3. 导数算符和测地线

导数算符定义

导数算符是一个从流形\(M\)的 \((k,l)\) 型张量场映射到 \((k,l+1)\)型张量场的一个映射，记作\(\nabla_a\), 并且得满足以下性质：

\(\nabla_a \mu T = \mu \nabla_a T\)，\(\mu\)为常数

\(\nabla_a T_1T_2 = T_1 \nabla_a T_2 + T_2 \nabla_a T_1\), \(T_1,T_2\)不一定同型。

\(C \cdot \nabla = \nabla \cdot C\), C表示缩并

\(v(f) = v^a \nabla_a(f)\)

\(\nabla_a\nabla_b T = \nabla_b\nabla_a T\)

显然由定义，有\(\forall \nabla_a\), \(\enspace df = \nabla_a f\)

克氏符

假设对偶矢量场 \(w_a\) 与对偶矢量场 \(w_a'\) 在p点取值相同，则往证：

\[ (\tilde{ \nabla}_a - \nabla_a) w_b|_p = (\tilde{ \nabla}_a - \nabla_a) w_b'|_p \]

证明：

\[\begin{aligned} \nabla_a (w_b-w_b')|_p &= \nabla_a \Omega_b|_p = \nabla_a (\Omega_\mu (dx)^\mu_b)|_p \\\\&= \Omega_\mu|_p \nabla_a ((dx)^\mu_b)|_p +(dx^\mu)_b|_p \nabla_a (\Omega_\mu )|_p \\\\&= 0 \nabla_a ((dx)^\mu_b)|_p + (dx^\mu)_b|_p \tilde{\nabla_a} (\Omega_\mu )|_p \\\\&= \tilde{\nabla_a} (w_b-w_b')|_p \end{aligned}\]

由于p任取，所以\(\tilde\nabla_a -\nabla_a\) 对应一个 (1,2) 型的张量。

任意导数算符 \(\nabla_a\), 它所对应的克氏符定义为 \(\Gamma^{\\;\\;\\;c}_{ab}w_c = \nabla_a w_b- \partial_aw_b\)

\(\partial_a\) 为普通导数算符，定义为（以二维张量\(T_{b}^{\\;c}\)为例）：

\(\partial_a T^{\;c}_{b}= (dx^\eta)_a(dx^\mu)_b(\frac{\partial}{\partial x^\nu})^c \frac{\partial T_{\mu}^{\;\nu}}{\partial x^\eta}\)

在展开 \(\partial_a T^{\;c}_{b}\)时，一般将分量\(\frac{\partial T_{\nu}^{\;\mu}}{\partial x^\eta}\) 记作 \(T_{,\eta\nu}^{\;\;\;\;\mu}\)

把对易算符可以用导数算符表示：

\[ [u,v]^a = u^b\nabla_b v^a - v^b \nabla_b v^a \]

导数算符的无挠性等价于\(\nabla_a \delta_{b}^{\\;\\;c}=0\)

平移

\(v^a\) 是沿着曲线 \(C(t)\) 的矢量场，若满足 \((\frac{\partial}{\partial t})^a \nabla_a v^b = 0\)，则称 \(v^a\) 矢量场是沿着 \(C(t)\) 平移的。

\((\frac{\partial}{\partial t})^a \nabla_a v^b = (\frac{\partial}{\partial t})^a (\partial_a v^b- \tau_{ac}^{\;\;b} v^c)\)

$ = ()^a (v_{,}^{;;;}(dx)a()^b - {}^{;;;} v^(dx)_a()^b) =0$

$ +_{}^{;;;}v = 0$

度规适配的导数算符

为使得平移的矢量内积保持不变，需要满足：

先写出一个标量场沿着曲线值不变的表达式：

\[\begin{aligned} &f(p) = const \\;p \in curve \\\\&\frac{d f \cdot C(t)}{dt} = 0 \\\\ &\frac{\partial}{\partial t} (f) = 0 \\\\ &(\frac{\partial}{\partial t})^a \nabla_a (f) = 0 \end{aligned}\]

