The Euclidean quantisation of Kerr-Newman-de Sitter black holes

We study the family of Einstein-Maxwell instantons associated with the Kerr-Newman metrics with a positive cosmological constant. This leads to a quantisation condition on the masses and charges of the resulting Euclidean solutions.


D SI units 1 Introduction
Euclidean counterparts of Lorentzian solutions play an important role in Euclidean Quantum Gravity [8,10]. It appears therefore of interest to find Euclidean versions of key Lorentzian solutions.
As such, Kerr-Newman solutions have a unique position in view of their uniqueness properties. The associated solutions with positive cosmological constant, discovered by Demiański and Plebański [18] and, independently, by Carter [2], are similarly expected to be unique under natural conditions. Surprisingly enough, their compact Euclidean counterparts do not seem to have been explored in the literature. The object of this paper is to fill this gap.
More precisely, we construct two new families of compact Riemannian fourdimensional manifolds satisfying the Einstein-Maxwell equations with a positive cosmological constant. The solutions are obtained by complex substitutions in the Kerr-Newman de Sitter metric. The requirement of smoothness and compactness of the underlying manifold leads to a quantisation condition on the mass and charge parameters of the associated Lorentzian manifold. We thus obtain our first family of metrics, on S 2 -and RP 2 -bundles over S 2 , parameterised by two integers (n 1 , n 2 ). The second family is parameterised by a single integer n ∈ N and is obtained by passing to a limit à la Page in the Euclidean Kerr-Newman de Sitter metrics. We determine several physical parameters associated with the Lorentzian equivalents of the solutions and study their asymptotics as one, or both, parameters tend to infinity. We calculate the associated Euclidean actions, which determine the contribution of our instantons to the Euclidean path integral in a saddle point approximation, as well as horizon entropies and temperatures.
Our Riemannian solutions ( 4 M , g ) have a clear quantum relevance. On a more mundane level, since the Maxwell energy-momentum tensor has vanishing trace, the metrics we have constructed provide time-symmetric initial data for the 4 + 1 vacuum Einstein equations with a positive cosmological constant, or for Einstein equations with matter (e.g., dust) having constant density on the initial data surface 4 M . Indeed, the four-dimensional Euclidean Einstein-Maxwell equations imply that the four-dimensional Riemannian metric g has constant positive scalar curvature. Therefore the initial data set ( 4 M , g , K = 0) satisfies the 4 + 1 vacuum time-symmetric constraint equations with a positive cosmological constant, or 4+1 time-symmetric constraint equations with dust which has constant density, or with a constant scalar field, or with a mixture of the above.
The solutions in our first family are uniquely parameterized by the already mentioned quantum numbers (n 1 , n 2 ) ∈ N 2 , 1 ≤ n 2 < n 1 , and the value of the cosmological constant Λ. It might be viewed as amusing, and perhaps not entirely unexpected, that after inserting the experimentally determined value of Λ, the masses of all Lorentzian solutions associated with our Euclidean ones are of the same order as some standard current estimates, based on the FLRW model, for the total mass of the visible universe.
The quantum numbers (n 1 , n 2 ), resulting from the requirement of regularity of the Riemannian manifold, lead to a quantisation of the mass, the angular momentum, and the combination p 2 − e 2 of the magnetic charge parameter p and electric charge parameter e. We show that the requirement of a welldefined test Dirac field with charge q 0 on the Riemannian manifold introduces two further quantum numbers (n 1 ,n 2 ), together with a quantisation of e, p and q 0 .

