Magnetic Reissner–Nordström Black Holes

• Magnetic Reissner–Nordström black holes are exact solutions with magnetic charges derived from the Einstein–Maxwell equations.
• The Harrison transformation immerses these black holes in external magnetic fields to achieve equilibrium and eliminate conical singularities.
• Dilaton generalizations and closed-form thermodynamic expressions demonstrate their significance in both gravitational theory and high-energy physics.

A Magnetic Reissner–Nordström black hole is a solution of the Einstein–Maxwell equations that generalizes the standard electrically charged Reissner–Nordström metric to configurations where the electromagnetic charge is magnetic or, more generally, where the spacetime is embedded in an external magnetic field. These objects play a central role in gravitational theory, especially in the mathematical classification of black holes, their equilibrium states, and their interaction with external fields. Recent work has also extended their construction to include multi-black-hole solutions, dilaton couplings, external gravitational potentials, and applications in the AdS/CFT correspondence.

1. Construction of the Magnetic Reissner–Nordström Solution

The Einstein–Maxwell equations admit both purely electric and purely magnetic black hole solutions; the latter are typically called magnetic Reissner–Nordström black holes. In the canonical spherically symmetric case, the metric takes the familiar form: $$ds^2 = -f(r) dt^2 + f(r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2),$$ with

$f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2},$

where $Q$ is the charge parameter, interpreted as a magnetic charge for the magnetic Reissner–Nordström solution. The Maxwell field is %%%%1%%%% corresponding to a magnetic monopole.

More generally, a magnetic analog can be constructed for the double-Reissner–Nordström solution: for two interacting black holes with magnetic charges $Q$ and $q$, the full solution is specified by a set of axisymmetric Ernst potentials

$$\mathcal{E} = \frac{A - B - iC}{A+B}, \quad \Phi = \frac{iC}{A+B},$$

where $A$, $B$, $C$ are specific axisymmetric potentials. The magnetic potential $A_\varphi$ (the $\varphi$-component of the gauge potential) is then explicitly constructed from the Kinnersley potential using the Sibgatullin method. The resulting $A_\varphi$ satisfies coupled first-order PDEs, and its explicit axis values are essential for analyzing regularity and balance conditions (see eq. (14) in (0811.2029)).

2. Harrison Transformation and Embedding in External Magnetic Fields

The Harrison transformation is a solution-generating technique that immerses a given Einstein–Maxwell solution in an external uniform magnetic field (of Bonnor–Melvin type). Applied to the seed metric $(\mathcal{E}_0, \Phi_0)$, the transformation modifies the potentials as: $$f' = \lambda f, \quad e^{2\gamma'} = \lambda^2 e^{2\gamma}, \quad A'_\varphi = 2B^{-1}[\lambda(1 + 2B A_\varphi) - 1],$$ with

$\lambda = (1 + B A_\varphi)^2 + B^2 \rho^2 f^{-1}.$

Here $B$ is the external magnetic field parameter. For the double–Reissner–Nordström solution, this procedure "dresses" the system, rendering the spacetime axially symmetric and enabling equilibrium conditions that would otherwise be impossible in purely electrovac backgrounds.

Balance (equilibrium) of two black holes requires the removal of conical singularities (struts) on the symmetry axis. This translates mathematically to vanishing of certain combinations of the metric function $\gamma$ and the Harrison scaling factor $\lambda$ along all axis segments. Explicitly, one finds: $$A_\varphi|_\text{I} = 0,\quad A_\varphi|_\text{II} = 2q,\quad A_\varphi|_\text{III} = 2(Q+q),$$ yielding the balance condition $Q + q = 0$; i.e., vanishing total magnetic charge. The necessary value of $B$ to achieve equilibrium is found from precise algebraic expressions (cf. eqs. (21) and (29)).

3. Asymmetric Black Diholes: General Properties and Horizon Thermodynamics

The most general subclass for which equilibrium is possible are "asymmetric black diholes": two non-extremal black holes with

• unequal Komar masses ($M \ne m$),
• charges $Q$ and $-Q$ of equal magnitude but opposite sign,
• and $Q_\text{tot} = 0$.

Such configurations are not only mathematically precise but also physically rich, providing a realization where mutual electromagnetic attraction/repulsion is exactly compensated by the externally imposed magnetic field. Conical singularities are avoided because the balance conditions are met globally.

The horizon areas and surface gravities possess elegant closed-form expressions. For the upper constituent $\Sigma$: $$A_{\Sigma} = \frac{4\pi[(R+M+m)(M+\Sigma) - Q(Q+q)]}{R + \Sigma - \sigma},\quad \kappa_\Sigma = \frac{1}{\sigma((R+\Sigma)-\sigma)^2} \left([(R+M+m)(M+\Sigma) - Q(Q+q)]^2\right),$$ with analogous expressions for the lower constituent $\sigma$ (see eqs. (10) and (11)). These formulas remain simple even in the generic interacting case, reflecting the integrability of the axisymmetric system.

