Non-minimal Wu-Yang wormhole
Non-minimal Wu-Yang wormhole
A. B. Balakin,1, ∗ S. V. Sushkov,1, 2, † and A. E. Zayats1, ‡
Department of General Relativity and Gravitation,
Kazan State University, Kremlevskaya str. 18, Kazan 420008, Russia
Department of Mathematics, Tatar State University of Humanities and Education,
Tatarstan str. 2, Kazan 420021, Russia
We discuss exact solutions of three-parameter non-minimal Einstein-Yang-Mills model, which
describe the wormholes of a new type. These wormholes are considered to be supported by SU(2)-
symmetric Yang-Mills field, non-minimally coupled to gravity, the Wu-Yang ansatz for the gauge
field being used. We distinguish between regular solutions, describing traversable non-minimal Wu-
Yang wormholes, and black wormholes possessing one or two event horizons. The relation between
the asymptotic mass of the regular traversable Wu-Yang wormhole and its throat radius is analysed.
PACS numbers: 04.20.Jb, 14.80.Hv, 04.20.Gz
Keywords: Einstein-Yang-Mills theory, non-minimal coupling, traversable wormhole
I. INTRODUCTION
Wormholes are topological handles in spacetime linking widely separated regions of a single universe, or “bridges”
joining two different spacetimes [1]. Recent interest in these configurations has been initiated by Morris and Thorne
[2]. These authors constructed and investigated a class of objects they referred to as “traversable wormholes”.
The central feature of wormhole physics is the fact that traversable wormholes are accompanied by an unavoidable
violation of the null energy condition, i.e., the matter threading the wormhole’s throat has to be possessed of “exotic”
properties [2, 3]. The known classical matter does satisfy the usual energy conditions, hence physical models providing
the existence of wormholes must include hypothetical forms of matter. Various models of such kind have been
considered in the literature, among them scalar fields [4]; wormhole solutions in semi-classical gravity [5]; solutions in
Brans-Dicke theory [6]; wormholes on the brane [7]; wormholes supported by matter with an exotic equation of state,
namely, phantom energy [8], the generalized Chaplygin gas [9], tachyon matter [10], etc [11, 12].
The electromagnetic field and the non-Abelian gauge field can also be considered as sources for wormholes when
they satisfy the necessary unusual energy conditions. Such possibility can in principle appear if one considers nonlinear
electrodynamics [13] or takes into account the non-minimal coupling of gravity with vector-type fields, i.e., with the
non-Abelian Yang-Mills field, or Maxwell field. The non-minimal Einstein-Maxwell theory has been elaborated in
detail in both linear (see, e.g., [14, 15] for a review) and non-linear (see, [16]) versions. As for the non-minimal
Einstein-Yang-Mills theory, two concepts to derive the master equations are known. The first one is a dimensional
reduction of the Gauss-Bonnet action [17], this model contains one coupling parameter. The second concept is a non-
Abelian generalization of the non-minimal non-linear Einstein-Maxwell theory [16]. We will follow the latter approach
and construct a three-parameter non-minimal model being linear in the curvature by analogy with the well-known
model proposed by Drummond and Hathrell for linear electrodynamics [18]. Three coupling constants q1, q2, and q3
of the model are shown to introduce a new specific radius associated with the radius a of the wormhole throat.
In this work we focus on the example of exact solution of the non-minimal three-parameter EYM model describing
the wormhole of a new type, namely, non-minimal wormhole. It can also be indicated as non-minimal Wu-Yang
wormhole, since the solution of the non-minimally extended Yang-Mills subsystem of the total self-consistent EYM
system of equations is the direct analog of the Wu-Yang monopole [19].
The paper is organized as follows. In Sec. II we briefly describe the formalism of three-parameter non-minimal
Einstein-Yang-Mills model. In Sec. III, Subsect. A we adapt this model for the case of static spherically symmetric
field configuration, present the exact solution of the Wu-Yang type to the gauge field equations and formulate two
key equations for two metric functions σ(r) and N(r). In Subsect. B, C and D we discuss the details of the three-
parameter family of exact solutions for the function σ(r). Sec. IV is devoted to the analysis of the solution describing
the non-minimal Wu-Yang wormhole. Conclusions are formulated in the last section.