第三行来自于\(v(f) = v^a\nabla_a f\)，为了让沿着曲线平移的任意向量保持内积不变，则条件写为：

\((\frac{\partial}{\partial t})^a\nabla_a (g_{bc} v^b u^c) = 0\)

\(=(\frac{\partial}{\partial t})^a (g_{bc} v^b \nabla_a (u^c)\)

\(+ u^c g_{bc} \nabla_a (v^b)+ u^c v^b\nabla_a (g_{bc})) =0\)

$ = g_{bc} v^b ()^a a (u^c) + u^c g{bc} ()^a_a v^b$

\(+ u^c v^b(\frac{\partial}{\partial t})^a\nabla_a (g_{bc}) =0\)

\(\nabla_a g_{bc} = 0\)

第三行由于\(u,v\)沿曲线平移，可证明在规定了度规后的流形上存在唯一导数算符与之对应满足内积不变。

测地线

自平移的曲线称为测地线，既\(\frac{\partial}{\partial t} \nabla \frac{\partial}{\partial t} =0\)，对应的曲线参数t称仿射参数

分量方程：

\[\begin{aligned} &\frac{dv^\nu}{dt}+\tau_{\mu\theta}^{\\;\\;\\;\nu}v^\theta \frac{dx^\mu}{dt} = 0 \\;\\;v^{\nu} = (\frac{\partial }{\partial t})^{\nu} = \frac{dx^\nu}{dt} \\\\&\Rightarrow \frac{d^2x^\nu}{dt^2}+\tau_{\mu\theta}^{\\;\\;\\;\nu}\frac{dx^\theta}{dt} \frac{dx^\mu}{dt} = 0 \end{aligned}\]

曲线\(\gamma(t)\)满足方程：\(T^b \nabla_b T^a = \alpha(t)T^a\) ，往证存在参数变换\(t'=t'(t)\), 使得\(\gamma'(t')\)是测地线：

\[\begin{aligned} &T^b \nabla_b T^a = \alpha(t)T^a \\\\&\because \frac{\partial}{\partial t'}^b = \frac{dt}{dt'}\frac{\partial}{\partial t}^b \\\\ &\frac{dt'}{dt}(T'^bT'^a\nabla_b(\frac{dt'}{dt})+T'^b\frac{dt'}{dt}\nabla_b T'^a)= \alpha(t)\frac{dt'}{dt}T'^a \end{aligned}\]

想要证明\(T'^b\nabla_b T'^a = 0^a\)，等价于证明\(\Rightarrow T'^b\nabla_b(\frac{dt'}{dt}) = \alpha(t)\)

\[ T'^b\nabla_b(\frac{dt'}{dt}) = T'^b(\frac{dt'}{dt}) = d^2t'/dt^2 = \alpha(t) \]

也就是说满足曲线起点相同并且满足最后一行的微分方程的变换的曲线能够满足测地线的定义。

此处实际拓宽了\(\frac{dt'}{dt}\)的定义域让它变成了一个M上的标量场。

3-3-4 一点一矢定一测，给定了一个流形上的点以及对应的一个矢量，可以从该点出发得到唯一的测地线

线长参数必为仿射参数

prove:

由于\(T\)沿曲线平移，\(g(T,T)\)在曲线上为常值，线长表达式：

\[ l =\int \sqrt{|g_{ab}T^aT^b|}dt \]

\(\frac{dl}{dt}=\sqrt{|g_{ab}T^aT^b|} = const \Rightarrow g_ab T^A T^b = const\)

该内积沿曲线不变，所以假设t是仿射参数，对应的切矢为\(T\)，\(l\)对应的切矢为\(T'\)，\(\frac{dl}{dt} T= T' = cT\)

\(cT^a\nabla_b cT_c = c^2 T^a\nabla_b T_c =0\)

##` 测地线是两点间的长度取极值的连接线

闵氏空间中测地线最长