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|Global Positioning System

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## 物理代写|广义相对论代写General Relativity代考|Global Positioning System

The global positioning system is in wide use, and GR provides an important correction. Before worrying about the role played by GR, let’s look at a simple-minded system where only SR needs to be worried about.

Imagine our observer does not know where he is, though he is actually at $d z_0$. Overhead at height $h \ll R_e$, many planes fly with speed $V$ in the $z$-direction. The planes are at rest with respect to each other. At high frequency $\nu$, they broadcast their time and position $\left(d t^{\prime}, d z^{\prime}\right)$. These are connected to times and positions in the observer’s frame by the Lorentz transform,
\begin{aligned} \gamma & =\left(1-V^2\right)^{-1 / 2} \approx 1+V^2 / 2 \ d t & =\gamma\left(d t^{\prime}+V d z^{\prime}\right), \quad d z=\gamma\left(d z^{\prime}+V d t^{\prime}\right) . \end{aligned}
The time of flight $d T$ of the electromagnetic wave to the observer is $d T=\left(h^2+\left[d z-d z_0\right]^2\right)^{1 / 2}$, and the time of arrival at the observer is $d t_0=d T+d t$. The observer has a system that interprets the signals so that two simultaneously arriving signals are deduced. Of course, this means that the two signals were not emitted simultaneously by the planes. Also simultaneous arrival really means arrival within a narrow time window, and that depends on the accuracy of your device. If two planes provide simultaneous arrivals, then
\begin{aligned} d T_A+d t_A & =d T_B+d t_B, \ \left(h^2+\left[d z_A-d z_0\right]^2\right)^{1 / 2}+d t_A & =\left(h^2+\left[d z_B-d z_0\right]^2\right)^{1 / 2}+d t_B . \end{aligned}

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

In a fully relativistic theory, energy is just one component of the momentum vector. Momentum and energy conservation is just conservation of each component of that vector. Suppose it holds in one frame where, for example, a muon, as illustrated in Fig. 2.6, decays into three lighter particles $\mu \rightarrow$ $\nu \bar{\nu} e$. In the muon rest frame $\mathrm{O}$, conservation of momentum is as follows:
$$0=P^{\bar{\beta}}(I)-P^{\bar{\beta}}(F)=P^{\bar{\beta}}(\mu)-P^{\bar{\beta}}(\nu)-P^{\bar{\beta}}(\bar{\nu})-P^{\bar{\beta}}(e) .$$
The Lorentz transform is applied to every term in the above equation. This yields the momenta in a frame $\mathrm{O}^{\prime}$, where the muon is moving. It is obvious that momentum conservation holds in $\mathrm{O}^{\prime}$,
$$0=x^{\bar{\alpha}^{\prime}}, \bar{\beta}\left(P^{\bar{\beta}}(I)-P^{\bar{\beta}}(F)\right)=P^{\bar{\alpha}^{\prime}}(I)-P^{\bar{\alpha}^{\prime}}(F)$$
It is necessary to write physical law as a tensor equation, as in Eq. (2.33). Then, if the law holds for one observer, it holds for all. Such a decay is always handled in SR, hence the bars over momentum indexes. The earth is freely falling in the metric due mainly to the sun. As will be seen, it is an inertial system. In the weak gravity of an earth laboratory, there is no noticeable earthly gravitational effect on these particles. Though the decay is handled in SR, the general principle holds for all frames.

## 物理代写|广义相对论代写General Relativity代考|Global Positioning System

\begin{aligned} \gamma & =\left(1-V^2\right)^{-1 / 2} \approx 1+V^2 / 2 \ d t & =\gamma\left(d t^{\prime}+V d z^{\prime}\right), \quad d z=\gamma\left(d z^{\prime}+V d t^{\prime}\right) . \end{aligned}

\begin{aligned} d T_A+d t_A & =d T_B+d t_B, \ \left(h^2+\left[d z_A-d z_0\right]^2\right)^{1 / 2}+d t_A & =\left(h^2+\left[d z_B-d z_0\right]^2\right)^{1 / 2}+d t_B . \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。