∗Electronic address: Alexander.Balakin@ksu.ru
†Electronic address: sergey˙sushkov@mail.ru; Sergey.Sushkov@ksu.ru
‡Electronic address: Alexei.Zayats@ksu.ru
http://arxiv.org/abs/0704.1224v2
mailto:Alexander.Balakin@ksu.ru
mailto:sergey_sushkov@mail.ru; Sergey.Sushkov@ksu.ru
mailto:Alexei.Zayats@ksu.ru
II. NON-MINIMAL EINSTEIN-YANG-MILLS MODEL
The action of the three-parameter non-minimal Einstein-Yang-Mills model has the form1
SNMEYM =
ik(a) +
χikmnF
, (1)
where g = det(gik) is the determinant of a metric tensor gik, and R is the Ricci scalar. The Latin indices without
parentheses run from 0 to 3, the summation with respect to the repeated group indices (a) is implied. The tensor
χikmn, indicated in [16] as non-minimal susceptibility tensor, is defined as follows:
χikmn ≡
R (gimgkn − gingkm) +
(Rimgkn −Ringkm +Rkngim −Rkmgin) + q3Rikmn . (2)
Here Rik and Rikmn are the Ricci and Riemann tensors, respectively, and q1, q2, q3 are the phenomenological param-
eters describing the non-minimal coupling of the Yang-Mills and gravitational fields. Following [20], we consider the
Yang-Mills field, Fmn, to take the values in the Lie algebra of the gauge group SU(2):
Fmn = −iGF (a)mnt(a) , Am = −iGA(a)m t(a) . (3)
Here t(a) are Hermitian traceless generators of SU(2) group, G is a constant of gauge interaction, and the group index
(a) runs from 1 to 3. The generators t(a) satisfy the commutation relations:
[t(a), t(b)] = i ε(a)(b)(c)t(c) , (4)
where ε(a)(b)(c) is the completely antisymmetric symbol with ε(1)(2)(3) = 1. The Yang-Mills field potential, Ai, and
strength field, Fik, are coupled by the relation
Fik = ∂iAk − ∂kAi + [Ai ,Ak] , (5)
which guarantees that the equation
D̂lFik + D̂kFli + D̂iFkl = 0 (6)
turns into identity. Here the symbol D̂k denotes the gauge invariant derivative
D̂i ≡ ∇i + [Ai , ] , (7)
and ∇m is a covariant spacetime derivative.
The variation of the action (1) with respect to Yang-Mills potential A
i yields
ik ≡ ∇kHik +
= 0 , Hik = Fik + χikmnFmn . (8)
The tensor Hik is a non-Abelian analog of the induction tensor known in the electrodynamics [21], and thus χikmn
can be considered as a non-minimal susceptibility tensor [16]. The variation of the action with respect to the metric
gik yields
Rik −
R gik = 8π T
(eff)
ik . (9)
The effective stress-energy tensor T
(eff)
ik can be divided into four parts:
(eff)
ik = T
ik + q1T
ik + q2T
ik + q3T
(III)
ik . (10)
The first term T
mn(a) − F (a)in F
k , (11)
1 Hereafter we use the units c = G = ~= 1.
is a stress-energy tensor of the pure Yang-Mills field. The definitions of other three tensors relate to the corresponding
coupling constants q1, q2, q3:
ik = RT
mn(a) +
D̂iD̂k − gikD̂lD̂l
F (a)mnF
mn(a)
, (12)
ik = −
D̂mD̂l
Fmn(a)F l (a)n
−RlmFmn(a)F l (a)n
− F ln(a)
kn +RklF
−RmnF (a)im F
D̂mD̂m
ln(a)
+ D̂k
ln(a)
, (13)
(III)
mnlsF (a)mnF
F ls(a)
i Rknls + F
k Rinls
D̂mD̂n
k + F
. (14)
The tensor T
(eff)
ik satisfies the conservation law ∇kT
(eff)
ik = 0. The self-consistent system of equations (8) and (9)
with (10)-(14) is a direct non-Abelian generalization of the three-parameter non-minimal Einstein-Maxwell model
discussed in [16]. This system can also be considered as one of the variants of a non-minimal generalization of the
Einstein-Yang-Mills model.
III. EXACT SOLUTIONS OF THE STATIC MODEL WITH SPHERICAL SYMMETRY
A. Master equations
Let us take the metric of a static spherically symmetric spacetime in the form to be especially convenient for
studying a wormhole geometry:
ds2 = σ2Ndt2 − dr
r2 + a2
dθ2 + sin2 θdϕ2
, (15)
where the metric functions σ andN depend only on r. The properties of traversable wormholes dictate some additional
requirements for the metric (15), which were in great detail discussed in [1, 2]. In particular, we note that
(i) the radial coordinate r runs from −∞ to +∞. Two asymptotical regions r = −∞ and r = +∞ are connected
by the wormhole’s throat which has the radius a and is located at r = 0.
