本論文中，我們報告我們發展關於數值相對論的程式，該程式建立在BSSN數學表述，移動式穿刺法和調適性網格細化的應用下。單一黑洞與雙黑洞系統的模擬結果證明此程式是可信賴的，並且可以應用研究在較真實的天文方案與數值相對論上。

在我們的宇宙裡，雙黑洞系統總是座落在由超重黑洞為中心主體的星系的位勢上。就我們所知，所有先前的雙黑洞系統的研究只考慮為孤立系統，而忽略環境背景的影響。我們感興趣這樣的影響，進而提出一微擾的方案來研究這樣的重力位勢對雙黑洞演化的影響，特別是在重力波的輻射上。雙黑洞本身可以被考慮頭對頭碰撞與互繞融合兩種，關於對超重黑洞考慮自由掉落與圓形軌道盤旋兩種極限。組合出四種劇本，我們模擬這四種劇本來研究重力位勢的背景對雙黑洞演化的影響。

高速自旋的黑洞可能存在一些星系的中心，然而模擬這樣的系統是有困難的。我們提出一新的座標來改寫Kerr度規在穿刺法的形式下。不像先前的一些座標，在這座標下視界半徑是有限，甚至是無關自旋大小的常數，即便在極限的自旋限制時。我們使用三個參數來控制座標的扭曲，透過合適的參數選取我們可以建構一個好的初始資料給一個極限自旋的黑洞。然而我們需要更細部地調整我們的程式來精準地模擬這樣極限自旋的黑洞。

In this thesis we report on our code for numerical relativity, in which it is based on the Baumgarte-Shapiro-Shibata-Nakamura(BSSN) formulation, and the moving puncture method is applied, and an adaptive/fixed mesh refinement(AMR) is implemented. The simulation results including a single and a binary black hole system demonstrate that this code is reliable and ready to be used in the study of more realistic astrophysical scenarios and of numerical relativity.

In our universe, binary black hole systems are always located in the potential of some super-massive black hole hosted in the center of galaxy. To our knowledge, all binary black hole systems have been treated in previous studies as isolated systems by ignoring the effects of the environment in which they are located. We are interesting in the effects, and propose a perturbational scheme for investigating the effects of these background potentials on the evolution of a binary black holes, especially the gravitational radiation which is of interest for gravitational wave detection. The binary black hole systems we considered include head-on colliding and inspiralling binaries. With respect to the super-massive black hole, the free-falling and circular orbiting case are considered as two limit cases of realistic systems.

High-spinning black holes could exist at the center of some galaxies; then there are difficulties in numerically simulating it. We propose a new radial coordinate to re-write the Kerr metric in a punctured form. Unlike some coordinates introduced previously, the horizon radius can be finite, even being a constant which is independent of the size of the spin, in our new coordinate with the extreme Kerr limit a/M→1. There are three parameters in our coordinate controlling the distortion of the coordinate. Through a suitable choice, we can construct good initial data with the extreme Kerr limit. However, we need to fine tune our code to accurately simulate the extremal Kerr black hole.

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