Yang–Mills Monopoles in
Extremal Reissner–Nordström Black Hole Metric

The concept of magnetic monopoles was first conceived by P. Curie [9] and later theoretically formulated by Dirac [10] based on electromagnetic duality of the Maxwell theory. A remarkable feature of this concept is that its existence could lead to an explanation why electric charges are multiples of a unit charge [10, 13, 25, 27, 39]. Such a charge quantization phenomenon was then extended by Schwinger [28] to dyons, namely, particles of both electric and magnetic charges. In modern development, thought experiments with monopoles immersed in a type-II superconductor led Mandelstam [19, 20], Nambu [23], and t’ Hooft [33, 33] to come up with a linear confinement mechanism for monopoles and their colored, non-Abelian or the Yang–Mills, field theory versions, which has propelled enormous activities and progress towards an understanding [29, 30] of the unsettled puzzle of quark confinement [14]. Like that in the classical Maxwell theory, a monopole in the pure Yang–Mills theory, for example, the celebrated Wu–Yang monopole [1, 13], is of infinite energy [16]. There are several well-recognized paths which may be taken to overcome this infinity or divergence problem. These include adopting the model of the Born–Infeld nonlinear electrodynamics [4, 5], coupling the Yang–Mills fields with matter fields [1, 13, 17, 24, 25, 32, 35, 38, 40], namely, considering the Yang–Mills–Higgs theory, and hosting the Yang–Mills fields over curved spacetime backgrounds due to gravity. In the context of this last approach, if a gravitational black hole is present and fields are considered outside the event horizon so that possible singularities leading to energy divergence are concealed inside the horizon, as in the scenario of the cosmic censorship hypothesis [22, 37], the total energy of the fields outside the horizon could remain finite, as a result. The present study belongs to this category of investigation. Specifically our work is motivated by the recent paper of Bizoń and Kahl [3] who considered spherically symmetric $SU(2)$ Yang–Mills fields in the exterior of the event horizon of the extremal Reissner–Nordström black hole and obtained a countable family of static regular solutions to the Yang–Mills equations, given in terms of a scalar field sequence $\{W_{n}\}$, say, among which the first two, $W_{1}$ and $W_{2}$, are in closed forms. These solutions resemble the Bartnick–McKinnon solutions [2, 31] featured as soliton-like configurations connecting vacuum states at infinity and oscillating about the unit-charge monopole state in local domains. The main result of our present work is to show that, viewed as a connecting orbit between different vacuum states which passes through the unit-charge monopole state midway, $W_{1}$ is the unique global energy minimizer of the problem. The solution $W_{2}$, however, enjoys a different signature – it exhibits itself as a local energy maximizer in a well-formulated sense. Besides, we shall also obtain a new closed-form solution which has two singular points (in fact, two singular shells), near which energy blow-up takes place. Furthermore, we adapt the formalism of [3] to consider interior monopoles (inside the event horizon) and derive a governing wave equation which may be viewed as a companion of the wave equation of Bizoń–Kahl [3] for exterior monopoles. For this interior monopole equation, we obtain three closed-form solutions, similar to $W_{1}$, $W_{2}$, and the third singular solution to the exterior monopole equation described above, among which two are regular everywhere away from the curvature singularity of the Reissner–Nordström spacetime and the third one is singular at one point (in fact, one shell). It is noted that this interior monopole equation possesses a family of oscillatory solutions as the exterior equation, as well.

The content of the rest of the paper is outlined as follows. In Section 2, we formulate the problems to be studied with a review of the Yang–Mills fields over an extremal Reissner–Nordström spacetime and the exterior monopole equation derived in [3], followed with an extension of such a formulation to get the interior, ‘accompanying’, monopole equation. Subsequently, we focus on static equations. In Section 3, we review the two closed-form regular solutions, $W_{1}$ and $W_{2}$, obtained in [3], and present a third, singular, one, for the exterior equation. We obtain the exact values of the energies of these two regular solutions to be further explored in later sections. We then present two closed-form regular solutions and a third, singular, one, for the interior equation, similar to those for the exterior equation. In Section 4, we study several issues of the exterior solutions based on some energy consideration. In particular, we formulate a domain-wall type minimization problem that associates a distinguishing significance to $W_{1}$ as the unique global minimizer of the problem. We also present a virial identity which displays an energy partition relation for finite-energy solutions. In Section 5, we first prove the existence of a solution to the domain-wall minimization problem formulated in Section 4. Note that, due to the behavior of the weight function in the energy functional, it is not a trivial matter to preserve the asymptotic limit of the weak limit of a minimizing sequence. Fortunately, such a difficulty may be overcome by a local convergence method leading to realizing the weak limit as a monotone solution to the exterior equation and then by an energy comparison. We next prove the uniqueness of a monotone solution to the exterior equation. Since the energy functional is not convex, such a uniqueness result is usually not ensured [11] and seems surprising in the present context. Consequently, we recognize that $W_{1}$ is actually this energy minimizer whose energy calculated in Section 3 is thus the energy minimum of the formulated problem. We also illustrate that $W_{2}$ is a local energy maximizer among a class of testing configurations, which seems to suggest that, energetically, the closed-form solution $W_{2}$ of the exterior monopole equation has a certain ‘significance’ associated as well. In Section 6, we briefly comment on how to obtain oscillatory solutions for both exterior and interior monopole equations using their polar-variable representations.

