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

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## 物理代写|广义相对论代写General Relativity代考|Reference Frames and Events

Reference Frames and Events. The study of geometry on a two-dimensional surface starts with the concept of a point: other geometric elements (lines, curves, triangles, and so on) are sets of points. The analogous concept for spacetime geometry is an event, a physical occurrence (e.g., the collision of two particles) that marks a suitably welldefined point in space and instant of time.

A reference frame allows one to quantify an event’s location in spacetime by providing a mechanism (real or hypothetical) for assigning spacetime coordinates (three spatial coordinates and one time coordinate) to every event. We can visualize a reference frame as being a rigid cubical lattice of measuring sticks with an event-detector and associated clock located at each intersection (see figure 2.1). Any event occurring within the lattice is registered by the nearest detector and is assigned the three spatial coordinates of that detector’s lattice position and the time coordinate registered by the detector’s associated clock. We can imagine making the spacing between lattice intersections as fine as needed to achieve any desired resolution. An observer collects and interprets the data generated by the frame’s detectors.

A cubical clock-lattice is usually too simplistic to be a practical technology. However, its simplicity is precisely its value as a visualization: it helps us think clearly about what a reference frame is and what it does. Any valid method of assigning coordinates must yield values equivalent to those obtained from a cubical clock-lattice.

We will often consider reference frames attached to specific objects. We will sometimes refer to a reference frame only obliquely, as in “an observer in the spaceship finds that …”: since an observer (by definition) interprets measurements obtained in a reference frame, such a phrase presumes a frame attached to the ship.

## 物理代写|广义相对论代写General Relativity代考|Inertial Reference Frames

Inertial Reference Frames. Depending on how it (or the object to which it is attached) moves through spacetime, a given reference frame may be either inertial or noninertial. An inertial reference frame (IRF) in both special and general relativity is defined to be a frame in which we observe a free object to move at a constant velocity (i.e., we find Newton’s first law to be obeyed) everywhere in the frame. All other reference frames are noninertial reference frames (NIRFs). Note that an observer in any frame can determine (by testing Newton’s first law) whether the frame is inertial without referring to anything outside the frame.

In general relativity, an object is “free” if it does not participate in non-gravitational interactions with other objects. The definition of an inertial reference frame then implies that the frames near a massive body that most closely approximate ideal IRFs are non-rotating freely falling reference frames (FFRFs), sometimes also called freefloat reference frames. We will see later that the tidal effects of gravity mean that FFRFs are not exactly inertial, but we can make this approximation as good as we need by limiting the frame to a sufficiently small region of spacetime.

In the contexts of both special and general relativity, you can easily prove (see box 2.1) that a reference frame will be inertial if and only if it moves at a constant velocity and does not rotate relative to an established IRF (as long as both frames are able to assign coordinates to events in a common region of spacetime).

## 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|Numerical Predictions for the GP-B Experiment

Figure 1 of the GP-B paper defines the experimental geometry. The gyroscope and its housing are launched in a close in, circular, polar orbit. The orbit rotation is counter clockwise, leading to an angular momentum vector that points out of the orbit plane and toward the reader. This is taken as the $\hat{e}{z}$ direction. Earth’s angular momentum is directed from south to north pole, and is in the orbit plane pointing up. This is taken as the $\hat{e}{x^{-}}$ direction. Then, $\hat{e}{y}=\hat{e}{z} \times \hat{e}{x}$ lies in the orbit plane, and is in the direction indicated for the initial spin. The position of the gyroscope is $$\vec{r}=\rho \hat{e}{\rho}=\rho\left(\hat{e}{x} \cos \phi+\hat{e}{y} \sin \phi\right), \quad \rho=1.1 R_{e},$$
where $\phi$ is the angle between $\vec{J}{e}$ and $\vec{r}$. Then, \begin{aligned} \vec{v} &=\left(M{e} / \rho\right)^{1 / 2} \hat{e}{z} \times \hat{e}{\rho}=\left(M_{e} / \rho\right)^{1 / 2} \hat{e}{\phi} \ M{e} &=4.431 \times 10^{-3} \mathrm{~m}, R_{e}=6.378 \times 10^{3} \mathrm{~m}, T_{e}=2.584 \times 10^{13} \mathrm{~m} \ J_{e} &=0.829\left(2 M_{e} R_{e}^{2} / 5\right) \omega_{e} \mathrm{~m}^{2}, \omega_{e}=2 \pi / T_{e} \end{aligned}

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

Motion in the equatorial plane $\theta=\pi / 2$ of a black hole is considered. Problem 7 shows this is a possible solution, even in Kerr space. The equations of motion for objects with finite rest mass $m$ are most easily obtained by noting $0=g_{\mu \nu, 0}=g_{\mu \nu}, 2$. So $U_{0}=P_{0} / m$ and $U_{2}=P_{2} / m$ are constant. The constants will be chosen so that the equations of motion match Eqs. (6.2)$(6.5)$, with Eq. (6.6), when $a=0$. Thus,
\begin{aligned} \frac{d \phi}{d \tau} &=U^{2}=g^{2 \nu} U_{\nu} \ &=g^{20} U_{0}+g^{22} U_{2}=\frac{g_{20} U_{0}-g_{00} U_{2}}{\Lambda} \end{aligned}
When $a=0$,
$$\frac{d \phi}{d \tau}=U_{2} \frac{-g_{00}}{\Lambda}=U_{2} \frac{(1-R / r)}{r^{2}(1-R / r)}=\frac{U_{2}}{r^{2}}$$

# 广义相对论

## 物理代写|广义相对论代考General Relativity代写|Numerical Predictions for the GP-B Experiment

GP-B 论文的图 1 定义了实验几何。陀螺仪及其外壳在一个近距离的圆形极地轨道上发射。轨道旋转为逆时针方向，导致角动量矢量

$$\vec{r}=\rho \hat{e} \rho=\rho(\hat{e} x \cos \phi+\hat{e} y \sin \phi), \quad \rho=1.1 R_{e},$$

$$\vec{v}=(M e / \rho)^{1 / 2} \hat{e} z \times \hat{e} \rho=\left(M_{e} / \rho\right)^{1 / 2} \hat{e} \phi M e \quad=4.431 \times 10^{-3} \mathrm{~m}, R_{e}=6.378 \times 10^{3} \mathrm{~m}, T_{e}=2.584 \times 10^{13} \mathrm{~m} J_{e}=0.829\left(2 M_{e} R_{e}^{2} / 5\right) \omega_{e} \mathrm{~m}^{2}, \omega_{e}=2 \pi / T_{e}$$

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

$$\frac{d \phi}{d \tau}=U^{2}=g^{2 \nu} U_{\nu} \quad=g^{20} U_{0}+g^{22} U_{2}=\frac{g_{20} U_{0}-g_{00} U_{2}}{\Lambda}$$

$$\frac{d \phi}{d \tau}=U_{2} \frac{-g_{00}}{\Lambda}=U_{2} \frac{(1-R / r)}{r^{2}(1-R / r)}=\frac{U_{2}}{r^{2}}$$

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## 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。