Gravitating Global k-monopole
Xing-hua Jin, Xin-zhou Li†and Dao-jun Liu
Shanghai United Center for Astrophysics(SUCA), Shanghai Normal University,
Shanghai 200234, China
School of Science, East China University of Science and Technology, 130 Meilong
Road, Shanghai, 200237, China
E-mail: †kychz@shnu.edu.cn
Abstract.
A gravitating global k-monopole produces a tiny gravitational field outside the
core in addition to a solid angular deficit in the k-field theory. As a new feature, the
gravitational field can be attractive or repulsive depending on the non-canonical kinetic
term.
PACS numbers: 11.27.+d, 11.10. Lm
http://arxiv.org/abs/0704.1685v2
Gravitating Global k-monopole 2
1. Introduction
The phase transition in the early universe could produce different kinds of topological
defects which have some important implications in cosmology[1]. Domain walls are two-
dimensional defects, and strings are one-dimensional defects. Point-like defects also arise
in same theories which undergo the spontaneous symmetry breaking, and they appears
as monopoles. The global monopole, which has divergent mass in flat spacetime, is one
of the most interesting defects. The idea that monopoles ought to exist has been proved
to be remarkably durable. Barriola and Vilenkin [2] firstly researched the characteristic
of global monopole in curved spacetime, or equivalently, its gravitational effects. When
one considers the gravity, the linearly divergent mass of global monopole has an effect
analogous to that of a deficit solid angle plus that of a tiny mass at the origin. Harari
and Loustò [3], and Shi and Li [4] have shown that this small gravitational potential
is actually repulsive. Furthermore, Li et al [5, 6, 7] have proposed a new class of cold
stars which are called D-stars (defect stars). One of the most important features of
such stars, comparing to Q-stars, is that the theory has monopole solutions when the
matter field is absent, which makes the D-stars behave very differently from the Q-
stars. The topological defects are also investigated in the Friedmann-Robertson-Walker
spacetime [8]. It is shown that the properties of global monopoles in asymptotically
dS/AdS spacetime [9] and the Brans-Dicke theory [10] are very different from those of
ordinary ones. The similar issue for the gravitational mass of composite monopole, i.e.,
global and local monopole has also been discussed [22].
The huge attractive force between global monopole M and antimonopole M̄
proposes that the monopole over-production problem does not exist, because the pair
annihilation is very efficient. Barriola and Vilenkin have shown that the radiative
lifetime of the pair is very short as they lose energy by Goldstone boson radiation [2]. No
serious attempt has made to develop an analytical model of the cosmological evolution of
a global monopole, so we are limited to the numerical simulations of evolution by Bennett
and Rhie [11]. In the σ-model approximation, the average number of monopoles per
horizon is NH ∼ 4. The gravitational field of global monopoles can lead to clustering
in matter, and later evolve into galaxies and clusters. The scale-invariant spectrum of
fluctuations has been given [11]. Furthermore, one can numerically obtain the microwave
background anisotropy (δT/T )rms patterns [12]. Comparing theoretical value to the
observed rms fluctuation, one can find the constraint of parameters in global monopole.
On the other hand, non-canonical kinetic terms are rather ordinary for effective field
theories. The k-field theory, in which the non-canonical kinetic terms are introduced in
the Lagrangian, have been recently investigated to serve as the inflaton in the inflation
scenario, which is so-called k-inflation [13], and to explain the current acceleration
of the universe and the cosmic coincidence problem, k-essence [14]. Armendariz-
Picon et al [15, 16] have discussed gravitationally bound static and spherically
symmetric configurations of k-essence fields. Another interesting application of k-
fields is topological defects, dubbed by k-defects [17]. Monopole [18] and vortex [19]
Gravitating Global k-monopole 3
of tachyon field, which as an example of k-field comes from string/M theory, have also
been investigated. The mass of global k-monopole diverges in flat spacetime, just as that
of standard global monopole, therefore, it is of more physical significance to consider
the gravitational effects of global k-monopole.
In this paper, we study the gravitational field of global k-monopole and derive the
solutions numerically and asymptotically. We find that the topological condition of
vacuum manifold for the formation of a k-monopole is identical to that of an ordinary
monopole, but their physical properties are disparate. Especially, we show that the
mass of k-monopole can be positive in some form of the non-canonical kinetic terms.
In other words, the gravitational field can be attractive or repulsive depending on the
non-canonical kinetic term.