(ii) Since the spacetime of a traversable wormhole has neither singularities nor event horizons, the metric components
gtt = σ
2N and −grr = 1/N should be regular and positive everywhere. Note that, in particular, this means
that N(r) is positive defined, both σ(r) and N(r) are finite, and neither σ(r) nor N(r) can take zero values.
(iii) In addition, one may demand the asymptotical flatness of the wormhole spacetime at r = ±∞. This is guaranteed
provided the following boundary conditions for the functions σ and N are satisfied:
σ2 (±∞) = 1 , N (±∞) = 1. (16)
Below we will search for solutions of the non-minimal Einstein-Yang-Mills model, which satisfy the listed requirements.
The non-minimal Yang-Mills equations (8) are satisfied identically, when the gauge field is parameterized as [22, 23]
A0 = Ar = 0 , Aθ = itϕ , Aϕ = −iν sin θ tθ , (17)
which is known to be the so-called Wu-Yang monopole solution [19]. The parameter ν is a non-vanishing integer, tr,
tθ and tϕ are the position-dependent generators of the SU(2) group:
tr = cos νϕ sin θ t(1) + sin νϕ sin θ t(2) + cos θ t(3),
tθ = ∂θtr, tϕ =
ν sin θ
∂ϕtr, (18)
which satisfy the following commutation rules
[tr, tθ] = i tϕ, [tθ, tϕ] = i tr, [tϕ, tr] = i tθ. (19)
The field strength tensor Fik has only one non-vanishing component
Fθϕ = iν sin θ tr . (20)
Since the effective energy-momentum tensor T
(eff)
ik is divergence-free, the Einstein equations for the spherical symmetric
metric (15) are known to be effectively reduced to the two key equations, say, for equations with i = k = 0 and
i = k = r. The components G 00 and G
r of the Einstein tensor G
i = R
i − 12δ
i R are
G 00 =
1− rN ′ −N
r2 + a2
− N a
(r2 + a2)2
, (21)
G rr =
1− rN ′ −N
r2 + a2
− 2rNσ
σ(r2 + a2)
(r2 + a2)2
. (22)
The corresponding components of the effective energy-momentum tensor (see (11)-(14)) take the form
T 00 =
2(r2 + a2)2
− 2q1Na
(r2 + a2)4
− q1N
′r + q1 + q2 + q3
(r2 + a2)3
(13q1 + 4q2 + q3)Nr
(r2 + a2)4
, (23)
T rr =
2(r2 + a2)2
(r2 + a2)3
′r + q1 + q2 + q3 +
2q1Nrσ
− (7q1 + 4q2 + q3)Nr
(r2 + a2)4
. (24)
The difference G 00−G rr=8π
T 00−T rr
of these equations can be transformed into the decoupled equation for σ(r) only
1− κq1
(r2 + a2)2
(r2 + a2)
(r2 + a2)3
(10q1 + 4q2 + q3)r
2 − q1a2
, (25)
where κ is a new charge parameter with the dimensionality of area, κ = 8πν2/G2. Solving this equation one can find
the function σ(r) in the explicit form for arbitrary values of parameters q1, q2 and q3:
σ(r) =
r2 + a2
1− κq1
(r2 + a2)2
] 10q1+4q2+q3
. (26)
Excluding the function σ from the Einstein equation with i = k = 0, we obtain the equation for N(r) only:
(r2 + a2)2
N ′ +N
(r2 + a2)
κ(13q1 + 4q2 + q3)
(r2 + a2)2
κa2(15q1 + 4q2 + q3)
(r2 + a2)3
= 1− κ
2(r2 + a2)
κ(q1 + q2 + q3)
(r2 + a2)2
. (27)
The solution of Eq. (27) can be clearly represented in quadratures. Note that the coefficient Θ(r) ≡
1− κq1(r2 + a2)−2
in front of the first (highest) derivative in both differential equations (25) and (27) can take,
in principle, zero values depending on the sign of the guiding parameter q1. Thus, searching for the solutions of these
equations, we have to distinguish three qualitatively different cases q1 < 0, q1 = 0 and q1 > 0.
