Second-Order Approximate Lie Symmetry Methods to Define Energy Content of Some Spacetimes

Abstract

Obtaining a well-defined formula for energy-momentum in General Relativity has been a challenge since Einstein’s time. Unfortunately, General Relativity does not provide a generally accepted definition of energy-momentum. The distribution of energy-momentum caused by matter, non-gravitational, and gravitational fields has been de-scribed by a variety of energy-momentum complexes whose physical significance has been questioned. Hence in this thesis, the second-order approximate Lie symmetry method is used to investigate the problem of energy in General Relativity. For this purpose, the gravitational energy of the Reissner-Nordström black hole and the charged-Kerr black holes surrounded by dark energy are studied using the approximate Lie symmetry approach. For this, three different dark energy scenarios are considered: cosmological constant, quintessence dark energy, and the frustrated network of cosmic strings. It is shown that due to the presence of the cosmological constant and quintessence dark energy, the energy of the Reissner-Nordström black hole surrounded by dark energy decreases but the equation of state parameter ωc = − 1 3 which corresponds to the frustrated network of cosmic strings, increases the energy of the Reissner-Nordström black hole surrounded by dark energy. For the charged-Kerr black hole surrounded by dark energy, it is observed that for cosmological constant and quintessence dark energy, the energy content of this black hole spacetime is reduced, whereas for ωc = − 1 3 , it is noted that the contribution of dark energy first decreases the total energy of the underlying spacetime for smaller values of radial coordinate r and then increases for comparatively larger values of r. Lastly, we investigated the effect of the magnetic field on the gravitational energy of the magnetized Reissner-Nordström black hole carrying a magnetic charge and an electric charge. It is noted that the gravitational energy of the underlying charged and static black holes increase due to the presence of the magnetic field.