2. Equations of Motion
We shall work within a particular model in units c = 1, where a global O(3) symmetry
is broken down to U(1) in the k-field theory. Its action is given by
M4K(X̃/M4)−
λ2(φ̃aφ̃a − σ̃2
, (1)
where κ = 8πG and λ is a dimensionless constant. In action (1), X̃ = 1
∂̃µφ̃
a∂̃µφ̃a
where φ̃a is the SO(3) triplet of goldstone field and σ̃0 is the symmetry breaking
scale with a dimension of mass. After setting the dimensionless quantities: x = Mx̃,
φa = φ̃a/M and σ0 = σ̃0/M , action (1) becomes
−g [K(X)− V (φ)] , (2)
where V (φ) = 1
λ2(φaφa − σ2
)2. The hedgedog configuration describing a global k-
monopole is
φa = σ0f(ρ)
, (3)
where xaxa = ρ2 and a = 1, 2, 3, so that we shall actually have a global k-monopole
solution if f → 1 at spatial infinity and f → 0 near the origin.
The static spherically symmetric metric can be written as
ds2 = B(ρ)dt2 −A(ρ)dρ2 − ρ2(dθ2 + sin2 θdϕ2) (4)
with the usual relation between the spherical coordinates ρ, θ, ϕ and the ”Cartesian”
coordinate xa. Introducing a dimensionless parameter r = σ0ρ, from (2) and (3), we
obtain the equations of motion for f as
f ′′ +
f ′ − 2
X ′f ′
− λ2f(f 2 − 1) = 0, (5)
where the prime denotes the derivative with respect to r, the dot denotes the derivative
with respect to X and X = −f2
− f ′2
. Since we only consider the static solution, positive
X and negative X are irrelevant each other. In this paper, we will assume K(X) to be
valid for negative X .
Gravitating Global k-monopole 4
The Einstein equation for k-monopole is
Gµν = κTµν (6)
where Tµν is the energy-momentum tensor for the action (2). The tt and rr components
of the Einstein equations now could be written as
= ǫ2T 0
= ǫ2T 1
, (8)
where
= −K + λ
(f 2 − 1)2 (9)
= −K + λ
(f 2 − 1)2 − K̇ f
and ǫ2 = κσ2
= 8πGσ2
is a dimensionless parameter.
3. k-monopole
Although the existence of global k-monopole, as well as the standard one, is guaranteed
by the symmetry-breaking potential, there exist the non-canonical kinetic term in k-
monopole which certainly leads to the appearance of a new scale in the action and the
mass parameter in the potential term. However, the non-canonical kinetic term is non-
trivial. At small gradients, it can be chosen to have a same asymptotical behavior with
that of the standard one, so that it ensures the standard manner of a small perturbations.
While at large gradient we choose it to have a different form from the standard one.
In the small X case, we assume that the kinetic term has the asymptotically
canonical behavior, which can avoid ”zero-kinetic problem”. If |X| ≪ 1, we have
K(X) ∼ Xα, and α < 1 then there is a singularity at X = 0; and α > 1 then the
system becomes non-dynamical at X = 0. For the monopole solution, it is easily found
that K(X) ∼ X at r ≫ 1. On the other hand, we assume that the modificatory kinetic
term K(X) ∼ Xα and α 6= 1 at |X| ≫ 1. One can easily obtain the equation of motion
inside the core of a global monopole after assuming that |X| ≫ 1 in the core of the
global monopole. The equations of motion are highly non-linear and cannot be solved
analytically. Next, we investigate the asymptotic behaviors of global monopole with
non-linear in X kinetic term. To be specific, we consider the following type of kinetic
K(X) = X − βX2, (11)
where β is a parameter of global k-monopole. It is easy to find that global k-monopole
will reduce to be the standard one when β = 0. It is easy to check whether the kinetic
term (11) satisfy the condition for the hyperbolicity [16, 20, 17]
2XK̈ + K̇
> 0, (12)
Gravitating Global k-monopole 5
which leads to a positive definite speed of sound for the small perturbations of the field.
The stability of solutions shows that for the case β < 0, the range 1
> X > 1
be excluded. However, this will not destroy the results which are carried out from the
case β > 0. We here only consider the cases for β > 0.
