Bei's Study Notes

General Relativity 02 - Manifold and tensor fields
Last updated: 2017-04-19 21:55:03 PDT.

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(This is from Charpter 2.2 of Wald)


In GR, we discuss about spaces that are not exactly flat. If we can split the space into parts and each part is continously corresponsive to a region of $\mathbb R^n$ , then it is called a manifold. $n$ is called toe dimension of the manifold.

Normally, a finite dimensional manifold can be embeded in a higher dimensional Euclidean space. Whereas such embedding might not be natural. In GR, the spacetime does not naturally live in a higher dimensional space, so an abstract definition of a manifold is necessary.

Definition (Manifold): An n-dimensional, $C^\infty$ , real manifold $M$ is a set together with a collection of subsets $\{O_\alpha\}$ satisfying the following properties:

  1. $\{O_\alpha\}$ is a open cover of $M$ .
  2. For each $\alpha$ , there is a one-to-one, onto, map $\psi_\alpha: O_\alpha \to U_\alpha$ where $U_\alpha$ is an open subset of $\mathbb R^n$ .
  3. For each $O_\alpha$ and $O_\beta$ that $O_\alpha \cap O_\beta \neq \emptyset$ , the map $\psi_\beta \circ \psi_\alpha^{-1}$ maps $\psi_\alpha[O_\alpha \cap O_\beta] \subset U_\alpha$ to $\psi_\beta[O_\alpha \cap O_\beta] \subset U_\beta$ . Then both sets must be open and the map must be $C^\infty$ .

Each map $\psi_\alpha$ is called a "chart" or a "coordinate system". The definition of $C^k$ manifold and complex manifold simply is the same with some natural changes.

Through out this book, the manifold involed are all assumed to be Hausdorff and paracomact.

NOTE By this definition, the charts, by no means, are required to provide a "straight coordinate" to $\mathbb R^n$ , by which I mean the tangent vector can change through out each $O_\alpha$ .

Example Euclidean space $\mathbb R^n$ with one chart and map to be identity function.

Example 2-sphere (a 2-dimensional spherical surface embedded in $\mathbb R^3$ ).

In SR, each coordinate system is applicable to the whole universe. but in GR, a coordinate system is only usable inside its corresponding open set.

With the mapping to $\mathbb R^n$ , we can easily define differentiability and smoothness of maps between manifolds.

Defintion (Diffeomorphism): Let $M$ and $M'$ to be manifolds and $\{\psi_\alpha\}$ and $\{\psi_\beta\}$ to be chart maps. A map $f:M\to M'$ is said to be $C^\infty$ if for each $\psi'_\beta \circ f \circ \psi_\alpha^{-1}$ is $C^\infty$ in Euclidean spaces. If $f$ is $C^\infty$ , one-to-one and onto, then it is called a diffeomorphism, and $M$ and $M'$ are said to be diffeomorphic.

NOTE diffeomorphism requires the manifolds to have the same dimension.

Tangent vector in a manifold (without embedding in $\mathbb R^n$ ):

We can define tangent vectors as directional derivatives. In $\mathbb R^n$ , the mapping between vectors and directional derivatives is one-to-one. $(v^1,...,v^n)$ defines derivative operator $\sum_\mu v^n(\partial/\partial x^\mu)$ . Directional derivatives are characterized by Leibnitz rules.

Let $\mathcal F$ denote the collections of all $C^\infty$ functions from manifold $M$ into $\mathbb R$ . We define a tangent vector $v$ at point $p \in M$ to be a map $v: \mathcal F \to \mathbb R$ which is (1) linear and (2) obeys Leibnitz rules: $\forall f,g \in \mathcal F, a, b \in R,$

  1. $v(af+bg)=av(f) + bv(g)$
  2. $v(fg) = f(p)v(g) + g(p)v(f)$

NOTE: Be very careful that the second rule also applies the function $f$ and $g$ at $p$ .

