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|Plane Waves

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## 物理代写|广义相对论代写General Relativity代考|Plane Waves

In empty space there is no source, so plane wave solutions are possible. With enough plane waves of different wave vectors and accompanying amplitudes, any wave shape can be accommodated by superposition. In the case of a single wave vector $k_\chi$ with amplitude $A_{\mu \nu}$, a complex constant, the wave function in rectangular coordinates is the real part of $\bar{h}{\mu \nu}=A{\mu \nu} \exp \left(i k_\chi x^\chi\right)$. The phase factor is an invariant. It is easily shown that
\begin{aligned} & \bar{h}{\mu \nu, \beta}=i k\beta \bar{h}{\mu \nu}, \ & \square \bar{h}{\mu \nu}=-\left(k_\beta k^\beta\right) \bar{h}{\mu \nu}=0, \quad k\beta k^\beta=0 . \end{aligned}
As with an electromagnetic wave, the relation between frequency $k^0 \equiv \omega$ and wave 3 -vector $\vec{k}$ is identical. In free space, there is no dispersion, so the phase and group velocities are unity. The direction of $\vec{k}$ is the direction of wave travel. The gauge condition forced $\bar{h}\mu^\nu,{ }\nu=0$. Thus,
$$k_\nu A_\mu^\nu=0$$
This is another restriction, an orthogonality restriction on $A_\mu^\nu$.
A more useful solution can be obtained, by again applying a gauge transformation with vector $\xi_\alpha$. The vector satisfies $\square \xi_\alpha=0$. It can produce a solution ${ }^{T T} \bar{h}{\mu \nu}$, with amplitude $\bar{A}{\mu \nu}$, that is traceless ${ }^{T T} \bar{h}\mu^\mu=\bar{A}\mu^\mu=0$. Using $\xi_\mu=B_\mu \exp \left(i k_\chi r^\chi\right)$ and results from Problem 2 ,
\begin{aligned} \bar{h}{\mu^{\prime} \nu^{\prime}} & \equiv T T \bar{h}{\mu \nu}=\bar{h}{\mu \nu}-\xi\mu,{ }\nu-\xi{\nu,{ }\mu}+\eta{\mu \nu} \xi^\chi,{ }\chi, \ \bar{A}{\mu \nu} & =A_{\mu \nu}-i\left(B_\mu k_\nu+k_\mu B_\nu-\eta_{\mu \nu} B^\chi k_\chi\right), \ \bar{A}\mu^\alpha & =A\mu^\alpha-i\left(B_\mu k^\alpha+k_\mu B^\alpha-\delta_\mu^\alpha B^\chi k_\chi\right), \ 0 & =\bar{A}\mu^\mu=A\mu^\mu-i\left(B_\mu k^\mu+k_\mu B^\mu-4 B^\chi k_\chi\right) \ & =A_\mu^\mu+2 i B_\mu k^\mu=A_\mu^\mu+2 i B^\mu k_\mu, \quad i B^\mu k_\mu=-A_\mu^\mu / 2 . \end{aligned}

