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# 物理代写|广义相对论代写General Relativity代考|PHYS760 Focus on Black Holes

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## 物理代写|广义相对论代写General Relativity代考|Focus on Black Holes

Focus on Black Holes. Let me emphasize again that the Schwarzschild metric applies only in the vacuum outside a spherical and static gravitating object, not inside such an object. Most gravitating objects (such as normal stars and galaxies) have surface $r$-coordinates that are large compared to their Schwarzschild radii $r_s=2 G M$, so we will not observe for such objects the strongly non-Newtonian behaviors that we have discussed that happen at small $r$-coordinates. In this and the next few chapters, however, we will focus specifically on the physics of black holes, i.e., objects which do not have a surface outside of $2 G M$. Black holes display most vividly the differences between general relativity and Newtonian gravitational theory. In this chapter, we will focus specifically on the strange physics of the surface at $r=2 G M$ that we call the Schwarzschild spacetime’s event horizon.
A Catalog of Pathologies at $r=2 G M$. The Schwarzschild metric is
$$d s^2=-\left(1-\frac{2 G M}{r}\right) d t^2+\left(1-\frac{2 G M}{r}\right)^{-1} d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$
We also saw in chapter 9 (see equation $9.12 b$ ) that the light emitted at $r$-coordinate $r_E$ and received at $r_R$ is red-shifted or blue-shifted according to
$$\frac{\lambda_R}{\lambda_E}=\sqrt{\frac{1-2 G M / r_R}{1-2 G M / r_E}}$$

## 物理代写|广义相对论代写General Relativity代考|Possible Roots of These Pathologies

Possible Roots of These Pathologies. These pathologies (particularly the infinities) signal that something bad is going on at $r=2 G M$. There are two possible explanations for what is going wrong:

1. The spacetime has a geometric pathology at $r=2 G M$.
2. The Schwarzschild coordinate system is broken at $r=2 G M$.
A geometric pathology occurs when the physical characteristics of the spacetime are such that we cannot describe it at all using the mathematics we have developed. One of the most fundamental assumptions we made in chapter 5 was that our spacetime was not so horribly curved that we could not model a sufficiently small patch around any point as being flat. The apex of a cone is an example of a geometric pathology: since we cannot model even an infinitesimal region around the apex as being flat, our mathematics breaks down and no coordinate system will adequately describe the surface of the cone at that point.

On the other hand, a coordinate pathology occurs when the underlying geometry of the spacetime is perfectly reasonable but we happen to be using a coordinate system that describes that geometry poorly at one or more events or locations. For example, the latitude-longitude coordinate system on the surface of a sphere exhibits coordinate pathologies at the poles, because the $g_{\phi \phi}$ component of the metric $d s^2=R^2 d \theta^2+R^2 \sin ^2 \theta d \phi^2$ goes to zero there. This is not because the pole is geometrically different than any other location on the spherical surface, but rather because in the coordinate system, all the lines of longitude come together at the poles, meaning that the pole has no well-defined $\phi$ coordinate.

## 物理代写|广义相对论代写General Relativity代考|Focus on Black Holes

$$d s^2=-\left(1-\frac{2 G M}{r}\right) d t^2+\left(1-\frac{2 G M}{r}\right)^{-1} d r^2+r^2 d \theta^2+r^2 \sin ^2 \theta d \phi^2$$

$$\frac{\lambda_R}{\lambda_E}=\sqrt{\frac{1-2 G M / r_R}{1-2 G M / r_E}}$$

## 物理代写|广义相对论代写General Relativity代考|Possible Roots of These Pathologies

1. 时空具有几何病理学 $r=2 G M$.
2. Schwarzschild 坐标系在 $r=2 G M$.
当时空的物理特性使我们无法使用我们开发的数学来描述它时，就会出现几何病理学。我们在第 5 章中 做出的最基本的假设之一是，我们的时空没有弯曲到我们无法将任何点周围的足够小的补丁建模为平坦 的程度。圆雉的顶点是几何病理学的一个例子: 由于我们无法将顶点周围的无穷小区域建模为平坦的， 因此我们的数学会崩溃，并且没有坐标系可以充分描述该点处的圆锥表面。
另一方面，当时空的基本几何结构完全合理但我们恰好使用的坐标系在一个或多个事件或位置处描述该几何结 构时，就会发生坐标病态。例如，球体表面的经纬度坐标系在两极表现出坐标病态，因为 $g_{\phi \phi} \phi^{\prime}$ 指标的组成部分 $d s^2=R^2 d \theta^2+R^2 \sin ^2 \theta d \phi^2$ 在那里变为雺。这并不是因为极点在几何上不同于球面上的任何其他位置，而 是因为在坐标系中，所有经线都汇集在极点处，这意味着极点没有明确定义 $\phi$ 协调。

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