### 物理代写|广义相对论代写General relativity代考|PHYSICS 7009

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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|Geodesics as Self-parallel Curves

We now know how to displace a vector to a nearby point so that it remains parallel to itself in the general sense defined in the preceding two sections. We may use this to define and study the idea of a generalized straight line or geodesic. Our definition of a geodesic stems naturally from classical Euclidean geometry and intuition. Suppose we have a curve $C$ specified by giving the coordinates as functions of some scalar parameter $p$ which labels points on $C$,
Curve $C: x^{\mu}=x^{\mu}(p)$
We call $C$ a geodesic if it is everywhere parallel to itself; this means that if we parallel displace a tangent vector along $C$ then it remains a tangent vector.

This definition leads to a differential equation for the geodesic. Call the tangent vector $t^{\alpha}(p)$ at $p$. We parallel displace it along the curve to a nearby point labeled $p^{\prime}$ at a coordinate distance $\mathrm{d} x^{\alpha}$ to obtain
$$t^{* \alpha}\left(p^{\prime}\right)=t^{\alpha}(p)-\Gamma_{\beta \gamma}^{\alpha} \mathrm{d} x^{\beta} t^{\gamma}(p)$$
The actual tangent at $p^{\prime}$ may be obtained from that at $p$ by a Taylor Series expansion
$$t^{\alpha}\left(p^{\prime}\right)=t^{\alpha}(p)+\frac{\mathrm{d} t^{\alpha}}{\mathrm{d} p} \mathrm{~d} p$$
By our above definition of a geodesic the parallel displaced tangent vector in (5.23) is to be equal to the actual tangent vector in $(5.24)$, so that
$$\frac{\mathrm{d} t^{\alpha}}{\mathrm{d} p} \mathrm{~d} p=-\Gamma_{\beta \gamma}^{\alpha} \mathrm{d} x^{\beta} t^{\gamma}$$
We may choose the curve parameter $p$ to be the curve length, that is $\mathrm{d} p=\mathrm{d} s$, and use the normalized position derivative as an obvious tangent vector, normalized to unity,
$$t^{\beta}=\frac{\mathrm{d} x^{\beta}}{\mathrm{d} s}$$
Substituting this into (5.25) we obtain a differential equation for the geodesic
$$\frac{\mathrm{d}^{2} x^{\alpha}}{\mathrm{d} s^{2}}+\Gamma_{\beta \gamma}^{\alpha} \frac{\mathrm{d} x^{\beta}}{\mathrm{d} s} \frac{\mathrm{d} x^{\gamma}}{\mathrm{d} s}=0$$

## 物理代写|广义相对论代写General relativity代考|Geodesics as Extremum Curves

The self-parallel definition of a geodesic is one of several equivalent ones. In Euclidean geometry a straight line is the shortest distance between two given points. This property can be generalized to give the following definition of a geodesic: let the curve $C$ have length $s$ between two fixed points; then $C$ is a geodesic if the length $s$ is an extremum, that is it is either the shortest or longest among all nearby curves. We will show that this definition leads to the differential equation (5.28) and is equivalent to the self-parallel definition. The extremum calculation is a problem in the calculus of variations, well-known in classical mechanics. If the reader is not familiar with such problems and the Euler-Lagrange method of solution he should first consult Appendix $2 .$
As before the curve $C$ is denoted by
$$\text { Curve } C: x^{\mu}=x^{\mu}(p) \text {. }$$
Here $p$ is an invariant parameter, which may be the arc length of the curve but need not be. This is shown schematically in Fig. $5.5$.
The line element along the curve and the arc length $s$ can be written as

$$\begin{gathered} \mathrm{d} s^{2}=g_{\alpha \beta} \mathrm{d} x^{\alpha} \mathrm{d} x^{\beta}=g_{\alpha \beta} \dot{x}^{\alpha} \dot{x}^{\beta} \mathrm{d} p^{2} \equiv T\left(x^{\lambda}, \dot{x}^{\kappa}\right) \mathrm{d} p^{2}, \quad \dot{x}^{\kappa} \equiv \frac{\mathrm{d} x^{\kappa}}{\mathrm{d} p} \ s=\int_{i}^{f} \sqrt{g_{\alpha \beta} \dot{x}^{\alpha} \dot{x}^{\beta}} \mathrm{d} p=\int_{i}^{f} \sqrt{T\left(x^{\lambda}, \dot{x}^{\kappa}\right)} \mathrm{d} p \end{gathered}$$
where we have assumed the line element $\mathrm{d} s^{2}$ is positive. Finding the extremum of this arc length integral is a standard problem in the calculus of variations and solvable by the Euler-Lagrange method. Indeed it is the analog of a classical mechanics problem with a Lagrangian
$$L=\sqrt{T\left(x^{\lambda}, \dot{x}^{x}\right)}, \quad T\left(x^{\lambda}, \dot{x}^{\kappa}\right) \equiv g_{\alpha \beta} \dot{x}^{\alpha} \dot{x}^{\beta} .$$

## 物理代写|广义相对论代写General relativity代考|Affine Connections, Abstract View

Let us see how we may motivate and interpret the coefficients of affine connection using the abstract view introduced in Sect. 4.3. Recall that a vector may be expanded in a coordinate basis, that is vectors aligned along the coordinate axes, according to
$$\vec{V}=V^{j} \vec{e}{j}, \quad \vec{e}{j}=\text { coordinate basis. }$$
If we think of moving the vector to a nearby point it will change due to a change in its components and also a change in the basis vectors,
$$\mathrm{d} \vec{V}=\vec{e}{i} \mathrm{~d} V^{i}+V^{j} \mathrm{~d} \vec{e}{j}$$
As we discussed previously the vector spaces associated with different points in a Riemann space are ab initio independent. As such it is necessary to postulate a way to relate them. This leads to the idea of vector transplantation and the specific version of transplantation called parallel displacement that we discussed in Sects. $5.1$ and 5.3. We can think of this in the present abstract view as giving an effective change in the coordinate basis, which we assume is a bilinear expression in the basis vectors and the coordinate displacement; it is a rather compelling assumption. That is we postulate
$$\mathrm{d} \vec{e}{j}=\Gamma{k j}^{i}\left(\vec{e}_{i} \mathrm{~d} x^{k}\right) .$$

## 物理代写|广义相对论代写General relativity代考|Geodesics as Self-parallel Curves

d吨一个dp dp=−ΓbC一个dXb吨C

d2X一个ds2+ΓbC一个dXbdsdXCds=0

## 物理代写|广义相对论代写General relativity代考|Geodesics as Extremum Curves

曲线 C:Xμ=Xμ(p).

ds2=G一个bdX一个dXb=G一个bX˙一个X˙bdp2≡吨(Xλ,X˙ķ)dp2,X˙ķ≡dXķdp s=∫一世FG一个bX˙一个X˙bdp=∫一世F吨(Xλ,X˙ķ)dp

## 物理代写|广义相对论代写General relativity代考|Affine Connections, Abstract View

d在→=和→一世 d在一世+在j d和→j

d和→j=Γķj一世(和→一世 dXķ).

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