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|Determining Distance

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|Determining Distance

For very close stars, the distance can be determined by parallax. As the earth orbits the sun, the star whose distance is to be measured, appears to move against the background of seemingly fixed distant stars, as indicated in Fig. 9.4. From the extreme wanderings that occur half a year apart, the parallax angle $p$ can be determined from the straight line light paths,
$$b / d_{|}=b / d=\tan p \approx p, \quad d_{|}=b / p$$
The origin is at the sun’s center, $b=1.496 \times 10^{11} \mathrm{~m}$ is the radial coordinate of earth, and $d$ is the radial coordinate of the star. In Newtonian physics, these coordinates are the distances. This formula is good enough because gravity is so weak, and only close stars are considered.

General Relativity (GR) requires taking gravity into consideration. The light paths are geodesics as indicated by the dashed curve in Fig. 9.4. Following the development of Weinberg (1972), light leaves the source at position $\vec{d}$, and eventually reaches us. In the coordinate system $x^{\mu^{\prime}}$, where the origin is at the light source, the tip of the ray path is at $\vec{r}^{\prime}=\hat{n} r^{\prime}$. Here $\hat{n}$ is a fixed unit vector and $r^{\prime}$ is a parameter describing positions along the path. In order to translate to the coordinate system in which the light source is at $\vec{d}$ and the origin is at the center of the sun, the quasi-translation Equation (9.4) must be used, so that the same metric holds for both observers. For this case, use $\vec{a}=\vec{d}$ and $x^i \leftrightarrow x^{i^{\prime}}$. Thus,
$$\vec{r}\left(r^{\prime}\right)=r^{\prime} \hat{n}+\vec{d}\left[\left(1-k r^2\right)^{1 / 2}-\left[1-\left(1-k d^2\right)^{1 / 2}\right] \frac{r^{\prime} \hat{n} \cdot \hat{d}}{d}\right] .$$

## 物理代写|广义相对论代写General Relativity代考|Red Shift Versus Distance Relation

Return to the conditions of Section 9.2 where the red shift was defined. An expansion of $Q$ about $t_0$ can provide a relation for $z$ in terms of the radiation travel time $u_1 \equiv t_0-t_1$ or the luminosity distance $d_L$. Assume, the expansion can neglect terms of third order or higher, in the expansion variable $u=t_0-t$
\begin{aligned} Q & =Q_0-u \frac{d Q_0}{d t}+\frac{u^2}{2} \frac{d^2 Q_0}{d t^2},\left.\quad \frac{d^2 Q_0}{d t^2} \equiv \frac{d^2 Q}{d t^2}\right|_{t_0} \ & =Q_0\left[1-\frac{u}{Q_0} \frac{d Q_0}{d t}+\frac{u^2}{2 Q_0} \frac{d^2 Q_0}{d t^2}\right] \ & \equiv Q_0\left[1-H_0 u-q_0 H_0^2 u^2 / 2\right] \end{aligned}
In the above equation,
$$H=\frac{1}{Q} \frac{d Q}{d t},-q=\frac{1}{H^2 Q} \frac{d^2 Q}{d t^2}=1+\frac{1}{H^2} \frac{d H}{d t},$$
where $H$ is the Hubble parameter with present value $H_0$ and $-q$ is the acceleration parameter with present value $-q_0$.

Evaluate Eq. (9.14) at time $t_1$, where $u=u_1$, and find the relation between the travel time and red shift,
\begin{aligned} 1 & =\frac{Q_0}{Q_1}\left[1-H_0 u_1-H_0^2 q_0 u_1^2 / 2\right] \ & =(1+z)\left[1-H_0 u_1-H_0^2 q_0 u_1^2 / 2\right], \ z & =\frac{H_0 u_1+H_0^2 q_0 u_1^2 / 2}{1-H_0 u_1-H_0^2 q_0 u_1^2 / 2} \ & \approx\left[H_0 u_1+H_0^2 q_0 u_1^2 / 2\right]\left[1+H_0 u_1\right] \ & \approx H_0 u_1+H_0^2 u_1^2\left(1+q_0 / 2\right) . \end{aligned}

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

$$b / d_{|}=b / d=\tan p \approx p, \quad d_{|}=b / p$$

$$\vec{r}\left(r^{\prime}\right)=r^{\prime} \hat{n}+\vec{d}\left[\left(1-k r^2\right)^{1 / 2}-\left[1-\left(1-k d^2\right)^{1 / 2}\right] \frac{r^{\prime} \hat{n} \cdot \hat{d}}{d}\right] .$$

## 物理代写|广义相对论代写General Relativity代考|Red Shift Versus Distance Relation

\begin{aligned} Q & =Q_0-u \frac{d Q_0}{d t}+\frac{u^2}{2} \frac{d^2 Q_0}{d t^2},\left.\quad \frac{d^2 Q_0}{d t^2} \equiv \frac{d^2 Q}{d t^2}\right|_{t_0} \ & =Q_0\left[1-\frac{u}{Q_0} \frac{d Q_0}{d t}+\frac{u^2}{2 Q_0} \frac{d^2 Q_0}{d t^2}\right] \ & \equiv Q_0\left[1-H_0 u-q_0 H_0^2 u^2 / 2\right] \end{aligned}

$$H=\frac{1}{Q} \frac{d Q}{d t},-q=\frac{1}{H^2 Q} \frac{d^2 Q}{d t^2}=1+\frac{1}{H^2} \frac{d H}{d t},$$

\begin{aligned} 1 & =\frac{Q_0}{Q_1}\left[1-H_0 u_1-H_0^2 q_0 u_1^2 / 2\right] \ & =(1+z)\left[1-H_0 u_1-H_0^2 q_0 u_1^2 / 2\right], \ z & =\frac{H_0 u_1+H_0^2 q_0 u_1^2 / 2}{1-H_0 u_1-H_0^2 q_0 u_1^2 / 2} \ & \approx\left[H_0 u_1+H_0^2 q_0 u_1^2 / 2\right]\left[1+H_0 u_1\right] \ & \approx H_0 u_1+H_0^2 u_1^2\left(1+q_0 / 2\right) . \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。