4. Dilatonic Generalizations: Double–Gibbons–Maeda Spacetimes

The magnetostatic solution admits further generalization within dilaton gravity, with Lagrangian

$$\mathcal{L} = \frac{1}{16\pi} \sqrt{-g}[R - 2(\nabla\phi)^2 - e^{-2\alpha\phi} F^2],$$

where $\alpha$ is the dilaton coupling parameter (ranging from 0, pure Einstein–Maxwell, to Kaluza–Klein or string-theoretic values). The double–Reissner–Nordström solution seeds a corresponding double–Gibbons–Maeda solution, with

$$ds^2 = f/b (dt - A_t)^2 - f^{-1} b (e^{2\gamma}(d\rho^2 + dz^2) + \rho^2 d\phi^2),$$

and

$$f = (f_0)^{1/(1+\alpha^2)} e^{-2\alpha\phi_0/(1+\alpha^2)},\quad A_t = [A_t]_0.$$

Here, $f_0, \gamma, A_t$ are inherited from the double–Reissner–Nordström solution, while the dilaton field $\phi_0$ is taken as a linear sum of the individual (single-black-hole) potentials. The horizon areas and surface gravities acquire modified exponents, but the final result remains structurally compact: $$A_\Sigma = \frac{4\pi (4\Sigma)^{2/(1+\alpha^2)}}{1 - \alpha^2} \left([(R+M+m)(M+\Sigma) - Q(Q+q)]\right)^{1/(1+\alpha^2)},$$ with the analogous formula for $A_\sigma$ (see eq. (37)). This tractability survives even with arbitrary dilaton coupling.

5. Physical and Mathematical Structure of the Solutions

The explicit construction of the magnetic analog hinges on the one-to-one correspondence between the electrostatic and magnetostatic classes of Einstein–Maxwell fields. The physical content is encoded in the Ernst potentials and in a nontrivial, fully analytic formula for $A_\varphi$ (eq. (14)), which regulates both the field configuration and the structure of the solution along the axis.

The Harrison transformation is indispensable for exploring the global structure—by tuning $B$, one attains physical equilibrium in nontrivial two-black-hole systems without the need for singular struts. The unique features of asymmetric black diholes make them exceptionally attractive as fully regular, externally stabilized multi-black-hole solutions.

Furthermore, the thermodynamic properties—areas and surface gravities—are determined in closed-form expressions that combine parameters associated to mass, charge, and coordinate separation, with further modification upon inclusion of dilaton coupling.

6. Significance and Implications

The magnetic Reissner–Nordström black holes and their multi-black-hole extensions serve as a mathematically exact laboratory for the interaction between gravity, electromagnetism, and (optionally) scalar fields. The ability to achieve equilibrium and regularity for two unequal black holes using a physically reasonable external magnetic field highlights the power of solution-generating techniques (notably the Harrison transformation) for constructing globally regular spacetimes beyond the traditional single-black-hole paradigm.

Dilaton generalizations provide frameworks for testing the robustness of black hole thermodynamics when extra scalar fields and couplings are present. The elegance of the expressions for thermodynamic quantities, even in these generalized contexts, makes such systems particularly useful for analytic and numerical investigations in both gravitational and string-theoretic domains.

The results also provide foundational input for further paper in areas such as the stability of multi-black-hole systems, their role in string theory compactifications, and their thermodynamic phase structures, as well as for comparative classification with other calibrated binary systems.

7. Summary Table: Key Features

Aspect	Electrostatic RN	Magnetic RN	Double (Dihole) Extension
Charge	$Q$ (electric)	$Q$ (magnetic)	$Q, -Q$ (opposite sign dihole)
External Field	None	None / External $B$	Harrison transformation introduces $B$
Equilibrium (strut-free)	Not possible without $B$	Not possible without $B$	$Q + q = 0$ ensures equilibrium
Area Formula	$4\pi r_+^2$	$4\pi r_+^2$	Explicit, see above (cf. eq. (10))
Generalization to Dilaton	Gibbons–Maeda black hole	Gibbons–Maeda (magnetic)	Double–Gibbons–Maeda spacetime
Balance Condition	N/A	N/A	$A_{\varphi,\text{I}}=0$, $A_{\varphi,\text{III}}=0$

The equilibrium of black diholes, analytic tractability of the magnetic analog, and compactness of thermodynamic quantities underscore the importance of these solutions in mathematical and physical black hole research (0811.2029).