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

In this work, we exploit a dictionary that maps solutions of the Einstein--Scalar--Maxwell theory to those of gauged Skyrme--Maxwell--Einstein models in $(3+1)$ dimensions, allowing gravitating scalar-field configurations in external electromagnetic backgrounds to be interpreted in terms of baryonic quantities. Using the definition of Baryonic charge as the integral of a topological density, we derive novel mass formulas that relate the spacetime mass parameter to the Baryonic charge, the external magnetic field, and other solution parameters. As a result, the mass and Baryonic charge are not independent. These closed analytic expressions encode physical information that would otherwise be difficult to extract. In particular, for the class of solutions considered here, the relation between mass and Baryonic charge is linear at large mass, while significant nonlinear deviations appear at intermediate values. This framework provides a useful tool for both extracting new physical insights from known solutions and constructing new baryonic configurations using the solution-generating techniques available for Einstein--Scalar--Maxwell theory.