The fields
The Kerr-Newman-de Sitter (KNdS) metric is a solution of the Einstein-Maxwell equations, where Λ is the cosmological constant (which we assume to be positive throughout this work), and where In Boyer-Lindquist coordinates, after the replacement a → i a, t → i t and e → i e the metric takes the form 1 g = Σ ∆ r d r 2 + Σ ∆ θ d θ 2 + sin 2 (θ) Ξ 2 Σ ∆ θ (adt + (r 2 − a 2 )dϕ) 2 where, setting λ = Λ/3, Σ = r 2 − a 2 cos 2 (θ) , ∆ r = (r 2 − a 2 ) 1 − λr 2 − 2M r + p 2 − e 2 , (2.4) ∆ θ = 1 − λa 2 cos 2 (θ) , Ξ = 1 − λa 2 . (2.5) The Maxwell potential reads where the one-forms σ i , i = 1, 2, are defined as Now, each metric (2.3) is determined uniquely by the parameters a, M , and the combination p 2 eff := p 2 − e 2 . (2.8) of the magnetic charge parameter p and the electric charge parameter e. The notation in (2.8) might appear to be misleading, because the right-hand side of this equation could be negative. However, it turns out to be mostly appropriate, in that we have not found any non-singular solutions with p 2 ≤ e 2 using our procedure below except in the Page limit discussed in Appendix F. We emphasise that any pairs (e, p) satisfying (2.8) are allowed. When transforming back to the Lorentzian regime, there is no ambiguity in determining the parameters M and a characterising the Lorentzian solution, which remain unchanged. On the other hand, if we denote by p L and e L the parameters characterising the Maxwell field on the Lorentzian side, then any values of p L and e L satisfying p 2 L + e 2 L = p 2 + e 2 (2.9) are compatible with the Einstein-Maxwell equations for the Lorentzian metric. The question thus arises whether, given a set (a, M , e, p) arising from a Riemannian metric, there is a preferred choice of p L and e L . A natural choice is p L = p , e L = e . (2.10) The condition p 2 eff > 0 and (2.8) imply that the simplest choice p L = 0 in (2.10) is not possible, except in the Page limit. The next simplest choice, e L = 0, leads then to purely magnetic solutions with a quantised magnetic charge. We emphasise that our quantisation mechanism of magnetic charge has nothing to do with the Dirac one, see Section 7 below.
Whether or not (2.10) is the right choice appears to be a matter of debate, see [6,11]. An alternative would be to decree that the Lorentzian solutions with p L = 0 and e L = 0 correspond to Riemannian solutions for whichÂ := e L r σ 2 /Σ is a vector potential for where µναβ is the totally antisymmetric tensor. In this case p 2 eff = e 2 L (compare [6]). This choice leads to a quantisation of electric charge.
It might be of interest to note that planar Lorentz transformations of (p, e) preserve p 2 eff , and can be thought of as the Euclidean counterparts of the usual duality transformations of the Maxwell field, which instead act as rotations of the (p, e) plane.
In any case, we wish to find ranges of parameters so that (2.3) is a Riemannian metric on a closed manifold M . This leads to the following obvious restrictions: First, compactness requires ϕ and t to be periodic, with a period which needs to be determined.
Further, compactness of M requires a range of the variable r , bounded by two first-order zeros r 1 < r 2 of ∆ r , so that (2.3) is Riemannian for ∀r ∈ (r 1 , r 2 ), θ ∈ (0, π). 2 In particular Σ ∆ r > 0 and Σ ∆ θ > 0 ∀r ∈ (r 1 , r 2 ), θ ∈ (0, π). (2.12) Equations (2.4) and (2.5) show that Σ and ∆ θ are positive on the equatorial plane, and we conclude that Now, if r 1 r 2 ≤ 0, then 0 ∈ [r 1 , r 2 ], and since Σ| r =0 < 0 this case will not lead to a regular Riemannian metric. Changing r to its negative, it remains to consider the case where 0 < r 1 < r 2 . Positivity of Σ leads then to r 1 > |a|, and positivity of ∆ θ imposes the restriction λ −1 > a 2 . Summarising: Given a Euclidean metric as above with e = 0, the corresponding Lorentzian metric with the same real values of λ, M , a, e = 0, and p will be called a partner solution. Note that the locations r i of the horizons of the partner solution will not coincide with the locations r i of the rotation axes of the associated Euclidean solutions; similarly for areas, surface gravities, etc.

