Initial conditions for a boosted black hole in the CTT

Abstract

From its inception, General Relativity has been subject of a wide range of experimental scrutiny. Among the most successful of which are: the deviation of light in the prolonged solar eclipse of Saros 1919 [DED20], the anomalous precession of Mercury's perihelion, the change of frequency of the binary system PSR B1913+16 (proving the existence of gravitational radiation [TW89]), recent gravitational probes [GP-a, GP-b] that have proven, to second order, the existence of spacetime as the underlying fabric of the Universe and the recent direct detection of gravitational waves [AAA+16]. At present, there is much work which bases its results on General Relativity: The Cosmological standard model, ACDM, and its variations. Baryonic acoustic oscillations (BAO). Structure formation in the Universe. The very existence of a cosmological constant and the associated cosmological acceleration. Lienearised General relativity and gravitational waves. all of which are related to solutions to Einsteins equations for the gravitational field. At the present level of accuracy there is a wealth of observations that are yet to be completely understood. For example, the fluctuations measured by WMAP of the order of 10-5 in the anisotropies of the BAO [BH10]. With the new era of gravitational wave detection and gravitational wave Astronomy, the need for second (or even third!) order results will be of great importance. As is well known, the field equations are a system of coupled nonlinear partial differential equations and therefore quantitative results will only be achieved by direct numerical solutions. The main goal of this work is to take us from a knowledge of the Theory of General Relativity (GR) to a basic level in Numerical Relativity (NR). This includes the basic background and knowledge needed to reproduce contemporary literature, in particular we reproduce [IB09]. This work discusses the problem of obtaining the initial conditions for a “Boosted Black Hole". This is, we find the spacetime geometry around a black hole at a given time, when the black hole moves with linear momentum P ⃗, in some reference frame. This may be of use when discussing the interaction of black holes, as in the colission of two stellar compact objects. Although the work does not represent original ideas on the theory of numerical relativity, it compiles some points of view from the referenced authors and contains many calculations left to the reader. At the same time, there is plenty of room to extend this work to further calculations. After this brief introduction, chapter 1 is a general introduction to Numerical Relativity, in particular 3+1 NR. Chapter 2 explains the use of a particular gauge and coordinate choice which is used frequently in the literature and chapter 3 represents the solution to the problem presented in [IB09]. In the last section we finish with some concluding remarks.