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

# 物理代写|广义相对论代写General Relativity代考|PHYS760 Orthonormal Coordinate Bases

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 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。