Regularity at the rotation axes
For r ∈ [r 1 , r 2 ] let us introduce two functions ρ i , i = 1, 2, defined as with 1 = 1, 2 = −1, and and with functions 1 1,i which are smooth near the origin and satisfy 1 1,i (0) = 1. The function ρ 1 will serve as a coordinate replacing r for r ∈ [r 1 , r 2 ), while ρ 2 will replace r for r ∈ (r 1 , r 2 ]. Inverting, it follows that with functions 1 2 , 1 3 which are smooth near the origin, with 1 2 (0) = 1 = 1 3 (0). In order to make sure that the metric is regular near the intersection of the axes {sin θ = 0} with the axes {∆ r = 0}, near θ = 0 and for r ∈ [r 1 , r 2 ) we use a coordinate system (ρ 1 , t 1 , θ, φ 1 ), with t = ω 1 t 1 and ϕ defined through the formula dϕ := α 1 dφ 1 + a for some constants α 1 , ω 1 ∈ R * which will be determined shortly by requiring 2π-periodicity of t 1 and φ 1 . In (3.4) the coefficient in front of dt has been chosen so that g t t | ρ 1 =0 = 0. In this coordinate system the metric takes the form for some smooth functions 1 4 and F , with 1 4 (0, y) = 1. As is well known, when (ρ 1 , t 1 ) are viewed as polar coordinates around ρ 1 = 0, the one form ρ 2 1 dt 1 and the quadratic form dρ 2 1 + ρ 2 1 dt 2 1 are smooth. Similarly when (θ, φ 1 ) are polar coordinates around θ = 0, the one form sin 2 (θ)dφ 1 and the quadratic form dθ 2 + sin(θ) 2 dφ 2 1 are smooth. It is then easily inferred that the requirements of 2π-periodicity of t 1 and φ 1 , together with implies smoothness both of the sum of the diagonal terms of the metric g and of the off-diagonal term g t 1 φ 1 dt 1 dφ 1 on Note that (3.6) is equivalent to The above calculations remain valid without changes near θ = π. It is, however, convenient, to use a different symbol for the resulting polar coordinates: When θ ∈ (0, π] we will uset 1 andφ 1 for the relevant angular coordinates, and ω 1 ,α 1 for the corresponding coefficients. Thus, for θ ∈ (0, π]: Identical considerations for r ∈ (r 1 , r 2 ], using coordinate systems (ρ 2 , In an overlap region where both t and t 1 are coordinates, the equation t = ω 1 t 1 implies that t must be exactly 2π|ω 1 |-periodic. Similarly, in any overlap region where both t and t 2 are defined and are coordinates, t must be exactly 2π|ω 2 |-periodic. A similar argument applies tot i . So, the periodicity requirements of t i andt i lead to The uppermost curve is a plot of r 2 , the middle one that of r 1 , the lowest curve is a plot of the mass parameter M .

a = 0
When a = 0, and imposing the regularity conditions above, the metric (3.13) simplifies considerably: The coordinate ρ 1 can be written explicitly in terms of elliptic integrals, which is not very enlightening. After scaling to λ = 1, the periodicity conditions (3.12) are verified by a oneparameter family of solutions parameterized by a continuous parameter p 2 eff ∈ [0, 1/16), see Figure 3.1. These solutions will not be discussed any further.

a = 0: the quantisation conditions
When a = 0, without loss of generality, replacing t and/or ϕ by their negatives if necessary, we require a > 0 , ω 1 = ω > 0 . (3.14) To avoid ambiguities: except for the analysis of the Page limit in Appendix F, in what follows we will assume that (r, t , θ, ϕ) form a smooth coordinate system away from the rotation axes, with t and ϕ periodic.
Increasing φ 1 from zero to 2π with (ρ 1 , t 1 , θ) fixed takes one back to the starting point. Equations (3.4) and (3.7) show that ϕ changes by ±2π, and therefore the minimal period of ϕ must be 2π/k for some k ∈ N * . But then, increasing ϕ from zero to 2π/k with (r, t , θ) fixed takes one to the same place. This results in an increase of φ 1 by ±2π/k, which implies that k = 1. Hence, ϕ is exactly 2π-periodic. Now, increasing t 1 from zero to 2π with ϕ 1 fixed again takes one to the same place. This implies that ϕ must have changed by an integer multiple of 2π.
The same argument applies to ϕ 2 and t 2 . We conclude that (3.15)

