has been concluded that the
universe nowadays is expanding rapidly. One idea to explain
this acceleration is that the universe is dominated by dark
energy that seems to be more abundant than the usual energy
[1,2].
A similar scenario with respect to the occurrence of
acceleration also appears in the primordial stages of the universe.
At that period, it experienced a great acceleration called
inflation [3,4], but in those times the universe was very hot and
dense and only at the end of this inflationary period, the
structures began to be formed [5,6].
Although the general relativity is undoubtedly the most
promising theory to explain most of the events occurring
in the universe, many questions still seem inexplicable, as
for example the accelerated expansion. Another theoretical
point is the fact that there is no a theory of quantum gravity
based on general relativity. Thus, some theories were
proposed to unify the formalism of the quantum field theory
with gravitational formalism, as for example, the Einstein–
Cartan theory. This theory, extends the theory of general
relativity by including a connection that takes into account its
antisymmetric part. It was in this period that the concept of
torsion in general relativity was born.The great achievement
of Poincar’s gravity gauge theory is that besides the source
of gravity corresponding to energy–momentum, the intrinsic
angular momentum or the spin of matter is also a source of
gravitational interactions. In this way, the resulting
spacetime geometry is endowed with curvature and torsion [7].
Although the effects of spin are negligible at low energies,
certainly, it was very important in the primordial universe
when spin density was very high [8].
We also know that local or global change of the universe
implies in phase transitions that can originate from breaks
of symmetry, in analogy to what occurs in thermodynamic
systems. One of the most important laboratories for physics
is the universe and one of the most intriguing objects
investigated are the black holes [9].
It can carry mass, charge, angular momentum and can be
described by thermodynamic laws [10–13]. The implications
of the studies related to this object can undoubtedly help us to
understand a greater number of issues. However, so far black
hole physics has been dependent on the models, the
observations can provide data that discards certain models and
reinforces others. For this reason the study of black holes in
various contexts is of paramount importance [14–16]. In
special, we will consider a black hole with a topological defect,
namely, global monopole. This topological has been formed
at phase transitions in the very early universe. The
gravitational field of a global monopole possesses a deficit solid
angle which is proportional to the energy scale of symmetry
breaking [17].
Today it is believed that the universe is de-Sitter (dS)
with a small positive cosmological constant. Many
investigations have appeared in the literature considering the
cosmological constant to be the negative pressure necessary to
make the universe to expand in accelerated way [18].
Taking into account the presence of a cosmological constant,
some other studies consider the black hole in Anti- de-Sitter
(AdS) spaces. Despite that for this space-time, the
cosmological constant is negative, it can be included from the point
of view of the relation between an alien world with a border
according to which it could represent our particle physics
[19]. This mechanism became known as AdS/CFT
correspondence [20–22]. From the point of view of physics of a
black hole physics with AdS spacetime, we can relate the
entropy with the area of the black hole surface [13]. In this
formalism, the Hawking–Page phase transitions can be
studied [23]. The black hole in an AdS spacetime give us the
correct relation between temperature and the specific heat
which is important to understand the black hole
thermodynamics.
In this work a different context will be proposed to study
black hole thermodynamics, in such a way that it will taken
into account that the space time besides curvature has also
torsion. The theory is called Poincaré, gravity because the
connections that make up the Riemann tensor are
considered as gauge fields and contain information arising from
the metric as well as from the torsion. For consistency of
the equations, in order to have a torsion that can propagate,
we will use the formalism where two quadratic terms are
introduced, one for the Riemann tensor and another pair for
the torsion tensor [24,25]. One of the advantages of working
with this model is the appearance of spin chains. These spin
currents come from the conception that the derivative of the
action in relation to torsion can give us spin chains. Thus,
it should be important to study the coherence of this system
[26]. Another advantage of this model is to better understand
the de-Sitter space, through a correct relation between
temperature and specific heat, analogous to what happens in the
Anti-de-Sitter case. In fact, in this model, one can consider
de-Sitter and Anti-Sitter as experiencing a Hawking–Page
type phase transitions.
This paper is organized as follows. In Sect. 2, we present
more global aspects Poincar gravity. In Sect. 3, we perform
an analysis of thermodynamics of the system under
consideration. In Sect. 4, we discuss the spin current. Finally, in
Sect. 5, we present the conclusions.