## 物理代写|广义相对论代写General Relativity代考|The Graviton

For electromagnetic waves, the wave function of the vector field $A_\mu$ can describe all of the physics. When this field is quantized, the quanta are photons with spin $s=1$. In quantum electrodynamics, the interactions to lowest order are the exchange of virtual photons. In GR, the wave function of the field describing the physics is a tensor of rank $2 \bar{h}{\mu \nu}$. Thus, a quantum theory of gravity has a exchange particle of $\operatorname{spin} s=2$, with zero rest mass, called the graviton. A transparent way to see this is to consider what happens to a transverse electromagnetic or transverse, traceless gravitational plane wave amplitude, under rotation. If the plane wave is traveling in the 3-direction, the only nonzero amplitudes are $A_j$ for the electromagnetic wave and $\bar{A}{j k}$ for the gravitational wave. Here $(j, k) \neq 3$. One can rotate these wave functions by angle $\phi$ about the axis along the propagation direction, using the information in Fig. 1.1. Another set of rectangular axes, where basis and unit vectors are the same, is obtained. The nonzero elements of the rotation matrix $x^i, j^{\prime}$ are: $R{1^{\prime}}^1=R{2^{\prime}}^2=\cos \phi, R_{2^{\prime}}^1=-R_{1^{\prime}}^2=\sin \phi, R_{3^{\prime}}^3=1$.
Using the rotation $A_{j^{\prime}}=R_{j^{\prime}}^k A_k$, the electromagnetic amplitudes become
\begin{aligned} & A_{1^{\prime}}=R_{1^{\prime}}^1 A_1+R_{1^{\prime}}^2 A_2=\cos \phi A_1-\sin \phi A_2 \ & A_{2^{\prime}}=R_{2^{\prime}}^1 A_1+R_{2^{\prime}}^2 A_2=\sin \phi A_1+\cos \phi A_2 \end{aligned}
These equations yield
\begin{aligned} A_{1^{\prime}} \pm i A_{2^{\prime}} & =(\cos \phi \pm i \sin \phi) A_1+(-\sin \phi \pm i \cos \phi) A_2 \ & =\exp ( \pm i \phi) A_1 \pm(\cos \phi \pm i \sin \phi) i A_2=\exp ( \pm i \phi)\left[A_1 \pm i A_2\right] \end{aligned}
Using the rotation $\bar{A}{j^{\prime} k^{\prime}}=R{j^{\prime}}^l R_{k^{\prime}}^n \bar{A}{l n}$ and Eq. (7.15), the gravitational amplitudes become \begin{aligned} \bar{A}{1^{\prime} 1^{\prime}} & =R_{1^{\prime}}^1 R_{1^{\prime}}^1 \bar{A}{11}+R{1^{\prime}}^1 R_{1^{\prime}}^2 \bar{A}{12}+R{1^{\prime}}^2 R_{1^{\prime}}^1 \bar{A}{21}+R{1^{\prime}}^2 R_{1^{\prime}}^2 \bar{A}{22} \ & =\cos ^2 \phi \bar{A}{11}-2 \sin \phi \cos \phi \bar{A}{12}-\sin ^2 \phi \bar{A}{11} \ & =\cos 2 \phi \bar{A}{11}-\sin 2 \phi \bar{A}{12} \ \bar{A}{2^{\prime} 2^{\prime}} & =R{2^{\prime}}^1 R_{2^{\prime}}^1 \bar{A}{11}+R{2^{\prime}}^1 R_{2^{\prime}}^2 \bar{A}{12}+R{2^{\prime}}^2 R_{2^{\prime}}^1 \bar{A}{21}+R{2^{\prime}}^2 R_{2^{\prime}}^2 \bar{A}{22} \ & =\sin ^2 \phi \bar{A}{11}+2 \sin \phi \cos \phi \bar{A}{12}-\cos ^2 \phi \bar{A}{11} \ & =-\cos 2 \phi \bar{A}{11}+\sin 2 \phi \bar{A}{12} \ \bar{A}{1^{\prime} 2^{\prime}} & =R{1^{\prime}}^1 R_{2^{\prime}}^1 \bar{A}{11}+R{1^{\prime}}^1 R_{2^{\prime}}^2 \bar{A}{12}+R{1^{\prime}}^2 R_{2^{\prime}}^1 \bar{A}{21}+R{1^{\prime}}^2 R_{2^{\prime}}^2 \bar{A}{22} \ & =\sin \phi \cos \phi \bar{A}{11}+\left(\cos ^2 \phi-\sin ^2 \phi\right) \bar{A}{12}+\sin \phi \cos \phi \bar{A}{11} \ & =\sin 2 \phi \bar{A}{11}+\cos 2 \phi \bar{A}{12} . \end{aligned}

## 物理代写|广义相对论代写General Relativity代考|The Graviton

\begin{aligned} & A_{1^{\prime}}=R_{1^{\prime}}^1 A_1+R_{1^{\prime}}^2 A_2=\cos \phi A_1-\sin \phi A_2 \ & A_{2^{\prime}}=R_{2^{\prime}}^1 A_1+R_{2^{\prime}}^2 A_2=\sin \phi A_1+\cos \phi A_2 \end{aligned}

\begin{aligned} A_{1^{\prime}} \pm i A_{2^{\prime}} & =(\cos \phi \pm i \sin \phi) A_1+(-\sin \phi \pm i \cos \phi) A_2 \ & =\exp ( \pm i \phi) A_1 \pm(\cos \phi \pm i \sin \phi) i A_2=\exp ( \pm i \phi)\left[A_1 \pm i A_2\right] \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。