Maxwell fields
Let us check that the Maxwell fields, defined as dA away from all axes of rotation, extend by continuity to smooth fields once the above constraints have been imposed. This can be done by inspection of the Maxwell potentials (2.6) (which, incidentally, are not regular at the rotation axes). We start with an analysis of the p-contribution to A which, using (3.4) and its equivalent with r 1 replaced by r 2 , can be rewritten as where the index i on r i , α i and φ i takes the values i ∈ {1, 2}. More precisely, the underbraced term in the last line of (3.16) is smooth away from r = r 2 when i = 1, and away from r = r 1 when i = 2. Near the axis cos θ = 1 the last, nonsmooth term can be rewritten as (3.17) which shows smoothness of the p-contribution to F = dA near θ = 0. In fact, we have proved that, for i = 1, 2, the vector potentials which are well-defined and smooth away from all axes of rotation, extend by continuity across θ = 0 and r = r i to smooth covector fields. An identical calculation near the axis cos θ = −1, with φ i replaced byφ i , shows that the offending term can be rewritten as which finishes the proof of smoothness of the p-contribution to F everywhere.
We also see that the potentials extend smoothly to the axis θ = π. We continue with the e-contribution to A: This finishes the proof of smoothness of F . Note that (3.21) shows that e r σ 2 /Σ extends smoothly across both θ = 0 and θ = π without further due as long as one stays away from the axes r ∈ {r 1 , r 2 }.
The question then arises, in how many ways can one glue the sets above to obtain smooth closed manifolds. We point out some possible constructions here. While we suspect that these are all possibilities, we have not made indepth attempts to analyse whether or not the list below is exhaustive. 3 Note that oriented manifolds are obtained if and only if ω 2 = α 1 α 2 ω.
A similar construction applies to the RP 2 bundles above.
3. Let (ρ 1 ) max be the maximal value of the coordinate ρ 1 , thus and suppose that the map is an isometry. This, however, occurs for the metrics considered here only in the Page limit, and is therefore only relevant to Appendix F. Then the identification of (ρ 1 , t 1 , θ, φ 1 ) with leads to a smooth compact manifold.

