### 物理代写|广义相对论代写General relativity代考|MATH4105

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|广义相对论代写General relativity代考|The Trouble with Absolute Time

The story of the discovery of special relativity is one of the most interesting in physics, and is covered in many books, including several by Einstein (Einstein 1923, 1934; Bergmann 1942; Rindler 1969; Weaver 1987). Accordingly we will here discuss only very briefly the ideas which led Einstein to special relativity.

In the late nineteenth century the two great theories of physics were Newton’s mechanics and gravitational theory, and Maxwell’s electromagnetism. It was widely believed that there might be no more basic physical theories to be discovered: quantum mechanics was of course decades in the future. However there was a flaw in the combination of these two theories, inherent in the classical concept of time. Mechanics was based on absolute time; as Newton phrased it in the Principia, “Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without reference to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.”

The transformation between Cartesian reference frames in uniform motion, called the Galilean transformation, is based on the notion of absolute time, and was universally accepted in the nineteenth century. For motion along the $x$ direction the situation is shown in Fig. 1.1; the primed system moves past the unprimed system at velocity $v$, with the origins coinciding at time zero.
The Galilean transformation between the two systems is
$$x^{\prime}=x-v t, \quad y^{\prime}=y, \quad z^{\prime}=z, \quad t^{\prime}=t=\text { absolute time. }$$

If a body moves with velocity $u$ in the $x$ direction in system $S$ then it will have a velocity in system $S^{\prime}$ given by differentiating this with respect to the absolute time,
$$u^{\prime}=\frac{\mathrm{d} x^{\prime}}{\mathrm{d} t}=\frac{\mathrm{d} x}{\mathrm{~d} t}-v=u-v, \quad u=u^{\prime}+v .$$
That is the velocities $u^{\prime}$ and $v$ simply add to give $u^{\prime}+v$. You may easily convince yourself that the general vector expression for the addition of velocities must be
$$\vec{u}=\vec{u}^{\prime}+\vec{v} .$$

## 物理代写|广义相对论代写General relativity代考|The Simplest Lorentz Transformation

Einstein’s 1905 approach to special relativity was based on the following two postulates:

I. The analytical form of physical laws is the same in all inertial reference frames as described by systems of Cartesian coordinates.
II. The speed of light in vacuum is a universal constant.
Postulate (I) is a criterion of elegance, while (II) was supported by experiments done before 1905, such as that of Michelson and Morley, and is now verified to very high accuracy.

We want to derive now a transformation of the space coordinates plus time, to replace the Galilean transformation discussed above, but in which the velocity of light is the same in both systems. This is called a Lorentz transformation; due to its fundamental importance our derivation will be detailed and based on the most elementary assumptions (Sard 1970).

To begin we modify the Galilean transformation (1.1) in as simple a way as we can. First, we suppose that $y$ and $z$ are not changed, that is $y^{\prime}=y$ and $z^{\prime}=z$ (You should think about this a little). We next assume that time may be different in the two systems, and that the transformation is linear in $x$ and $t$. That is we assume
$$c t^{\prime}=a_{11} c t+a_{12} x, \quad x^{\prime}=a_{21} c t+a_{22} x .$$
In equivalent matrix form,
$$\left(\begin{array}{c} c t^{\prime} \ x^{\prime} \end{array}\right)=\left(\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right)\left(\begin{array}{c} c t \ x \end{array}\right), \quad A(v) \equiv\left(\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right) .$$
The matrix elements $a_{i j}$ must, of course, depend only on the velocity $v$. The notable property of this transformation is that time is allowed to be different in the two systems, which is the fundamental break with classical ideas made by Einstein. It is this which allows $c$ to be a universal constant. The use of $c t$ instead of $t$ in (1.4a) is for dimensional convenience, since $c t$ and $x$ both have dimensions of distance. There are 4 parameters in the transformation matrix $A$, which we must determine. We will make four physical demands based on the above two postulates that determine them uniquely.

## 物理代写|广义相对论代写General relativity代考|Some Elementary Properties and Applications

Many of the most interesting results of special relativity theory can be obtained using only the simple Lorentz transformation above (Taylor 1963). We will give a rather cursory discussion of some of the more important features, appropriate to a review: time dilation of a moving clock, length contraction of a moving rod, and the Doppler shift of light emitted by a moving object. The interested reader may consult the references for much more material.

First note that the Lorentz transformation contains the factor $\gamma$, which is greater than 1. If $\gamma$ is not to be infinite or imaginary then the velocity parameter $\beta$ must be less than 1 ; thus systems and objects cannot move faster than $c$, a famous result of relativity.

Time dilation in a moving system is an effect peculiar to relativity, which distinguishes it sharply from classical theory with its absolute time. Suppose a clock at rest at the origin in the moving system $S^{\prime}$ ticks at $t^{\prime}=0$ and again at $t^{\prime}=\Delta t^{\prime}$. Then in the system $S$, where we suppose our lab to be, it is seen to tick at $t=0$ at $x=0$ and again at $t=\Delta t$ at $x=v \Delta t$. With the Lorentz transformation in (1.18) we may relate these time intervals,
$$c \Delta t^{\prime}=\gamma c \Delta t-\beta \gamma \Delta x=\gamma c \Delta t-\beta \gamma v \Delta t=c \Delta t / \gamma \quad \text { or } \quad \Delta t=\gamma \Delta t^{\prime}$$
Thus, since $\gamma \geq 1$, the moving clock appears to run slower as seen in the lab in $S$. We refer to the system in which a clock is at rest as its rest system or proper system or rest frame. Time in the proper system is usually called proper time and often denoted by $\tau$.

## 物理代写|广义相对论代写General relativity代考|The Trouble with Absolute Time

X′=X−在吨,是′=是,和′=和,吨′=吨= 绝对时间。

## 物理代写|广义相对论代写General relativity代考|The Simplest Lorentz Transformation

C吨′=一个11C吨+一个12X,X′=一个21C吨+一个22X.

(C吨′ X′)=(一个11一个12 一个21一个22)(C吨 X),一个(在)≡(一个11一个12 一个21一个22).

## 物理代写|广义相对论代写General relativity代考|Some Elementary Properties and Applications

CΔ吨′=CCΔ吨−bCΔX=CCΔ吨−bC在Δ吨=CΔ吨/C 或者 Δ吨=CΔ吨′

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