2 Description of the Poincaré-gravity framework
In this section we present the basic ingredients of the Poincaré
gravity, including two quadratic terms, one of these is the
torsion and the other is the curvature. The full action that
contains both gauge fields are given by
S = Sg + S ,
where the Poincaré gravity action is written as
1 √
Sg = κ
−g d4x R − α4 RμaνbRμaνb − β4 Qaμν Qaμν ,
with α and β being constants and μ, ν = 0, 1, 2, 3 and a,
b = 0, 1, 2, 3 being spacetime indices and tetrad indices,
respectively. The quantities Rμνabeμeaν and gμν = eμaeν ηab
b
b
are, respectively, the curvature scalar and the metric
associated with the gravitational field. The Riemann and the torsion
tensor field are given by
Rμνab = ∂[μ ν]ab +
a a
Q μν = ∂[μeν] +
μ c cν b,
a
μa beν .
b
In the approach considered, these fields are gauge field
strengths related with the connection ν ab and the vierbein
eμa, as explicitly shown in Eqs. (
3
) and (
4
).
The general form of the time independent metric with
spherical symmetry in (3 + 1) dimensions is given by
ds2 = −e2μdt 2 + e2ν dr 2 + r 2(dθ 2 + sin θ 2dφ2),
(
5
)
where μ = μ(r ) and ν = ν(r ), which means that these
functions depend only on the radial coordinate.
In this case, the non-vanishing independent components
of torsion tensor are given by
0
Q 01 = f (r ),
1
Q 10 = h(r ),
2 3
Q 20 = Q 30 = k(r ),
2 3
Q 21 = Q 31 = −g(r ),
where f (r ), g(r ) h(r ) and k(r ) are function to be determined
and the Riemann-Cartan affine connection, ˜ , is given by
2 ˜ i k j = − i k j + Qik j , where i k j corresponds to the
Riemann part and Qik j to the contribution of the torsion.
(
1
)
(
2
)
(
3
)
(
4
)
(
6
)
≡ B(r ),
e−ν
r
The nonvanishing independent curvature components
Ri jkl are given by
2 1 2
R 323 = r 2 + k −
r1 e−ν − g
R0110 = (μ eμ−ν + eμ f ) e−μ−ν ≡ A(r ),
2
R0202 = R0303 = −( f + μ e−ν )
− g
≡ C (r ), (
9
)
R1212 = R1313 = r1 e−ν (r g − e−μ) + hk ≡ D(r ),
R1202 = R1303 = (ν e−ν + f )k ≡ E (r ),
R0212 = R0313 = (r k) e−rν − h r1 e−ν− g
≡ −F (r ).
In order to write equations in a simplest way, we are
identifying the components of the Riemann tensor with the
functions A(r), B(r), ….
The equations of the motion written in terms of the
vierbein, are given by
1
Gi j = Ri j − 2 ηi j R = −κ Ti j − αTi j − βQi j ,
where
1 1
Ti j = − 2 Rilmn R jlmn + 8 ηi j Rilmn Rilmn,
1 n m 1 n im .
Qi j = − 2 Qim Q j n + 8 ηi j Qim Q n
The action of matter corresponding to the defect is S . An
interesting type of defect that can be related with the black
hole is the global monopole with a O(
3
) symmetry broken.
The action associated to the matter field coupled with gravity
that represents the global monopole is
S = 21 d4x √−g (∂μφa )(∂μφa ) − 21 λ(φa φa − η2)2 ,
where λ is the self-interaction constant, η is the scale of
a gauge-symmetry breaking and the triplet field that will
result in a monopole configuration can be described by
φa = η ϕr(2r) x a , with a = 1, 2, 3 and x a x a = r 2. The
function ϕ(r ) is dimensionless and constrained by the condition
ϕ(0) = 0 and ϕ(r > η) ∼ 1.
The energy momentum tensor is given by
Tμi = 2√1−g δδeSiμ .
Considering that the monopole is formed in the black hole
background, we have that the energy momentum tensor is
given by
(
7
)
(
8
)
(
10
)
(
11
)
(
12
)
(
13
)
(
14
)
(
15
)
(
16
)
(
17
)
(
18
)
η2
T00 = −T11 ∼ r 2 = κr 2 ,
We are considering that the Einstein tensor components
are function of the metric and torsion. The contribution of
the torsion can be understood if we analyze the Riemann
tensor given by Eq. (
4
). The connection ν ab has two parts:
one related to the metric and the other to the torsion. The
non-vanishing components of the Einstein tensor are G00,
G11, G22 = G33, in which case there are off-diagonal terms
due to the presence of the torsion. This fact implies that
in energy–momentum tensor has also off-diagonal
componentes, namely, G01 and G10.
