{'fraction_non_alphanumeric': 0.03603918824352694, 'fraction_numerical': 0.009972008397480756, 'mean_word_length': 4.35503512880562, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Properties of global monopoles with an event horizon

We investigate the properties of global monopoles with an event horizon. We find that there is an unstable circular orbit even if a particle does not have an angular momentum when the core mass is negative. We also obtain the asymptotic form of solutions when the event horizon is much larger than the core radius of the monopole, and discuss if they could be a model of galactic halos.


I. INTRODUCTION
In unified theories, various kinds of topological defects have been predicted and may appear in cosmological phase transitions in the early universe. Therefore it is important to investigate such defects both theoretically and observationally. Concerning global monopoles, important facts were found recently: (i) Solutions with an event horizon exist [1,2,3]. (ii) Regular global monopoles coupled nonminimally to gravity have a stable circular orbit and may explain rotation curves in spiral galaxies, as shown by Nucamendi et al. (hereafter, NSS) [4].
The result (i) is interesting since the static solutions of regular global monopoles are always repulsive [5]. Moreover, black hole solutions are stable though topological charge is lost in the strict sense [6] and there are solutions with zero mass which are somewhat pathological [2]. Thus, we need to understand their properties. In particular, it would be interesting to investigate a particle motion around the horizon. This is one of our main concerns in this paper.
Possibility (ii) shows that global monopoles can be locally attractive in the nonminimally coupled theory of gravity. Although there have been many attempts to explain rotation curves in spiral galaxies, there is no definite one at present. Among solitonic objects, global monopoles have a remarkable property that energy density decreases with the distance r −2 [7], which may be desirable to explain flatness of rotation curves. To remove the unprefereble repulsive property of global monopoles, NSS introduced the nonminimal coupling and succeeded to obtain locally attractive solutions.
Both (i) and (ii) taking into consideration, we notice a possibility that global monopoles with an event horizon can explain rotation curves since they would be attractive. There are several advantages in this model com-pared with the NSS model. First, we need not require the nonminimal coupling, which are constrained astrophysically [8]. Secondly, they would also be model black holes in the central galaxies. Thirdly, the core mass can be chosen to be astronomically large, contrary to the NSS model, where the core mass is necessarily microscopic. Therefore it is important to study the properties of such global monopoles, and discuss whether or not they can be a realistic candidate as galactic halos, taking astrophysical bounds into account [9] . This paper is written as follows. In Sec. II, we explain our model and basic equations. In Sec. III, we investigate global monopoles with an event horizon in two situations separately: In Sec. III-A, we consider the case where the size of event horizon is comparable to the core radius of the monopole to compare with regular monopoles. In Sec. III-B, we consider the case where the size of an event horizon is astrophysically large. In Sec. IV, we denote concluding remarks and discuss problems concerning the restriction from observation.

II. BASIC EQUATIONS FOR NUMERICAL ANALYSIS
We begin with the action where G and Φ a are the gravitational constant and the real triplet Higgs field, respectively. The theoretical parameters v and λ are the vacuum expectation value and the self-coupling constant of the Higgs field, respectively. We assume that a space-time is static and spherically symmetric, in which the metric is written as where f (r) := 1 − 2Gm(r)/r. We adopt the hedgehog ansatz given by where r a is a unit radial vector. Under the above assumptions, the basic equations arē where ′ = d/dr and We have introduced the following dimensionless variables:r We assume the regular event horizon at r = r H .
The variables with subscript H are evaluated at the horizon.
We introduce the variablē and assume the boundary conditions at spatial infinity asm which means that the space-time is asymptotically "flat" with deficit angle. HereM corresponds to a core mass of the monopole, which determines a particle motion we will see below. We will obtain the black hole solutions numerically by solving Eqs.(4) -(6) iteratively with the boundary conditions (9), (10) and (12).

III. PROPERTIES
Space-time structure of global monopole black holes depends on the expectation value of the Higgs field [1]. Forv 2 < 1/2, there is a solution with asymptotically "flat" space-time. We concentrate on this realistic case.

