任意助推下视视界的面积不变性
Area Invariance of Apparent Horizons under Arbitrary Boosts
摘要：
广义相对论中一个著名的分析结果是，黑洞视视界的2维面积与观测者的运动无关，并且实际上与$ t=constant $切片无关，而这在广义相对论中可以是任意的。尽管如此，在某些框架中（由于度量的坐标形式而复杂化），地平线面积的显式计算通常比在其他框架中困难得多。在这里，我们给出了一个明确的证明非常有限的度量形式的（史瓦西和克尔）真空黑洞。在这些时空的Kerr-Schild坐标表达式中，它们有明确的洛伦兹不变量形式。我们考虑黑洞在坐标系中运动的{ \it boosted}版本。由于这是稳态黑洞时空，其视视界是黑洞视界的二维截面，因此我们用（boosted）$ t = constant $计算了boosted空间中的视视界面积，得到了与非boosted情形相同的结果.注意，虽然面积不变性是一般的，但我们只处理克尔-席德形式的黑洞，只考虑一个特别简单的切片变化，这相当于一个提升。即使有这些限制，我们发现，结果照亮了物理的地平线作为一个零表面，并提供了一个有用的教学工具。据我们所知，这是第一个明确的计算，这种类型的证明面积不变性的地平线。此外，这些计算与移动黑洞的计算表示中出现的变换直接相关。我们提出了一个应用程序，这一结果的初始数据增强黑洞。
英文摘要：
It is a well known analytic result in general relativity that the 2-dimensional area of the apparent horizon of a black hole remains invariant regardless of the motion of the observer, and in fact is independent of the $ t=constant $ slice, which can be quite arbitrary in general relativity. Nonetheless the explicit computation of horizon area is often substantially more difficult in some frames (complicated by the coordinate form of the metric), than in other frames. Here we give an explicit demonstration for very restricted metric forms of (Schwarzschild and Kerr) vacuum black holes. In the Kerr-Schild coordinate expression for these spacetimes they have an explicit Lorentz-invariant form. We consider {\it boosted} versions with the black hole moving through the coordinate system. Since these are stationary black hole spacetimes, the apparent horizons are two dimensional cross sections of their event horizons, so we compute the areas of apparent horizons in the boosted space with (boosted) $ t = constant $, and obtain the same result as in the unboosted case. Note that while the invariance of area is generic, we deal only with black holes in the Kerr-Schild form, and consider only one particularly simple change of slicing which amounts to a boost. Even with these restrictions we find that the results illuminate the physics of the horizon as a null surface and provide a useful pedagogical tool. As far as we can determine, this is the first explicit calculation of this type demonstrating the area invariance of horizons. Further, these calculations are directly relevant to transformations that arise in computational representation of moving black holes. We present an application of this result to initial data for boosted black holes.
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