If we analyze these components we can observe that when
the torsion components are zero, then, the off-diagonal
components of the Einstein tensor are zero, and as a consequence,
the components of the Einstein tensor become the ones
corresponding to the usual global monopole in a black hole
background. The non-vanishing quadratic terms of the energy–
momentum tensor associate to the curvature are T00, T11,
T22.
With torsion fields the off-diagonal components of the
energy–momentum tensor are not zero, even when the
metric is diagonal. This fact produces important consequences
on the geodesics of the test particles in the spacetime
generated by the system formed by the black hole and the defect,
specially when the torsion is taken into account. The
offdiagonal components of the energy–momentum tensor are
T10 = T01 when k = h = 0 and the off-diagonal
components of the energy–momentum tensor become zero.
The Einstein equations in weak field approximation are
given by
G00 = −2D(r ) − B(r ) = −α(2 x y + uv)
1
− 2 β[ f
2 − h2 − 2(k2 + g2)] − r 2 ,
G11 = B(r ) − 2C (r ) = −α(2 x y − uv)
− β(k2 + g2) + r 2 ,
1
G22 = G33 = A − y = −α uv + 2 β( f 2 − h2),
G01 = G10 = 4α E y − 2βk,
(
19
)
(
20
)
(
21
)
(
22
)
with E (r ) = −F (r ), x = C (r ) + D(r ), y = C (r ) − D(r ),
u = A(r ) + B(r ) and v = A(r ) − B(r ), where A(r), B(r),
C(r), D(r), E(r), F(r) are given by Eqs. (
7
)–(
12
).
In this work we assume that the functions which appear in
Eq. (
6
) are such that g = k, h = − f , and the metric
parameters, μ obey the usual relation ν μ = −ν. The condition
y = − 41α in the Eq. (
22
) gives us
μ e2μ − heμ = − 23 βkeμ,
(
23
)
and using the relation E = −F , we have
k h
− r − k + r = μ k.
Using these assumptions and results together with
Einstein’s equations given by Eqs. (
19
) and (
20
), we get
u = 4y + r 2 = 0,
which supplies the following relation
(μ e2μ) +2( f eμ) +
4keμ
,
with C1 and C2 being integration constants. For simplicity,
we will work with the metric in the form
ds2 = −B(r )dt 2 + B(r )−1dr 2 +r 2(dθ 2 + sin θ 2dφ2), (31)
where
B(r ) = 1 −
−
2G M
r
+
r 2
3 + N r 2 ln r.
G M is the energy of the system, = ± L32 is the
cosmological constant and the term with metric constant parameter N
is the new contribution arising from the torsion. This
metric represents the spacetime generated by the system we are
considering, which is formed by a black hole with a global
monopole in a scenario in which the torsion field is present.
3 Thermodynamics analysis
In this section, we analyze the behavior of the energy of the
system corresponding the source we are considering, formed
by a black hole and a global monopole, as a function of those
parameters presented in the last section. The dependence of
the energy, GM, as a function of the horizon rH , is given by
where the minus sign in the third term represents the case with
positive cosmological constant, which becomes
asymptotically de Sitter (dS) when r → 1 or β = 3. In the case where
this sign is positive, the cosmological constant is negative,
and thus when r → 1 or β = 3, we have the Anti-de-Sitter
case.
The behavior of GM, for different N’s, as a function of
the entropy is shown in Fig. 1, where the entropy is given by
S = πr H2 .
The torsion is codified by the parameters N and β. When
N = 0, the metric is, locally dS (up panel) and AdS (down
panel), both represented by the dotted curve. In the upper
panel, we can verify that the curve grows with the entropy up
to a certain value and then decreases. This behavior is very
different from the lower panel, in which case the entropy
always grows. To analyze the stability of the black hole we
need to study the behavior of temperature, specific heat and
free energy, which can be done simply by verifying how these
magnitudes evolve with the radius of the horizon.
It can be seen that the tangent lines to these curves give
us the temperature that satisfies the second law of
thermodynamics. From now on we can calculate the Hawking
temperature, TH , of the system by taking the derivative of the
energy with respect to the entropy of the black hole. From
the second law of thermodynamics we have that the Hawking
temperature is obtained from d G M = TH d S, and then we
find that
TH =
d S
drH
−1 d G M
drH
rH
= 4π
1 −
r H2
3
∓ L2 + N +3N ln(rH ) .
