### 物理代写|广义相对论代写General relativity代考|PHYC90012

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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|Affine Connections, Component View

Most of our considerations in Chap. 4 involved vectors and tensors associated with a single point. Now we study how to compare vectors and tensors at different points in a Riemann space, and how to move them (Misner 1973; Adler 1975; Schutz 2009). This is necessary in order to study tensor fields, that is tensors defined as functions of position in regions of space; these fields may be denoted for example as
$$\phi\left(x^{\mu}\right) \text { scalar, } V^{\propto}\left(x^{\mu}\right) \text { vector, } T^{\propto \beta}\left(x^{\mu}\right) 2 \text { nd rank tensor. }$$
This is not a trivial process since vector spaces at different points in a Riemann space are a priori independent and any connection between them requires analysis. The key concept is that of affine connections, for which we will motivate a definition; then on the basis of the definition we may obtain their transformation law.

Much of the work in this chapter is based on the classic component view, but we will relate it to the abstract view in the last section.

Consider first a constant vector field in Euclidean 3-space with Cartesian coordinates; the definition of such a constant vector field is obviously that the components are constant, as shown in Fig. 5.1a. But it is also clear that in spherical coordinates constant components do not correspond to what we think of as a constant vector field, as in Fig. 5.1b. Clearly, we should not define a constant vector field as one with constant components. The terms “constant” and “parallel” remain to be defined precisely. The proper definition of a constant vector field will introduce the concept of affine connections as an elegant generalization of the notion of parallel vectors.

## 物理代写|广义相对论代写General relativity代考|Transformation of the Affine Connections

The law of vector transplantation introduced in (5.5) is extremely general since there are no restrictions on the connections. Remarkably, from only the above definition

of vector transplantation we may obtain the transformation law for the connections, which we will find are not tensors. Moreover several theorems that result from the transformation law are basic and important for both mathematics and physics.

To find the transformation law we make the natural demand that a vector remains a vector as it is transplanted to a nearby point: that is it must obey the transformation law (4.14) at both $P$ and $P^{\prime}$. The transplanted vector, at $P^{\prime}$ in the barred and unbarred coordinate systems, is
$$V^{* i}=V^{i}-\Gamma_{p q}^{i} \mathrm{~d} x^{p} V^{q}, \quad \bar{V}^{* j}=\bar{V}^{j}-\bar{\Gamma}{m n}^{j} \mathrm{~d} \bar{x}^{m} \bar{V}^{n}$$ Here the vector and connections on the right side of $(5.6)$ are evaluated at $P$. The transformation matrix at $P^{\prime}$ may be gotten with a Taylor series expansion from that at $P$ $$\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P}=\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P}+\frac{\partial}{\partial x^{l}}\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P} \mathrm{~d} x^{l}=\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P}+\left(\frac{\partial^{2} \bar{x}^{j}}{\partial x^{l} \partial x^{i}}\right){P} \mathrm{~d} x^{l} .$$
We use these expressions and impose the vector transformation law on the vector at $P^{\prime}$
\begin{aligned} &\bar{V}^{* j}=\left(\frac{\partial \bar{x}^{j}}{\partial x^{l}}\right){P^{\prime}} V^{* l} \text { so } \ &\bar{V}^{j}-\bar{\Gamma}{m n}^{j} \mathrm{~d} \bar{x}^{m} \bar{V}^{n}=\left[\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P}+\left(\frac{\partial^{2} \bar{x}^{j}}{\partial x^{l} \partial x^{i}}\right){P} \mathrm{~d} x^{l}\right]\left[V^{i}-\Gamma_{p q}^{i} \mathrm{~d} x^{p} V^{q}\right] \ &=\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P} V^{i}-\left(\frac{\partial \bar{x}^{j}}{\partial x^{i}}\right){P}^{\Gamma_{p q}^{i}} \mathrm{~d} x^{p} V^{q}+\left(\frac{\partial^{2} \bar{x}^{j}}{\partial x^{l} \partial x^{i}}\right)_{P} \mathrm{~d} x^{l} V^{i} \end{aligned}

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

The law of vector transplantation (5.5) introduced in the preceding section provides a way to compare vectors at different nearby points in space. By repeated iterations we could also compare vectors at widely separated points. Our considerations have so far been quite general and we made no assumptions about how the connections might be specified. We now specialize to obtain the specific connections used in relativity theory; this provides a strikingly elegant generalization of the idea of moving a vector parallel to itself in Euclidean geometry, and is called parallel displacement. Parallel displacement is basic to the idea and definition of space curvature that we will develop in Chap. 8. It also allows us to define geodesic curves, which are the generalization of straight lines to general Riemann spaces.

Suppose that we transplant two vectors to a nearby point using the law of vector transplantation. There is no a priori reason that the inner product of the two will remain unchanged; however we may consider this to be a naturally compelling demand to be imposed so as to make the transplantation analogous to the parallel displacement of vectors in Euclidean geometry. In the special case of Euclidean space the demand for parallelism implies that the lengths of various vectors and the angles between them remain unchanged as they are transplanted. We thus impose this demand and refer to this special case of vector transplantation as generalized parallel displacement, or simply parallel displacement for brevity. Remarkably, the connections are then

determined uniquely by the metric. Figure $5.3$ shows the scenario for the parallel displacement of two vectors.

The derivation of the affine connections is conceptually simple and involves only slightly tedious algebra. The demand that the inner product of the two vectors be unchanged under vector transplantation may be expressed as
$$\mathrm{d}\left(\xi^{j} \eta^{k} g_{j k}\right)^{*}=0$$

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

φ(Xμ) 标量， 在∝(Xμ) 向量， 吨∝b(Xμ)2 nd 秩张量。

## 物理代写|广义相对论代写General relativity代考|Transformation of the Affine Connections

（5.5）中引入的向量移植定律非常普遍，因为对连接没有限制。值得注意的是，仅从上述定义

(∂X¯j∂X一世)磷=(∂X¯j∂X一世)磷+∂∂Xl(∂X¯j∂X一世)磷 dXl=(∂X¯j∂X一世)磷+(∂2X¯j∂Xl∂X一世)磷 dXl.

d(Xj这ķGjķ)∗=0

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