B. The case q1 < 0
For negative q1 the coefficient Θ(r) has only one root, r = 0. In this case, the function σ(r) takes the form
σ(r) =
r2 + a2
κ|q1|
(r2 + a2)2
] 10|q1|−4q2−q3
4|q1|
. (28)
Notice that σ(r) given by Eq. (28) turns into zero at r = 0, i.e., σ(0) = 0. This violates the condition (ii), and so this
solution cannot describe a traversable wormhole. Note also that
−g(0) = 0, where g = −σ2(r2 + a2)2 sin2 θ is the
determinant of gik. This means that the chosen coordinate system is ill-defined at r = 0, and well-defined only in the
range (0,+∞) (or, equivalently, in (−∞, 0)).
C. The case q1 = 0
In this case Θ(r) = r, and so r = 0 is the only root of Θ(r) as in the case q1 < 0. The solution for σ(r) transforms
now into
σ(r) =
r2 + a2
−κ(4q2 + q3)
(r2 + a2)2
. (29)
The function σ(r) given by Eq. (29) turns into zero at r = 0, i.e., σ(0) = 0. This means that the case q1 = 0 does
not admit the existence of traversable wormholes.
D. The case q1 > 0
For positive q1 the number of real roots of Θ(r) depends on the value β ≡ (κq1)1/4. In case β > a, Θ(r) has three
real roots, namely, r = 0, r = ±r∗, where
β2 − a2 . (30)
For β = a the roots r = ±r∗ coincide with r = 0, and for β < a one has only one real root r = 0. Below we consider
each case separately.
I. β < a. Rewrite the solution (26) as follows
σ(r) =
r2 + a2
(r2 + a2 − β2)(r2 + a2 + β2)
(r2 + a2)2
] 10q1+4q2+q3
. (31)
It is clear that due to the condition β < a the expression in square brackets in Eq. (31) is positive for all r. Thus, for
all values of the power parameter (10q1 + 4q2 + q3)/4q1 the sign of the function σ(r) inherits the sign of r, and turns
into zero at r = 0, i.e., σ(0) = 0. As in previous cases, this means that traversable wormholes do not exist.
II. β > a. In this case the expression in square brackets in Eq. (31) vanishes, when r = r∗ ≡ (β2 − a2)1/2. Now,
depending on the sign of the power parameter (10q1 + 4q2 + q3)/4q1, the solution σ(r) turns into zero or tends to
infinity at r∗. When 10q1+4q2+q3 = 0, one has again σ(0) = 0. Thus, the case β > a also does not admit traversable
wormholes.
III. β = a (or, equivalently, q1 = a
4/κ). It will be convenient to rewrite the solution (31) for σ(r) in the following
form:
σ(r) =
r2 + a2
r2 + 2a2
r2(r2 + 2a2)
(r2 + a2)2
] 12q1+4q2+q3
. (32)
Now the critical point of interest is r = 0. The behavior of σ(r) near r = 0 essentially depends on the sign of the
new power parameter, namely, 12q1 + 4q2 + q3. In particular, for 12q1 + 4q2 + q3 > 0 one has σ(0) = 0, and for
12q1 + 4q2 + q3 < 0 one has σ(0) = ∞. Such behavior of σ(r) excludes traversable wormholes. Consider the last
particular case, when this parameter vanishes, 12q1 + 4q2 + q3 = 0. Now we obtain
σ(r) =
r2 + a2
r2 + 2a2
. (33)
The function σ(r) given by Eq. (33) is regular and positive in the whole interval (−∞,+∞), moreover, σ(±∞) = 1.
Thus, σ(r) given by Eq. (33) satisfies the necessary conditions (i-iii) and the corresponding field configuration can be
considered as a candidate in searching for traversable wormholes. In the next section we will complete the solution
for σ(r) by the solution for N(r) and discuss the properties of the non-minimal Wu-Yang wormhole solution.
IV. NON-MINIMAL WU-YANG WORMHOLE
In this section we consider in more details the special case corresponding to the following choice of the non-minimal
coupling parameters q1, q2, q3:
q1 = a
4/κ, 12q1 + 4q2 + q3 = 0. (34)
Then, the equation (27) can be easily integrated in the quadratures to give
N(r) =
(r2 + a2)3/2
r2 + 2a2
(x2 + a2)3/2
x2 + 2a2
x4 + 2x2
a2 − κ
10a4 +
+ 3κq2
 , (35)
where C is a constant of integration. Note that for arbitrary values of a, q2, and C the function N(r) given by Eq.