The solutions
The question then arises to find values of (M , p 2 eff , a) so that It follows from (3.7) and (3.11) that the above equations are equivalent to In addition we need to fulfill ∆ r (r i ) = 0, leading to the system of polynomial equations for (r 1 , r 2 , n 1 , n 2 , a, M , p 2 eff ).
Note that n 1 > n 2 ≥ 1 in view of (5.7). Moreover the solutions have to satisfy the constraints We note that we also need ∀r ∈ (r 1 , r 2 ) : ∆ r (r ) > 0, but this follows from the fact that ∆ r (r 1 ) is positive by (5.5) and ∆ r (r 2 ) is negative by (5.6).
We also note that equations (5.3)-(5.7) involve neither ζ nor the α i 's as in (4.7)-(4.8), which can thus be arbitrarily chosen once a solution has been found.
Our strategy is to prescribe λ ∈ R * + , n 1 , n 2 ∈ N * so that (5.3)-(5.7) become a system of five polynomials in the variables (r 1 , r 2 , p 2 eff , M , a). We use MATH-EMATICA to compute a Gröbner basis of the system. This provides a simpler equivalent system to solve. It turns out that one is led to a hierarchic system of polynomial equations, the first one depending only on p 2 eff , the second one only on p 2 eff and a, and so forth. An example is provided in Appendix A. Our MATHEMATICA calculations show the following: Let n max = 50 . (5.8) Then: 1. There exist no solutions with (n 1 , n 2 ) ∈ N×N with 1 ≤ n 2 < n 1 ≤ n max and p 2 eff ≤ 0. In particular there are no vacuum solutions with the properties set forth above.
2. For every pair (n 1 , n 2 ) ∈ N×N with 1 ≤ n 2 < n 1 ≤ n max there exists exactly one solution satisfying our constraints.
3. The physical parameters (see Appendix B) of the Lorentzian partner solutions are all bounded, cf. Table 5.1. In particular the physical mass of the Lorentzian partners is strictly positive, bounded away from zero, and bounded from above.
It should be emphasised that the existence of the solutions of the system as above is a rigorous result, derived by exact computer algebra. While numerics is used to check whether the joint zeros of the Gröbner basis satisfy the desired inequalities, this is again a rigorous statement, as the numerical errors introduced when checking the inequalities are well below the gaps occurring in the inequalities.
We expect that the threshold (5.8) is irrelevant, and indeed we have randomly sampled many further values of (n 1 , n 2 ), including e.g. are bounded, and that the values of the parameters approach affine correlations as both n 1 and n 2 tend to infinity. This is explained in Section 6 below, where exact bounds and the asymptotically affine relations are derived.
6 The limit n 1 → ∞ An interesting case arises when we require r = a to be a double zero of ∆ r . While in this case the geometry is not compact anymore, the resulting manifold provides a description of the geometry which is approached when n 1 tends to infinity with n 2 kept fixed. The values of the parameters (a, m, p 2 eff ) which arise in this case correspond to the limiting curves which arise in the plots showing the correlations between the parameters.
We have ∆ r | r =a = 2 − 10a 2 , so that ∆ r is positive for 0 < a < r 2 , with a simple zero at r = r 2 , if and only if 0 < a < 1 5 . (6.8) Inspection of (2.3) shows that the metric g is complete, with a smooth axis of rotation at the other zero r = r 2 of ∆ r when n 2 ∈ Z. The set r = a is infinitely far away, with the region r → a displaying an interesting geometry: While the circles of constant t , r and θ ∈ {0, π} shrink to zero as r tends to a, the metric on the spheres of constant ϕ and r is stretched along the meridians and approaches a smooth Riemannian metric on a cylinder obtained by removing the north and south pole from S 2 .
M phys attains its maximum at n 2 = 1 2 799 + 565 2 + 10 2 + 14 ≈ 56.409, at which point it equals 1/4. Closer inspection, taking into account that we are only interested in integer values of n 2 , gives 0.20361015 ≈ M phys | n 2 =1 ≤ M phys ≤ M phys | n 2 =56 ≈ 0.24999998 , (6.20) with the bounds being optimal. All quantities have an asymptotic expansion, as n 2 tends to infinity, in terms of negative powers of n 2 . This leads to simple relations between various quantities for n 1 and n 2 large, as follows: for large n 2 we have the approximate relations From this one obtains various approximately affine relations between the quantities above for 1 n 1 n 2 , e.g.   One can similarly make a second-order approximation in 1/n, by expanding the quantities of interest up to o(n −1 ) and eliminating n from the equations. As an example, near the maximum value of |q phys | we obtain the relation The exact solution and the curve resulting from the second order approximation in 1/n can be seen in Figure 6.3. In Figure 6.4 we plot the dependence on the continuous variable n 2 , in the n 1 → ∞ limit, of the area of the cross section of the horizon A + and the surface gravity κ + in the partner Lorentzian solutions.
In Appendix D the reader will find a translation of some of the numerical values above to SI units.

