General Relativity/Differentiable manifolds

<General Relativity

A smooth ${\displaystyle n}$-dimensional manifold ${\displaystyle \mathrm {M} ^{n}}$ is a set together with a collection of subsets ${\displaystyle \{O_{\alpha }\}}$ with the following properties:

1. Each ${\displaystyle p\in \mathrm {M} }$ lies in at least one ${\displaystyle O_{\alpha }}$, that is ${\displaystyle \mathrm {M} =\cup _{\alpha }O_{\alpha }}$.
2. For each ${\displaystyle \alpha }$, there is a bijection ${\displaystyle \psi _{\alpha }:O_{\alpha }\longrightarrow U_{\alpha }}$, where ${\displaystyle U_{\alpha }}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$
3. If ${\displaystyle O_{\alpha }\cap O_{\beta }}$ is non-empty, then the map ${\displaystyle \psi _{\alpha }\circ \psi _{\beta }^{-1}:\psi _{\beta }[O_{\alpha }\cap O_{\beta }]\longrightarrow \psi _{\alpha }[O_{\alpha }\cap O_{\beta }]}$ is smooth.

The bijections are called charts or coordinate systems. The collection of charts is called an atlas. The atlas induces a topology on M such that the charts are continuous. The domains ${\displaystyle O_{\alpha }}$ of the charts are called coordinate regions.

• Euclidean space, ${\displaystyle \mathbb {R} ^{n}}$ with a single chart (${\displaystyle O=\mathbb {R} ^{n},\psi =}$ identity map) is a trivial example of a manifold.
• 2-sphere ${\displaystyle S^{2}=\{(x,y,z)\in \mathbb {R} ^{3}|x^{2}+y^{2}+z^{2}=1\}}$.

Notice that ${\displaystyle S^{2}}$ is not an open subset of ${\displaystyle \mathbb {R} ^{3}}$. The identity map on ${\displaystyle \mathbb {R} ^{3}}$ restricted to ${\displaystyle S^{2}}$ does not satisfy the requirements of a chart since its range is not open in ${\displaystyle \mathbb {R} ^{3}.}$

The usual spherical coordinates map ${\displaystyle S^{2}}$ to a region in ${\displaystyle \mathbb {R} ^{2}}$, but again the range is not open in ${\displaystyle \mathbb {R} ^{2}.}$ Instead, one can define two charts each defined on a subset of ${\displaystyle S^{2}}$ that omits a half-circle. If these two half-circles do not intersect, the union of the domains of the two charts is all of ${\displaystyle S^{2}}$. With these two charts, ${\displaystyle S^{2}}$ becomes a 2-dimensional manifold. It can be shown that no single chart can possibly cover all of ${\displaystyle S^{2}}$ if the topology of ${\displaystyle S^{2}}$ is to be the usual one.