Energy content of a regular black hole via approximate Lie symmetries
The definition of energy and momentum is a longstanding problem in General Relativity, as there is no widely accepted definition of energy and momentum. In this paper, we investigate the gravitational mass (energy) of the charged Hayward black hole using the approach of the approximate Lie symmetries of the second-order. To analyze the gravitational energy for the charged Hayward black hole, we build a system of second-order perturbed geodesic equations, primarily by considering the black hole mass $M$ and charge $Q$ in terms of small perturbation parameter $\epsilon$. We find that the exact symmetries are recovered as second-order approximate trivial symmetries. These trivial approximate symmetries result in the rescaling of the arc length parameter $s$ in this spacetime, indicating that the energy in the underlying spacetime must be re-scaled by a factor that is dependent on the charge $Q$, mass $M$, the radial coordinate $r$ and the parameter $l$ present in the Hayward black hole. This rescaling factor is compared to the energy rescaling factor of the Reissner-Nordström black hole given in [Hussain et al. SIGMA (2007)]. We note that the energy in the charged Hayward black hole decreases due to the presence of the Hayward parameter $l$.