Differential Geometry¶

Metric¶

Definitions¶

Denote the basis in use as $$\hat e_\mu$$, then the metric can be written as

$g_{\mu\nu}=\hat e_\mu \hat \cdot e_\nu$

if the basis satisfies

Inversed metric

$g_{\mu\lambda}g^{\lambda\nu}=\delta_\mu^\nu = g_\mu^\nu$

How to calculate the metric¶

Let’s check the definition of metric again.

If we choose a basis $$\hat e_\mu$$, then a vector (at one certain point) in this coordinate system is

$x^a=x^\mu \hat e_\mu$

Then we can construct the expression of metric of this point under this coordinate system,

$g_{\mu\nu}=\hat e_\mu\cdot \hat e_\nu$

For example, in spherical coordinate system,

(1)$\vec x=r\sin \theta\cos\phi \hat e_x+r\sin\theta\sin\phi \hat e_y+r\cos\theta \hat e_z$

Now we have to find the basis under spherical coordinate system. Assume the basis is $$\hat e_r, \hat e_\theta, \hat e_\phi$$. Choose some scale factors $$h_r=1, h_\theta=r, h_\phi=r\sin\theta$$. Then the basis is

$\hat e_r=\frac{\partial \vec x}{h_r\partial r}=\hat e_x \sin\theta\cos\phi+\hat e_y \sin\theta\sin\phi+\hat e_z \cos\theta,$

etc. Then collect the terms in formula (1) is we get $$\vec x=r\hat e_r$$, this is incomplete. So we check the derivative.

\begin{align}\begin{aligned}\mathrm d\vec x = \hat e_x (\mathrm dr \sin\theta\cos\phi+r\cos\theta\cos\phi\mathrm d\theta-r\sin\theta\sin\phi\mathrm d\phi)\\\hat e_y (\mathrm dr\sin\theta\sin\phi+r\cos\theta\sin\phi\mathrm d\theta+r\sin\theta\cos\phi\mathrm d\phi)\\\hat e_z (\mathrm dr\cos\theta-r\sin\theta\mathrm d\theta)\\ = \mathrm dr(\hat e_x\sin\theta\cos\phi +\hat e_y \sin\theta\sin\phi -\hat e_z \cos\theta)\\\mathrm d\theta (\hat e_x\cos\theta\cos\phi +\hat e_y \cos\theta\sin\phi - \hat e_z \sin\theta)r\\\mathrm d\phi (-\hat e_x\sin\phi +\hat e_y \cos\phi)r\sin\theta\\=\hat e_r\mathrm dr+\hat e_\theta r\mathrm d\theta +\hat e_\phi r\sin\theta\mathrm d \phi\end{aligned}\end{align}

Once we reach here, the component ($$e_r ,e_\theta, e_\phi$$) of the point under the spherical coordinates system basis ($$\hat e_r, \hat e_\theta, \hat e_\phi$$) at this point are clear, i.e.,

$\begin{split}\mathrm d\vec x = \hat e_r\mathrm d r+\hat e_\theta r\mathrm d \theta+\hat e_\phi r\sin\theta \mathrm d\phi \\ = e_r\mathrm d r+e_\theta \mathrm d\theta+e_\phi \mathrm d\phi\end{split}$

In this way, the metric tensor for spherical coordinates is

$\begin{split}g_{\mu\nu}=(e_\mu\cdot e_\nu) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2\theta \end{pmatrix}\end{split}$

Connection¶

First class connection can be calculated

$\Gamma^\mu_{\phantom{\mu}\nu\lambda}=\hat e^\mu\cdot \hat e_{\mu,\lambda}$

Second class connection isfootnote{Kevin E. Cahill}

$[\mu\nu,\iota]=g_{\iota\mu}\Gamma^\mu_{\phantom{\mu}\nu\lambda}$

$T^b_{\phantom bc;a}= \nabla_aT^b_{\phantom bc}=T^b_{\phantom bc,a}+\Gamma^b_{ad}T^d_{\phantom dc}-\Gamma^d_{ac}T^b_{\phantom bd}$

Curl¶

For an anti-symmetric tensor, $$a_{\mu\nu}=-a_{\nu\mu}$$

$\begin{split}\mathrm{Curl}_{\mu\nu\tau}(a_{\mu\nu}) \equiv a_{\mu\nu;\tau}+a_{\nu\tau;\mu}+a_{\tau\mu;\nu} \\ = a_{\mu\nu,\tau}+a_{\nu\tau,\mu}+a_{\tau\mu,\nu}\end{split}$

Divergence¶

$\begin{split}\mathrm{div}_\nu(a^{\mu\nu})&\equiv a^{\mu\nu}_{\phantom{\mu\nu};\nu} \\ & = \frac{\partial a^{\mu\nu}}{\partial x^\nu}+\Gamma^\mu_{\nu\tau}a^{\tau\nu}+\Gamma^\nu_{\nu\tau}a^{\mu\tau} \\ & = \frac1{\sqrt{-g}}\frac{\partial}{\partial x^\nu}(\sqrt{-g}a^{\mu\nu})+\Gamma^\mu_{\nu\lambda}a^{\nu\lambda}\end{split}$

For an anti-symmetric tensor

$\mathrm {div}(a^{\mu\nu})=\frac1{\sqrt{-g}}\frac{\partial}{\partial x^\nu}(\sqrt{-g}a^{\mu\nu})$

Annotation Using the relation $$g=g_{\mu\nu}A_{\mu\nu}$$, $$A_{\mu\nu}$$ is the algebraic complement, we can prove the following two equalities.

$\Gamma^\mu_{\mu\nu}=\partial_\nu\ln{\sqrt{-g}}$
$V^\mu_{\phantom\mu;\mu}=\frac1{\sqrt{-g}}\frac{\partial}{\partial x^\mu}(\sqrt{-g}V^\mu)$

In some simple case, all the three kind of operation can be demonstrated by different applications of the del operator, which $$\nabla\equiv \hat x\partial_x+\hat y\partial_y+\hat z \partial_z$$.

• Gradient, $$\nabla f$$, in which $$f$$ is a scalar.
• Divergence, $$\nabla\cdot \vec v$$
• Curl, $$\nabla \times \vec v$$
• Laplacian, $$\Delta\equiv \nabla\cdot\nabla\equiv \nabla^2$$

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