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

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|A Special Coordinate System

Recall that in Chap. 4 we stated the Signature Theorem, that at any point $P$ there exists a special coordinate system in which the metric is diagonal and has diagonal elements equal to 1 or $-1$ or 0 . The special system may be reached by a linear transformation. This form of the metric is called the Cayley-Sylvester canonical form. We proved the theorem for the case of two dimensions in Appendix 4.1 (Perlis 1952).

In this chapter we obtained another special coordinate system, the geodesic system, in which the affine connections vanish at any given point $P$. If the connections are zero, then from the definition of the Christoffel connections (5.19) this clearly means that the first derivatives of the metric must also be zero. We can in fact combine these transformations and for any given point $P$ find a coordinate system in which the metric has the Cayley-Sylvester canonical form and also has vanishing first derivatives and thus vanishing connections. To do this we merely apply the two transformations together with the point $P$ taken to be the origin,
$$\bar{x}^{j}=L_{k}^{j} x^{k}+\frac{1}{2} A_{j l}^{i}\left(L_{n}^{j} x^{n}\right)\left(L_{m}^{l} x^{m}\right) \text {. }$$
The $L$ array makes the transformation to the system in which the metric has the Cayley-Sylvester canonical form, and the $A$ array specifies the transformation to the geodesic system. The coordinate system thus obtained is very special: the axes are orthogonal, the metric is Lorentz, and the connections vanish, so physics is locally much like that of special relativity, but of course only in a vanishingly small region near $P$.

物理代写|广义相对论代写General relativity代考|The Extremum Problem

For completeness we briefly review one of the most important problems in the calculus of variations, one which is familiar to most physicists from the Lagrangian formulation of classical mechanics (Goldstein 1980). The Lagrangian is assumed to be a given function of the coordinates and generalized velocities, $L\left(x^{\lambda}, \dot{x}^{\alpha}\right)$. A quantity $S$ called the action is then defined as the integral of the Lagrangian along some curve from a fixed initial point $i$ to a fixed final point $f$,
$$S=\int_{i}^{f} L\left(x^{\lambda}, \dot{x}^{\alpha}\right) \mathrm{d} p, \dot{x}^{\alpha} \equiv \frac{\mathrm{d} x^{\alpha}}{\mathrm{d} p}$$
That is, the action is a functional of the Lagrangian. The Euler-Lagrange method of extremizing the action is to calculate the variation in $S$ as the path $x^{\mu}(p)$ is varied by a small amount $\delta x^{\mu}(p)$ as shown in Fig. $5.5$; the extremum path is characterized by the vanishing of the variation, precisely analogous to the vanishing of a derivative of a function at its extremum. The variation in $S$ is calculated in a straight-forward way as follows,
\begin{aligned} \delta S &=\int_{i}^{f}\left[\frac{\partial L}{\partial x^{\alpha}} \delta x^{\alpha}+\frac{\partial L}{\partial \dot{x}^{\alpha}} \delta \dot{x}^{\alpha}\right] \mathrm{d} p \ &=\int_{i}^{f}\left[\frac{\partial L}{\partial x^{\alpha}} \delta x^{\alpha}+\frac{\mathrm{d}}{\mathrm{d} p}\left(\frac{\partial L}{\partial \dot{x}^{\alpha}} \delta x^{\alpha}\right)-\delta x^{\alpha} \frac{\mathrm{d}}{\mathrm{d} p}\left(\frac{\partial L}{\partial \dot{x}^{\alpha}}\right)\right] \mathrm{d} p \ &=\int_{i}^{f}\left[\frac{\partial L}{\partial x^{\alpha}}-\frac{\mathrm{d}}{\mathrm{d} p}\left(\frac{\partial L}{\partial \dot{x}^{\alpha}}\right)\right] \delta x^{\alpha} \mathrm{d} p+\left(\frac{\partial L}{\partial \dot{x}^{\alpha}} \delta \dot{x}^{\alpha}\right)_{i}^{f} \end{aligned}
where we have integrated by parts and used $\delta \dot{x}^{\propto}=\mathrm{d}\left(\delta x^{\alpha}\right) / \mathrm{d} p$. Since we consider only paths between fixed endpoints the last term in the last line above is zero. Since we consider any small variation $\delta x^{\alpha}$ the bracket in the integral must be identically zero, so we conclude
$$\frac{\mathrm{d}}{\mathrm{d} p}\left(\frac{\partial L}{\partial \dot{x}^{\alpha}}\right)-\frac{\partial L}{\partial x^{\alpha}}=0$$
These differential equations are called the Euler-Lagrange equations, and yield a curve for which the action is extremum.