(34)
In the case where the cosmological constant is negative
we still have the possibility of obtaining a black hole either
for N = 0 or with N = 0. However, for the positive
cosmological constant we have the possibility of the existence of
stable black hole only when N = 0. This is the most
important feature of this model. We will study in the next sections,
in details, both cases, namely, for negative and positive
cosmological constants.
In Fig. 2, the behavior of the temperature with the radius
of the horizon is showed, when the cosmological constant is
positive = L32 . In the panel on the upper part two black
holes appear when torsion is taken into account. The
Hawking temperature decreases as the horizon radius increases,
until reaching a minimum value, and than, starts to increase
with the radius of the horizon, for = 0 and = 0.5. In
the cases where = 1.0 and = 1.5, the Hawking
temperature increases with the radius of the horizon. When the black
hole is in the presence of a global monopole there appears
the phase transition for 0 ≤ ≤ 1.5. It can be visualized that
the behavior of the temperature as a function of the horizon is
sensible to the variation of the parameter of the deficit solid
angle associated with of the global monopole.
Analyzing this behavior we find that the deficit solid angle
has to be less than one. At these limits the temperature
decreases to a minimum, showing in this analysis the
compatibility of the system. When ≥ 1, the Hawking temperature
is negative, which is incompatible with any thermodynamic
system. In the lower panel one has the graph of the
temperature as a function of the radius of the horizon for the case
without torsion, N = 0. It shows that for small values of
the deficit solid angle, the Hawking temperature is positive
in a certain region and decreases as the horizon radius rH
us use the following relations
increases. However, it does not present a minimum value,
and thus the black hole is not stable.
Initially we will analyze the critical points of the system,
by calculating the minimum of the Hawking temperature,
namely, by solving the equation ∂TH = 0. To do this, let
∂rH rHmin
−3N LambertW 23 (1− )e 23 ∓ N+4N ∓2 +8N
N
THmin =
1
rHmin = e− 6
1 −
2πrHmin
+ 23 N r H2min ,
where the minimum temperature is such that THmin =
TH (rHmin ) and indicates the existence of a thermal phase
transitions between the black holes with the same temperature.
The interesting effects occur when N = 0 which is given
by ln(rH ). For temperatures T < Tmin, there are no black
holes, but only radiation. For T > Tmin there are multiply
black holes, whose horizon, rH , associated to them, can be
determined by Eq. (34), which gives us the following result
N − L32 + 3N ln(rH ) r H2 − 4π TH rH + 1 −
= 0. (36)
The general expression for the temperature is given by Eq.
(34) and the expression for the heat capacity by
(37)
(38)
d G M
drH ·
d T
drH
−1
,
+ N ∓ L32 + 3N ln(rH ) r H2 ,
− 1 +
3
4N ∓ L2 + 3N ln(rH ) r H2 .
C =
d G M
d TH
where
d(G M )
drH
d T
drH
1
= 2
1
= 4πr H2
=
1 −
This result was obtained using the second law of
thermodynamics d(GM) = TdS. In the general case, the positive sign
corresponds to the positive cosmological constant, while the
negative sign corresponds to the negative cosmological
constant. In this scenario, we find that
⎡ 1 −
C = 2πr H2 ⎣
+
− 1 +
Figure 3 shows the heat capacity for different values of
= κη2. It can be seen that for 0 ≤ κη2 < 1 the black
hole is stable because the condition T > 0 and C > 0, are
satisfied.
Note that we have T > Tmin, when rH > r min, and thus
H
the temperature and the heat capacity are positive and we
say that the black hole is stable. In the case where rH <
rHmin there is a region where the temperature is positive and
the heat capacity is also positive, but in this case the black
hole experiences a transition for the equilibrium point. In
this region there is a small black hole. The region with the
negative heat capacity is the forbidden region.
Then we have an interesting black hole with positive
cosmological constant when the torsion is present which is
analogue to the AdS black hole. The stable black hole has
the temperature increasing with the horizon radius rH . This
behavior is interesting and means that when the black hole
gains energy, as the radius of the horizon rH increases, the
temperature also increases.