Using Eqs.(5)-(10), we get the asymptotic expression for A(r), B(r), and f(r) which
is valid near r = 0,
f(r) = f0r +
2 λ2 (−3 + ǫ2) + +42β ǫ2 f04
1 + 5β f0
) r3 (13)
(9 + 7β λ2) ǫ2 f0
2 + 36β2 ǫ2 f0
1 + 5β f0
) r3 +O(r4)
A(r) = 1 +
λ2 + 6 f0
2 + 9β f0
r2 +O(r3) (14)
B(r) = 1 +
λ2 − 9β f04
r2 +O(r3), (15)
where the undetermined coefficient f0 is characterized as the mass of k-monopole, which
can be determined in the numerical calculation.
In the region of r ≫ 1, similarly we can expand f(r), A(r) and B(r) as
f(r) = 1− 1
− 3− 2 ǫ
2 + 4β λ2
+O(r−5) (16)
A(r) =
1− ǫ2
(1− ǫ2)2
ǫ2(1− βλ2)
(1− ǫ2)2 λ2
(1− ǫ2)3
+ O(r−3)
B(r) = (1− ǫ2) +M∞
2 (1− βλ2)
+O(r−3), (18)
where the constant M∞ will be discussed in the following.
Using shooting method for boundary value problems, we get the numerical results
of the function f(r) which describes the configuration of global k-monopole. In Fig.1
we show the function f(r) for β = 0, β = 1, β = 5 and β = 10 respectively and for
given values of λ and ǫ. Obviously, the configuration of field f is not impressible to the
choice of the parameter β.
From Eqs.(13) and (16), it is easy to construct global k-monopole which has the
same asymptotic condition with standard global monopole, i.e., f will approach to zero
when r ≪ 1 and approach to unity when r ≫ 1.
Actually there is a general solution to Einstein equation with energy-momentum
tensor Tµν which takes the form as (9) and (10) for spherically symmetric metric (4)
A(r)−1 = 1−
[−K +
(f 2 − 1)2]r2dr (19)
B(r) = A(r)−1 exp
(K̇f ′2r)dr
. (20)
Gravitating Global k-monopole 6
0 5 10 15 20
Figure 1. The plot of f(r) as a function of r. Here we choose λ = 1, G = 1 and
ǫ = 0.001. The four curved lines are plotted when β = 0, β = 1, β = 5 and β = 10
respectively.
In terms of the dimensionless quantity ǫ, the metric coefficient A(r) and B(r) can be
formally integrated and read as
A(r)−1 = 1− ǫ2 −
2Gσ0MA(r)
B(r) = 1− ǫ2 − 2Gσ0MB(r)
. (22)
The small dimensionless parameter ǫ arise naturally from Einstein equations and clearly
ǫ2 describes a solid angular deficit of space-time.
A global k-monopole solution f should approach unity when r ≫ 1. If this
convergence is fast enough, then MA(r) and MB(r) will also rapidly converge to finite
values. Therefore, from Eqs.(16)-(18) we have the asymptotic expansions:
MA(r) = M∞ + 4πσ0
8πσ0 (−1 + ǫ2 + 2 β λ2)
+ O(r−5)
MB(r) = M∞ + 4πσ0
4πσ0 (−1 + ǫ2 − 4 β λ2)
+ O(r−5),
where M∞ ≡ limr→∞MA(r). One can easily find that the dependence on ǫ of the
asymptotic expansion for f(r) is very weak, in other words, the asymptotic behavior is
quite independent of the scale of symmetry breakdown σ0 up to value as large as the
planck scale. On the contrary, MA(r) depends obviously on σ0.
Gravitating Global k-monopole 7
0 5 10 15 20
Figure 2. The plot of MA(r)/σ0 as a function of r. Here we choose λ = 1, G = 1 and
ǫ = 0.001. The four curved lines are plotted when β = 0, β = 1, β = 5 and β = 10
respectively.
