[论文解读] Evolving Geometries in General Relativity

Anastasios Taliotis|arXiv (Cornell University)|Jul 8, 2010

Black Holes and Theoretical Physics参考文献 28被引用 5

一句话总结

该论文提出一种微扰方法，用于在四维闵可夫斯基时空内建模点状冲击波的非对称碰撞，通过在平坦时空周围展开度规并施加因果边界条件。结果表明，度规以因果方式演化，δ-函数型剖面从每个冲击波中心以光速径向扩展，并在零碰撞参数时因守恒律破坏而显示出可能形成黑洞的迹象。

ABSTRACT

The problem of collisions of shockwaves in gravity is well known and has been studied extensively in the literature. Recently, the interest in this area has been revived trough the anti-de-Sitter space/Conformal Field Theory correspondence (AdS/CFT) with the difference that in this case the background geometry is Anti de Sitter in five dimensions. In a recent project that we have completed in the context of AdS/CFT, we have gained insight in the problem of shockwaves and our goal in this work is to apply the technique we have developed there in the case of ordinary gravity. In the current project, each of the shockwaves correspond to a point-like Stress-Energy tensor that moves with the speed of light while the collision is asymmetric and involves an impact parameter (b). Our method is to expand the metric $(g_{μν})$ in the background of flat space-time in the presence of the two shockwaves and compute corrections that satisfy causal boundary conditions taking into account back-reactions of the Stress-Energy tensor of the two point-like particles. Our solution respects causality as expected but this casual dependence takes place in an intuitive way. In particular, $g_{μν}$ at any given point $\vec{r}$ on the transverse plane at fixed $τ$ evolves according from whether the propagation from the center of each of the shockwaves or from both shockwaves has enough proper time ($τ$) to reach the point under consideration or not. Simultaneously around the center of each shockwave, the future metric develops a $δ$-function profile with radius $τ$; therefore this profile expands outwards from the centers (of the shockwaves) with the speed of light. Finally, we discuss the case of the zero impact parameter collision which results to the violation of conservation and we argue that this might be a signal for the formation of a black hole.

研究动机与目标

• 将AdS/CFT中冲击波研究的技术扩展至四维闵可夫斯基时空。
• 在非零碰撞参数（b）下，建模两个点状冲击波的非对称碰撞。
• 将冲击波应力-能量张量的反作用效应纳入时空几何。
• 通过从每个冲击波中心施加正确的时间演化，确保解满足因果性。
• 研究零碰撞参数极限及其对黑洞形成的影响。

提出的方法

• 将时空度规 $ g_{\mu\nu} $ 按冲击波强度的幂级数展开，围绕平坦闵可夫斯基时空。
• 将每个冲击波建模为以光速运动的点状应力-能量张量。
• 微扰求解爱因斯坦方程，包含来自冲击波源的反作用效应。
• 通过追踪从每个冲击波中心到横向平面上任意点的固有时 $ \tau $，施加因果边界条件。
• 构建度规，使得每个冲击波的影响以光速向外传播，形成半径为 $ \tau $ 的 $ \delta $-函数剖面。
• 通过叠加单个冲击波解，描述具有碰撞参数 $ b $ 的非对称碰撞几何。

实验结果

研究问题

• RQ1 在闵可夫斯基空间中，两个非对称碰撞的冲击波如何导致时空度规的因果演化？
• RQ2 由两个点状、以光速运动的粒子引起的度规修正具有何种结构？
• RQ3 碰撞参数 $ b $ 如何影响几何的因果发展？
• RQ4 在零碰撞参数极限下，几何结构如何变化？是否预示着黑洞形成？
• RQ5 能否通过每个冲击波中心的固有时一致地描述解的因果结构？

主要发现

• 横向平面上任意点的度规演化取决于从一个或两个冲击波中心出发的固有时 $ \tau $ 是否已到达该点。
• 在每个冲击波中心附近，未来度规发展出一个以光速径向向外扩展的 $ \delta $-函数剖面，形成类似光锥的结构。
• 解以一种物理上直观的方式尊重因果性，影响仅从冲击波源的过去光锥传播。
• 在零碰撞参数时，解表现出能量-动量守恒的破坏，提示可能存在黑洞形成的信号。
• 微扰框架成功捕捉了点状、超相对论性源在平坦时空中的引力效应的因果传播。