Differential Geometry ======================= Metric -------- Definitions ~~~~~~~~~~~~ Denote the basis in use as :math:\hat e_\mu, then the metric can be written as .. math:: g_{\mu\nu}=\hat e_\mu \hat \cdot e_\nu if the basis satisfies Inversed metric .. math:: 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 :math:\hat e_\mu, then a vector (at one certain point) in this coordinate system is .. math:: x^a=x^\mu \hat e_\mu Then we can construct the expression of metric of this point under this coordinate system, .. math:: g_{\mu\nu}=\hat e_\mu\cdot \hat e_\nu For example, in spherical coordinate system, .. math:: \vec x=r\sin \theta\cos\phi \hat e_x+r\sin\theta\sin\phi \hat e_y+r\cos\theta \hat e_z :label: EQrelativityMetricPoint Now we have to find the basis under spherical coordinate system. Assume the basis is :math:\hat e_r, \hat e_\theta, \hat e_\phi. Choose some scale factors :math:h_r=1, h_\theta=r, h_\phi=r\sin\theta. Then the basis is .. math:: \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 :eq:EQrelativityMetricPoint is we get :math:\vec x=r\hat e_r, this is incomplete. So we check the derivative. .. math:: \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 Once we reach here, the component (:math:e_r ,e_\theta, e_\phi) of the point under the spherical coordinates system basis (:math:\hat e_r, \hat e_\theta, \hat e_\phi) at this point are clear, i.e., .. math:: \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 In this way, the metric tensor for spherical coordinates is .. math:: 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} Connection ----------- First class connection can be calculated .. math:: \Gamma^\mu_{\phantom{\mu}\nu\lambda}=\hat e^\mu\cdot \hat e_{\mu,\lambda} Second class connection is\footnote{Kevin E. Cahill} .. math:: [\mu\nu,\iota]=g_{\iota\mu}\Gamma^\mu_{\phantom{\mu}\nu\lambda} Gradient, Curl, Divergence, etc --------------------------------- Gradient ~~~~~~~~~~~ .. math:: 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, :math:a_{\mu\nu}=-a_{\nu\mu} .. math:: \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} Divergence ~~~~~~~~~~~~~ .. math:: \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} For an anti-symmetric tensor .. math:: \mathrm {div}(a^{\mu\nu})=\frac1{\sqrt{-g}}\frac{\partial}{\partial x^\nu}(\sqrt{-g}a^{\mu\nu}) **Annotation** Using the relation :math:g=g_{\mu\nu}A_{\mu\nu}, :math:A_{\mu\nu} is the algebraic complement, we can prove the following two equalities. .. math:: \Gamma^\mu_{\mu\nu}=\partial_\nu\ln{\sqrt{-g}} .. math:: 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 :math:\nabla\equiv \hat x\partial_x+\hat y\partial_y+\hat z \partial_z. * Gradient, :math:\nabla f, in which :math:f is a scalar. * Divergence, :math:\nabla\cdot \vec v * Curl, :math:\nabla \times \vec v * Laplacian, :math:\Delta\equiv \nabla\cdot\nabla\equiv \nabla^2