[论文解读] Local Invariants Vanishing on Stationary Horizons

Andrey A. Shoom|arXiv (Cornell University)|Jan 14, 2015

Black Holes and Theoretical Physics参考文献 9被引用 1

一句话总结

本文提出了一种方法，通过局部标量多项式曲率不变量来定位黑洞时空中的非奇点静止视界。通过构造与时空共维数相等的独立曲率不变量的梯度，其楔积的平方范数在视界处精确为零，从而为数值相对论中的视界检测提供了一种局部、几何的判据。

ABSTRACT

For a general black hole spacetime, the location of its surface (the event horizon, which is the boundary of the region from which causal curves can go to asymptotic future null infinity) depends on the future evolution of the spacetime and is not determined locally. However, for black holes that settle down to become stationary, one might ask whether one can find a local invariant that is generically nonzero off the horizon but vanishes on the horizon. In particular, one might look for a scalar polynomial curvature invariant [1–6], which is a scalar obtained by complete contraction of all the indices of a polynomial in the Riemann curvature tensor and its covariant derivatives. For example, Anders Karlhede, Ulf Lindstrom and Jan Aman [7] showed that RRαβγδ;ǫ crosses zero and switches sign as one crosses the horizon of the Schwarzschild metric, and one can easily show that this is also true for any smooth static spherically symmetric horizon. An invariant that vanishes on more general stationary horizons could be useful in numerical relativity for finding the approximate location of the horizon once the spacetime has settled down to become approximately stationary. Majd Abdelqader and Kayll Lake [8] have recently found a local scalar polynomial curvature invariant that vanishes at the horizon of the Kerr black hole. After casting this invariant into a simpler form that is proportional to the squared norm of the wedge product of two curvature-invariant gradients, we realized that the procedure generalizes to give a way to locate any nonsingular stationary horizon in terms of a local curvature invariant. Essentially, if one constructs as many gradients of independent curvature invariants as the cohomogeneity of a stationary spacetime, at a generic point in the spacetime these gradients will be linearly independent and spacelike, but at an horizon, a linear combination will become null. This implies that the squared norm of the wedge product of the gradients will vanish at a stationary horizon. Abdelqader and Lake [8] give the following six curvature invariants for the Kerr metric, which we here copy

研究动机与目标

• 确定是否存在一种局部标量多项式曲率不变量，其在静止视界上为零，而在视界之外非零。
• 解决事件视界为非局部量且依赖于未来时空演化，导致在数值模拟中难以定位的问题。
• 将现有结果（如史瓦西和克尔黑洞的结果）推广为适用于任意静止时空的统一框架。
• 为数值相对论中提供一种实用的、局部的几何判据，利用曲率不变量实现视界检测。

提出的方法

• 构造一组标量多项式曲率不变量，其数量等于静止时空的共维数。
• 在时空中一般点计算这些不变量的梯度，这些梯度通常线性无关且为类空。
• 形成这些梯度一形式的楔积，并计算其平方范数作为标量不变量。
• 证明该平方范数在静止视界处精确为零，此时梯度的线性组合变为类光。
• 通过已知情形（如克尔度规）验证该方法，将不变量简化为与楔积范数平方成正比的形式。
• 将该构造推广至任意非奇点静止时空，无论其对称性或具体解的形式如何。

实验结果

研究问题

• RQ1 能否构造一种局部标量多项式曲率不变量，使其在静止视界上精确为零？
• RQ2 与时空中一般点相比，曲率不变量梯度的几何行为在静止视界处如何变化？
• RQ3 是否存在一种系统性方法，仅利用曲率不变量在数值相对论中定位视界，而无需依赖全局因果结构？
• RQ4 时空的共维数与检测其视界所需曲率不变量的数量之间存在何种关系？
• RQ5 该方法能否超越已知解（如克尔或史瓦西解）推广至任意静止时空？

主要发现

• 任意非奇点静止时空的曲率不变量梯度楔积的平方范数在视界处为零。
• 在视界之外的一般点，曲率不变量的梯度线性无关且为类空，确保该不变量非零。
• 在视界处，这些梯度的线性组合变为类光，导致楔积范数为零。
• 该方法推广了已知的克尔黑洞结果，此前已发现特定曲率不变量在视界处为零。
• 该构造与度规的具体形式无关，适用于任意共维数有限的静止时空。
• 该方法为数值相对论模拟中提供了局部、几何且计算上可实现的视界检测判据。