The numerical results of MA(r)/σ0 are shown in Fig.2 by shooting method for
boundary value problems where we choose λ = 1, G = 1 and ǫ = 0.001. From the
figure, we find that the mass of global k-monopole decrease to a negative asymptotic
value when r approaches infinity in the case that β = 0 and β = 1. While the mass will
be positive if β = 5 or β = 10. The asymptotic mass for the cases above are −19.15σ0,
−13.83σ0, 3.62σ0 and 22.14σ0 respectively. It is clear that the presence of parameter β,
which measures the degree of deviation of kinetic term from canonical one, affects the
effective mass of the global k-monopole significantly. It is not difficult to understand
this property. From Eqs.(11), (19) and (21), the mass function MA(r) can be expressed
explicitly as
MA(r)
= −r +
βX2 −X +
(f 2 − 1)2
r2dr. (25)
Obviously, the β-term in the integration has the positive contribution for the mass
function. From Fig.1, f (then X) is not sensitive to the value of β, so the greater
parameter β is chosen, the larger value MA(r) takes for a given r, and if β is greater
than some value, MA(r) will be positive for large r as Fig.2 shows. However, inside
the core, X varies slowly but f varies fast with respect to r, therefore, λ2-term in the
integration will become dominant as r decreases. This leads to two characteristics of the
mass curves which is also shown in Fig.2: (i) the mass curves with different β converge
gradually in the region near r = 0; (ii) in the case that β is large enough, MA(r) has a
minimum.
Gravitating Global k-monopole 8
To show the effect of solid defect angle, we then investigate the motion test particles
around a global k-monopole. It is a good approximation to take MA(r) as a constant in
the region far away from the core of the global k-monopole, since the the mass MA(r)
approaches very quickly to its asymptotic value. Therefore, we can consider the geodesic
equation in the metric (4) with
A(r)−1 = B(r) = 1− ǫ2 −
where M = σ0M∞. Solving the geodesic equation and introducing a dimensionless
quantaty u = GM/r, one will get the second order differentiating equation of u with
respect to ϕ [9, 21]
+ (1− ǫ2)u =
+ 3u2, (27)
where L is the angular momentum per unit of mass. When
ϕ ≪ 1, one have the
approximate solution of u
1− ǫ2
+ e cos
1− 3√
(1− ǫ2)3
, (28)
where e denotes the eccentricity. When a test particle rotates one loop around the global
k-monopole, the precession of it will be
∆ϕ = 6π
(1− ǫ2)3
ǫ2. (29)
The last term in Eq.(29) is the modificaiton comparing this result with that for the
precession around an ordinary star.
4. Conclusion
In summary, k-monopole could arise during the phase transition in the early universe.
We calculate the asymptotic solutions of global k-monopole in static spherically
symmetric spacetime, and find that the behavior of a k-monopole is similar to that
of a standard one. Although the choice of the parameter β, which measures the degree
of deviation of kinetic term from canonical one, have little influence on the configuration
of k-field φ, the effective mass of global k-monopole is affected significantly. The mass
might be negative or positive when different parameters β are chosen. This shows
that the gravitational field of the global k-monopole could be attractive or repulsive
depending on the different non-canonical kinetic term.
The configuration of a global k-monopole is more complicated than that of a
standard one. As for its cosmological evolution, we should not attempt to get the
analytical mode, instead we can only use numerical simulation. However, the energy
dominance of global k-monopole is in the region outside the core. We can roughly
estimate that global k-monopoles will result in the clustering in matter and evolve into
galaxies and clusters in a way similar to that of standard monopoles.
Gravitating Global k-monopole 9
Acknowledgement
This work is supported in part by National Natural Science Foundation of China under
Grant No. 10473007 and No. 10503002 and Shanghai Commission of Science and
Technology under Grant No. 06QA14039.
References
[1] Vilenkin A and Shellard E P S, 1994 Cosmic Strings and Other Topological Defects (Cambridge
Unversity Press, Cambridge, England)
[2] Barriola M and Vilenkin A, 1989 Phys. Rev. Lett. 63, 341
[3] Harari D and Loustò C, 1990 Phys. Rev.D42, 2626
[4] Shi X and Li X Z, 1991 Class. Quantum Grav. 8, 761
[5] Li X Z and Zhai X H, 1995 Phys. Lett. B364, 212
[6] Li J M and Li X Z, 1998 Chin. Phys. Lett.15, 3; Li X Z, Liu D J and Hao J G, 2002 Science in
China A45, 520
[7] Li X Z, Zhai X H and Chen G, 2000 Astropart. Phys. 13, 245
[8] Basu R, Guth A H and Vilenkin A, 1991 Phys. Rev. D44 340; Basu R and Vilenkin A, 1994 Phys.