A. small horizon
Typically, it is supposed that a global monopole has a nontrivial structure in the core r r core := 2/v √ λ [5], while the field is almost constant, h ∼ = 1, outside the core r r core . Therefore, we expect that a monopole black hole with a small horizon r H ∼ = r core (i.e.,r H ∼ = 1) may have new properties. First, we show field distributions of black hole solutions in Figs. 1 (a)r-h and (b)r-m core . We choosev = 0.6, 0.2 andr H = 0.3. In Fig. 1 (a), we find that the Higgs field has a nontrivial structure extending to ∼ 10 and it does not depend on the expectation value of the Higgs field, which is important in the later analysis. In this solution, h increases monotonically withr. Although the solutions where h is not monotonic exist [3], here we only consider monotonic solutions for simplicity. Fig. 1 (b) shows thatm core forv = 0.6 decreases withr much faster than that forv = 0.2. This is natural because of the factorv 2r in (11). The important point is thatm core becomes negative forv = 0.6 and positive forv = 0.2 in the asymptotic region. As we will discuss below, the sign ofm core at large r (i.e.,M ) determines the qualitative behavior of particle motions around the monopole.
Let us consider the geodesic equation of a test particle in the equatorial plane. In our coordinate system (2), this is expressed as where˙= d/dt and E is the energy of the particle per unit rest mass. V ef f is defined as where L is the angular momentum of the particle per unit rest mass.
On the other hand, the potential maximum cannot be evaluated from the asymptotic form of the solution because it is near the horizon if it exists, as shown by Fig.  2. However, we can discuss its existence as follows. If M < 0, V ef f decreases withr and approaches 1 − 2v 2 Asymptotically; since V ef f = 0 at the horizon, there is at least one local maximum at somer. IfM > 0, on the other hand, V ef f increases withr and approaches 1 − 2v 2 asymptotically; whether or not the potential maximum appear depends onL as in the Schwarzschild black hole. Thus, the sign ofM is important to determine the particle motion around the black hole.
Such a convex form of the potential for a L = 0 particle motion is characteristic of a global monopole black hole, and does not appear neither in a Schwarzschild black hole nor in a regular global monopole. In the case of a regular global monopole, the whole space-time is repulsive [5], which means that V ef f decreases monotonically. Fig. 3 shows the relation betweenr H andM for various v.M is negative in the limitr H → 0 as it is expected from the regular solutions. Asv increases, the region of M < 0 extends to largerr H . As it was pointed out in Ref. [2], there are solutions withM = 0. In this case, a potential minimumr − goes to infinity and a test particle cannot feel a black hole.

B. large horizon
NSS argued that global monopoles with the nonminimally coupled gravity may explain rotation curves in spiral galaxies [4]. However, the nonminimally coupled gravity has been constrained astrophysically. Moreover, in the NSS model the bound orbits exist only in the microscopic region r r core . Thus, it is desirable to seek for other possibilities.
On the other hand, Wetterich discussed the possibility that a massless scalar field may explain rotation curves of galactic halos [10]. In his solutions, however, physical boundary conditions were not taken into account. If we assume regularity at the center or an event horizon of a black hole, only a trivial solution remains. Nonexistence of nontrivial black hole solutions are guaranteed by no hair theorem [11]. In this sense, his model is also unrealistic.
Then, we turn to a global monopole with an event horizon. This model is free from the above difficulties existing in the NSS model. Furthermore, the existence of an event horizon is realistic because massive black holes are observed in the central regions of galaxies.
Let us consider astrophysical bounds from the mass density of monopoles in the universe at first. If we demand that mass density of monopoles should be less than 10 times of critical density [9], we have where n is the number density of monopoles. For definiteness, we choosev = 0.2 × 10 −4 and discuss the case where the event horizon is cosmological size. To see the structure of the Higgs field forr H ≫ 1, we plot r H -(1−h H ) in Fig. 4. We find h H ∼ = 1 and (1−h H ) ∼ = 1/r 2 H . We can check this relation analytically as follows. In the asymptotic region, supposing the asymptotic form, we have C 1 = 0 and C 2 = −1, which are consistent with our numerical results above. Because of this asymptotic behavior,M can be estimated by substituting h = 1 into Eq. (4). Then, we have the asymptotic relation, We show the relationr H -M in Fig. 5 forv = 0.2 × 10 −4 andv = 0.2, which confirms the relation (18). Actually, Fig. 3 shows that this approximation is fairly good even forr H ∼ 1 whenv is small. Let us consider particle motions. Setting h = 1, we have Sincev is small as constrained by (16), we find that the particle motion is practically the same as that in Schwarzschild black hole. As long as we consider an astronomical-sized event horizon, the effect of scalar fields on particle motions is negligible.

IV. CONCLUSION AND DISCUSSION
We investigated properties of global monopoles with an event horizon and revealed interesting features which had not been known so far. The main features of test particle motions are determined by the sign of the core mass; if it is negative and if the event horizon is as small as the core radius, there is an unstable circular orbit even for a particle with zero angular momentum. We also found the asymptotic form of the solutions when the event horizon is much larger than the core radius; the qualitative features of the monopole black hole is the same as that of the Schwarzschild black hole.
Although our model does not explain observed rotation curves very well, we obtain some lessons here. Our results indicates that massive scalar fields would encounter with the same difficulty as in our model. A typical mass scale of particle physics is so large that it generally contributes only to microscopic structure, whose size is of order of the inverse of the mass. In this sense, a massless scalar field considered by Wetterich [10] might be useful. Although his model itself does not satisfy physical boundary conditions neither of a black hole nor of a regular solution, interaction with matter may be a key ingredient to solve this problem. Including this possibility, we also want to consider other soliton-like models such as boson-fermion stars in future [12].