(35) satisfies the boundary condition N(±∞) = 1. Near r = 0 the solution N(r) is, generally speaking, divergent.
Such behavior of N(r) is unsuitable for description of traversable wormholes. However, there are special values of
parameters q2 and C, namely:
C = 0 , q2 = −
, (36)
for which the solution (35) transforms into
N(r) =
(r2 + a2)3/2
r2 + 2a2
J(r) , (37)
where
J(r) =
(x2 + a2)3/2
x2 + 2a2
x2 + 2a2 − κ
is a function of r and two guiding parameters, a and κ. Note that near r = 0 the function N(r), given by Eq. (37),
behaves as
N(r) ≃ (3a2)−1(a2 − κ/4) +O(r2) . (39)
It is seen that N(r) can be positive, negative, or zero at r = 0 depending on the relation between two parameters:
a (the wormhole throat radius) and κ (the charge parameter). It will be convenient further to use a dimensionless
parameter α = aκ−1/2. The behavior of N(r) depending on α is illustrated in the Fig.1.
Taking into account the relations gtt = σ
2N and −grr = 1/N and using the solutions (33) and (37) for σ(r) and
N(r) we finally obtain the following metric, which presents the new exact solution of the non-minimally extended
Einstein-Yang-Mills equations:
ds2 =
(r2 + a2)5/2
r3(r2 + 2a2)3/2
J(r)dt2 − r
3(r2 + 2a2)1/2
(r2 + a2)3/2
− (r2 + a2) (dθ2 + sin2 θdϕ2) . (40)
This metric describes a regular (i.e., without singularities) spacetime containing two asymptotically flat regions r =
±∞ connected by a throat located at r = 0. Thus, the metric (40) describes a wormhole, which we will hereafter call
as a non-minimal Wu-Yang wormhole.
The spacetime structure of the Wu-Yang wormhole essentially depends on the value of the dimensionless parameter
α = aκ−1/2. We note that for α > 1/2 the function N(r) is positive defined (see Fig. 1), and so the metric components
gtt = σ
2N and −grr = 1/N are finite and positive in the whole region (−∞,+∞). This means that the spacetime
has no event horizons, thus in this case the Wu-Yang wormhole is traversable.
FIG. 1: Graphs of the function N(r) given for α ≡ aκ−1/2 > 1/2, α = 1/2, and α < 1/2 from up to down, respectively.
In case α < 1/2 the function N(r) changes the sign. It is positive for |r| > rh, negative for |r| < rh, and zero at
|r| = rh, i.e., N(±rh) = 0 (rh is some parameter, which can be easily found numerically for every α < 1/2). In the
vicinity of |r| = rh one has gtt ∼ (r − rh) and grr ∼ (r − rh)−1. This means that the points |r| = rh are nothing
but two event horizons of Schwarzschild-like type in the wormhole spacetime, and rh is the radius of horizons. In
the accepted nomenclature, the regions |r| > rh with N(r) > 0 and |r| < rh with N(r) < 0 are R- and T-regions,
respectively. Thus, in the case α < 1/2 the throat of Wu-Yang wormhole turns out to be hidden in the T-region
behind the horizons. Such a wormhole is non-traversable from the point of view of a distant observer. By analogy
with black holes one may call such objects as black wormholes.
Note that for α = 1/2 two event horizons |r| = rh merge with each other and form an event horizon located at the
wormhole’s throat r = 0. Now, in the vicinity of r = 0 one has gtt ∼ r2 and grr ∼ r−2, and this means that r = 0 is
an extremal horizon. In this case the T-region is absent, and the event horizon divides two R-regions.
Now let us discuss a formula for an asymptotic mass of the Wu-Yang wormhole measured by a distant observer. A
mass of a static spherically symmetric configuration is defined as M = 1
limr→±∞
|r| (1− gtt(r))
. Using the metric
(40) we can obtain after some algebra the following expression for the mass of the non-minimal Wu-Yang wormhole:
≡ M̃(α) = π
, (41)
where Γ(z) is gamma function, and α = aκ−1/2 and M̃ = Mκ−1/2 are dimensionless quantities. The graph of M̃(α)
is given in Fig. 2. It is worth to note that M̃(α) is positive defined, M̃(α) > 0. Moreover, the function M̃(α) has a
minimum M̃min ≈ 0.653 at α = αmin ≈ 0.545.
