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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。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 物理代写|广义相对论代写General Relativity代考|Orthonormal Coordinate Bases

Orthonormal Coordinate Bases. As stated, equations 12.1 and 12.3 (or 12.4) allow us to plot the trajectory of a photon in Schwarzschild coordinates $r$ and $\phi$ as a function of the Schwarzschild time coordinate $t$. But how would an observer at rest in Schwarzschild spacetime characterize the photon’s direction of motion?

Answering a question like this requires thinking carefully about how observers in a gravitational field will measure quantities. This is a good opportunity to digress for a moment from studying the motion of photons in particular to address the more general question of how an ordinary observer would interpret four-vector quantities whose components we know in a coordinate system like Schwarzschild coordinates. This general approach will be valuable to us in a number of contexts in the future.

Just as we can use cartesian coordinates on any sufficiently small patch of a curved surface (such as the surface of the earth), we can set up a local cartesian-like coordinate system in any sufficiently small region of spacetime. Indeed, any observer who is trying to measure physical quantities in his or her local region will do precisely this if they set up a clock lattice (like the one described in chapter 2 ) or something equivalent. Observers in a gravitational field will have trouble synchronizing lattice clocks because clocks higher in the field run faster than those lower, but such problems will be negligible in a sufficiently small frame.

## 物理代写|广义相对论代写General Relativity代考|The Four-Momentum of Light

The Four-Momentum of Light. In the case of photons, the four-vector of interest is the photon’s four-momentum $\boldsymbol{p}$, so let’s find that four-momentum. Now, the Schwarzschild components of $\boldsymbol{p}$ for a particle with nonzero rest mass $m$ are
$$p^\mu=m \frac{d x^\mu}{d \tau}=m \frac{d t}{d \tau} \frac{d x^\mu}{d t}=\frac{m e}{1-2 G M / r} \frac{d x^\mu}{d t}=\frac{E}{1-2 G M / r} \frac{d x^\mu}{d t}$$
where I have used equation 10.5 to eliminate the $d t / d \tau$ term and $E=m e$ is the particles’s relativistic energy at infinity. The last expression in equation 12.11 is well-defined in the limit $m \rightarrow 0$, so we will take it to be the definition of the Schwarzschild four-momentum components for a photon having energy $E$ at infinity. If the photon moves in the equatorial plane, then equations 12.1 and 12.3 imply that
$$\begin{gathered} p^t=\frac{E}{(1-2 G M / r)}, \quad p^\phi=\frac{E}{(1-2 G M / r)} \frac{d \phi}{d t}=E \frac{b}{r^2} \ p^\theta=0, \quad \text { and } p^r=\frac{E}{(1-2 G M / r)} \frac{d r}{d t}= \pm E \sqrt{1-\frac{b^2}{r^2}\left(1-\frac{2 G M}{r}\right)} \end{gathered}$$
The last is positive for an outgoing photon, negative for an ingoing photon.

## 物理代写|广义相对论代写General Relativity代考|The Four-Momentum of Light

$$p^\mu=m \frac{d x^\mu}{d \tau}=m \frac{d t}{d \tau} \frac{d x^\mu}{d t}=\frac{m e}{1-2 G M / r} \frac{d x^\mu}{d t}=\frac{E}{1-2 G M / r} \frac{d x^\mu}{d t}$$

$$p^t=\frac{E}{(1-2 G M / r)}, \quad p^\phi=\frac{E}{(1-2 G M / r)} \frac{d \phi}{d t}=E \frac{b}{r^2} p^\theta=0, \quad \text { and } p^r=\frac{E}{(1-2 G M / r)} \frac{d r}{d t}= \pm E \sqrt{1-\frac{b^2}{r^2}\left(1-\frac{2 G M}{r}\right)}$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:General Relativity, 广义相对论, 物理代写

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## 物理代写|广义相对论代考General Relativity代写|Expansion of the Metric Elements

In this section, the calculation of the predictions for the GP-B precessions will be carried out. It is too difficult to work with the full metric. One can expand the metric elements to higher than the lowest order correction explored in Chapter 5 , and obtain meaningful results. Such calculations are the essence of what is called post-Newtonian celestial mechanics. In making these calculations, I have been guided, in part, by S. Weinberg (1972).
In Chapter 5, the weak gravity lowest order correction to the metric, for a static spherically symmetric source of gravitation, gave the Newtonian potential $\Psi_{G}$,
\begin{aligned} g_{00} &=\eta_{00}+h_{00}=-1-2 \Psi_{G}=-1+2 M^{\prime} / r, \ g_{i i} &=1+h_{i i}=1-2 \Psi_{G}, \ \bar{v} & \approx\left(M^{\prime} / r\right)^{1 / 2} \ll 1 . \end{aligned}

## 物理代写|广义相对论代考General Relativity代写|The C Symbols

The $\mathrm{C}$ symbols involve terms with a partial derivative with respect to time. It’s important to note that as for powers of $\bar{v}$,
$$\frac{\partial}{\partial t} \propto \bar{v} / r .$$
This result and Eqs. (8.18)-(8.20) are required to make sure Eq. (8.21) is satisfied. The $\mathrm{C}$ symbols are obtained from
$$\Gamma_{\mu \nu}^{\xi}=g^{\xi \chi}\left(g_{\mu \chi}, \nu+g_{\nu \chi}, \mu-g_{\mu \nu}, \chi\right) / 2 .$$

# 广义相对论

## 物理代写|广义相对论代考 General Relativity代写|Expansion of the Metric Elements

$$g_{00}=\eta_{00}+h_{00}=-1-2 \Psi_{G}=-1+2 M^{\prime} / r, g_{i i} \quad=1+h_{i i}=1-2 \Psi_{G}, \bar{v} \approx\left(M^{\prime} / r\right)^{1 / 2} \ll 1 .$$

## 物理代写|广义相对论代考General Relativity代写|The C Symbols

avatest.org 为您提供可靠及专业的论文代写服务以便帮助您完成您学术上的需求，让您重新掌握您的人生。我们将尽力给您提供完美的论文，并且保证质量以及准时交稿。除了承诺的奉献精神，我们的专业写手、研究人员和校对员都经过非常严格的招聘流程。所有写手都必须证明自己的分析和沟通能力以及英文水平，并通过由我们的资深研究人员和校对员组织的面试。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。