Effect of magnetic field on the gravitational energy of static and charged black holes

We investigate the effect of the magnetic field on the gravitational energy of the magnetized Reissner–Nordström black hole (RN-BH) carrying a magnetic charge and an electric charge, by applying approximate Lie symmetry methods. For this purpose, second-order perturbed geodesic equations are constructed by considering the mass, charge, and magnetic field in terms of a small perturbation parameter ϵ. It is shown that in the process of calculating the second-order approximate Lie symmetries, energy must be rescaled by some factor. The obtained energy-rescaling factors depend on the parameters of the black hole and the magnetic field. The rescaling factors obtained here are compared with the energy rescaling factor of the RN-BH investigated in Hussain et al. (2007) and the effects of the magnetic field on it are discussed. It is observed that the effect of the magnetic field results in increasing the gravitational energy of the underlying static and charged black holes.