### 物理代写|广义相对论代写General relativity代考|PHYS6203

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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|Riemann Spaces

We now make the transition from the Minkowski spacetime of special relativity to more general spaces and coordinate systems and the mathematical objects in them, vectors and tensors and forms. There are standard reference texts dating over many years (Pauli 1958; Bergmann 1942; Rindler 1969; Weinberg 1972; Misner 1973; Adler 1975 ; Kenyon 1990). Here we will rely heavily on examples to illustrate the basic ideas.

Think of a physical space such as the surface of a blackboard or sphere or torus as in Fig. 4.1, or the Euclidean 3 -space of classical physics. In particular include the spacetime of special relativity that we studied in Part I. We imagine a marker system or labeling system or coordinate system to specify the points in the space with a set of real numbers. In general there will be many ways to set up such a marker system, and we assume that there will be transformations between them. An excellent example to remember is the Euclidean 3 -space of classical geometry and physics, labeled by Cartesian or spherical coordinates. We denote the transformation between two coordinate systems, denote them as unprimed and primed, by a set of functions
$$x^{\prime j}=f^{j}\left(x^{n}\right)$$
The functions $f^{j}$ are assumed to be continuous monotonic one-to-one and differentiable as often as needed. The transformation therefore has an inverse. For brevity we usually denote the transformation and its inverse in shorthand notation,

$$x^{\prime j}=x^{\prime j}\left(x^{n}\right), \quad x^{k}=x^{k}\left(x^{\prime j}\right)$$
The square array of derivatives we denote as
$$\frac{\partial x^{\prime j}}{\partial x^{n}}, \quad \frac{\partial x^{j}}{\partial x^{\prime /}}$$
These are the transformation or Jacobian matrices familiar from elementary calculus. Loosely speaking a space coordinatized in several different ways by $n$ real numbers as we use here is called an $n$-dimensional manifold. A manifold is defined as a space which locally resembles a Euclidean space and in which we can perform the usual analytic operations as in Euclidean space. Thus we can for example set up systems of differential equations in a manifold. See Appendix 1 for a more detailed discussion of the manifold idea.

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

We will first discuss mathematical objects in Riemann space from the point of view of their components, the view mainly used in the early part of the twentieth century for the invention and the early development of general relativity by Einstein and others (Pauli 1958; Bergmann 1942; Rindler 1969; Adler 1975). This is the approach we used in special relativity in Part I for Minkowski spacetime but now applied to a Riemann space. In Sect. 4.3, we will relate the component view to the more modern invariant abstract view, which became popular and fashionable in the later twentieth century (Misner 1973; Schutz 2009).

As we have already noted the component view may be termed the classic view, and is most useful for calculations such as finding solutions to the Einstein field equations and solving for the trajectories of moving objects. The abstract view can give a different perspective on the mathematics. The reader interested only in applications such as cosmology might choose to skim or skip the sections on the abstract view but could benefit from being exposed to both views.

The line element in $(4.4)$ is the archetype of an invariant, a crucially important mathematical object; it is postulated to be the same in all coordinate systems. That is an invariant or scalar is defined as any quantity which has the same value in all coordinate systems, for example for an unprimed and a primed system,
$$\phi^{\prime}=\phi \text { scalar or invariant. }$$
The concept of an invariant is one of the most fundamental in relativity and all of physics. Virtually everything that theory predicts should be expressed as an invariant for comparing with experimental measurement since nature does not know or care about our choice of coordinates.

Note that in this chapter we generally will not limit ourselves to any specific number of dimensions or metric signature, and the indices that we will use may be either Latin or Greek.

## 物理代写|广义相对论代写General relativity代考|Vectors and 1-Forms, Abstract View

We can connect the above idea of vectors as component $n$-tuples with the idea of intrinsic or abstract vectors, often represented in physics by arrows, and in the process introduce a definition of a metric. We may think of such vectors as intrinsic or abstract, but the word physical is also appropriate since they are taken to exist independently of the coordinate system and are invariant. These abstract vectors are taken to exist in an idealized physical world, whereas component vectors only exist when we represent them in terms of a specific coordinate system.

Look at a single point $P$ in a Reimann space. We introduce a set of basis vectors $\vec{e}_{k}$ along the grid lines illustrated in Fig. $4.5$ for two dimensions, but we do not assume the basis is orthonormal. The basis set spans a vector space associated with that point. Such a basis is naturally called a coordinate basis.

A small displacement $\mathrm{d} \vec{s}$ along a curve or in some specified direction is given by
$$\mathrm{d} \vec{s}=\vec{e}{j} \mathrm{~d} x^{j},$$ and its square is given by \begin{aligned} &\mathrm{d} \vec{s}^{2}=\mathrm{d} s^{2}=\left(\vec{e}{j} \mathrm{~d} x^{j}\right) \cdot\left(\vec{e}{k} \mathrm{~d} x^{k}\right)=\left(\vec{e}{j} \cdot \vec{e}{k}\right) \mathrm{d} x^{j} \mathrm{~d} x^{k}=g{j k} \mathrm{~d} x^{j} \mathrm{~d} x^{k} \ &g_{j k}=\vec{e}{j} \cdot \vec{e}{k} . \end{aligned}

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

X′j=Fj(Xn)

X′j=X′j(Xn),Xķ=Xķ(X′j)

∂X′j∂Xn,∂Xj∂X′/

φ′=φ 标量或不变量。

## 物理代写|广义相对论代写General relativity代考|Vectors and 1-Forms, Abstract View

ds→=和→j dXj,它的平方由下式给出

ds→2=ds2=(和→j dXj)⋅(和→ķ dXķ)=(和→j⋅和→ķ)dXj dXķ=Gjķ dXj dXķ Gjķ=和→j⋅和→ķ.

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## MATLAB代写

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