### 物理代写|广义相对论代写General relativity代考|MATH7105

statistics-lab™ 为您的留学生涯保驾护航 在代写广义相对论General relativity方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写广义相对论General relativity代写方面经验极为丰富，各种代写广义相对论General relativity相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|广义相对论代写General relativity代考|The Lorentz Group

We have obtained the Lorentz transformation for motion in the $x$ direction and discussed some elementary applications. Now we are going to look at such transformations from a more sophisticated mathematical viewpoint, and with a more elegant notation (Schutz 2009). This chapter is intended to orient you towards the geometric viewpoint of general relativity, and to show that the notation can do much of the algebraic work for you. Only cartesian coordinates will be used in this chapter.
We will first derive a more general definition of a Lorentz transformation. Recall that special relativity is based on the following principles (Schwartz 1968).

I. The analytical form of physical laws is the same in all inertial reference frames as described by systems of Cartesian coordinates.
II. The speed of light in vacuum is a universal constant.
A more sophisticated way to state principle II is that we wish to make the equation of an expanding spherical wave front of light invariant under the relevant transformation of the space and time coordinates. We write the wave front as
$$c^{2} t^{2}-\vec{x}^{2}=0$$
and show a picture in Fig. $2.1$ with the $z$ coordinate suppressed. Because of the shape of the surface in this picture it is called a light cone. Events in an inertial system are points in four-dimensional spacetime or Minkowski space. They are labeled by $x^{\mu}=(c t, x, y, z)$, with $c t$ taken as the zeroth coordinate. The set of coordinates is also called the position 4-vector. We wish to find a transformation between such coordinates in two systems, with the linear form

$$x^{\prime \mu}=\sum_{0}^{3} a_{v}^{\mu} x^{\nu}=a_{v}^{\mu} x^{v}$$
Notice that in (2.2) we simply omitted the summation sign with the understanding that repeated indices are to be summed over. This is the famous Einstein summation comvention which we will use henceforth; it makes the equations look much simpler. The light cone equation (2.1) may be written in matrix notation as.
$$(c t, x, y, z)\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \end{array}\right)\left(\begin{array}{c} c t \ x \ y \ z \end{array}\right)=0$$

## 物理代写|广义相对论代写General relativity代考|Four-Vectors and Tensors

We have called the set of coordinates of an event in spacetime the position 4-vector; the position 4 -vector is the archetype of a contravariant 4 -vector, which we now define in general as any set of 4 quantities which transform under a Lorentz transformation as
$$\bar{V}^{\alpha}=a^{\alpha}{ }_{\tau} V^{\tau}$$
That is, a contravariant 4 -vector is a set of quantities that transforms like the coordinates. We will often refer to a contravariant 4 -vector as simply a 4 -vector.

We define another 4-component object with a lower index using the Lorentz metric,
$$V_{\alpha}=g_{\mu v} V^{v},$$
which we call a covariant 4 -vector. For example, the covariant position 4-vector is.
$$x_{\mu}=(c t,-x,-y,-z)$$
The operation in $(2.10)$ is called lowering an index. An index may be raised similarly with the inverse of the Lorentz metric, which we denote as $g^{\mu v}$,
$$V^{\alpha}=g^{\alpha v} V_{v}, \quad g^{\alpha \lambda} g_{\lambda \omega}=\delta_{\alpha v^{*}}^{\alpha}$$
You may easily verify that $(2.10)$ and $(2.11)$ are consistent. From the specific form of the Lorentz metric it is easy to see that the inverse of the Lorentz metric is simply the Lorentz metric itself, which is a convenient fact,
$$g^{\alpha \lambda}=\left(\begin{array}{cccc} -1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 \ 0 & 0 & -1 & 0 \ 0 & 0 & 0 & -1 \end{array}\right)$$

## 物理代写|广义相对论代写General relativity代考|Energy and Momentum

The previous chapter contained a lot of formalism and little discussion of the physical world. Now it is time to see that the formalism we have developed can make physics more clear and easier (Schwartz 1968; Taylor 1963). We will consider some examples of 4 -vectors in physics. As in classical mechanics we first consider the trajectory of a particle. Its position can be described by giving the functions of time $x(t), y(t), z(t)$ in some inertial lab frame; we thereby have the position 4 -vector $(c t, x(t), y(t), z(t))$ as a function of time in that frame. The trajectory is a curve in four-dimensional spacetime and is also called the world-line of the particle. We illustrate it for two space dimensions in Fig. 3.1. Since the particle moves at less than the velocity of light the trajectory lies inside a light cone with vertex on any point of the trajectory, called the local light cone.

First consider an inertial coordinate system centered on a uniformly moving particle; recall that it is called the proper or rest frame of the particle. In this frame the position 4-vector is $x^{\mu}=(c \tau, 0,0,0)$, where $\tau$ is the time that a clock attached to the particle would measure, which we call the proper time. However, since $x^{\mu} x_{\mu}$ is an invariant we may write a relation that gives the proper time in any frame
$$c^{2} \tau^{2}=x^{\mu} x_{\mu}=c^{2} t^{2}-\vec{x}^{2}$$
We emphasize that the proper time is an invariant, as is obvious from this expression!

## 物理代写|广义相对论代写General relativity代考|The Lorentz Group

C2吨2−X→2=0

X′μ=∑03一个在μXν=一个在μX在

(C吨,X,是,和)(1000 0−100 00−10 000−1)(C吨 X 是 和)=0

## 物理代写|广义相对论代写General relativity代考|Four-Vectors and Tensors

Xμ=(C吨,−X,−是,−和)

G一个λ=(−1000 0−100 00−10 000−1)

## 物理代写|广义相对论代写General relativity代考|Energy and Momentum

C2τ2=XμXμ=C2吨2−X→2

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。