# 基灵矢量场

## 数学定义

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

${\displaystyle g(\nabla _{Y}X,Z)+g(Y,\nabla _{Z}X)=0\,}$

${\displaystyle \nabla _{\mu }X_{\nu }+\nabla _{\nu }X_{\mu }=0\,.}$

• 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。

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

## 广义时空几何中的对称性和守恒律

${\displaystyle ds^{2}=\eta _{\mu \nu }dx^{\mu }dx^{\nu }\,}$

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

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

### 平移对称性和动量守恒

${\displaystyle p^{\lambda }\nabla _{\lambda }p^{\mu }=0}$

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

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

{\displaystyle {\begin{aligned}\Gamma _{\lambda \mu }^{\sigma }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{aligned}}}

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

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

## 基灵矢量

${\displaystyle {\boldsymbol {K}}=\partial _{\sigma }\,}$

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

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

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

{\displaystyle {\begin{aligned}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{aligned}}}

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

${\displaystyle \nabla _{(\mu }K_{\nu _{1}\nu _{2}...\nu _{l})}=0\,}$

${\displaystyle l\,}$ 阶张量${\displaystyle K_{\nu _{1}\nu _{2}...\nu _{l}}\,}$ 对应有守恒量${\displaystyle {K}_{\nu _{1}\nu _{2}...\nu _{l}}p^{\nu _{1}\nu _{2}...\nu _{l}}\,}$

${\displaystyle p^{\mu }\nabla _{\mu }\left({K}_{\nu _{1}\nu _{2}...\nu _{l}}p^{\nu _{1}\nu _{2}...\nu _{l}}\right)=0\,}$

### 性质

${\displaystyle \nabla _{\mu }\nabla _{\sigma }K^{\rho }=R_{\sigma \mu \nu }^{\rho }K^{\nu }\,}$

${\displaystyle \nabla _{\mu }\nabla _{\sigma }K^{\mu }=R_{\sigma \nu }K^{\nu }\,}$

${\displaystyle K^{\lambda }\nabla _{\lambda }R=0\,}$

### 类时的基灵矢量

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

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

${\displaystyle K_{\nu }\,}$ 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${\displaystyle \Sigma \,}$ 内积分从而定义时空中的总能量：

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

## 参考资料

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