Dirac strings
Similarly to [5], the existence of charged spinor fields on the Euclidean manifold leads to further constraints on the parameters of the solution. Indeed, comparing (3.18) with (3.20) shows that the transition from a gauge potential which is regular near {cos(θ) = 1 , r = r i } to a gauge potential which is regular near {cos(θ) = −1 , r = r i } requires a gauge transformation If a Dirac field ψ carries a charge q 0 , such a gauge transformation induces a transformation Recall that ϕ is 2π/k-periodic, with k = 1 except in the Page limit where k = 2 can arise and which needs to be analysed separately in any case, see Section F.3 below. Thus in the remainder of this section we assume that ϕ is 2π periodic. The requirement of single-valuedness of ψ results in the condition 2q 0 p ħΞ =:n 1 ∈ Z . (7.1) Next, (3.18) and (3.21) show that the gauge potentials are regular near r = r i and θ = 0. Passing from one to the other requires a gauge transformation In this way we are led to a discrete family of solutions parameterised by four integers (n 1 , n 2 ,n 1 ,n 2 ) subject to the constraints (7.5) and (7.8).

A A typical solution
We rescale the metric so that λ = 1. We choose n 1 = 10, n 2 = 9. With this choice the system (5.3-5.7) takes the explicit form together with an identical equation for r 1 .
The structure of the equations is typical in the following sense: Since MATH-EMATICA does not manage to find a Gröbner basis when n 1 and n 2 are left as general parameters, our procedure is to provide the values of n 1 and n 2 and then seek the basis. All the resulting polynomials that we have inspected have then a structure identical to the one above.
It can be seen that solving the system (A.2) in the manner described above requires only solving polynomial equations in a single variable of at most forth order, and so explicit analytic expressions can be given. However, the expressions obtained, especially for r 1 and r 2 , become very unwieldy. Therefore, instead of the full analytic expressions, we give only the first five nontrivial digits after the decimal point of the parameters for the solution of (A.1) that fulfills the constraints: The "surface gravity" of the zeros of the ∂ t -Killing vector, located at r 1 and r 2 , reads Since ∂ ϕ and ∂ t are Killing fields and we obtain the following formula for the areas of the zero-set of ∂ t , located at r 1 and r 2 , and for the volume of the manifold The action of the Einstein-Maxwell system is given by Let S G be the gravitational action, we have A MATHEMATICA calculation gives leading to (B.7) Together this yields The minimum of the action is attained at (n 1 , n 2 ) = (2, 1), and equals S min ≈ −2.357. Since r 1 → n 1 →∞ a and p 2 eff → n 1 →∞ 0.32 (see (6.11)), the action is unbounded from above. It follows from the analysis in Section 6 that S G is bounded from above by −π/2, so only the Maxwell action grows without bound. Now, if r 2 is close to r 1 , then the Maxwell action is very small. One expects this to be true when both n 1 and n 2 are very large. This suggests very strongly that the set of pairs (n 1 , n 2 ), for which the Maxwell action S EM is very small compared to the gravitational one, is unbounded. Numerics shows that this is indeed the case for all large numbers n 1 that we have looked at.
In particular solutions with very large values of n 1 − n 2 are strongly suppressed when path-integral arguments are invoked.

B.2 Lorentzian case
In this section we consider the Lorentzian solutions with e = 0 and with the value of a, M and p 2 eff arising from a smooth compact Euclidean solution with e = 0. To avoid ambiguities, we write ∆ Lor := (r 2 + a 2 )(1 − λr 2 ) − 2M r + p 2 + e 2 and Ξ Lor := 1 + λa 2 . (B.9) In all solutions that we have found the function ∆ Lor has precisely two real first-order zeros, with exactly one positive, denoted by r + . The associated horizon is usually referred to as the cosmological horizon. The global structure of the Lorentzian solution is shown in Figure B.1.
As already pointed-out, there is an ambiguity in the definition of total mass of the associated Lorentzian space-time. In a Hamiltonian approach this ambiguity is related to the choice of the Killing vector field for which we calculate the Hamiltonian [3]. In any case, the physical mass M phys and the angular momentum J phys are usually calculated using the formulae Lor . (B.10) (The above mass of the Lorentzian solution is obtained by calculating the Hamiltonian associated with the Killing vector field Ξ Lor ∂ t +3 −1 Λa∂ ϕ , while the total angular momentum is the Hamiltonian associated with ∂ ϕ .) Figure B.1: A projection diagram for the Kerr-Newman -de Sitter metrics with exactly two distinct real first-order zeros of ∆ r , r − < 0 < r + , from [4]. Outside of the shaded regions, which contain the singular rings and the time-machines with boundaries atr ± , the diagram represents accurately (within the limitations of a twodimensional projection) the global structure of the space-time. Here r − and r + indicate the radii of the Lorentzian horizons, not to be confused with the Euclidean rotation axes from the body of the paper.
The area of the cross-section of the horizon located at r + is given by and is usually interpreted as the entropy of the cosmological horizon [7]. The surface gravity of the horizon r = r + associated with the Killing vector X := ∂ t + Ω∂ ϕ , where Ω is chosen so that X is tangent to the generators of the horizon, is ∆ Lor | r =r + . (B.12)