Prop: $h\in\mathcal F,\ \forall q \in M, h(q) = c \text{ (constant) } \implies v(h) = 0$


$$\begin{align*} p \mapsto h(p)h(p) &= p \mapsto ch(p) \implies hh = ch\\ v(hh) &= 2h(p)v(h) = 2cv(h)\quad,\, \text{whereas}\\ v(ch) &= cv(h)\quad,\text{therefore} \\ cv(h) &= 0 \\ \end{align*}$$

If $c = 0$ , $v(h) = 0$ since linearlity, otherwise divide the equation by $c$ , and $v(h) = 0$ . $\square$

The maps of a tangent vectors of a point $p$ forms a vector space by adding this addition law: $\forall a \in \mathbb R, v_1 + av_2 \equiv h \mapsto v_1(h) + av_2(h)$ .


Theorem 2.2.1: Let $M$ be an $n$ -dimensional manifold. Let $p\in M$ and let $V_p$ denote the tangent space at $p$ , then $\mathrm {dim}(V_p) = n$ .

Proof. Given a chart $\psi$ of open set $O$ where $p \in O$ , If $f \in \mathcal F$ , then $f\circ \psi^{-1} \to \mathbb R$ is $C^\infty$ . For $\mu = 1, ..., n$ define functional $X_{p,\mu}: \mathcal F \to \mathbb R$ by

$$\begin{align*} X_{p,\mu}[f] &= \frac{\partial}{\partial x^\mu}\left[f\circ\psi^{-1}\right](\psi(p)) \\ \end{align*}$$

This means, the $\mu$ th component of $X$ is a functional that takes the $\mu$ th partial derivative of function $f\circ\psi^{-1}$ in $\mathbb R^n$ , and apply the point $\psi(p)$ to it. It is clear that $X_{p,\mu}$ is a derivative from the chain rule. Now we need to prove that $V_p = \mathrm{span}\left\{X_{p,\mu}\right\}$ .

For any function $F: \mathbb R^n \to \mathbb R$ , if $F$ is $C^\infty$ ,

$$ \forall a\in\mathbb R^n, \exists H_{a,\mu}(x\in \mathbb R^n) \in C^\infty, s.t.\\ F(x) = F(a) + \sum^n_{\mu=1}(x^\mu-a^\mu)H_{a,\mu}(x)\quad, $$

especially, we have

$$\begin{align*} H_{a,\mu}(a) &= \lim_{x\to a} H_{a,\mu}(x)\\ &= \frac{\partial}{\partial x^\mu}[F](a)\quad. \end{align*}$$

Let $F= f\circ \psi^{-1}$ , and $a=\psi(p)$ we have

$$\begin{align*} f(q) &= f(p) + \sum^n_{\mu=1}(x^\mu \circ \psi(q)-x^\mu \circ \psi(p)) H_{\psi(p),\mu}(\psi(q))\quad.\\ H_{\psi(p),\mu}(\psi(p)) &= \frac{\partial}{\partial x^\mu}[F](\psi(p)) \\ &= \frac{\partial}{\partial x^\mu}[f \circ \psi^{-1}](\psi(p)) \\ &= X_{p,\mu}[f]\quad, \end{align*}$$

where $x^\mu \circ \psi$ denotes $(x\mapsto x^\mu) \circ \psi$ , that is, picking the $\mu$ th element of the result.

For any $v \in V_p$ , apply the functional,

$$\begin{align*} v[f]&=\left.\cancel{v[f(p)]} + \sum^n_{\mu=1}v[x^\mu \circ \psi-\cancel{x^\mu \circ \psi(p)}]\cdot (H_{\psi(p),\mu}\circ\psi)(p) \\ + \sum^n_{\mu=1}\cancel{\left(x^\mu \circ \psi(p)-x^\mu \circ \psi(p)\right)} v[H_{\psi(p),\mu}\circ\psi]\right.\quad, \\ &= \sum^n_{\mu=1}v[x^\mu \circ \psi]X_{p,\mu}[f]\quad. \end{align*}$$

