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|GRAV ITAT IONAL F IELD

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|GRAV ITAT IONAL F IELD

The gravitational field is described by a tetrad $e_a^i(x)$ (to be compared with the Maxwell field $\left.A_a(x)\right)$ or equivalently by the $4 \mathrm{~d}$ Lorentzian metric
$$g_{a b}(x)=\eta_{i j} e_a^i(x) e_b^j(x)$$
The geometry of spacetime is determined by the line element
$$d s^2=g_{a b}(x) d x^a d x^b$$
At each spacetime point $p$ with coordinates $x_p^a$, the tetrad determines local Cartesian coordinates $X^i=e_a^i(\mathrm{x})\left(\mathrm{x}^a-\mathrm{x}_p^a\right)$ that define a ‘free falling’ local reference frame, where physics is just as it is in Minkowski space up to the second order in $X^i$. The existence of this reference system, which put Einstein on the right track to find the theory, is called the ‘equivalence principle’.

Since coordinates are arbitrary, it makes little sense to attribute physical dimensions to them. Hence we see from the last equation that the gravitational field $g_{a b}(x)$ has the natural dimension of Length $^2$ (we have $c=1$ ). A different convention can also be found in the literature: to have $g_{a b}(x)$ dimensionless and to insist on having dimension-full coordinates. This comes naturally when $g_{a b}(x)$ is the Minkowski metric. Far less so, say, using polar coordinates.

From the equivalence principle, it follows that a clock moving along a finite timelike line $\gamma: \tau \mapsto x^a(\tau)$ measures the time
$$T=\int_\gamma \sqrt{-g_{a b} \dot{X}^a \dot{X}^b} d \tau$$

## 物理代写|广义相对论代写General Relativity代考|EFFECTS OF GRAV ITY

The following derives immediately from the equivalence principle. Massive particles (not subjected to forces other than gravity) move along geodesics. That is, in the parametrisation of their world line $x^a(\tau)$ where $|\dot{x}|^2=-1$, the equation of motion of a massive particle is the geodesic equation
$$\ddot{\mathbf{x}}^d+\Gamma_{a b}^d \dot{x}^a \dot{\mathrm{x}}^b=0$$
(to be compared with the Lorentz force equation $\ddot{X}^d-\frac{e}{m} F^d{ }_a \dot{\mathrm{X}}^a=0$ ) where the Levi-Civita connection is defined in (3.58), which I copy here for completeness:

$$\Gamma_{a b}^d=\frac{1}{2} g^{d c}\left(\partial_a g_{c b}+\partial_b g_{c a}-\partial_c g_{a b}\right) .$$
The second term in the geodesic equation (4.5) can be viewed as the ‘gravitational force term’, or as the effect of the spacetime geometry on the motion. The two notions are identified. A particle moving along a geodesic is called in ‘free fall’. If other forces act on the particle, they add to (4.5). For instance, if the particle is charged,
$$\ddot{\mathrm{X}}^d+\Gamma_{a b}^d \dot{\mathrm{X}}^a \dot{\mathrm{X}}^b-\frac{e}{m} F_a^d \dot{\mathrm{X}}^a=0,$$
where $F_{a b}$ is the electromagnetic field.
The trajectory of light rays is simply determined by $d s=0$. That is, light rays (electromagnetic wavefront trajectories in the highfrequency limit) move along null lines: their world lines $x^a(\tau)$ satisfy
$$\frac{d s}{d \tau}=|\dot{X}|=\sqrt{g_{a b}(x) \dot{X}^a \dot{\mathbf{X}}^b}=0$$

## 物理代写|广义相对论代写General Relativity代考|GRAV ITAT IONAL F IELD

$$g_{a b}(x)=\eta_{i j} e_a^i(x) e_b^j(x)$$

$$d s^2=g_{a b}(x) d x^a d x^b$$

$$T=\int_\gamma \sqrt{-g_{a b} \dot{X}^a \dot{X}^b} d \tau$$

## 物理代写|广义相对论代写General Relativity代考|EFFECTS OF GRAV ITY

$$\ddot{\mathbf{x}}^d+\Gamma_{a b}^d \dot{x}^a \dot{\mathbf{x}}^b=0$$
(与洛伦兹力方程比较 $\left.\ddot{X}^d-\frac{e}{m} F^d{ }a \dot{\mathrm{X}}^a=0\right)$ 其中 Levi-Civita 连接在 (3.58) 中定义，为了完整起见，我将 其复制到此处: $$\Gamma{a b}^d=\frac{1}{2} g^{d c}\left(\partial_a g_{c b}+\partial_b g_{c a}-\partial_c g_{a b}\right)$$

$$\ddot{\mathrm{X}}^d+\Gamma_{a b}^d \dot{\mathrm{X}}^a \dot{\mathrm{X}}^b-\frac{e}{m} F_a^d \dot{\mathrm{X}}^a=0$$

$$\frac{d s}{d \tau}=|\dot{X}|=\sqrt{g_{a b}(x) \dot{X}^a \dot{\mathbf{X}}^b}=0$$

## 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。