0.545
0.653
FIG. 2: Wormhole mass fM(α). The shaded region corresponds to α < 1/2.
V. CONCLUSIONS
In this paper we have considered the non-minimally extended Einstein-Yang-Mills model given by the action (1).
The model contains three phenomenological parameters q1, q2 and q3, which determine the non-minimal coupling of
the Yang-Mills and gravitational fields. In the framework of this model we have studied static spherically symmetric
configurations with the Yang-Mills field possessing the SU(2) symmetry. Basing on the Wu-Yang ansatz for the gauge
field we have obtained a three-parameter family of the explicit exact solutions to the non-linear Einstein-Yang-Mills
equations. Only one solution from this family is regular and belongs to the class of wormhole spacetimes. We have
denoted this solution as a non-minimal Wu-Yang wormhole (see Eq. (40)). Let us emphasize some of its properties.
1. The non-minimal Wu-Yang wormhole corresponds to the specific choice of coupling parameters q1, q2, q3, namely,
, q2 = −
, q3 =
. (42)
Thus, the Wu-Yang wormhole geometry turns out to be completely determined by two model parameters: the
wormhole throat radius a, and the charge parameter κ = 8πν2/G2, or, equivalently, by a and the dimensionless
parameter α ≡ aκ−1/2. Note that in the minimal limit, when q1 = q2 = q3 = 0, the relations (42) yield a = 0,
i.e., this wormhole does not exist. In other words, the obtained exact solution is essentially non-minimal.
2. The parameter α can be treated as guiding one. Indeed, in case α > 1/2 the spacetime of Wu-Yang wormhole
has no event horizons, and so it is traversable in principle. The condition α > 1/2 equivalently reads a > 1
κ1/2,
that is the throat’s radius a of traversable Wu-Yang wormholes is necessary greater than 1
κ1/2. In case α < 1/2
(a < 1
κ1/2) the wormhole spacetime (40) possesses two Schwarzschild-type event horizons at |r| = rh, where
rh is an event horizon radius given by the equation σ
2N(rh) = 0. The presence of event horizons means the
Wu-Yang wormhole is non-traversable from the point of view of a distant observer. It is worth to note that in
this case the wormhole throat located at r = 0 turns out to be hidden behind the horizons. For this reason one
can call such objects as black wormholes. For the particular value α = 1/2 (a = 1
κ1/2) two event horizons merge
with each other and form a single event horizon at the throat r = 0. Now in the vicinity of r = 0 the metric
functions behave as gtt ∼ r2 and grr ∼ r−2, and so the metric (40) behaves near the horizon as the extreme
Reissner-Nordström metric.
3. For a distant observer the Wu-Yang wormhole manifests itself through its asymptotical massM . It is determined
by the charge parameter κ and expressed through the wormhole throat radius a (see Eq. (41) and Fig. 2). Is it
possible for the observer to reconstruct the invisible throat radius using the estimated mass? In principle, yes,
but the procedure is ambiguous, since two values of a correspond to one appropriate value of the mass.
4. The important feature is that there exists the lower limit for the mass of the non-minimal Wu-Yang wormhole.
In other words, the wormhole mass cannot be less than some minimal value Mmin ≈ 0.653 κ1/2, i.e., M ≥ Mmin.
To make estimations we assume that the monopole magnetic charge ν is equal to one, ν = 1, and the square
of the constant of gauge interaction is given by G2 = 4παem, where αem = e2/~c ≈ 1/137 is the fine structure
constant. Then, in the dimensional units we have Mmin ≈ 10.8Mpl, amin ≈ 9Lpl, where Mpl and Lpl are the
Planck mass and the Planck length, respectively.
Recently Kirill Bronnikov attracted our attention to the papers [24, 25], where the authors discuss solutions they
refer to as regular black holes. He also emphasized that black wormholes obtained in our paper represent the kind of
regular black holes.
Acknowledgments
This work was partially supported by the Deutsche Forschungsgemeinschaft through the project No.
436RUS113/487/0-5 and the Russian Foundation for Basic Research grant No. 05-02-17344.
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http://arxiv.org/abs/gr-qc/0611022
	Introduction
	Non-minimal Einstein-Yang-Mills model
	Exact solutions of the static model with spherical symmetry
	Master equations
	The case q1<0
	The case q1=0
	The case q1>0
	Non-minimal Wu-Yang wormhole
	Conclusions
	Acknowledgments
	References
