Melvin--Bonnor and Bertotti--Robinson spacetimes with Baryonic charge

TL;DR

The paper tackles the problem of obtaining analytic black hole solutions carrying Baryonic charge in strong gravitational fields with external magnetic backgrounds. It leverages a mapping between Gauged Skyrme–Maxwell–Einstein theory (GSMT) and Einstein–Scalar–Maxwell theory to transfer scalar-field seeds into the GSMT sector, enabling configurations with nonzero Baryonic charge via the Callan–Witten density. A closed analytic relation between the black hole mass parameter $M$, the Baryonic charge $Q_B$, and the background magnetic field $B$ is derived for magnetized seeds, with linear behavior for large $M$ and significant nonlinear corrections at intermediate masses; in the Melvin–Bonnor case $Q_B$ is nonzero, while in Bertotti–Robinson the net charge vanishes but shows polarization-like density. The work provides analytic control over the interplay between Baryonic charge, magnetic fields, and gravity, and suggests future studies of magnetic susceptibilities, thermodynamics, and additional seed solutions to explore the phase structure of these novel configurations.

Abstract

Recently, a novel dictionary relating solutions of the Einstein--Scalar--Maxwell theory to solutions of gauged Skyrme--Maxwell--Einstein models in $(3+1)$ dimensions has been established. This development provides a clear and systematic route to constructing new configurations with nontrivial Baryonic charge and magnetic field, leveraging the fact that the Einstein--Scalar--Maxwell system is considerably more tractable, thanks to powerful solution-generating techniques. In this work, we exploit the framework that allows compact sources dressed by scalar fields to be consistently embedded in external electromagnetic backgrounds, and we construct their dual counterparts carrying Baryonic charge in the Skyrme sector. The resulting Baryonic charge is expressed directly in terms of the parameters characterizing the seed spacetime, and a corresponding quantization condition involving these parameters is explicitly derived. Consequently, the mass and the Baryonic charge are not independent parameters. These results provide a closed analytic formula for the black hole mass parameter in terms of the Baryonic charge and the magnetic field. This relation between the mass parameter and the Baryonic charge is linear for large values of the mass, while significant deviations from linearity arise if the mass takes intermediate values.

Figures (6)

• Figure 1: In panel (a), a $v=\mathrm{const.}$ slice of the Schwarzschild--Melvin configuration is displayed. Panel (b) shows a slice at $u_y=0$. Panels (c) and (d) present the same sections for the Melvin background solution $(M=0)$. Regions of higher color intensity correspond to a higher Baryonic charge density.
• Figure 2: Baryonic charge density obtained after subtracting the contribution of the Melvin background. The darkest regions indicate where the Baryonic charge is most strongly concentrated.
• Figure 3: The mass $M$ as a function of the Baryonic charge $Q_B$.
• Figure 4: The behavior of the change of $M$ with respect to $Q_B$. For small $B$, $\partial M/\partial Q_B$ becomes constant (dashdotted curve).
• Figure 5: A $z=\rm{const.}$ slice of $\rho_B$ is displayed, where the charge density is highly concentrated near the event horizon along the $\rho$-direction and decays monotonically as $\rho \to \infty$.