重力場的對稱性與穩定性分析與研究

Abstract

1977年起，Hawking 等人提出一個 no-hair conjecture，推論只要有一個宇宙常數 項，宇宙演化最終會趨近於 de Sitter space。1983年，Wald 提出一個廣泛的證明， 證明只要系統滿足強能量條件 (SEC) 和主要能量條件 (DEC) ，則這個有宇宙常 數項的宇宙，最終會演化程 de Sitter space。但是，陸續有人提出不滿足兩種能量 條件的不均向宇宙演化解，宣稱這些解違背Hawking的推論。我們發展出一套簡 化微擾的模式，陸續證明這些解都是不穩定的。延續這些進展，我們計畫研究是 否可以提出類似 Wald 所提出的廣泛證明，並試圖降低宇宙演化所需的條件限 制。我們也計畫先對不均勻微擾所可以得到的訊息，做廣泛而深入的分析。我們 也將對黑洞解的穩定性進行廣泛與一般的分析與研究，試圖探討黑洞解和宇宙演 化解之間的相互關連，與期穩定性之間的對應關係。希望藉由這個分析，探討最 低能量狀態下，這些解析解的穩定性與期對稱性間的可能關連。

Hawking et. al. proposed a no-hair conjecture in 1977, now known as Hawking conjecture of the cosmological evolution. Hawking believes that de Sitter space is the inevitable final state of any cosmological evolution if there exists a cosmological constant term. One of the success in the pursuit of the proof of this conjecture came in 1983 when Wald provides a solid proof: de Sitter space is indeed the only final state of any cosmological system provided that the system obeys strong energy condition (SEC) and dominant energy condition (DEC) in the presence of a cosmological constant. In the meantime, many counter-examples are shown to violate both the SEC and DEC. These analytic anisotropic solutions are claimed to be stable, not willing to go to the de Sitter final state. We have been able to show that these solutions are in fact unstable by a new and simple perturbation method. We plan to generalize the general proof of Wald to support and prove more rigorously the Hawking conjecture. We also propose to study the relations between symmetry of the system, symmetry of vacuum, and the stability of the local and lowest energy solutions of the system.