Consider the Reissner–Nordström metric [22, 37]

describing a charged black hole of mass $M$ and charge $Q$ in the Schwarzschild spherical coordinates. At the critical situation

the metric (2.1) assumes the form

known as the extremal Reissner–Nordström black hole metric with its event horizon at $r=M$. We will be interested in both the exterior region, $r>M$, and interior one, $r<M$, possibly extended to its curvature singularity at $r=0$ as exhibited by the associated Kretschmann invariant given by

composed from the Riemann tensor induced by the gravitational metric form (2.1).

In [3], the following two nontrivial exact solutions to the exterior Yang–Mills monopole equation (2.17) are obtained:

Here we find that there is a third exact solution,

which is apparently singular at

giving rise to the corresponding singular radii

in the original radial variable, which are distributed about the Schwarzschild radius $r_{\mbox{\small s}}$ following

For the interior Yang–Mills monopole equation (2.23), we likewise find the following three nontrivial exact solutions:

for $-\infty<x<0$. The solutions $w_{1}$ and $w_{2}$ given in (3.6) and (3.7), respectively, are regular but $w_{3}$ in (3.6) is singular at

whereby the irrelevant positive root is discarded.

We now compute the energies of these solutions.

Recall that the energy-momentum tensor induced from (2.13) is

which via ${\cal H}=T^{0}_{0}=g^{00}T_{00}$ gives rise to the Hamiltonian energy density associated with (2.6)–(2.7) and (2.12) to be

Thus, evaluating the static total energy

we obtain the exact value

for the solution $W_{1}$ given in (3.1). For the solution $W_{2}$ given in (3.1), utilizing MAPLE, we obtain the exact value

which is approximately $2\pi(0.9664005272)$. These two exact results are consistent with those found in [3] based on numerical evaluation. Not surprisingly, the singular solution $W_{3}(x)$ is of infinite energy. In other words, this solution behaves singularly in both point-wise and energy-wise manners at $x_{\pm}$.

We now turn our attention to the interior solutions listed in (3.6)–(3.8). In view of (3.10), (2.12), and (2.19)–(2.20), we get the Hamiltonian energy density

which renders the total energy for a static interior solution in the full region $0<r<M$ to be

All the solutions given in (3.6)–(3.8) are of infinity energy due to the divergence of the energy density (3.15) as a consequence of the curvature singularity of the Reissner–Nordström black hole spacetime at $r=0$ displayed by (2.4). Nevertheless, we may assume that we are in a situation that gravity is generated from a bulk of matter which occupies a finite region within the event horizon,

Then we can focus on the ‘regularized’ region

instead. It is seen that the solutions (3.6) and (3.7) are of finite energies over (3.18) for any $0<r_{\mbox{\tiny B}}<M$ and the solution (3.8) is of finite energy only when $x_{\mbox{\tiny B}}$ is below its singular point $x_{0}$ spelled out in (3.9). Thus, we are led to the finite-energy condition

or equivalently,

Over a given interval (3.18), the energy of the solution (3.6) is generically much greater than that of the solution (3.7), since the former is singular at its right boundary point, $x=0$, and the latter is everywhere regular. As a comparison, we list our calculation of the energy

for $x_{\mbox{\tiny B}}=-1$ for the solutions $w_{1}$ and $w_{2}$, given in (3.6) and (3.7), respectively, as follows:

We now investigate the problem whether there is a nontrivial energetically stable exterior monopole solution linking two vacuum states, $W=\pm 1$, characterized by the boundary condition $W(-\infty)=\pm 1$ and $W(\infty)=\pm 1$. It is clear that the problem of linking the same vacuum state with $W(-\infty)=W(\infty)=\pm 1$ is not well posed without some further elaboration since the vacuum states themselves minimize the energy and may be approached arbitrarily by nontrivial field configurations. Thus it remains to consider the problem of linking two different vacuum states, say,

and ask whether this minimization problem has a solution, where $E(W)$ is as given in (3.12). This is typically a domain wall problem [6, 7, 8, 26, 36], interpolating two ground state domains, $W=\pm 1$. However, it is not hard to see that this problem is not well posed either.

