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|GR Equations of Motion

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|GR Equations of Motion

In the locally inertial frame specified by $x^{\bar{\mu}}$, all objects travel in a straight line without acceleration. The equation of motion of a particle with rest mass is as follows:
$$\frac{d^2 x^{\bar{\mu}}}{d \tau^2}=\frac{d U^{\bar{\mu}}}{d \tau}=0 .$$

Using, Eq. (3.13), the above, in another frame becomes
\begin{aligned} 0 & =\frac{d U^{\bar{\alpha}}}{d \tau}=\frac{d\left(x^{\bar{\alpha}}, \mu U^\mu\right)}{d \tau}=x^{\bar{\alpha}}, \frac{d U^\mu}{d \tau}+U^\mu \frac{d}{d \tau} x^{\bar{\alpha}}, \mu \ & =x^{\bar{\alpha}},\mu \frac{d U^\mu}{d \tau}+U^\mu x^{\bar{\alpha}}, \mu, \nu \frac{d x^\nu}{d \tau} \ & =x^{\bar{\alpha}},\mu \frac{d U^\mu}{d \tau}+U^\mu x^{\bar{\alpha}}, \mu, \nu U^\nu \ & =x^\beta,{\bar{\alpha}}\left(x^{\bar{\alpha}}, \mu \frac{d U^\mu}{d \tau}+U^\mu x^{\bar{\alpha}}, \mu, \nu U^\nu\right)=\delta^\beta \frac{d U^\mu}{d \tau}+U^\mu U^\nu \Gamma{\mu \nu}^\beta \ & =\frac{d U^\beta}{d \tau}+U^\mu U^\nu \Gamma_{\mu \nu}^\beta \end{aligned}
The equation of motion involves the $\mathrm{C}$ symbols, which depend on the metric tensor.

Since a photon has $d \tau=0$, the above cannot be used as its equation of motion. One must substitute another parameter, say $d q$. Allowable parameters are called affine parameters. For example, $d \tau=k d q$ yields an allowable $d q$, where for photons $k=0$. The parameter $q$ describes the path, such that as the photon moves $\frac{d x^\mu}{d q}=W^\mu$ is the tangent vector, with the property,
\begin{aligned} W_\mu W^\mu & =g_{\mu \nu} W^\mu W^\nu=g_{\mu \nu} \frac{d x^\mu}{d q} \frac{d x^\nu}{d q}=\frac{g_{\mu \nu} d x^\mu d x^\nu}{d q d q} \ & =\left(\frac{d \tau}{d q}\right)^2=0 \end{aligned}

## 物理代写|广义相对论代写General Relativity代考|Geodesics

The thought experiments of Einstein, illustrated in Fig. 4.1, show that in the presence of gravity light moves in a curved path. The top of the figure shows two equivalent observers $\mathrm{O}$, one in gravity free space, and the other freely falling in a region of uniform gravity. They observe that a horizontally traveling light ray enters and exits their capsule a distance $L$ above the floor. The bottom of the figure shows an observer $\mathrm{O}^{\prime}$, not freely falling due to gravity. $\mathrm{O}^{\prime}$ also observes, that for the freely falling capsule, light entered and exited the same distance above the floor. However, according to $\mathrm{O}^{\prime}$ the exit point will have fallen in the time light crossed the capsule. Thus, the light also must have fallen or moved in a downward-curved path. The conclusion to be drawn is that gravity affects light, a break from Newtonian physics.

In the geometry of flat space, geodesics are the paths of minimum distance between two points, for motion with constant velocity, or the paths that minimize the travel time. Light in empty space certainly fits this case and before GR the phrase, “light travels in straight lines,” was often heard. This is because gravity is so weak, that the deviation from a straight line path, was too small to be observed. GR knowledgeable observers where gravity acts, but not freely falling, know that nothing can make the trip between two points faster than light. Thus, the “straight lines” or geodesics, are actually curved paths.

## 物理代写|广义相对论代写General Relativity代考|GR Equations of Motion

$$\frac{d^2 x^{\bar{\mu}}}{d \tau^2}=\frac{d U^{\bar{\mu}}}{d \tau}=0 .$$

\begin{aligned} 0 & =\frac{d U^{\bar{\alpha}}}{d \tau}=\frac{d\left(x^{\bar{\alpha}}, \mu U^\mu\right)}{d \tau}=x^{\bar{\alpha}}, \frac{d U^\mu}{d \tau}+U^\mu \frac{d}{d \tau} x^{\bar{\alpha}}, \mu \ & =x^{\bar{\alpha}},\mu \frac{d U^\mu}{d \tau}+U^\mu x^{\bar{\alpha}}, \mu, \nu \frac{d x^\nu}{d \tau} \ & =x^{\bar{\alpha}},\mu \frac{d U^\mu}{d \tau}+U^\mu x^{\bar{\alpha}}, \mu, \nu U^\nu \ & =x^\beta,{\bar{\alpha}}\left(x^{\bar{\alpha}}, \mu \frac{d U^\mu}{d \tau}+U^\mu x^{\bar{\alpha}}, \mu, \nu U^\nu\right)=\delta^\beta \frac{d U^\mu}{d \tau}+U^\mu U^\nu \Gamma{\mu \nu}^\beta \ & =\frac{d U^\beta}{d \tau}+U^\mu U^\nu \Gamma_{\mu \nu}^\beta \end{aligned}

\begin{aligned} W_\mu W^\mu & =g_{\mu \nu} W^\mu W^\nu=g_{\mu \nu} \frac{d x^\mu}{d q} \frac{d x^\nu}{d q}=\frac{g_{\mu \nu} d x^\mu d x^\nu}{d q d q} \ & =\left(\frac{d \tau}{d q}\right)^2=0 \end{aligned}

## 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。