Rev. D50 7150; Chen C, Cheng H, Li X Z and Zhai X H, 1996 Class. Quantum Grav. 13, 701
[9] Li X Z and Hao J G, 2002 Phys. Rev. D66, 107701; Hao J G and Li X Z, 2003 Class. Quantum
Grav. 20, 1703
[10] Li X Z and Lu J Z, 2000 Phys. Rev. D62, 107501
[11] Bennett D P and Rhie S H, 1990 Phys. Rev. Lett. 65, 1709
[12] Bennett D P and Rhie S H, 1993 Astrophys. J. 406, L7
[13] Armendariz-Picon C, Damour T, Mukhanov V, 1999 Phys.Lett. B458, 209
[14] Armendariz-Picon C, Mukhanov V, Steinhardt P J, 2000 Phys.Rev.Lett.85, 4438; Armendariz-
Picon C, Mukhanov V, Steinhardt P J, 2001 Phys. Rev. D63: 103510
[15] Armendariz-Picon C and Lim E A, 2005 JCAP 0508, 007
[16] Bilic N, Tupper G B and Viollier R D, 2006 JCAP 0602 013; Diez-Tejedor A and Feinstein A,
2006 Phys. Rev. D74 023530; Nucamendi U, Salgado M and Sudarsky D, 2000 Phys. Rev. Lett.
84 3037; Nucamendi U, Salgado M and Sudarsky D, 2001 Phys. Rev. D63 125016.
[17] Babichev E, 2006 Phys. Rev. D74 085004
[18] Li X Z and Liu D J, 2005 Int. J. Mod. Phys. A20, 5491
[19] Liu D J and Li X Z, 2003 Chin. Phys. Lett. 20, 1678
[20] Rendall A D, 2006 Class.Quant.Grav. 23, 1557.
[21] Wald R M, 1984 General Ralitivity (The university of Chicago Press, Chicago)
[22] Spinelly J, de Freitas U and Bezerra de Mello E R, 2002 Phys.Rev. D66, 024018.
	Introduction
	Equations of Motion
	k-monopole
	Conclusion

  A gravitating global k-monopole produces a tiny gravitational field outside
the core in addition to a solid angular deficit in the k-field theory. As a new
feature, the gravitational field can be attractive or repulsive depending on
the non-canonical kinetic term.

Introduction
The phase transition in the early universe could produce different kinds of topological
defects which have some important implications in cosmology[1]. Domain walls are two-
dimensional defects, and strings are one-dimensional defects. Point-like defects also arise
in same theories which undergo the spontaneous symmetry breaking, and they appears
as monopoles. The global monopole, which has divergent mass in flat spacetime, is one
of the most interesting defects. The idea that monopoles ought to exist has been proved
to be remarkably durable. Barriola and Vilenkin [2] firstly researched the characteristic
of global monopole in curved spacetime, or equivalently, its gravitational effects. When
one considers the gravity, the linearly divergent mass of global monopole has an effect
analogous to that of a deficit solid angle plus that of a tiny mass at the origin. Harari
and Loustò [3], and Shi and Li [4] have shown that this small gravitational potential
is actually repulsive. Furthermore, Li et al [5, 6, 7] have proposed a new class of cold
stars which are called D-stars (defect stars). One of the most important features of
such stars, comparing to Q-stars, is that the theory has monopole solutions when the
matter field is absent, which makes the D-stars behave very differently from the Q-
stars. The topological defects are also investigated in the Friedmann-Robertson-Walker
spacetime [8]. It is shown that the properties of global monopoles in asymptotically
dS/AdS spacetime [9] and the Brans-Dicke theory [10] are very different from those of
ordinary ones. The similar issue for the gravitational mass of composite monopole, i.e.,
global and local monopole has also been discussed [22].
The huge attractive force between global monopole M and antimonopole M̄
proposes that the monopole over-production problem does not exist, because the pair
annihilation is very efficient. Barriola and Vilenkin have shown that the radiative
lifetime of the pair is very short as they lose energy by Goldstone boson radiation [2]. No
serious attempt has made to develop an analytical model of the cosmological evolution of
a global monopole, so we are limited to the numerical simulations of evolution by Bennett
and Rhie [11]. In the σ-model approximation, the average number of monopoles per
horizon is NH ∼ 4. The gravitational field of global monopoles can lead to clustering
in matter, and later evolve into galaxies and clusters. The scale-invariant spectrum of
fluctuations has been given [11]. Furthermore, one can numerically obtain the microwave
background anisotropy (δT/T )rms patterns [12]. Comparing theoretical value to the
observed rms fluctuation, one can find the constraint of parameters in global monopole.
