Approximate Solutions of Einstein Field Equations
Mousa Saaed Mohammed Emter
موسى سعيد محمد مطير
Einstein eld equations (EFEs) play an important role in understanding the theory of general relativity and related phenomena such as gravitational waves. Since, in general, it is almost impossible to nd analytical solutions of EFEs, it is necessary to solve these equations numerically (approximately). In this work, we derive the Einstein eld equations (EFEs) and the standard ADM (Arnowitt, Deser and Misner) equations form of EFEs. The ADM form consists of constraint equations and evolution equations for the raw spatial metric and extrinsic curvature tensors. The corner stone in the derivation of this form is \3+1 formalism", where one splits spacetime into three-dimensional space on the one hand, and time on the other. We study the BSSN (Baumgarte, Shapiro, Shibata and Nakamura) formulation. In this formulation the ADM equations were modi ed by factoring out the conformal factor and introducing three connections. The evolution equations can then be reduced to wave equations for the conformal metric components, which are coupled to evolution equations for the connection functions. Small amplitude gravitational waves were evolved and a direct comparison of the numerical performance of the modi ed ADM equations with the standard ADM equations was made. The results demonstrate that the standard implementation of the ADM system of equations, consisting of evolution equations for the bare metric and extrinsic curvature variables, is more susceptible to numerical instabilities than the modi ed form of the equations based on a conformal decomposition as suggested by Shibata and Nakamura. Further, in this work, we consider the problem of specifying Cauchy initial data in the case 3+1 formalism. We also apply the Optimal Homotopy Asymptotic Method, OHAM, and solving the Einstein eld equations corresponding to Schwarzschild geometry, i.e. we determine the Schwarzschild solution using OHAM.