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# 物理代写|广义相对论代写General Relativity代考|The need for a theory of gravity

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## 物理代写|广义相对论代写General Relativity代考|The need for a theory of gravity

Newton’s theory of gravitation is a spectacularly successful theory. For centuries it has been used by astronomers to calculate the motions of the planets, with a staggering success rate. It has, however, the fatal flaw that it is inconsistent with Special Relativity. We begin by showing this.

As every reader of this book knows, Newton’s law of gravitation states that the force exerted on a mass $m$ by a mass $M$ is
$$\mathbf{F}=-\frac{M m G}{r^3} \mathbf{r}$$
Here $M$ and $m$ are not necessarily point masses; $r$ is the distance between their centres of mass. The vector $\mathbf{r}$ has a direction from $M$ to $m$. Now suppose that the mass $M$ depends on time. The above formula will then become
$$\mathbf{F}(t)=-\frac{M(t) m G}{r^3} \mathbf{r}$$
This means that the force felt by the mass $m$ at a time $t$ depends on the value of the mass $M$ at the same time t. There is no allowance for time delay, as Special Relativity would require. From our experience of advanced and retarded potentials in electrodynamics, we can say that Special Relativity would be satisfied if, in the above equation, $M(t)$ were modified to $M(t \quad r / c)$. This would reflect the fact that the force felt by the small mass at time $t$ depended on the value of the large mass at an earlier time $t \quad r / c$; assuming, that is, that the relevant gravitational ‘information’ travelled at the speed of light. But this would then not be Newton’s law. Newton’s law is Equation (1.2) which allows for no time delay, and therefore implicitly suggests that the information that the mass $M$ is changing travels with infinite velocity, since the effect of a changing $M$ is felt at the same instant by the mass $m$. Since Special Relativity implies that nothing can travel faster than light, Equations (1.1) and (1.2) are incompatible with it. If two theories are incompatible, at least one of them must be wrong. The only possible attitude to adopt is that Special Relativity must be kept intact, so Newton’s law has to be changed.

## 物理代写|广义相对论代写General Relativity代考|Gravitation and inertia: the Equivalence Principle in mechanics

Einstein’s new approach to gravity sprang from the work of Galileo (1564-1642; he was born in the same year as Shakespeare and died the year Newton was born). Galileo conducted a series of experiments rolling spheres down ramps. He varied the angle of inclination of the ramp and timed the spheres with a water clock. Physicists commonly portray Galileo as dropping masses from the Leaning Tower of Pisa and timing their descent to the ground. Historians cast doubt on whether this happened, but for our purposes it hardly matters whether it did or didn’t; what matters is the conclusion Galileo drew. By extrapolating to the limit in which the ramps down which the spheres rolled became vertical, and therefore that the spheres fell freely, he concluded that all bodies fall at the same rate in a gravitational field. This, for Einstein, was a crucially important finding. To investigate it further consider the following ‘thought-experiment’, which I refer to as ‘Einstein’s box’. A box is placed in a gravitational field, say on the Earth’s surface (Fig. 1.1(a)). An experimenter in the box releases two objects, made of different materials, from the same height, and measures the times of their fall in the gravitational field g. He finds, as Galileo found, that they reach the floor of the box at the same time. Now consider the box in free space, completely out of the reach of any gravitational influences of planets or stars, but subject to an acceleration a (Fig. 1.1(b)). Suppose an experimenter in this box also releases two objects at the same time and measures the time which elapses before they reach the floor. He will find, of course, that they take the same time to reach the floor; he must find this, because when the two objects are released, they are then subject to no force, because no acceleration, and it is the floor of the box that accelerates up to meet them. It clearly reaches them at the same time. We conclude that this experimenter, by releasing objects and timing their fall, will not be able to tell whether he is in a gravitational field or being accelerated through empty space. The experiments will give identical results. A gravitational field is therefore equivalent to an accelerating frame of reference – at least, as measured in this experiment. This, according to Einstein, is the significance of Galileo’s experiments, and it is known as the Equivalence Principle. Stated in a more general way, the Equivalence Principle says that no experiment in mechanics can distinguish between a gravitational field and an accel erating frame of reference. This formulation, the reader will note, already goes beyond Galileo’s experiments; the claim is made that all experiments in mechanics will yield the same results in an accelerating frame and in a gravitational field. Let us now analyse the consequences of this.

We begin by considering a particle subject to an acceleration a. According to Newton’s second law of motion, in order to make a particle accelerate it is necessary to apply a force to it. We write
$$\mathbf{F}=m_{\mathrm{i}} \mathbf{a}$$
Here $m_{\mathrm{i}}$ is the inertial mass of the particle. The above law states that the reason a particle needs a force to accelerate it is that the particles possesses inertia. A very closely related idea is that acceleration is absolute; (constant) velocity, on the other hand, is relative. Now consider a particle falling in a gravitational field g. It will experience a force (see (1.2) and (1.3) above) given by
$$\mathbf{F}=m_{\mathrm{g}} \mathbf{g} .$$

## 物理代写|广义相对论代写General Relativity代考|The need for a theory of gravity

$$\mathbf{F}=-\frac{M m G}{r^3} \mathbf{r}$$

$$\mathbf{F}(t)=-\frac{M(t) m G}{r^3} \mathbf{r}$$

## 物理代写|广义相对论代写General Relativity代考|Gravitation and inertia: the Equivalence Principle in mechanics

$$\mathbf{F}=m_{\mathrm{i}} \mathbf{a}$$

$$\mathbf{F}=m_{\mathrm{g}} \mathbf{g} .$$

## 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。