Taub-NUT from the Dirac monopole

Writing the metric of an asymptotically flat spacetime in Bondi coordinates provides an elegant way of formulating the Einstein equation as a characteristic value problem. In this setting, we find that a specific class of asymptotically flat spacetimes, including stationary solutions, contains a Maxwell gauge field as free data. Choosing this gauge field to correspond to the Dirac monopole, we derive the Taub-NUT solution in Bondi coordinates.


Introduction
Asymptotically flat spacetimes have been extensively studied over the last 50 years, particularly, recently, in the context of gravitational wave detections [1] and asymptotic symmetry groups [2,3]. In this paper, motivated by recent work on dual gravitational charges [4][5][6][7], we show how treating asymptotically flat spacetimes in terms of a characteristic value problem [8] provides an intriguing way of viewing the Dirac magnetic monopole as a progenitor of the Taub-NUT spacetime. 1 Our starting point is to consider a general class of asymptotically flat metrics, written in a Bondi coordinate system (u, r, x I = {θ, φ}), such that the metric takes the form 2 with the metric functions satisfying the following fall-off conditions at large r: C I (u, r, x I ) = C I 0 (u, x I ) h IJ (u, r, x I ) = ω IJ + C IJ (u, x I ) r + C 2 ω IJ 4r 2 + D IJ (u, x I ) r 3 + E IJ (u, x I ) r 4 + o(r −4 ), (1.2) where ω IJ is the standard metric on the round 2-sphere with coordinates x I = {θ, φ} and C 2 ≡ C IJ C IJ . Moreover, a residual gauge freedom allows us to require that where h ≡ det(h IJ ), and ω ≡ det(ω IJ ) = sin θ. Following Ref. [7], we do not impose on the fields defined above any regularity conditions on the 2-sphere.
One may introduce a complex null frame of vector fields e µ a = (ℓ a , n a , m a ,m a ) with inverse E µ a , and Given the choice of Bondi coordinates, the formalism most adapted to the problem of constructing solutions from initial data is the characteristic approach [8]. In the characteristic value problem, such as that defined by Bondi coordinates, the spacetime is viewed as a foliation of null hypersurfaces, called the characteristic surfaces, corresponding to the level sets of u (see figure 1). u = c o n s t a n t h y p e r s u r f a c e u r Figure 1: Hypersurfaces defining a characteristic value problem.
The vacuum 3 Einstein equation may then be divided into three types of equations [8]: 1. Hypersurface equations: these are equations that hold in each u = constant hypersurface and are of the form where F denotes the set of hypersurface variables, which in Bondi coordinates correspond to {β, F, C I }, while G denotes evolution variables, which in Bondi coordinates corresponds to h IJ . The operator H F , as well as H G defined below, is non-linear in derivatives with respect to the three hypersurface coordinates. The structure of the hypersurface equations is such that the r-dependence of the right hand side is always explicit, and therefore the hypersurface variables may simply be determined by integrating with respect to r.
2. Evolution equations: these are of the form Note that there are no second order derivatives in time.
3. Conservation equations: these are satisfied on r = constant hypersurfaces transverse to the characteristics, and are of the form where h F is some non-linear operator in the two angular coordinates in the u = constant hypersurfaces. These are to be thought of as conservation equations rather than evolution equations because their structure is such that once they are satisfied on a particular r = constant hypersurface, they are guaranteed to hold for all values of r.
Note that the m µmν G µν = 0 component is automatically satisfied if we assume the other components to hold [12]. Since we are assuming specific expansions in inverse powers of r for the metric components, 4 at least for the first few orders, the corresponding hypersurface equations simplify to algebraic equations at fixed u for all scalar and vector fields on the right hand side of the fall-off conditions for {β, F, C I }, except for the fields F 0 and C I 1 , which as we will explain later constitute initial data, whose evolutions are determined by the conservation equations. The evolution equations then determine the evolution of D IJ and E IJ (as well as the other terms at higher powers in the 1/r expansion of h IJ ), given prescribed initial data. Rather unusually, C IJ (u, x I ) constitutes free data. Furthermore, note that the initial data are unconstrained, in contrast to a Cauchy formulation where they would satisfy elliptic constraints. The explicit Einstein equations for the metric components are listed in section 2.2 of Ref. [11].
In summary, the Einstein equation is solved by prescribing initial data where the ellipses denote terms at higher powers in the 1/r expansion of h IJ , and an arbitrary trace-free symmetric tensor C IJ (u, x I ). The hypersurface equations then determine all scalar, vector and tensor fields in the expansions (1.2) on the initial hypersurface. Now, the evolution and conservation equations can be integrated to determine {F 0 , C I 1 , D IJ , E IJ , . . .} at a time step u 0 + ∆u, before iterating the above procedure to find the full solution at this time step and so on.
One of the hypersurface equations, arising at order 1/r 3 of the ℓ µ m ν G µν = 0 equation (see equation (2.16) of Ref. [11]), gives 5 which determines C I 0 given the data C IJ (u, x I ). Alternatively, we may view this equation as determining C IJ given a choice of C I 0 , up to functions of integration which may themselves be arbitrarily chosen. That is to say, we may exchange the freedom to choose a trace-free tensor C IJ (u, x I ) with the freedom to choose a vector C I 0 . It will be useful in what follows to take this perspective. Therefore, the data for the characteristic value problem are given by where we emphasise that the choice for C I 0 is not only an initial value choice as with the other fields, but that we have the freedom to choose the vector completely, as a function also of u.
Let us consider, briefly, the physical significance of these quantities. The Bondi mass and angular momentum are given by [12,13] where is the angular momentum aspect, so that if L φ is constant then J B = L φ . Moreover, we have that the dual Bondi mass or Bondi NUT charge is given by [7] where C 0 = C 0 I dx I . Thus F 0 , C I 0 and C I 1 constitute the data determining the Bondi mass, NUT charge and angular momentum. Furthermore, the other data given by terms at higher powers in the 1/r expansion of h IJ correspond to subleading BMS charges [11], [5]. For example, D IJ determines the BMS charge at order 1/r 2 .
In this paper, our focus will be on constructing stationary solutions, meaning that all the scalar, vector and tensor fields in the expansions (1.2) will be independent of the retarded time u. This means that the evolution and conservation equations (1.11) and (1.12) will now give rise to genuine constraints on the initial data. Thus, all stationary asymptotically flat solutions are given by prescribing the following data subject to the constraints implied by equations (1.11) and (1.12).