Another point that can be analyzed, in the graphs of Fig. 3,
is the behavior of the curves of heat capacity for different
Fig. 3 The Graphs represent C × rH considering the positive
cosmological constant. In the upper panel N = 0.5 and in the lower panel
shows the heat capacity for = 0.5
values of the deficit solid angle. Comparing the heat capacity
of the upper and lower panels we can see that it is possible to
obtain the same behavior by varying the values of the deficit
solid angle or the contribution due to the torsion.
We can verify that the curve N = 0 behaves similarly to
= 1.5, while N = 2 is similar to = 1 and N = 0.5 and 1
are similar to = 0 and 0.5, respectively.
3.1 The Hawking–Page phase transitions
Another important quantity to analyze the stability of the
black hole is the Hawking–Page phase transition. This
transition, at a constant volume, occurs at a point where the
Helmholtz free energy, given by F = G M − T S, is exactly
equal to zero.
This transition can be studied by keeping the temperature
constant for different values of . With this methodology we
obtained the curves of Fig. 4, for several values of .
Performing the same procedure of the last section by
calculating the roots of the Helmholtz free energy equation and
solving numerically the problem of finding one of this energy
that is unique and that touches the rH axis, we construct the
graph of the Fig. 5.
In Fig. 6 it can be seen that in the absence of the global
monopole, we obtain that the specific heat C > 0 and the
temperature T > 0, and therefore, exists a stable black hole
This behavior is consistent with the case when the energy
increases more rapidly than the temperature, so that the
specific heat increases. In this situation, a new phase transition
appears, even in the case of the temperature T > Tmin, at
rH = rHc . For rHmin < rH ≤ rHc , the specific heat decreases
with rH , as the temperature increases.
These results are similar in both cases, namely, with
positive or negative cosmological constant. These are no
asymptotically neither dS nor AdS, due to the presence of the
logarithm term.or equivalently, due to the presence of the torsion.
3.2 The AdS limit for black holes
In order to better understand the phase transitions in black
holes systems, we initially worked with the case where we
have a locally AdS geometry. We will study now, besides the
phase transitions, the role of the global monopole. We can
easily see in metric (31) that the existence of the torsion can
break the AdS geometry when β = 3, due to the presence of
the logarithm. For this reason, to study the locally AdS case
with torsion, we will initially consider β = 3. In this case
N = 0 in the metric (31).
The dependence of the energy (GM) in function of the
horizon radius rH , is
The Hawking temperature is, therefore, the derivative of
the energy with respect to the entropy of the black hole. From
the first law of thermodynamics we have that the Hawking
temperature is given by d G M = T d S and then we find
1
T = 4πrH
1 −
We will now examine the heat capacity, which can be
calculated from the energy by using Eq. (37), resulting the
following expression
(40)
(41)
(42)
(43)
(44)
(45)
(46)
Fig. 7 Graphs of the temperature T × rH upper panel and the specific
heat capacity (lower panel) considering the Ad S4 case. In the upper for
N = 0 and L = 5
rmin =
√
1 −
√3
L ,
Tmin = √3 2π √11−− L . (47)
In Fig. 7 we analyze all the important points in the graphs of
the temperature as a function of the horizon radius.
In general, in the case of a negative cosmological constant,
a black hole in an asymptotically flat space has a spherical
event horizon. If the cosmological constant is negative, the
black hole has no longer a spherical horizon. This type of
black hole is called a topological black hole.
In Fig. 7, we show the temperature as a function of the
horizon (upper panel). For a temperature T < Tmin there are
no black holes. It is a pass of pure radiation. For T > Tmin
there are two black holes with rH given by Eq. (46).
.
.
1 √
TH P = Lπ
√
rH P =
1 −
1 − ,
L .
If we use the Eq. (34), we find
L32 r +2 − 4π T r+ + 1 −
= 0.
C = 2πr H2
1 −
+ L32 r H2
− 1 + L32 r H2
Now let us consider the energy of the Hawking–Page
phase transition that occurs when the Helmholtz free energy
is zero. Giving that this e-energy is
F = G M − T S
1
= 2 rH 1 −
+ L12 r H2
− π T r H2 ,
thus, the Hawking–Page temperature and the horizon radius
are expressed as
This result gives us a pair of BH’s (large/small) with the radii
given by
2 1
rLarge/Small = 3 π T L ± 3
4L2π 2 + 3 − 3L .
