### 物理代写|广义相对论代写General relativity代考|ASP3051

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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|Tensors, Component View

We continue in this section with the classic component view of vectors and tensors as indexed arrays. This section consists largely of a set of theorems which are proved by a relatively simple algebraic process often called index juggling. It should become clear that after some practice the balancing of the indices does much of the work for us.

To define a tensor we generalize the idea of a vector as defined as an $n$-tuple with a well-defined transformation between coordinate systems: a tensor is defined as a set of quantities with any number of indices, which transforms according to
$$\bar{T}{m \ldots}^{l \ldots}=\frac{\partial \bar{x}^{l}}{\partial x^{q}} \ldots \frac{\partial x^{n}}{\partial \bar{x}^{m}} \ldots T^{q \ldots . . .}, \quad \text { tensor components. }$$ The total number of indices is referred to as the rank; some of the indices may be upper, or contravariant, and others may be lower, or covariant. The number of such indices is written as $(M, N)$. Thus for example a vector is a first rank tensor and (1, 0 ). Another example is $V^{j} W{q}$, which is a second rank tensor and $(1,1)$.

From this tensor definition many simple but powerful theorems follow. We have already introduced and proved two of them in Sect. 4.2: Theorem 1 concerned the Jacobian matrices and Theorem 2 the invariance of the inner product of vectors. Let us continue to more such theorems.

Theorem 3 To contract a tensor we set an upper index equal to a lower index and sum, which gives another tensor; for example one contraction of $T^{\alpha \beta}{ }{\lambda y}$ is $T{\beta \gamma}^{\alpha \beta}=S^{\alpha}{ }{\gamma}$. Contraction of a rank $r$ tensor produces a rank $r-2$ tensor. Consider the above 4th rank tensor as an example. Then the contracted object transforms as $\bar{T}{\beta \gamma}^{\alpha \beta}=\frac{\partial \bar{x}^{\alpha}}{\partial x^{\omega}} \frac{\partial \bar{x}^{\beta}}{\partial x^{\sigma}} \frac{\partial x^{\lambda}}{\partial \bar{x}^{\beta}} \frac{\partial x^{\eta}}{\partial \bar{x}^{\gamma}} T^{\omega \sigma}{ }{\lambda \eta}=\frac{\partial \bar{x}^{\alpha}}{\partial x^{\omega}} \delta{\sigma}^{\lambda} \frac{\partial x^{\eta}}{\partial \bar{x}^{\gamma}} T^{\omega \omega \sigma}{ }{\lambda \eta}^{\partial x^{\eta}}$ $=\frac{\partial \bar{x}^{\alpha}}{\partial x^{\omega \omega}} \frac{\partial x^{\eta}}{\partial \bar{x}^{\gamma}} T{\sigma \eta}^{\omega \omega} .$

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

As with vectors we may view tensors as abstract objects instead of from the classic component point of view discussed in the previous section. In this abstract approach an $(M, N)$ tensor is defined to linearly map $M$ vectors and $N$-forms to the reals. For example a $(0,2)$ tensor $T$ operates as a linear map on vectors $\vec{V}, \vec{W}$ as follows
$$T(\vec{V}, \vec{W})=T\left(V^{\beta} \vec{e}{\beta}, W^{\mu} \vec{e}{\mu}\right)=V^{\beta} W^{\mu} T\left(\vec{e}{\beta}, \vec{e}{\mu}\right) \equiv V^{\beta} W^{\mu} T_{\beta \mu}$$
The components $T_{\beta \mu}$ defined here are the same as we discussed in the previous section. Thus a vector is also a $(1,0)$ tensor and a 1 -form is also a $(0,1)$ tensor. The metric is the most important special case of a $(0,2)$ tensor, so we explicitly note its operation in terms of components
$$g(\vec{V}, \vec{W})=V^{\beta} W^{\mu} g_{\beta \mu}$$
Let’s look at another example of a $(0,2)$ tensor. Define the direct product of two 1 -forms as something that operates linearly on two vectors to give a real in the following natural way,
$$\tilde{p} \otimes \tilde{q}=\text { direct product of 1-forms, } \tilde{p} \otimes \tilde{q}(\vec{V}, \vec{W})=\tilde{p}(\vec{V}) \tilde{q}(\vec{W})$$
That is, the first factor in the direct product operates on the first vector and the second factor in the direct product operates on the second vector. The direct product in (4.56) is thus a $(0,2)$ tensor. It should be clear that we can extend the definition to the direct product of any number $M$ of 1 -form factors to produce a $(0, M)$ tensor and so forth.
Recall that we discussed in Sect. $4.3$ a basis for 1-forms which we denoted as $\tilde{\omega}^{\alpha}$. We can similarly show that there exists a basis for the product of two 1 -forms or ( 0 , 2) tensors. Indeed the basis is a linear combination of the direct product of the $\tilde{\omega}^{\alpha}$. We write that linear combination as
$$f=f_{\alpha \beta} \tilde{\omega}^{\alpha} \otimes \tilde{\omega}^{\beta} .$$

In general relativity we often find it useful to use tetrads, a set of four basis vectors that forms an orthonormal basis as in special relativity. This sets up a reference frame at a point that is analogous to the reference frame of special relativity. The tetrad differs from the set of coordinate basis vectors in that it is normalized and need not align with the coordinate axes. More generally, in $n$ dimensions we define an $n$-trad, a set of $n$ basis vectors $\vec{e}{a}$ oriented and normalized so that $$\vec{e}{a} \cdot \vec{e}{b}=\eta{a b}$$
where the $\eta_{a b}$ matrix is chosen for convenience. It is usually taken to be the constant Lorentz metric in relativity theory but may be any constant matrix such as the Kronecker delta as needed in other situations; we refer to it as the $n$-trad metric. In this section the $n$-trads will be labeled with lower Latin indices early in the alphabet like $b$, and the space indices will usually be Greek.

Notice that the local Lorentz frame we previously discussed is essentially the same as the frame provided by the tetrads. Indeed it is possible to develop the theory of tetrads based on the transformation to the local Lorentz frame, although we will not do that here (Lawrie 1990).

In this section we will denote the coordinate basis as $\vec{g}{\beta}$ to distinguish it from the $n$-trad basis $\vec{e}{a}$, and it will be labeled with Greek indices. The $n$-trad may be expanded in terms of the coordinate basis as
$$\vec{e}{a}=e{a}^{\beta} \vec{g}{\beta}, \quad e{a}^{\beta}=n \text {-trad components in coordinate basis. }$$
This gives a beautiful relation for the $n$-trad metric in terms of the metric,
\begin{aligned} &\eta_{a b}=\vec{e}{a} \cdot \vec{e}{b}=\left(e_{a}^{\beta} \vec{g}{\beta}\right) \cdot\left(e{b}^{\mu} \vec{g}{\mu}\right)=e{a}^{\beta} e_{b}^{\mu}\left(\vec{g}{\beta} \cdot \vec{g}{\mu}\right)=e_{a}^{\beta} e_{b}^{\mu} g_{\beta \mu}, \ &\eta_{a b}=e_{a}^{\beta} e_{b}^{\mu} g_{\beta \mu} \end{aligned}

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

G(在→,在→)=在b在μGbμ

p~⊗q~= 1-形式的直接乘积， p~⊗q~(在→,在→)=p~(在→)q~(在→)

F=F一个bω~一个⊗ω~b.

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