In fact, let $u(x)$ be a smooth function satisfying

Then, using $u_{(a)}=u(x-a)$ as a testing function, we have

Here $K_{0}>0$ is a constant depending on the properties of $u$ over $[-1,1]$. As a consequence, this shows the quantity $\eta$ in (4.1) is zero which is therefore not attainable.

On the other hand, since the boundary condition in (4.1) dictates that the unit magnetic charge monopole configuration, $W=0$, is to occur somewhere as an intermediate state, we may conveniently impose the following additional ‘intermediate state condition’

It is interesting that this condition implies that the potential part of the energy density

is maximized at the spot where the intermediate state occurs, $x=0$. In Figure 1, we plot the energy density (4.5) for the solutions $W_{1}$ and $W_{2}$ given in (3.1) as an illustration. It is interesting to notice that the former peaks sharply at $x=0$ where the solution passes the intermediate state and the latter also peaks at the same spot which is not where the solution passes the intermediate state but is the midpoint between the two spots,

where the solution passes the intermediate state. Such a property of energy concentration is typical for solitons in field theories.

These pictures coincide with what we know about a domain wall soliton configuration in general.

Thus, we are led to modifying (4.1) into the minimization problem

We shall show that $W_{1}$ given in (3.1) is the unique solution to this problem. Consequently, as a by-product, we have

in view of (3.13).

Let $W$ be a finite-energy critical point of the energy functional (3.12). Set $W_{\lambda}(x)=W(\lambda x)$. Then, from $\frac{\mathrm{d}}{\mathrm{d}\lambda}E(W_{\lambda})|_{\lambda=1}=0$, we get the virial identity

which is an energy partition relation. This identity demonstrates that the ‘potential energy’ is much greater than the ‘elastic energy’ for an exterior monopole. For example, for the exact solution $W=W_{1}$ in (3.1), we have

and, for $W=W_{2}$ given in (3.1), these quantities are approximately $0.8765154575$ and $0.08988507022$, respectively.

The symmetry of the functional (3.12) indicates that the problem (4.7) amounts to considering the minimization problem

over the set of admissible functions which are absolutely continuous on all compact subintervals of the half line $[0,\infty)$.

Let $\{{\cal W}_{n}\}$ be a minimizing sequence of (5.1) satisfying

By modifying the sequence $\{{\cal W}_{n}\}$ if necessary, we may assume that $\{{\cal W}_{n}\}$ enjoys the property that each function ${\cal W}_{n}$ satisfies $0\leq{\cal W}_{n}(x)\leq 1$. Furthermore, for each ${\cal W}_{n}$, we may modify it such that it minimizes the partial functional

among functions fulfilling the boundary condition $W(0)=0,W(n)={\cal W}_{n}(n)$ since the functional is differentiable and weakly lower semicontinuous over the Sobolev space $W^{1,2}(0,n)$, say (cf. Theorem 2 on page 448 in Evans[11] specifically as well as the studies in [12, 35, 40] in other contexts). Therefore ${\cal W}_{n}$ when restricted to the interval $[0,n]$ is a critical point of $I_{n}$. As a consequence, we have

Extracting a suitable subsequence if necessary, for example, using a diagonal subsequence argument, we may assume that $\{{\cal W}_{n}\}$ is weakly convergent in the Sobolev space $W^{1,2}(0,a)$, with the weighted measure $C^{-2}(x)\,\mathrm{d}x$, for any $a>0$. Let ${{\cal W}}$ denote such a weak limit which is well defined over $[0,\infty)$. Fix $a>0$. Neglecting the first few terms of the sequence if necessary, we may also assume $n>a$. Thus (5.4) gives us

Using the compact embedding from the Sobolev space into the space $C[0,a]$ and letting $n\to\infty$ in (5.5), we arrive at

which leads us to conclude by the standard elliptic regularity theory [12, 18] that ${{\cal W}}$ is a classical solution to the Euler–Lagrange equation of the functional $I$. Hence ${\cal W}$ satisfies

Since $\{{\cal W}_{n}\}$ is a sequence with values confined in $[0,1]$, so is its limit ${\cal W}$. However, since ${\cal W}$ satisfies (5.7), we see that ${\cal W}\equiv 0$ or $0<{\cal W}(x)<1$ for $x>0$ because ${\cal W}\equiv 0$ and ${\cal W}\equiv 1$ are two equilibria of the differential equation in (5.7). We now show that the former does not occur.

In fact, by using weak convergence, we have

for any $a>0$. Thus, letting $a\to\infty$ in (5.8), we get

Recall the result

Besides, we also have $I(0)=1$. Hence ${\cal W}\not\equiv 0$. Consequently, ${\cal W}$ satisfies $0<{\cal W}(x)<1$ for $x>0$.

