Charged nutty black holes are hairy

We uncover the physical nature of the electric and magnetic monopoles discovered by McGuire and Ruffini on Misner strings accompanying charged nutty black holes, showing that these strings carry singular, nonuniform flows of electric and magnetic fields. These fields inevitably have nonzero divergence, thereby simulating the effective electric and magnetic charge densities along the strings. The latter create a complex short-range electromagnetic hair zone around the horizon, making the combined Misner-Dirac strings classically observable. Typical features of this new type of hair are presented. We also note that rotation can act as a hair generator even in the absence of NUT.

💡 Research Summary

The paper investigates the long‑standing puzzle of electric and magnetic monopoles that appear on Misner strings in charged “nutty” black‑hole solutions. By extending Bonnor’s interpretation of Misner strings as physical singularities and combining it with Dirac’s description of magnetic monopoles, the authors demonstrate that the strings carry singular, non‑uniform flows of electric and magnetic fields. These flows have a non‑zero divergence, which can be interpreted as effective electric and magnetic charge densities localized on the strings.

Starting from the Brill solution (a charged Taub‑NUT metric) the authors write the electromagnetic potential A and its dual B, compute the Maxwell two‑form F and its dual ˜F, and show that at the polar axis (θ=0,π) the exterior derivative of the azimuthal one‑form dφ yields a delta‑function term. This term produces singular contributions to F and ˜F (Eqs. 5 and 7) that reduce to the usual Dirac‑string fluxes when the NUT parameter n vanishes, but for n≠0 the fluxes decay with radius, making the string’s field non‑uniform. Consequently ∇·F and ∇·˜F are non‑zero, giving rise to electric and magnetic current densities J_e and J_m (Eqs. 8‑10). These densities are not true charges; they are effective sources generated by the gravitationally induced non‑uniformity of the field.

Integrating the flux through a sphere of radius r (Gauss’ law) yields effective charges Q(r) and P(r) (Eqs. 12‑13) that differ from the asymptotic charges q and p at finite r. The functions Q(r) and P(r) cross zero at radii r_e and r_m, respectively, signalling a change of sign of the effective charge density along the strings. The authors identify three distinct “hair” structures: (i) SS‑hair, where field lines start on one string segment and end on the opposite segment; (ii) SH‑hair, where lines are confined between the horizon and the string; and (iii) HH‑hair, which appears only when rotation is present and consists of closed loops connecting oppositely charged regions on the horizon itself.

The paper then extends the analysis to the rotating Kerr‑Newman‑NUT solution. The rotation parameter a introduces additional terms in the effective densities (Eq. 16) and leads to a polarized horizon (Eq. 17) where the sign of the charge density varies with polar angle. This gives rise to HH‑hair even when the NUT parameter is zero, showing that rotation alone can generate hair.

To visualise the field‑line structure, the authors define electric and magnetic fields measured by a static observer (E^μ = u_t F^{tμ}, H^μ = u_t ˜F^{tμ}) and plot integral curves of these fields in a compactified radial coordinate arctan(r/k). A series of figures (2‑17) illustrate how varying the parameters (mass m, NUT n, electric charge q, magnetic charge p, rotation a) produces five basic patterns: pure outward lines, pure confined lines, mixed configurations, and the three hair types. In particular, symmetric dyonic configurations (q=p) with n>0 exhibit both SH and SS hair, while rotating configurations display HH hair on the horizon.

Finally, the authors apply the same distributional method to a seven‑parameter N=4 supergravity solution (the Gal’tsov‑Kechkin solution). The presence of a complex axidilaton charge d modifies the effective densities (Eq. A‑1) and introduces additional roots where the densities vanish, leading to richer hair patterns. The supergravity case confirms that the mechanism is not limited to Einstein‑Maxwell theory; any theory where gravity couples to gauge fields in a non‑trivial way will exhibit similar string‑induced hair.

In conclusion, the work resolves the paradox of “charge without charge” on Misner‑Dirac strings by showing that the strings carry effective, gravity‑induced charge densities that produce observable short‑range electromagnetic structures—hair—around the black‑hole horizon. Rotation and the NUT parameter act as independent hair generators, and the analysis extends naturally to supersymmetric settings. This provides a concrete physical picture of how singular string structures can become classically observable features of black holes, enriching our understanding of black‑hole uniqueness theorems, membrane‑paradigm interpretations, and the interplay between gravity and gauge fields.