C 0 I as a Maxwell gauge field
As can be verified from the form of the BMS transformations in the notation of Ref. [11] that we are using here, under a supertranslation with diffeomorphism parameter ξ = s(θ, φ) ∂u+ · · · , we have which is the case for stationary solutions, but also more generally, then Thus, in a very real sense, one can think of a u-independent 1-form C 0 as being analogous to a Maxwell gauge potential.
This interesting observation is one reason why we chose to view C 0 I rather than C IJ as characteristic data in the previous section.

Dirac monopole
Given the interpretation of a u-independent C 0 as a Maxwell gauge field, it is natural to consider the case where this 1-form describes a Dirac magnetic monopole on the 2-sphere, with C 0 of the form where p is a constant. This 1-form is singular at the north and the south poles of the sphere. We shall look for solutions that are stationary and axisymmetric, 7 so all the metric functions will be assumed to be independent of u and of φ. Of course, in order to specify a particular solution we must also prescribe initial data for the other fields. However, since gives A = −4p sin 2 θ 2 dφ, which is singular only at the south pole, whilst taking Λ = 2p φ gives A = 4p cos 2 θ 2 dφ, which is singular only at the north pole. By comparing with equation (2.3), and taking s to be a constant multiple of the azimuthal coordinate φ, we can perform precisely the kinds of gauge transformation we described above for the Dirac monopole. 8 We may take as our starting point the slight generalisation of (3.1) where Using the fact that C IJ is symmetric, trace-free and depends only on θ, by the axisymmetry assumption, we can substitute (3.2) into (1.13) and solve to find where c 1 and c 2 are constants of integration. These constants will contribute to subleading BMS charges, and as is evident from the definition (1.15) of the angular momentum aspect L I and the form of (3.2), c 1 will also contribute to the angular momentum. Note that D I D J C IJ = 0, and therefore c 1 and c 2 do not contribute to the Bondi mass given in (1.14).
It is interesting to note that although in general this solution for C IJ is singular at one or both of the poles of the sphere, we can find a non-singular solution in the special case where we choose c 1 = 0, c 2 = −2p and k = 0, for which the only non-vanishing components of the symmetric C IJ are specified just by Of course, since our starting point was C 0 I having the form of a Dirac monopole, which is necessarily singular somewhere on the sphere, there is no particular reason why we should 8 The function φ is of course singular on the sphere. However, since we are already entertaining the idea of using a monopole configuration for C0 I that is singular on the sphere, there is no longer any reason to restrict ourselves to non-singular supertranslation parameters. expect or require C IJ to be non-singular. Indeed, as we shall see below, there are reasons to prefer different assignments for (c 1 , c 2 , k) for which C IJ does have singularities.
As explained before, the other hypersurface equations are algebraic equations that determine the form of other fields given initial data. Therefore, we need not concern ourselves with those just yet. However, the evolution and conservation equations are now non-trivial equations, constraining the data. Therefore, we focus on these equations.
The n µ n ν G µν = 0 projection of the Einstein equation at order 1/r 2 implies the conservation equation [11] With C 0 and C IJ given by (3.2) and (3.4), respectively, we may now seek to solve for C 1 I (θ).
Since the solutions are a little complicated in general we shall not present them here, but just remark that in general they involve terms that have power-law singularities at θ = 0 and θ = π, and also terms proportional to log cot θ 2 with logarithmic singularities at the poles of the sphere. The solution for C 1 I will be free of logarithmic singularities if and only if we choose the c 1 and c 2 integration constants in (3.4) to be The solution is then given by C 1 θ = c 3 sin θ + 3k cos 4θ + 4(k 2 + p 2 ) cos 3θ − 12kp cos 2θ − 36(k 2 + p 2 ) cos θ − 55kp 4 sin 5 θ , where c 3 and c 4 are constants of integration.