Since ${\cal W}$ is bounded, we deduce that there is a sequence $\{x_{n}\}$, $x_{n}\to\infty$ as $n\to\infty$, such that ${\cal W}^{\prime}(x_{n})\to 0$ as $n\to\infty$. Using this result as the boundary condition and integrating the differential equation in (5.7), we have

Hence ${\cal W}^{\prime}(x)>0$ for $x>0$. In particular,

for some $L\in(0,1]$. Thus, if $L<1$ in (5.12), we infer from (5.11) that

which is false. Therefore $L=1$ in (5.12). This establishes that ${\cal W}$ lies in the admissible space of the problem (5.1). Thus $I({\cal W})\geq I_{0}$. Combining this result with (5.9), we arrive at

In other words, the existence of a solution to the minimization problem (5.1) follows.

We next prove that, actually,

Thus, in particular, equality in (5.10) holds. We achieve this goal by showing that a positive solution to the initial value problem (5.7), with a limiting value at infinity, is unique.

In fact, let ${\cal W}_{1}$ and ${\cal W}_{2}$ be two such solutions. Then $w={\cal W}_{1}-{\cal W}_{2}$ satisfies

On the other hand, for any differentiable function $h$, we have

Now substituting $h=\frac{w}{{\cal W}_{1}}$ into (5), we have

Moreover, using the Cauchy–Kovalevskaya theorem, we know that the solution of (5.7) has the asymptotic form

As a consequence of (5.19), we may integrate (5) and drop the resulting boundary terms to arrive at

However, by virtue of (5.16), we may rewrite the left-hand side of (5) as

Combining (5) and (5), we conclude with $w\equiv 0$.

Hence the existence and uniqueness of an exterior energy-minimizing monopole solution linking different vacuum states at $x\pm\infty$ and passes through the unit-charge monopole state at $x=0$ is established which is the exact solution $W_{1}$ stated in (3.1) and found in [3] which determines the minimum energy, $E_{\min}=\frac{8\pi}{5}$, as stated in (3.13).

It is interesting to note that a rich family of even and odd solutions of (2.17) linking the same and different vacuum states is obtained in [3] which are characterized by the number $n$ of zeros they possess. When $n=1$, their calculated energy coincides with our exact value; when $n\geq 2$, their calculated energies are all significantly greater than that with $n=1$.

Motivated by the construction in this section and the even solutions with multiple zeros linking to the same vacuum state, $W(-\infty)=W(\infty)=\pm 1$, obtained in [3], we may impose the following minimization problem

over the admissible set of functions which are absolutely continuous on all compact subintervals of the real line. Thus, with the function (5.15) as a testing function, i.e., choosing $W(x)=|\tanh(\frac{x}{2})|$ for all $x$, we have the estimate

With the problem (5.22) in mind, we consider the exact solution $W_{2}$ given in (3.1) whose value at $x=0$ is

and we ask what this ‘height’ $h_{0}$ means with regard to the problem (5.22). So we take the trial function

which satisfies $W_{h}(0)=-h$ and is suggested by the form of $W_{2}(x)$ in (3.1). Hence we can compute the energy $E(W_{h})$. Surprisingly, we find that $E(W_{h})$ is maximized at $h=h_{0}$ as illustrated in Figure 2.

Therefore, the quantity $h_{0}$ may be regarded as a ‘canonical height’ at which the energy confined over the profile family of functions of the form (5.25) is maximized and brings forth a critical point of the energy functional.

In [3], a family of solutions of (2.17) with prescribed numbers of zeros are obtained. In this section, we revisit such a construction of oscillatory solutions from the viewpoint of recasting the equation into a form of its ‘polar variable’ representation such that the appearance of multiple zeros of the solutions becomes somewhat more transparent as we now show.

For our purpose, we use the following polar variable anzatz

to represent a solution of (2.17). Thus, we see that the pair $(\rho,{\omega})$ satisfies the coupled system of equations

To justify this representation, we observe in view of the uniqueness theorem for solutions of the initial value problems of ordinary differential equations that if $\rho$ satisfying (6.2) is positive at a point then it will stay positive where it exists. So, we may impose the initial condition

Thus, in particular, to produce an even solution of (2.17), we take $b=0$, and an odd solution, we impose $b=\frac{\pi}{2}$. In the subsequent discussion, we fix $b$ and adjust $a$ to achieve multiple zeros for the field $W$. Accordingly, in view of (6.1), such a goal may be achieved by making the angular variable ${\omega}$ circulate through the full circle multiple times, in a sense to be made precise below.

Indeed, we fix $b,K_{0}>0$. Then the theorem of continuous dependence of solutions on the initial data allows us to get some number $a_{0}>0$ such that the solution to (6.2)–(6.4) satisfies