The corresponding Hawking temperature TH is given by the
derivative of Hawking temperature as a function of rH . Their
minimum values are thus obtained from
Note that, we have T > Tmin when rH < rH min . Thus,
the temperature is always positive, as we can see in lower
panel of Fig. 1, The heat capacity is negative, C < 0, and we
have an unstable black hole that we call small black hole. In
the case where rH > rH min the temperature is positive and
according to Fig. 4 (lower panel) corresponds to the positive
heat capacity, C > 0. In this case we have a stable black hole
that we call big black hole.
The importance of working with a topological defect in
the AdS geometry is that the existence of a topological defect
in the AdS bulk is an amount conserved at the boundary.
Although the defect remains at the border, this does not break
the CFT locally, because only the deficit solid angle is felt
when we have a velocity relative to the defect [27]. This fact
can bring us many applications of this metric to systems of
condensed matter, where the vortex has an important role
for the determination of the properties of the system and the
fact that the vortex introduces a mass gap, responsible for the
mass of the fermionic fields. As in the case of the AdS frontier
being a conformal field theory, the presence of vortices can
provide a mechanism to generate mass.
4 The spin current
The discovery of new materials such as graphene and
topological insulators present to us new behaviors emerged in
relation to the role played by the spin, dynamizing the area
called spintronics. The theory developed in this paper permit
us to get conclusions about how is the spin current behavior
induced by torsion and curvature. The spin current tensor is
given by
μ
S i j = √
1
−g δ μ
δ SM
i j .
Thus, considering the part of the quadratic terms of the
curvature and torsion, the spin current can be written as
κ Sμi j = − T¯ i j + α Rμiνj||ν ,
μ
where T¯ μij and Rμiνj||ν are given by
T¯ i j = K μi j − βQ i j
μ μ
Rμiνj||ν = Rμiνj;ν −
λ
− λσ
= (1 − β)Qμi j −Qmmi δ μj −Qmjm δiμ,
k νi Rμkνj − k ν j Rμiνk
Rμiσj − σμν Rμiσj .
By performing these calculations we can find that the
components of the spin current density are S001, S101, S202 = S303
and S212 = S313. The contribution of the torsion to the spin
μ
current is given by the following components of T¯ i j
(52)
(53)
(54)
(55)
(56)
(57)
1
We will focus on the case β = 3, where we have: ± L2 =
21α , with α being positive or negative. In the negative case
we have de-Sitter space with cosmological constant =
2|3α| and in the positive case the Anti-de-Sitter, with negative
cosmological constant = − 2|3α| , where a = (1 − κη2 −
2GrM + Lr22 )1/2 and the quantities k and h are given by
In this way, using Eqs. (49) and (52) for β = 3, we obtain
the graphs of Fig. 8 with s1t (r ) = κ(S001)t = −κ(S101)t =
−h + 2k, s2t (r ) = κ(S212)t = κ(S303)t = −h, s3t (r ) =
κ(S202)t = κ(S303)t = h.
In Fig. 8 the behavior of the components of the spin current
density can be analyzed taken into account the torsion. This
figure shows that the spin current falls to zero when r → ∞
and as the test particles approach the black hole, the spin
current density increases. The contribution of the monopole
can act as a fine regulator of the spin current, which has the
role of attenuating its density. The twist contributes weakly
to the spin current if we compare it with the contribution of
the curvature, as can be seen in Fig. 9. Note that close to
the BH this contribution may be relevant. The importance of
calculating in the Anti-Sitter case is that it may be relevant
to understanding the quantum nature of gravitation at the
boundary of that space via the correspondence Ad S4/C F T3.
The contributions coming from the curvature are given by the
expression (51), or more explicitly,
κ S101 = −α R1 −(a −h)R1 +2k R2
− 2
a R1
2a + 2 r − 3 r
κ S212 = κ S303 = −α 2k R5 + 2
+h R1) ,
κ S202 = κ S303 = −α R5 −
a
r −k R3
R4
a
a + 2 r − h
R3
− h R2 −
−
a 3
2 a + 2 r − r
a
r − k (R6 + R4) − k R2
R5 ,
3
- 0
1
k i
xj 1.0
κS 0.0
2.0
0
2
6
8
Fig. 8 These Graphs show the contribution of the torsion to the
components of the spin current density on the upper panel, s1t is shown. The
lower panel, shows s2t and s3t , with κη2 = 0.3 e 0.4 and L = 103
κ S212 = κ S313 = −α R4 + (a − h) R4 + h R5
− k R6 +
+
a
r − k
a 3
2 a + 2 r − r
R4 + k R3
R4 ,
(58)
with R1 = a a h − a h , R2 = −a k − a k/r + a h/r − h k,
R3 = −(−h + a )(a/r − k), R4 = −(a + h) k, R5 =
a /r (k + r k − a ) + hk and R6 = 1/r 2 + k2 (1/r a − k)2.
