First law of black holes in the gravitational electromagnetic system with magnetic charge

The quantum corrections in Einstein-Maxwell theory are typically described by introducing higher-curvature terms and non-minimally coupled electromagnetic fields. Our paper investigates the first law of dyonic black holes in the four-dimensional gravitational electromagnetic system with the Lagrangian L ( g a b , R a b c d , F a b ) . In previous studies, the magnetic “potential-charge” term in the first law of dyonic black holes was primarily obtained directly from the analytic solutions of black holes or through Komar integral variations in Einstein gravity, with limited applicability. Our analysis reveals that the failure of the Iyer-Wald method to derive the magnetic “potential-charge” term is mainly due to the assumption that the electromagnetic vector potential is regular at the axial poles. Consequently, without requiring regularity of the electromagnetic vector potential at the future Killing horizon and axial poles, we generally derive the first law of asymptotically flat stationary-axisymmetric black holes, which includes both electric and magnetic “potential-charge” terms. Finally, we apply our findings to the general quadric corrected Einstein-Maxwell theory. Our results indicate that the first law with the magnetic charge of black holes might be valid for the Einstein-Maxwell theory with some quantum corrections in the effective region.