This means $v[f]$ is a linear combination of $\{X_{p,\mu}[f]\}$ . $\square$

The basis $\{X_\mu\}$ is called a coordinate basis, frequently denoted as simply $\partial/\partial x^\mu$ . For each different chart $\psi'$ chosen, there is a different coordinate basis $\{X'_\nu\}$ , and

$$\begin{align*} X'_{p,\nu}[f] &= \sum^n_{\nu=1} X'_{p,\nu}[x^\mu \circ \psi] X_{p,\mu}[f] \\ &= \sum^n_{\nu=1} \frac{\partial}{\partial x'^\nu}[x^\mu \circ \psi\circ\psi'^{-1}](\psi'(p)) X_{p,\mu}[f] \\ &= \sum^n_{\nu=1} \frac{\partial x^\mu}{\partial x'^\nu}(\psi'(p)) X_{p,\mu}[f]\quad, \\ X_{p,\mu} &= \sum^n_{\nu=1} \frac{\partial x'^\nu}{\partial x^\mu}(\psi(p)) X'_{p,\nu}\quad. \\ \end{align*}$$

We can also get the vector transformation law from it:

$$ v'^\nu=\sum^n_{\mu=1}v^\mu\frac{\partial x'^\nu}{\partial x^\mu}\quad. $$

A smooth curve, $C$ on a manifold $M$ is a $C^\infty$ map of $\mathbb R$ into $M$ , $C: \mathbb R \to M$ . At each point $p \in M \cap C$ , there is a tangent vector $T \in V_p$ associated with $C$ as follows.

$$\begin{align*} T[f]&=\frac{d}{dt}[f\circ C] \\ &=\sum_{\mu=1}^n \frac{d x^\mu}{dt}\frac{\partial}{\partial x^\mu}[f\circ \psi^{-1}] \\ &=\sum_{\mu=1}^n\frac {dx^\mu}{dt} X_\mu(f)\quad.\\ \end{align*}$$

Therefore the components of $T$ is given by

$$ T^\mu = \frac{dx^\mu}{dt}\quad. $$

If $p$ and $q$ are on the manifold, there is no way to correlate them in a general manifold. Another construct ("connection", or "parallel transportation") must be introduced to do so. However, if the curvation is nonzero, the identification of $V_p$ with $V_q$ obtained in this manner will depend on the choice of curve.

A tangent field, $v$ , on a manifold $M$ is an assignment of a tangent vector, $v\vert_p \in V_p$ for each point $p \in M$ . Despite the fact that the tangent spaces $V_p$ and $V_q$ at different points are different vector spaces, there is a natural notion of what it means for $v$ to vary smoothy from point to point.

A one-parameter group of diffeomorphisms $\phi_t$ is a $C^\infty$ map from $\mathbb R \times M \to M$ . In particular:

  1. $\forall \in\mathbb R, \phi_t$ is a diffeomorphism, and
  2. $\forall s, t \in \mathbb R, \phi_{s+t} = \phi_s \circ \phi_t$ .

At each point $p$ , $\phi_t(p)$ is a curve, called the orbit of $\phi_t$ which passes through $p$ at $t=0$ . Define $v|_p$ to be the tangent vector at $t=0$ . Thus we can consider the vector field $v$ to be the generator of such a group.

Conversely, we can ask a question, that if given a vector field $v$ , can we find a family of curves s.t. for each point $p \in M$ , there is one and only one curve that passes through the point with the tangent vector equals to $v\vert_p$ . The answer is yes.

Therefore we have a one-to-one mapping between a tangent field and a diffeomorphism. We can thus consider a tangent field to be a mapping of type $M \to M$ :

$$ v|_p = \left.\frac{\mathrm d\phi_t(p)}{\mathrm dt}\right|_{t=0} $$

Given two smooth vector fields $v$ and $w$ , we can define the commutator field $[v, w]$ as follows:

$$ [v,w](f) = v(w(f)) - w(v(f)) $$

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