It can be seen in Fig. 9 that the contribution to the spin
current due to the curvature is greater if we consider a smaller
negative cosmological constant. This result is compatible
with what we find today in the universe, but in the
primordial universe at high densities, this contribution may be of
the order of the components arising from the twist. The
contribution of the defect in this case can change its behavior,
attenuating the current in certain regions and amplifying in
others.
The obtained components of the spin current density
without torsion is proportional to the inverse of the powers of the
radius. In this way, when r → ∞, that is, for regions far from
the black hole, the spin current is too small to be observed.
The study of the spin chains in an AdS gravitation together
with the holographic principle is important to understand
this quantity in new materials that are governed by the Dirac
equation as topological insulators and graphene.
5 Conclusions
In this work we obtained the solution of a system
corresponding to a black hole with a global monopole in the
framework of the Poincaré gravity gauge theory. The topological
defect induces the existence of a deficit solid angle that is
responsible for several interesting effects, and these could
be investigated in the future. The resulting metric presents a
logarithmic correction which is important in the vicinity of
the system.
Both terms, r 2 and r 2 ln(r ), can be interpreted as
confinement terms mediated by the connection coefficients. The
confinement potential may play an important role for strong
gravitational fields and also in the strong interaction regime.
The thermodynamics behavior was studied considering
different cases. Firstly,the Hawking temperature and the
specific heat were studied in the general case containing
logarithmic corrections in the metric. Although in this case we do
not have a de-Sitter or Anti-de-Sitter geometry, these
corrections cause the stability of the black hole in the case where
the cosmological constant is positive. This stability was
studied using the comparison between specific heat and Hawking
temperature and the Hawking–Page phase transition was also
analyzed. In this analysis some critical points were found.
One of those is the minimum temperature, Tmin , that
separates the regions were the black hole is stable from that were
it is unstable.
Both phases have temperature and heat capacity positives.
For r H < rmin , there is a region where the temperature drops
rapidly to Tmin . This we have a small black hole. Otherwise
when r H > rmin we have a big black hole. There are another
phase transition, to heat capacity, when for r > rmin that
passes for two regimes, in one decrease while the temperature
increase and in the other the heat capacity increase while the
temperature increase. The behavior of the spin current density
in the presence of strong gravitational fields was also studied.
Due to the asymptotic behavior, these currents were studied
in the AdS and dS spacetimes which are conceptualized with
the holographic formalism.
It was found the coupling constant of the square
curvature represented by the Riemann tensor Rμaνb with the
cosmological constant. This shows that for small values of the
cosmological constant there is a contribution of spin current
density due to the curvature, which is more relevant than the
contribution arising from the twist, in the neighborhood of
the black hole. In spite of this behavior if we consider that
the cosmological constant may have a different value in the
primordial universe, we have that both the contribution from
the torsion and from the curvature are of the same order of
magnitude. Although spin content is negligible at low
energies, in the vicinity of high gravitational fields where the spin
current density is relevant or in the primordial universe, when
spin current density, certainly, played a very important role,
such spin content may be crucial for understanding the
quantum properties of these systems. In this sense, the theory used,
which extends the geometrical scheme to inclusion the
intrinsic angular momentum, is very interesting because it unifies
spin and gravitation through the antisymmetric connection
that is related to the torsion, defined within the framework of
a Riemann–Cartan manifold.
In a near future, we plan to study the holographic content
of high spin densities due to its importance to understand the
quantum nature of systems subject to strong gravitational
fields.