C A sample
We list in Table C.1 the defining parameters of all solutions for λ = 1, ζ = −1 , n 1 , n 2 ∈ {−10, 10} , n 1 > n 2 , fulfilling the constraints, as well as some associated physical quantities. The constraints a < r 1 < r 2 and a 2 < 1 are clearly seen to be fulfilled. The physical quantities M phys , |J phys |, |q phys | are defined in (B.10), while S denotes the Euclidean action of the solutions.

D SI units
Recall that λ := Λ/3. The replacements  It is easy to check that if is a solution of the system Eq.(5.3-5.7) for λ = 1, then provides a solution of this system with an arbitrary value λ.
In SI-units we have where G is the gravitational constant, c the speed of light and 0 the electric constant. Then the physical angular momentum in SI-units can be computed as Putting all this together we obtain Since Ξ Lor and ∆ Lor are invariant under rescaling, it follows The black hole temperature in SI-units T SI reads The results are given in Table D.2.

E Lorentzian partner solutions
Consider a set of parameters n 1 , n 2 , M , a, and p 2 eff that solve, together with the positive zeros of ∆ r , the system (5.3-5.7) and fulfill the constraints. For this set of parameters we calculate the zeros of the Lorentzian partner ∆ Lor given by (B.9) of the Euclidean function ∆ r . As already mentioned, for all (n 1 , n 2 ) that we have investigated the function ∆ Lor has only two real first-order zeros, with exactly one positive zero r + . minimal physical mass /charge

E.1 Geometric units
In Table E.1 we list the values of r + , the surface gravity ("temperature") and the area ("entropy") of the horizon.
The reader will find some physical quantities of interest associated with our solutions in Tables E.2 and E.3.
To close this section, let us assume that the above universe consists of protons, neutrons, and hydrogen atoms. This means that for the range of values, as given above, we have n items ≈ M phys /M proton ≈ 2 × 10 79 items. On the other hand n protons = |q phys |/e − ≈ 2 × 10 61 particles are required to produce the required charge. As a consequence, every 10 18 -th item carries a charge.

F Page limit
The aim of this appendix is to discuss the charged solutions obtained by Page's limiting procedure [16]. (These solutions have been already been discussed in [13,Section 7.4, Equations (135)-(136)] from a rather different perspective; compare [14].) Recall that Page's approach is the following: Let r 0 be a zero of ∆ r , and let be a small parameter. We define new coordinates (χ,φ, η) as where η andφ are 2π-periodic, and ω 0 is a constant to be determined. We choose the parameters (M , a, p 2 eff ) so that for a suitable constant C = C ( ). After taking the limit → 0 the metric takes the form An Euclidean signature will be obtained if Note that the transformation η → −η has the effect of changing the sign of ar 0 , so without loss of generality we can assume that ar 0 > 0. Since a simultaneous change of sign of a and r 0 leaves the metric invariant, we can assume that a ≥ 0 and r 0 > 0 .
Near χ = 0 we introduce a new coordinate φ, 2π-periodic, chosen so that g ηη | χ=0 = 0: for some constant α ∈ R * . Standard considerations show that the metric will be smooth if The constant ω P of (F.9) coincides with Page's constant ω Page , when p 2 eff = 0 and when the requirement that r 0 is a double zero of ∆, which is implicit in the construction here, is taken into account.) When a = 0, the metric is now a product of two round metrics, with possibly different curvatures, on S 2 × S 2 . From now on we only consider the case a > 0 .
The constraints (F.7) then become We continue with the caseē > 0.