This is the most general solution for C 1 I without logarithmic singularities, starting from the Dirac monopole connection (3.2). We may now evaluate the angular momentum given in (1.14), obtaining However, the Komar angular momentum with j = ∂/∂φ is divergent. This is easy to see, since (⋆dj ♭ ) θφ = 2[2k sin θ + p sin 2θ] r + [2(c 4 − 6km) sin θ − 6pm sin 2θ] + O(r −1 ). (3.15) Therefore, before sending r to infinity, the right hand side of the expression for J K gives A simple explanation for why the divergence above vanishes for k = 0 is that the wire singularity provides a divergent contribution, however, when k = 0 the divergent contributions from the wire singularities at the north and south poles cancel. There has been a proposal to resolve this divergence issue for general k in Ref. [15], by redefining the Komar integral.
However, given that k parameterises a large gauge transformation, we may simply avoid these difficulties by choosing to set k = 0. (Recall that k is the gauge parameter appearing in (3.2).) Now, we obtain Thus, c 4 is a Kerr-like angular momentum parameter, which, for simplicity, we shall for now set to zero. The integration constant c 3 will contribute to subleading BMS charges [11], [5], and we shall also set this to zero for simplicity.
With c 3 = c 4 = 0, and making the gauge choice k = 0 as discussed above, the expressions for C 1 I in (3.12) become The previous expressions (3.2) and (3.4) for C 0 I and C IJ then give Substituting these expressions into the ℓ µ n ν G µν = 0 component of the Einstein equation at order 1/r 4 then gives Other choices of the integration constants will give more general solutions, generically presumably with more severe singular behaviour on the sphere, such as the logarithmic behaviour in C 1 I that we avoided above by the judicious choice (3.11) for the constants of integration c 1 and c 2 .
We have already seen, (3.17), that the constant of integration c 4 in the expression (3.12) for C 1 φ is related to angular momentum. In fact, if we turn off the Dirac monopole altogether by choosing p = 0 (and k = 0), i.e. setting C 0 I = 0, and letting c 3 = 0 and c 4 = 2ma, we find that our solution (3.12) becomes As can be seen from the expressions in appendix A of [7], the C 1 I above is precisely that of

Discussion
We showed that, in a stationary setting, starting from a choice of the gauge field C 0 as the Dirac monopole, we can integrate the vacuum Einstein equation to recover the Taub-NUT solution in Bondi coordinates. Assuming the solution to be stationary, the evolution and conservation equations transformed into constraint equations on the other characteristic data. For example, we found that F 0 must be constant, giving a Bondi mass. Solving another of the constraint equations, we found C I 1 up to two constants of integration. One of these constants corresponded to a Kerr-like angular momentum parameter. Working through the other constraint equations would also generate other constants, which contribute to subleading BMS charges. Therefore, what the Dirac monopole actually generates is a family of Taub-NUT-like solutions, of which the usual Taub-NUT solution is one member. In general, the other members of the family would correspond to other stationary axisymmetric Weyl solutions with a non-trivial NUT charge.
The 1-form C 0 can in fact be regarded as a gauge connection more generally, namely in any spacetime, even a time-dependent one, provided that C 0 itself is still time-independent.
Therefore the iterative procedure that we carried out, starting with a Dirac monopole con-figuration, can be repeated in a more general non-stationary setting. It would be interesting to see if dynamical solutions with Taub-NUT charge can be constructed this way.
Moreover, in this work, for simplicity, we have assumed a 1/r-expansion of the Bondi metric, at least for the first few orders. However, this is not necessary and only simplifies the hypersurface, (1.10), and evolution, (1.11), equations. Therefore, one could consider the more general system of equations with weaker fall-off conditions, which would presumably be necessary when considering time-dependent solutions. θ = θ+ 2ℓ 2 cos θ r 2 sin 3 θ + 2ℓ 4 (sin 4 θ+10 sin 2 θ−15) cos θ 3r 4 sin 7 θ + 4mℓ 4 cos 3 θ r 5 sin 5 θ +· · · . (A.2) We have actually worked to a higher order than the terms presented here, sufficient for our later purposes. Using these expansions, we then obtain the Taub