On the other hand, non-canonical kinetic terms are rather ordinary for effective field
theories. The k-field theory, in which the non-canonical kinetic terms are introduced in
the Lagrangian, have been recently investigated to serve as the inflaton in the inflation
scenario, which is so-called k-inflation [13], and to explain the current acceleration
of the universe and the cosmic coincidence problem, k-essence [14]. Armendariz-
Picon et al [15, 16] have discussed gravitationally bound static and spherically
symmetric configurations of k-essence fields. Another interesting application of k-
fields is topological defects, dubbed by k-defects [17]. Monopole [18] and vortex [19]
Gravitating Global k-monopole 3
of tachyon field, which as an example of k-field comes from string/M theory, have also
been investigated. The mass of global k-monopole diverges in flat spacetime, just as that
of standard global monopole, therefore, it is of more physical significance to consider
the gravitational effects of global k-monopole.
In this paper, we study the gravitational field of global k-monopole and derive the
solutions numerically and asymptotically. We find that the topological condition of
vacuum manifold for the formation of a k-monopole is identical to that of an ordinary
monopole, but their physical properties are disparate. Especially, we show that the
mass of k-monopole can be positive in some form of the non-canonical kinetic terms.
In other words, the gravitational field can be attractive or repulsive depending on the
non-canonical kinetic term.
2. Equations of Motion
We shall work within a particular model in units c = 1, where a global O(3) symmetry
is broken down to U(1) in the k-field theory. Its action is given by
M4K(X̃/M4)−
λ2(φ̃aφ̃a − σ̃2
, (1)
where κ = 8πG and λ is a dimensionless constant. In action (1), X̃ = 1
∂̃µφ̃
a∂̃µφ̃a
where φ̃a is the SO(3) triplet of goldstone field and σ̃0 is the symmetry breaking
scale with a dimension of mass. After setting the dimensionless quantities: x = Mx̃,
φa = φ̃a/M and σ0 = σ̃0/M , action (1) becomes
−g [K(X)− V (φ)] , (2)
where V (φ) = 1
λ2(φaφa − σ2
)2. The hedgedog configuration describing a global k-
monopole is
φa = σ0f(ρ)
, (3)
where xaxa = ρ2 and a = 1, 2, 3, so that we shall actually have a global k-monopole
solution if f → 1 at spatial infinity and f → 0 near the origin.
The static spherically symmetric metric can be written as
ds2 = B(ρ)dt2 −A(ρ)dρ2 − ρ2(dθ2 + sin2 θdϕ2) (4)
with the usual relation between the spherical coordinates ρ, θ, ϕ and the ”Cartesian”
coordinate xa. Introducing a dimensionless parameter r = σ0ρ, from (2) and (3), we
obtain the equations of motion for f as
f ′′ +
f ′ − 2
X ′f ′
− λ2f(f 2 − 1) = 0, (5)
where the prime denotes the derivative with respect to r, the dot denotes the derivative
with respect to X and X = −f2
− f ′2
. Since we only consider the static solution, positive
X and negative X are irrelevant each other. In this paper, we will assume K(X) to be
valid for negative X .
Gravitating Global k-monopole 4
The Einstein equation for k-monopole is
Gµν = κTµν (6)
where Tµν is the energy-momentum tensor for the action (2). The tt and rr components
of the Einstein equations now could be written as
= ǫ2T 0
= ǫ2T 1
, (8)
where
= −K + λ
(f 2 − 1)2 (9)
= −K + λ
(f 2 − 1)2 − K̇ f
and ǫ2 = κσ2
= 8πGσ2
is a dimensionless parameter.
3. k-monopole
Although the existence of global k-monopole, as well as the standard one, is guaranteed
by the symmetry-breaking potential, there exist the non-canonical kinetic term in k-
monopole which certainly leads to the appearance of a new scale in the action and the
mass parameter in the potential term. However, the non-canonical kinetic term is non-
trivial. At small gradients, it can be chosen to have a same asymptotical behavior with
that of the standard one, so that it ensures the standard manner of a small perturbations.
While at large gradient we choose it to have a different form from the standard one.