F.1.2ē > 0
The addition of a positive charge parameterē increases the right-hand side of the second inequality in (F.24) ∀ν ∈ (0, 1). Thus from the analysis of the uncharged case, we can conclude that this constraint holds as well in the charged case. The right-hand side of the third inequality in (F.24) is monotonously increasing for ν ∈ (0, 1). It follows that the infimum and supremum are attained at ν = 0 and ν = 1 respectively. From this we can conclude the following: • The inequalityē < 8 5 is a necessary criterion to obtain an Euclidean signature, otherwise the third constraint in (F.24) is nowhere satisfied for ν ∈ (0, 1).
• For 1 2 <ē < 8 5 the right-hand side of the third inequality in (F.24) has a simple zero at some value ν * ∈ (0, 1), thus the constraints (F.24) are not fulfilled on (0, ν * ). Futhermore (F.20) is required. As the third inequality in (F.24) is a quadratic in the variable ν 2 , it is easy to verify that this condition holds on (ν * , 1). For the interval (ν * , 1) it follows from a simple analysis that the function which at fixedē assigns to ν the right-hand side of (F.23) attains every value in N above some threshold n min (ē) and that the constraints are fulfilled. The zeros of the first derivative of (F.23) lead to a fifth order polynomial. Thus the minimum value can only be determined numerically. The result is illustrated in Figure F occurring "quantum number" forē ∈ 1 2 , 8 5 .

F.1.3ē < 0
The addition of a negative charge increases the right-hand side of the third inequality in (F.24) ∀ν ∈ (0, 1). Thus from the analysis of the uncharged case, we can conclude that this constraint holds as well in the charged case. The right-hand side of the second inequality in (F.24) is monotonously decreasing in the uncharged case for ν ∈ (0, 1) and attains a zero at ν = 1. The addition of a negative charge increases the rate of decreasing. From this it follows that there exists a zero of (F.15) located at ν * ∈ (0, 1). Thus the constraints are fulfilled, for a given negative charge parameter, if and only if ν ∈ (0, ν * ).
The numerator of the n-function (F.15) has no zeros on (0, ν * ), which follows from the second constraint in (F.7). Thus it suffices to determine if, for a given parameterē, the maximum n max (ē) of the function of ν defined by the right-hand side of (F.23), for ν ∈ (0, ν * ), is greater than or equal to one. This analysis can be carried out numerically. The result is illustrated in the plot F.2.
From the numerical analysis we conclude, thatē −0.5 is a necessary criterion for the existence of a solution, and that n = 1 is the only possibility whenē ≤ 0.

F.2 The Maxwell fields in the Page limit
In this section we analyse the regularity of the one-form (2.6) after passage to the limit → 0. The coordinate transformations (F.1)-(F.3) yield the following form for the p-contribution of (2.6) in (η, χ, θ,φ) coordinates: The closed part has no limit as goes to zero but can be discarded without affecting the Maxwell field. Keeping the same symbol A (e) for the four-potential obtained after removing the singular term and taking the limit → 0, we find 1 − 2r 2 0 Σ r 0 dη + a r 0 Σ r 0 sin 2 (θ) dφ .
Near χ = 0 we use the 2π-periodic coordinate φ, as introduced in the analysis of the regularity of the metric, with corresponding coordinate differential (F.8). This yields is smooth for χ < π. An analogous analysis near χ = π, using the coordinateφ of (F.12), shows that the four-potential is smooth for χ > 0.

F.3 Dirac strings
The results of Section F.2 can be summarised as follows: the potential