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
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|PHYC90012

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 $P^{\prime}$
&\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}

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|PHYC90012


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

我们在第一章中的大部分考虑。4 涉及与单个点相关的向量和张量。现在我们研究如何比较黎曼空间中不同点的向量和张量,以及如何移动它们(Misner 1973;Adler 1975;Schutz 2009)。这对于研究张量场是必要的,张量被定义为空间区域中位置的函数;例如,这些字段可以表示为

φ(Xμ) 标量, 在∝(Xμ) 向量, 吨∝b(Xμ)2 nd 秩张量。 


首先考虑具有笛卡尔坐标的欧几里得 3 空间中的常数向量场;这样一个常数向量场的定义显然是分量是常数,如图 5.1a 所示。但同样清楚的是,在球坐标系中,常数分量并不对应于我们认为的常数矢量场,如图 5.1b 所示。显然,我们不应该将恒定矢量场定义为具有恒定分量的场。术语“恒定”和“平行”仍有待精确定义。常数向量场的正确定义将引入仿射连接的概念作为并行向量概念的优雅概括。

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



为了找到变换定律,我们提出一个向量在移植到附近点时仍然是向量的自然要求:即它必须在两个点都遵守变换定律 (4.14)磷和磷′. 移植的载体,在磷′在 barred 和 unbarred 坐标系中,是

在∗一世=在一世−Γpq一世 dXp在q,在¯∗j=在¯j−Γ¯米nj dX¯米在¯n这里右边的向量和连接(5.6)被评估在磷. 变换矩阵磷′可以通过泰勒级数展开得到磷

(∂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.

在¯∗j=(∂X¯j∂Xl)磷′在∗l 所以  在¯j−Γ¯米nj dX¯米在¯n=[(∂X¯j∂X一世)磷+(∂2X¯j∂Xl∂X一世)磷 dXl][在一世−Γpq一世 dXp在q] =(∂X¯j∂X一世)磷在一世−(∂X¯j∂X一世)磷Γpq一世 dXp在q+(∂2X¯j∂Xl∂X一世)磷 dXl在一世

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

上一节介绍的向量移植定律(5.5)提供了一种比较空间中不同附近点的向量的方法。通过重复迭代,我们还可以比较相距较远的点的向量。到目前为止,我们的考虑非常笼统,我们没有对如何指定连接做出任何假设。我们现在专门获取相对论中使用的特定联系;这为在欧几里得几何中平行于自身移动向量的想法提供了一个非常优雅的概括,称为平行位移。平行位移是空间曲率概念和定义的基础,我们将在第 1 章中阐述。8. 它还允许我们定义测地线曲线,即直线到一般黎曼空间的推广。





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有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





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多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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