Analytic solutions for generic electric-magnetic EMD electrovacua

Derive analytic solutions of the coupled ordinary differential equations governing cohomogeneity-one Einstein-Maxwell-dilaton electrovacua with independent electric and magnetic field parameters for general dilaton coupling constants.

Background

The authors reduce the equations for electrovacua containing independent electric and magnetic fields to two second-order ordinary differential equations and a Hamiltonian constraint. Closed-form solutions are obtained for the special couplings a=1 and a=sqrt(3), whereas the general-coupling case is left to numerical treatment.

References

For general dilaton coupling constants with generic $(E,B)$, analytic solutions are unaccessible, and one must resort to numerical methods, which we shall not discuss in this paper.

— Schwarzschild Black Hole in Uniform Magnetic Field in Einstein-Maxwell-Dilaton Theories  (2609.19261 - Ma et al., 16 Sep 2026) in Section 5.3, Special analytical solutions

However, we have been unable to introduce the blackening parameter $\mu$ and construct the Schwarzschild black hole immersed in such electrovacua for general dilaton couplings, even at the level of ansatz.

— Schwarzschild Black Hole in Uniform Magnetic Field in Einstein-Maxwell-Dilaton Theories  (2609.19261 - Ma et al., 16 Sep 2026) in Section 5.5, Schwarzschild black hole in external electromagnetic fields

Consequently, we are unable to generalize the construction to arbitrary dilaton coupling constants beyond $a=1$ and $a=\sqrt3$.

— Schwarzschild Black Hole in Uniform Magnetic Field in Einstein-Maxwell-Dilaton Theories  (2609.19261 - Ma et al., 16 Sep 2026) in Section 5.5, Schwarzschild black hole in external electromagnetic fields