物理代写|广义相对论代写General relativity代考|Christoffel Connections as Fictitious Forces

The Christoffel connections are actually familiar objects in classical mechanics, but they are seldom identified as such explicitly or seen from the geometrical point of view. They give rise to the well-known fictitious forces encountered in non-cartesian coordinate systems, rotating systems being a favorite example. To illustrate how this works we will study the motion of a particle in a potential in 3 -dimensional space with a general coordinate system using the Lagrangian formulation of classical mechanics. The manipulations are similar to those used in the preceding appendix and for discussing geodesics in the text.

Let the particle have a trajectory in three dimensions, with the position is given as a function of absolute (invariant) time by $x^{j}(t)$ in some coordinate system. Along this trajectory the line element represents the Euclidean distance
$$\mathrm{ds}{ }^{2}=g_{i j} \mathrm{~d} x^{i} \mathrm{~d} x^{j} .$$
Thus we may write the square of the velocity as
$$v^{2}=g_{i j} \dot{x}^{i} \dot{x}^{j}, \quad \dot{x}^{i} \equiv \frac{\mathrm{d} x^{i}}{\mathrm{~d} t} .$$
For a particle moving in a potential field the Lagrangian is generally taken to be the kinetic energy minus the potential energy,
$$L=\frac{m}{2} v^{2}-V\left(x^{k}\right)=\frac{m}{2} g_{i j} \dot{x}^{i} \dot{x}^{j}-V\left(x^{k}\right) .$$
Note the similarity of this to the function $T$ which we used in discussing geodesics. Lagrangian mechanics is based on the postulate that the action, the integral of $L$, is extremized for the correct trajectory. That is
$$\delta S=0, \quad S=\int_{i}^{f} L \mathrm{~d} t=\int_{i}^{f}\left[\frac{m}{2} g_{i j} \dot{x}^{i} \dot{x}^{j}-V\left(x^{k}\right)\right] \mathrm{d} t .$$
Extremizing the action we are led to the Euler-Lagrange equations as in our derivation of the geodesic equation, but now we also have a potential energy term. The EulerLagrange equations are obtained as usual, and are,

$\frac{\partial L}{\partial \dot{x}^{i}}=m g_{i j} \dot{x}^{j}, \quad \frac{\mathrm{d}}{\mathrm{d} t} \frac{\partial L}{\partial \dot{x}^{i}}=m\left(g_{i j} \ddot{x}^{j}+g_{i j, k} \dot{x}^{j} \dot{x}^{k}\right)$
$\frac{\partial L}{\partial x^{i}}=\frac{m}{2} g_{i j, k} \dot{x}^{j} \dot{x}^{k}-\frac{\partial V}{\partial x^{i}}$
$m\left(g_{i j} \ddot{x}^{j}+g_{i j, k} \dot{x}^{j} \dot{x}^{k}-\frac{1}{2} g_{j k, i} \dot{x}^{j} \dot{x}^{k}\right)+\frac{\partial V}{\partial x^{i}}=0$

物理代写|广义相对论代写General relativity代考|A Special Coordinate System

X¯j=大号ķjXķ+12一个jl一世(大号njXn)(大号米lX米).

物理代写|广义相对论代写General relativity代考|The Extremum Problem

d小号=∫一世F[∂大号∂X一个dX一个+∂大号∂X˙一个dX˙一个]dp =∫一世F[∂大号∂X一个dX一个+ddp(∂大号∂X˙一个dX一个)−dX一个ddp(∂大号∂X˙一个)]dp =∫一世F[∂大号∂X一个−ddp(∂大号∂X˙一个)]dX一个dp+(∂大号∂X˙一个dX˙一个)一世F

ddp(∂大号∂X˙一个)−∂大号∂X一个=0

物理代写|广义相对论代写General relativity代考|Christoffel Connections as Fictitious Forces

Christoffel 连接实际上是经典力学中熟悉的对象，但它们很少被明确地识别为此类对象或从几何角度来看。它们产生了在非笛卡尔坐标系中遇到的众所周知的虚拟力，旋转系统是一个最喜欢的例子。为了说明这是如何工作的，我们将使用经典力学的拉格朗日公式来研究粒子在具有一般坐标系的 3 维空间中的势能运动。操作类似于前面附录中使用的操作以及用于讨论文本中的测地线。

ds2=G一世j dX一世 dXj.

d小号=0,小号=∫一世F大号 d吨=∫一世F[米2G一世jX˙一世X˙j−在(Xķ)]d吨.

∂大号∂X˙一世=米G一世jX˙j,dd吨∂大号∂X˙一世=米(G一世jX¨j+G一世j,ķX˙jX˙ķ)
∂大号∂X一世=米2G一世j,ķX˙jX˙ķ−∂在∂X一世

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