In the small X case, we assume that the kinetic term has the asymptotically
canonical behavior, which can avoid ”zero-kinetic problem”. If |X| ≪ 1, we have
K(X) ∼ Xα, and α < 1 then there is a singularity at X = 0; and α > 1 then the
system becomes non-dynamical at X = 0. For the monopole solution, it is easily found
that K(X) ∼ X at r ≫ 1. On the other hand, we assume that the modificatory kinetic
term K(X) ∼ Xα and α 6= 1 at |X| ≫ 1. One can easily obtain the equation of motion
inside the core of a global monopole after assuming that |X| ≫ 1 in the core of the
global monopole. The equations of motion are highly non-linear and cannot be solved
analytically. Next, we investigate the asymptotic behaviors of global monopole with
non-linear in X kinetic term. To be specific, we consider the following type of kinetic
K(X) = X − βX2, (11)
where β is a parameter of global k-monopole. It is easy to find that global k-monopole
will reduce to be the standard one when β = 0. It is easy to check whether the kinetic
term (11) satisfy the condition for the hyperbolicity [16, 20, 17]
2XK̈ + K̇
> 0, (12)
Gravitating Global k-monopole 5
which leads to a positive definite speed of sound for the small perturbations of the field.
The stability of solutions shows that for the case β < 0, the range 1
> X > 1
be excluded. However, this will not destroy the results which are carried out from the
case β > 0. We here only consider the cases for β > 0.
Using Eqs.(5)-(10), we get the asymptotic expression for A(r), B(r), and f(r) which
is valid near r = 0,
f(r) = f0r +
2 λ2 (−3 + ǫ2) + +42β ǫ2 f04
1 + 5β f0
) r3 (13)
(9 + 7β λ2) ǫ2 f0
2 + 36β2 ǫ2 f0
1 + 5β f0
) r3 +O(r4)
A(r) = 1 +
λ2 + 6 f0
2 + 9β f0
r2 +O(r3) (14)
B(r) = 1 +
λ2 − 9β f04
r2 +O(r3), (15)
where the undetermined coefficient f0 is characterized as the mass of k-monopole, which
can be determined in the numerical calculation.
In the region of r ≫ 1, similarly we can expand f(r), A(r) and B(r) as
f(r) = 1− 1
− 3− 2 ǫ
2 + 4β λ2
+O(r−5) (16)
A(r) =
1− ǫ2
(1− ǫ2)2
ǫ2(1− βλ2)
(1− ǫ2)2 λ2
(1− ǫ2)3
+ O(r−3)
B(r) = (1− ǫ2) +M∞
2 (1− βλ2)
+O(r−3), (18)
where the constant M∞ will be discussed in the following.
Using shooting method for boundary value problems, we get the numerical results
of the function f(r) which describes the configuration of global k-monopole. In Fig.1
we show the function f(r) for β = 0, β = 1, β = 5 and β = 10 respectively and for
given values of λ and ǫ. Obviously, the configuration of field f is not impressible to the
choice of the parameter β.
From Eqs.(13) and (16), it is easy to construct global k-monopole which has the
same asymptotic condition with standard global monopole, i.e., f will approach to zero
when r ≪ 1 and approach to unity when r ≫ 1.
Actually there is a general solution to Einstein equation with energy-momentum
tensor Tµν which takes the form as (9) and (10) for spherically symmetric metric (4)
A(r)−1 = 1−
[−K +
(f 2 − 1)2]r2dr (19)
B(r) = A(r)−1 exp
(K̇f ′2r)dr
. (20)
Gravitating Global k-monopole 6
0 5 10 15 20
Figure 1. The plot of f(r) as a function of r. Here we choose λ = 1, G = 1 and
ǫ = 0.001. The four curved lines are plotted when β = 0, β = 1, β = 5 and β = 10
respectively.
In terms of the dimensionless quantity ǫ, the metric coefficient A(r) and B(r) can be
formally integrated and read as
A(r)−1 = 1− ǫ2 −
2Gσ0MA(r)
B(r) = 1− ǫ2 − 2Gσ0MB(r)
. (22)
The small dimensionless parameter ǫ arise naturally from Einstein equations and clearly
ǫ2 describes a solid angular deficit of space-time.
A global k-monopole solution f should approach unity when r ≫ 1. If this
convergence is fast enough, then MA(r) and MB(r) will also rapidly converge to finite
values. Therefore, from Eqs.(16)-(18) we have the asymptotic expansions:
MA(r) = M∞ + 4πσ0
8πσ0 (−1 + ǫ2 + 2 β λ2)
+ O(r−5)
MB(r) = M∞ + 4πσ0
4πσ0 (−1 + ǫ2 − 4 β λ2)
+ O(r−5),
where M∞ ≡ limr→∞MA(r). One can easily find that the dependence on ǫ of the
asymptotic expansion for f(r) is very weak, in other words, the asymptotic behavior is
quite independent of the scale of symmetry breakdown σ0 up to value as large as the
planck scale. On the contrary, MA(r) depends obviously on σ0.
