# 基灵矢量场

## 数学定义

$\mathcal{L}_{X} g = 0 \,.$

$g(\nabla_{Y} X, Z) + g(Y, \nabla_{Z} X) = 0 \,$

$\nabla_{\mu} X_{\nu} + \nabla_{\nu} X_{\mu} = 0 \,.$

• 里奇曲率意味着不存在非平凡基灵场。
• 非正里奇曲率，意味着任何基灵场都是平行的，即沿着任何向量场的共变导数恒为零。
• 如果截面曲率为正且M维数为偶，一个基灵场一定有零点。

$\mathcal{L}_{X} g = \lambda g \,$

## 广义时空几何中的对称性和守恒律

$ds^2 = \eta_{\mu\nu}dx^{\mu}dx^{\nu}\,$

$x^{\nu} \to x^{\nu} + a^{\nu}\,$平移对称性
$x^{\nu} \to \Lambda^{\nu}_{\mu}x^{\nu}\,$洛伦兹对称性

$\partial_{\sigma}g_{\mu\nu} = 0 \qquad \Rightarrow \qquad x^{\sigma} \to x^{\sigma} + a^{\sigma}\,$

### 平移对称性和动量守恒

$p^{\lambda}\nabla_{\lambda}p^{\mu} = 0$

$p^{\lambda}\partial_{\lambda}p_{\mu} - \Gamma^{\sigma}_{\lambda \mu}p^{\lambda}p_{\sigma} = 0\,$

$p^{\lambda}\partial_{\lambda}p_{\mu} = m\frac{dx^{\lambda}}{d\tau}\partial_{\lambda}p_{\mu} = m\frac{dp_{\mu}}{d\tau}\,$

\begin{align} \Gamma^{\sigma}_{\lambda \mu}p^{\lambda}p_{\sigma} & = \frac{1}{2}g^{\sigma\nu} \left( \partial_{\lambda}g_{\mu\nu} + \partial_{\mu}g_{\nu\lambda} - \partial_{\nu}g_{\lambda\mu} \right) p^{\lambda}p_{\sigma}\\ & = \frac{1}{2}\left( \partial_{\lambda}g_{\mu\nu} + \partial_{\mu}g_{\nu\lambda} - \partial_{\nu}g_{\lambda\mu} \right) p^{\lambda}p^{\nu}\\ & = \frac{1}{2}\left( \partial_{\mu}g_{\nu\lambda} \right) p^{\lambda}p^{\nu} \end{align}

$m\frac{dp_{\mu}}{d\tau} = \frac{1}{2}\left( \partial_{\mu}g_{\nu\lambda} \right) p^{\lambda}p^{\nu}\,$

$\partial_{\sigma}g_{\mu\nu} = 0 \qquad \Rightarrow \qquad \frac{dp_{\sigma}}{d\tau} = 0 \,$

## 基灵矢量

$\boldsymbol{K} = \partial_{\sigma}\,$

${K}^{\mu} = \left( \partial_{\sigma} \right)^{\mu} = \delta^{\mu}_{\sigma}\,$

$p_{\sigma} = {K}^{\nu} p_{\nu}\,$

$\frac{dp_{\sigma}}{d\tau} = 0 \qquad \Leftrightarrow \qquad p^{\mu}\nabla_{\mu} \left({K}_{\nu} p^{\nu} \right) = 0 \,$

\begin{align} p^{\mu}\nabla_{\mu} \left({K}_{\nu} p^{\nu} \right) & = p^{\mu}\nabla_{\mu} {K}_{\nu} p^{\nu} + p^{\mu}p^{\nu}\nabla_{\mu}K_{\nu}\\ & = p^{\mu}p^{\nu}\nabla_{\mu}K_{\nu} \\ & = p^{\mu}p^{\nu}\nabla_{( \mu}K_{\nu)} \end{align}

$\nabla_{( \mu}K_{\nu)} = 0 \qquad \Rightarrow \qquad p^{\mu}\nabla_{\mu} \left({K}_{\nu} p^{\nu} \right) = 0\,$

$\nabla_{( \mu}K_{\nu_1 \nu_2 ... \nu_l)} = 0\,$

$l\,$阶张量$K_{\nu_1 \nu_2 ... \nu_l}\,$对应有守恒量${K}_{\nu_1 \nu_2 ... \nu_l} p^{\nu_1 \nu_2 ... \nu_l}\,$

$p^{\mu}\nabla_{\mu} \left({K}_{\nu_1 \nu_2 ... \nu_l} p^{\nu_1 \nu_2 ... \nu_l} \right) = 0\,$

### 性质

$\nabla_{\mu}\nabla_{\sigma}K^{\rho} = R^{\rho}_{\sigma\mu\nu}K^{\nu}\,$

$\nabla_{\mu}\nabla_{\sigma}K^{\mu} = R_{\sigma\nu}K^{\nu}\,$

$K^{\lambda}\nabla_{\lambda}R = 0\,$

### 类时的基灵矢量

$J^{\mu} = K_{\nu}T^{\mu\nu}\,$

$\nabla_{\mu}J^{\mu} = \left( \nabla_{\mu}K_{\nu} \right) T^{\mu\nu} + K_{\nu} \left( \nabla_{\mu} T^{\mu\nu} \right) = 0\,$

$K_{\nu}\,$是一个类时的基灵矢量时，可以通过对这个守恒流在整个类空超平面$\Sigma\,$内积分从而定义时空中的总能量：

$E = \int_{\Sigma} J^{\mu}n_{\mu}\sqrt{\gamma}\,d^3x\,$

## 参考资料

• Sean M. Carroll. Spacetime and Geometry: An Introduction to General Relativity (Hardcover). Benjamin Cummings. 2003. ISBN 978-0805387322 （英文）.
• Jost, Jurgen. Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag. 2002. ISBN 3-540-42627-2 （英文）..
• Adler, Ronald; Bazin, Maurice & Schiffer, Menahem. Introduction to General Relativity (Second Edition). New York: McGraw-Hill. 1975. ISBN 0-07-000423-4 （英文）. 见第三章和第九章
• Misner, Thorne, Wheeler. Gravitation. W H Freeman and Company. 1973. ISBN 0-7167-0344-0 （英文）.