Gravitational energy of the spherically symmetric static and charged black holes encircled by the dark energy through approximate Lie symmetries

Abstract

Abstract We investigate the gravitational energy of the Reissner-Nordström black hole encircled by dark energy using the approach of approximate Lie symmetries of second-order. We confine our calculations to following values of the equation of state parameter ω: (i) ω q = − 1 2 (quintessence regime of dark energy), (ii) ω Λ = −1 (the pure cosmological constant regime), (iii) ω c = − 1 3 (frustrated cosmic strings network). In order to examine the gravitational energy for the Reissner-Nordström black hole spacetime encircled by the dark energy, we construct a system of second-order perturbed geodesic equations, which is done primarily by considering the mass M and charge Q of the black hole in terms of some small perturbation parameter ϵ. It is observed that we do not obtain any non-trivial approximate symmetry generator for any value of the state parameter ω, we get only trivial second-order approximate symmetry generators. As a result of these trivial approximate symmetries we get a rescaling of the proper time s which gives the rescaling of energy in the Reissner-Nordström spacetime encircled by the dark energy. It is found that for ω q = − 1 2 and ω Λ = −1, the rescaling factor for energy is dependent on r and for ω c = − 1 3 , the rescaling factor is independent of the radial distance r. Furthermore, we investigate that the effect of cosmological constant and quintessence dark energy decreases the energy of the underlying spacetime while the equation of state parameter ω c = − 1 3 , for which the Universe is static, increases the energy of the Reissner-Nordström black hole spacetime encircled by the dark energy.