# 物理代写|广义相对论代写General relativity代考|The Equivalence Principle and optics

#### Doug I. Jones

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## 物理代写|广义相对论代写General relativity代考|The Equivalence Principle and optics

The Equivalence Principle is a principle of indistinguishability; it is impossible, using any experiment in mechanics, to distinguish between a gravitational field and an accelerating frame of reference. To this extent it is a symmetry principle. If a symmetry of nature is exact,this means that various situations are experimentally indistinguishable. If, for example, parity were an exact symmetry of the world (which it is not, because of beta decay), it would be impossible to distinguish left from right. The fact that it $i s$ possible to distinguish them is a direct indication of the breaking of the symmetry.

No experiment in mechanics, then, can distinguish a gravitational field from an accelerating frame. What about other areas of physics? Let us generalise the Equivalence Principle to optics, and consider the idea that no experiment in optics could distinguish a gravitational field from an accelerating frame. ${ }^3$ To make this concrete, return to the Einstein box and consider the following simple two experiments. The first one is to release monochromatic light (of frequency $v$ ) from the ceiling of the accelerating box, and receive it on the floor (Fig. 1.5). The light is released from the source $S$ at $t=0$ towards the observer $O$. At the same instant $t=0$ the box begins to accelerate upwards with acceleration $a$. The box is of height $h$. Light from $S$ reaches $O$ after a time interval $t=h / c$, at which time $O$ is moving upwards with speed $u=a t=a h / c$.

Now consider the emission of two successive crests of light from $S$. Let the time interval between the emission of these crests be $\mathrm{d} t$ in the frame of $S$. Then
$$\mathrm{d} t=\frac{1}{v} \quad \text { in frame } S$$
where $v$ is the frequency of the light in frame $S$. Arguing non relativistically, the time interval between the reception of these crests at $O$ is
$$\mathrm{d} t^{\prime}=\mathrm{d} t-\Delta t=\mathrm{d} t-u \frac{\mathrm{d} t}{c}=\mathrm{d} t\left(1-\frac{u}{c}\right)=\frac{1}{v^{\prime}}$$
hence
$$\frac{v^{\prime}}{v}=\frac{1}{1-u / c}=1+\frac{u}{c}+\mathrm{O}\left(\frac{u}{c}\right)^2>1 ;$$
the light is Doppler (blue) shifted. With $v^{\prime}=v+\Delta v$ we have
$$\frac{\Delta v}{v}=\frac{u}{c}+\mathrm{O}\left(\frac{u}{c}\right)^2=\frac{a h}{c^2}+\mathrm{O}\left(\frac{a h}{c^2}\right)^2$$

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

A surface is a 2-dimensional space. It has the distinct advantage that we can imagine it easily, because we see it (as the mathematicians say) ’embedded’ in a 3-dimensional space – which also happens to be flat (I mean, of course, Euclidean 3-space). What I want to demonstrate, however, is that there are measurements intrinsic to a surface that may be performed to see whether it is flat or not. It is not necessary to embed a surface in a 3-dimensional space in order to see whether or not the surface is curved; we can tell just by performing measurements on the surface itself. The reader will appreciate that this is a necessary exercise; for if we are to make the statement that 3 dimensional space is curved, this statement must have an intrinsic meaning. There is no fourth dimension into which our 3 -dimensional space may be embedded (time does not count here).

To begin, consider the three surfaces illustrated in Fig. 1.8. They are a plane, a sphere and a saddle. On each surface draw a circle of radius $a$ and measure its circumference $C$ and area $A$. On the plane, of course, $C=2 \pi a$ and $A=\pi a^2$, but our claim is that these relations do not hold on the curved surfaces. In fact we have
$\begin{array}{llll}\text { Plane: } & C=2 \pi a & A=\pi a^2 & \text { flat (zero curvature), } \ \text { Sphere: } & C<2 \pi a & A<\pi a^2 & \text { curved (positive curvature), } \\ \text { Saddle: } & C>2 \pi a & A>\pi a^2 & \text { curved (negative curvature). }\end{array}$ In the case of the sphere, for example, to see that $C<2 \pi a$ imagine cutting out the circular shape which acts as a ‘cap’ to the sphere. This shape cannot be pressed flat. In order to make it flat some radial incisions must be inserted, but this then has the consequence that the total circumference $C$ of the dotted circle, which is equal to the sum of the arcs of all the incisions in the diagram below, is less than $2 \pi a$. In the case of the saddle the opposite thing happens; in order to get the circular area to lie flat, we have to fold parts of it back on itself, so that the true circumference $C$ is greater than $2 \pi a$.

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|The Equivalence Principle and optics

$$\mathrm{d} t=\frac{1}{v} \quad \text { in frame } S$$

$$\mathrm{d} t^{\prime}=\mathrm{d} t-\Delta t=\mathrm{d} t-u \frac{\mathrm{d} t}{c}=\mathrm{d} t\left(1-\frac{u}{c}\right)=\frac{1}{v^{\prime}}$$

$$\frac{v^{\prime}}{v}=\frac{1}{1-u / c}=1+\frac{u}{c}+\mathrm{O}\left(\frac{u}{c}\right)^2>1 ;$$

$$\frac{\Delta v}{v}=\frac{u}{c}+\mathrm{O}\left(\frac{u}{c}\right)^2=\frac{a h}{c^2}+\mathrm{O}\left(\frac{a h}{c^2}\right)^2$$

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

$\begin{array}{llll}\text { Plane: } & C=2 \pi a & A=\pi a^2 & \text { flat (zero curvature), } \ \text { Sphere: } & C<2 \pi a & A<\pi a^2 & \text { curved (positive curvature), } \\ \text { Saddle: } & C>2 \pi a & A>\pi a^2 & \text { curved (negative curvature). }\end{array}$以球体为例，要看到$C<2 \pi a$，想象切割出作为球体“帽”的圆形。这种形状不能压平。为了使它平坦，必须插入一些径向切口，但这样做的结果是，虚线圆的总周长$C$等于下图中所有切口的弧线之和，小于$2 \pi a$。马鞍的情况正好相反;为了使圆形区域平放，我们必须把它的一部分折叠起来，这样真正的周长$C$大于$2 \pi a$。

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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。

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