球對稱時空中愛因斯坦純量場方程數值模擬之研究

Abstract

在文獻[7]中，Choptuik 提出了球對稱時空中愛因斯坦純量場方程塌縮的數值模擬，他發現存在一個臨界值p*，可以把愛因斯坦方程的解 S[p] 被分成兩組，一組會產生黑洞，另一組則否。他另外發現所產生的黑洞質量會滿足此關係 M_{BH}∝|p-p*|^γ，其中 γ≈0.37。這篇論文將會整理出研究上述問題的數值方法，而我們將用這些方法來實現愛因斯坦純量場的演化模擬，其中我們會使用3+1的方法將四維時空拆成時間和空間，再使用iterative Crank-Nicolson方法實現模擬。除此之外，還有一些模擬的細節也會呈現在此論文中。這篇論文將不僅確認了Choptuik的數值結果，也整理了一些有用的數值方法。

In [7], Choptuik presented a numerical study of spherically symmetric collapse of Einstein scalar field equations. He discovered that a family of solutions S[p] with the property that a critical parameter value, p*, separates solutions containing black holes from those which do not. He also conjectured that the masses of black holes satisfy a power law M_{BH}∝|p-p*|^γ, where γ≈0.37 is a universal exponent. This thesis is a summary of numerical techniques which could be useful for the study of such problems. We use them to simulate the evolution of Einstein scalar field equations with spherical symmetry. The 3+1 formalism for numerical simulation is used and treatment of dynamical system is iterative Crank­-Nicolson method. Moreover, the details of several numerical algorithm are presented in this thesis. This thesis not only confirms the simulative results of Choptuik but also summarize useful numerical techniques.