6.killing场

killing场和超曲面

killing场定义：

假设流形\((M,g_{ab})\)上\(v\)矢量场对应的单参微分同胚映射是保度规映射，既\(\phi_t^* g_{ab} = g_{ab}\)，则称\(v\)为killing场

theorem1:

\(v\)是killing场等价于\(g_{ab}\)关于\(v\)的李导数等于0等价于killing方程：

\[ \nabla_av_b +\nabla_bv_a = 0 \]

证明：

由李导数的定义

\(\mathscr{L}_v g_{ab} = \lim_{t \rightarrow 0}\frac{\phi_t^*g_{ab}-g_{ab}}{t}\)

显然\(v\)是killing场可直接推出\(g_{ab}\)关于\(v\)的李导数等于0，反过来，由李导数等于0，选取适配坐标系，得到：\(\frac{\partial g_{\mu\nu}}{\partial x_1}=0\)，所以\(g_{\mu\nu}\)在积分曲线上积分曲线相连的两点值相等。

设\(\phi\)从\(p\)点映射到\(q\)点

\[\begin{aligned} g_{\mu\nu}\mid_q &= g_{ab}(\frac{\partial}{\partial x^\mu})^a (\frac{\partial}{\partial x^\nu})^b\mid_q \\\\\phi_t^* g_{\mu\nu}\mid_q &= \phi_t^* (g_{ab}(\frac{\partial}{\partial x^\mu})^a (\frac{\partial}{\partial x^\nu})^b)\mid_q \end{aligned}\]

$ = _a^c _b^d ( _t^*g_{ab}_t^*()^c _t^*()^d)_p$

\[\begin{aligned} &\phi_t^*(\frac{\partial}{\partial x^\nu})^c \mid_p f(p) \\\\=& \phi_{-t*}(\frac{\partial}{\partial x^\nu})^c \mid_p f(p) \\\\ =& (\frac{\partial}{\partial x'^\nu})^c\mid_q f(\phi_{-t}(q)) \end{aligned}\]

由适配坐标系的定义: \(f(\phi_{-t}(q)) = F'(x'^{1}-t,x'^{2},x'^{3}...x'^{n}) = F(x^1,x^2,...x^n)\)

所以

\[\begin{aligned} &\phi_t^* (\frac{\partial}{\partial x'^\nu})^c\mid_p f(p) \\\\& = (\frac{\partial}{\partial x'^\nu})^c\mid_q f'(q) = \frac{\partial F'}{\partial x'^\nu} = \frac{\partial F}{\partial x^\nu} \\\\& = (\frac{\partial}{\partial x^\nu})^c\mid_p f(p) \end{aligned}\]

得到$_t^*()^c _p =()^c _q $，所以：

$ _a^c _b^d ( _t^*g_{ab}_t^*()^c _t^*()^d)_p$

\(= \delta_a^c \delta_b^d ( \phi_t^*g_{ab}\otimes (\frac{\partial}{\partial x^\mu})^c \otimes (\frac{\partial}{\partial x^\nu})^d)\mid_p\)

\(g_{\mu\nu} \mid_q = \phi_t^*g_{\mu\nu} \mid_p\)

最后由在积分曲线上\(g_{\mu\nu}\)为常数得到\(\phi_t^* g_{ab} = g_{ab}\)，一二等价证毕

由李导数作用于二阶张量的运算法则由：

\(\mathscr{L}_v(g_{ab}) = v^c \nabla_c g_{ab} + g_{ac} \nabla_b v^c + g_{cb} \nabla_a v^c\)

$ = _b v_a + _a v_b = 0 $

theorem 2:

如果度规张量的所有分量关于某个坐标\(x\)的导数为0，则\(x\)的坐标基矢是killing矢量场。

选取该坐标基矢场\((\frac{\partial }{\partial x})^a\)的适配坐标系，则得

\(\mathscr{L}_{v}g_{ab}=0\)

theorem 3:

\(T\)是测地线的切矢，\(v\)是killing矢量场，有：\(T^a\nabla_a (T^bv_b) = 0\)，也就是说killing场和切矢的内积沿测地线不变

\[\begin{aligned} &T^a\nabla_a (T^bv_b) \\\\ &= v_b T^a\nabla_a T^b +T^b T^a\nabla_a v_b \\\\ &= T^{(b} T^{a)}\nabla_a v_b \\\\ &= T^{(b} T^{a)}\nabla_{[a} v_{b]} \\\\ &= 0 \end{aligned}\]

第一行是莱布尼兹律，第二行来源于测地线定义，第三行是同一个向量不同抽象记号可以加对称记号，第四行是因为killing矢量场\(\nabla_a v_b + \nabla_b v_a = 0 \Rightarrow \nabla_{[a} v_{b]} = \nabla_{a} v_{b}\)

theorem 4:

一个n维流形最多有\(\frac{n(n+1)}{2}\)个killing矢量场。（未证）

theorem 5:

证明二维闵氏空间的boost-killing矢量场与洛伦兹变换对应。

boost矢量场：\(v= t(\frac{\partial}{\partial x})^a +x(\frac{\partial}{\partial t})^a\)

首先证明它在二维闵氏空间是killing矢量场。

先给出一对新坐标，\((\phi,\eta)\)，st, \(x = \eta \cosh \phi, t = \eta \sinh \phi\), 通过坐标变换轻易将\(g_{ab}\)展开为\((d\eta)_a(d\eta)_b - \eta^2 (d\phi)_a(d\phi)_b\)，由theorem2，得到：\(\phi\)的坐标基矢场是killing矢量场。通过坐标变换的关系易得

\[ (\frac{\partial}{\partial \phi})^a= t(\frac{\partial}{\partial x})^a +x(\frac{\partial}{\partial t})^a \]

假设点\(q(t_0,x_0) or (phi_0,\eta_0 )\)，通过boost-killing矢量场的单参微分同胚群元\(\phi_\Delta\)映射到了\(q\)，则由被动观点可知，\(q\)的坐标可以视作\(p\)的新坐标，欲求解q点为起点的积分曲线，由切矢在坐标基矢的展开式表达式：

\[ (\frac{\partial}{\partial t})^a =\frac{d x^\mu}{dt} (\frac{\partial}{\partial x^\mu})^a \]

得到：

\[\begin{aligned} &\frac{d x}{d\phi} = t \enspace \frac{d t}{d\phi} = x \\\\ & x(\phi) = A \cosh \phi + B \sinh \phi \\\\ &t = A \sinh \phi + B \cosh \phi \end{aligned}\]

再带入初始条件得到\(A=x_0\)，\(B=t_0\)

令\(tanh=v\)，则\(sinh= (v^{-2}-1)^{-\frac{1}{2}}\)，\(cosh=(1-v^2)^{-\frac{1}{2}}\)

化简即为：

\[\begin{aligned} x = \frac{1}{\sqrt{1-v^2}} (x_0+vt_0) \\\\ t = \frac{1}{\sqrt{1-v^2}} (t_0 +vx_0) \end{aligned}\]