Gravitating Global k-monopole 7
0 5 10 15 20
Figure 2. The plot of MA(r)/σ0 as a function of r. Here we choose λ = 1, G = 1 and
ǫ = 0.001. The four curved lines are plotted when β = 0, β = 1, β = 5 and β = 10
respectively.
The numerical results of MA(r)/σ0 are shown in Fig.2 by shooting method for
boundary value problems where we choose λ = 1, G = 1 and ǫ = 0.001. From the
figure, we find that the mass of global k-monopole decrease to a negative asymptotic
value when r approaches infinity in the case that β = 0 and β = 1. While the mass will
be positive if β = 5 or β = 10. The asymptotic mass for the cases above are −19.15σ0,
−13.83σ0, 3.62σ0 and 22.14σ0 respectively. It is clear that the presence of parameter β,
which measures the degree of deviation of kinetic term from canonical one, affects the
effective mass of the global k-monopole significantly. It is not difficult to understand
this property. From Eqs.(11), (19) and (21), the mass function MA(r) can be expressed
explicitly as
MA(r)
= −r +
βX2 −X +
(f 2 − 1)2
r2dr. (25)
Obviously, the β-term in the integration has the positive contribution for the mass
function. From Fig.1, f (then X) is not sensitive to the value of β, so the greater
parameter β is chosen, the larger value MA(r) takes for a given r, and if β is greater
than some value, MA(r) will be positive for large r as Fig.2 shows. However, inside
the core, X varies slowly but f varies fast with respect to r, therefore, λ2-term in the
integration will become dominant as r decreases. This leads to two characteristics of the
mass curves which is also shown in Fig.2: (i) the mass curves with different β converge
gradually in the region near r = 0; (ii) in the case that β is large enough, MA(r) has a
minimum.
Gravitating Global k-monopole 8
To show the effect of solid defect angle, we then investigate the motion test particles
around a global k-monopole. It is a good approximation to take MA(r) as a constant in
the region far away from the core of the global k-monopole, since the the mass MA(r)
approaches very quickly to its asymptotic value. Therefore, we can consider the geodesic
equation in the metric (4) with
A(r)−1 = B(r) = 1− ǫ2 −
where M = σ0M∞. Solving the geodesic equation and introducing a dimensionless
quantaty u = GM/r, one will get the second order differentiating equation of u with
respect to ϕ [9, 21]
+ (1− ǫ2)u =
+ 3u2, (27)
where L is the angular momentum per unit of mass. When
ϕ ≪ 1, one have the
approximate solution of u
1− ǫ2
+ e cos
1− 3√
(1− ǫ2)3
, (28)
where e denotes the eccentricity. When a test particle rotates one loop around the global
k-monopole, the precession of it will be
∆ϕ = 6π
(1− ǫ2)3
ǫ2. (29)
The last term in Eq.(29) is the modificaiton comparing this result with that for the
precession around an ordinary star.
4. Conclusion
In summary, k-monopole could arise during the phase transition in the early universe.
We calculate the asymptotic solutions of global k-monopole in static spherically
symmetric spacetime, and find that the behavior of a k-monopole is similar to that
of a standard one. Although the choice of the parameter β, which measures the degree
of deviation of kinetic term from canonical one, have little influence on the configuration
of k-field φ, the effective mass of global k-monopole is affected significantly. The mass
might be negative or positive when different parameters β are chosen. This shows
that the gravitational field of the global k-monopole could be attractive or repulsive
depending on the different non-canonical kinetic term.
The configuration of a global k-monopole is more complicated than that of a
standard one. As for its cosmological evolution, we should not attempt to get the
analytical mode, instead we can only use numerical simulation. However, the energy
dominance of global k-monopole is in the region outside the core. We can roughly
estimate that global k-monopoles will result in the clustering in matter and evolve into
galaxies and clusters in a way similar to that of standard monopoles.
Gravitating Global k-monopole 9
Acknowledgement
This work is supported in part by National Natural Science Foundation of China under
Grant No. 10473007 and No. 10503002 and Shanghai Commission of Science and
Technology under Grant No. 06QA14039.
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	Introduction
	Equations of Motion
	k-monopole
	Conclusion
