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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|The Affine Connection in Riemannian Geometry

We now enforce two additional conditions on the affine connection $\Gamma_{k l}^i$ as
\begin{aligned} \Gamma_{k l}^i &=\Gamma_{l k,}^i, \ g_{i k, l} &=0 . \end{aligned}
The affine connection obeying these conditions is said to be Riemannian connection and the corresponding geometry under consideration is called the Riemannian Geometry.
Now, formulae (1.16) and (1.32), yield
$$\Gamma_{i l}^m g_{m k}+\Gamma_{k l}^m g_{i m}=g_{i k, l},$$
or
$$\Gamma_{i l k}+\Gamma_{k l i}=g_{i k, l} .$$
Using Eq. (1.31), we have
$$\Gamma_{i l k} \equiv g_{m k} \Gamma_{i l}^m=g_{m k} \Gamma_{l i}^m=\Gamma_{l i k^*}$$

Now, making changes in a cyclic order in $i, k, l$ in (1.33), we get two other relations as
\begin{aligned} &\Gamma_{k i l}+\Gamma_{l i k}=g_{k l, i}, \ &\Gamma_{l k i}+\Gamma_{i k l}=g_{l i, k} \end{aligned}
Performing (1.35) $+(1.36)-(1.33)$ and using (1.34), we get
$$\Gamma_{i k l}=\frac{1}{2}\left[g_{i i, k}+g_{k l, i}-g_{i k, l}\right]$$

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

In Euclidean space, the metric tensors are constant throughout the space and hence all Christoffel symbols vanish. This is not true in Riemannian space. However, there exists a coordinate system known as Geodesic coordinate system $x^i$ such that all the Christoffel symbols vanish at a particular point $P$ which is known as pole.
$$\Gamma_{j k}^i=0=\Gamma_{i j k} \text { at point } P .$$
In the geodesic system, all the Christoffel symbols vanish but their derivatives do not necessarily become zero.

An important property of geodesic coordinate system is given as “the covariant derivative of a tensor at the pole equal to the corresponding partial derivative” (since, Christoffel symbols vanish at the pole).

According to equivalence principle, the local properties of curved spacetime and flat spacetime are indistinguishable. Therefore, it is possible to introduce new coordinates $x_P^{\mu \prime}$ in every point $\mathrm{P}$ of the spacetime such that
$$g_{\alpha \beta}\left(x_P^{\mu \prime}\right)=\eta_{\alpha \beta},$$
where $\eta_{\alpha \beta}=\operatorname{diag}(1,-1,-1,-1)$ is the flat Minkowski metric. In this $x_P^{\mu^{\prime}}$, the first derivatives of the metric vanish, i.e.,
$$\left.\frac{\partial g_{\alpha \beta}}{\partial x^{\mu \prime}}\right|_{x=x_p^{x^{\prime}}}=0 .$$
This new coordinate system, satisfying (i) and (ii) at a point, is known as local inertial coordinate system. This new coordinate reference frame is very similar to the inertial frame of flat spacetime, which acts only in an infinitesimal neighborhood of a single point $P$.

# 广义相对论代考

## 物理代写|广义相对论代写General relativity代考|The Affine Connection in Riemannian Geometry

$$\Gamma_{k l}^i=\Gamma_{l k,}^i, g_{i k, l} \quad=0 .$$

$$\Gamma_{i l}^m g_{m k}+\Gamma_{k l}^m g_{i m}=g_{i k, l},$$

$$\Gamma_{i l k}+\Gamma_{k l i}=g_{i k, l} .$$

$$\Gamma_{i l k} \equiv g_{m k} \Gamma_{i l}^m=g_{m k} \Gamma_{l i}^m=\Gamma_{l i k^*}$$

$$\Gamma_{k i l}+\Gamma_{l i k}=g_{k l, i}, \quad \Gamma_{l k i}+\Gamma_{i k l}=g_{l i, k}$$

$$\Gamma_{i k l}=\frac{1}{2}\left[g_{i i, k}+g_{k l, i}-g_{i k, l}\right]$$

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

$$\Gamma_{j k}^i=0=\Gamma_{i j k} \text { at point } P .$$

$$g_{\alpha \beta}\left(x_P^{\mu \prime}\right)=\eta_{\alpha \beta},$$

$$\left.\frac{\partial g_{\alpha \beta}}{\partial x^\mu}\right|_{x=x_p^{x^{\prime}}}=0 .$$

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assignmentutor™您的专属作业导师
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