Acknowledgements VBB thanks the CNPq for a partial financial
support and CNF thanks the Universidade Federal da Paraíba for the
financial support and hospitality during the preparation of this work.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
1. N. Haba , H. Ishida , R. Takahashi , Y. Yamaguchi , JHEP 1602 , 058 ( 2016 ). https://doi.org/10.1007/JHEP02( 2016 ) 058 . arXiv: 1511 .02107 [hep-ph]
2. P. Cox , A. Kusenko , O. Sumensari , T.T. Yanagida, JHEP 1703 , 035 ( 2017 ). https://doi.org/10.1007/JHEP03( 2017 ) 035 . arXiv: 1612 .03923 [hep-ph]
3. W. Hu , I. Sawicki , Phys. Rev. D 76 , 064004 ( 2007 ). arXiv: 0705 .1158 [astro-ph]
4. A.D. Linde , Rep. Prog. Phys. 47 , 925 ( 1984 ). https://doi.org/10. 1088/ 0034 -4885/47/8/002
5. D. Wang , X.H. Meng , Phys. Rev. D 95 , 023508 ( 2017 ). https://doi. org/10.1103/PhysRevD.95.023508
6. J. Dutta , W. Khyllep , N. Tamanini , Phys. Rev. D 95 , 023515 ( 2017 ). https://doi.org/10.1103/PhysRevD.95.023515. arXiv: 1701 .00744 [gr-qc]
7. F.W. Hehl , P. von der Heyde, Rev. Mod. Phys . 48 , 393 ( 1976 ). https://doi.org/10.1007/JHEP03( 2017 ) 035 . arXiv: 1612 .03923 [hep-ph]
8. S. Akhshabi , E. Qorani , F. Khajenabi , EPL 119 , 29002 ( 2017 ). https://doi.org/10.1209/ 0295 -5075/119/29002. arXiv: 1705 .04931 [gr-qc]
9. J.D. Bekenstein , Phys. Rev. D 7 , 949 ( 1973 )
10. J.D. Bekenstein , Lett. Nuovo Cim . 4 , 737 ( 1972 )
11. J.D. Bekenstein , Phys. Rev. D 7 , 2333 ( 1973 )
12. J.D. Bekenstein , Phys. Rev. D 9 , 3292 ( 1974 )
13. S.W. Hawking , Commun. Math. Phys. 43 , 199 ( 1975 ) [Erratumibid. 46 , 206 ( 1976 )]
14. R.-G. Cai, L. -M. Cao , N. Ohta , JHEP 1004 , 082 ( 2010 ). arXiv: 0911 .4379 [hep-th]
15. Y.S. Myung , Y.-W. Kim , Y.-J. Park , Phys. Rev. D 78 , 084002 ( 2008 ). arXiv: 0805 .0187 [gr-qc]
16. R. Biswas , S. Chakraborty , Gen. Relativ. Gravit. 43 , 41 ( 2011 )
17. M. Barriola , A. Vilenkin , Phys. Rev. Lett . 63 , 341 ( 1989 )
18. H.H. Zhao , L.C. Zhang , F. Liu, R. Zhao , Thermodynamics and critical behaviors of topological dS black hole with nonlinear sources . arXiv:1704 .05167 [hep-th]
19. C. A . Ballon Bayona , C.N. Ferreira , Phys. Rev. D 78 , 026004 ( 2008 ). https://doi.org/10.1103/PhysRevD.78.026004. arXiv: 0801 .0305 [hep-th]
20. J.M. Maldacena , Adv. Theor. Math. Phys. 2 , 231 ( 1998 ). arXiv:hep-th/9711200
21. J.M. Maldacena , Int. J. Theor. Phys . 38 , 1113 ( 1999 )
22. O. Aharony , S.S. Gubser , J.M. Maldacena , H. Ooguri , Y. Oz , Phys. Rep . 323 , 183 ( 2000 ). arXiv:hep-th/9905111
23. S. Hawking , D.N. Page , Commun. Math. Phys. 87 , 577 ( 1983 )
24. P. Baekler , F.W. Hehl , Class. Quantum Gravity 28 , 215017 ( 2011 ). https://doi.org/10.1088/ 0264 -9381/28/21/215017. arXiv: 1105 .3504 [gr-qc]
25. P. Baekler , F.W. Hehl , J.M. Nester , Phys. Rev. D 83 , 024001 ( 2011 ). https://doi.org/10.1103/PhysRevD.83.024001. arXiv: 1009 .5112 [gr-qc]
26. K. Hashimoto , N. Iizuka , T. Kimura, Phys. Rev. D 91 , 086003 ( 2015 ). https://doi.org/10.1103/PhysRevD.91.086003. arXiv: 1304 .3126 [hep-th]
27. C.A.B. Bayona , C.N. Ferreira , V.J.V. Otoya , Class. Quantum Gravity 28 , 015011 ( 2011 ). https://doi.org/10.1088/ 0264 -9381/28/1/ 015011. arXiv: 1003 .5396 [hep-th]