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# 物理代写|相对论代写Theory of relativity代考|CRN9074 GEODESICS

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## 物理代写|相对论代写Theory of relativity代考|GEODESICS

Introduction. A geodesic through spacetime can be defined in one of two ways: (1) as the worldline of longest proper time between a given pair of events, or (2) as the straightest possible worldline. In this section, we will focus on the first definition, and derive from that definition an expression for calculating geodesics in any arbitrary coordinate system.

A geodesic in an ordinary space would be the path of shortest distance between two points. So why is a geodesic in spacetime the worldline of longest proper time? The ultimate reason is the minus sign in the metric equation, which often screws up our expectations. However, here is a clue that this is correct. In the flat spacetime of special relativity, a free particle follows an inertial worldline (that is, it moves at a constant speed in a straight line). Moreover, in special relativity, an inertial clock passing through two events measures the longest possible proper time between those events (see box $8.1$ for a discussion). If the geodesic hypothesis is true (i.e., if free particles follow geodesics), then in flat spacetime at least, the geodesic between two events is thus indeed the worldline of longest proper time between those events. We are simply extending this equivalence to possibly curved spacetimies.

## 物理代写|相对论代写Theory of relativity代考|Timelike Geodesics

Timelike Geodesics. Consider two events $A$ and $B$ that have a timelike separation, and consider possible timelike worldlines between them. We can describe such a worldline by specifying the spacetime coordinates of events along it as a function of a parameter $\sigma$ that goes from 0 at event $A$ to 1 at event $B$, i.e., by specifying $x^\mu(\sigma)$ (see figure 8.1). The proper time measured along this worldline is then
$$\tau_{A B}=\int \sqrt{-d s^2}=\int_0^1 \sqrt{-g_{\mu \nu}\left(x^\alpha(\sigma)\right) \frac{d x^\mu}{d \sigma} \frac{d x^\nu}{d \sigma}} d \sigma$$
where the notation $g_{\mu \nu}\left(x^\alpha(\sigma)\right)$ reminds us that the metric is a function of the coordinates of events along the worldline, which in turn are functions of the parameter $\sigma$.
Finding the worldline of longest proper time is analogous to the goal of finding the trajectory $q_i(t)$ of extreme action using the variational principle in mechanics. In that case, we define a system’s action $S$ to be the time-integral of a Lagrangian $L$ :
$$S \equiv \int_{t_A}^{t_B} L\left(q_i, \dot{q}_i\right) d t$$
where $L\left(q_i, \dot{q}_i\right)$ indicates that the Lagrangian is a function of the generalized position coordinates $q_i$ and the generalized velocities $\dot{q}_i$ (where $i$ ranges over however many such coordinates you have). In classical mechanics we found that the trajectory $q_i(t)$ physically followed by the system was the one that made the action $S$ extreme, which in turn was the trajectory that satisfied the Euler-Lagrange equations
$$0=\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_i}\right)-\frac{\partial L}{\partial q_i}$$

## 物理代写|相对论代写THEORY OF RELATIVITY代考|GEODESICS

\tau_{A B}=\int \sqrt{-d s^2}=\int_0^1 \sqrt{-g_{\mu \nu}\left(x^\alpha(\sigma)\right) \frac{d x^\mu}{d \sigma} \frac{d x^\nu}{d \sigma}} d \sigma
$$符昊在哪里 g_{\mu \nu}\left(x^\alpha(\sigma)\right) 提醒我们度量是世界线上事件坐标的函数，而世界线又是参数的函数 \sigma. 寻找最长本征时间的世界线类似于寻找轨迹的目标 q_i(t) 使用力学中的变分原理进行极端动作。在那种情况下，我们定义一个系统的动作 S 是拉格朗日量的时间积分 L :$$
S \equiv \int_{t_A}^{t_B} L\left(q_i, \dot{q}_i\right) d t
$$在哪里 L\left(q_i, \dot{q}_i\right) 表示拉格朗日量是广义位置坐标的函数 q_i 和广义速度 \dot{q}_i where\i\rangesoverhowevermanysuchcoordinatesyouhave. 在经典力学中我们发现 轨迹 q_i(t) 系统实际跟随的是做出动作的人 S 极端，这反过来又是满足欧拉-拉格朗日方程的轨迹$$
0=\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_i}\right)-\frac{\partial L}{\partial q_i}


## 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。