对称性与守恒律

Symmetry and conservation law

\(\phi \in \mathrm{Diff}(M), \, \phi_* T = T\)，由矢量场 \(K = K^\mu \partial_\mu\) 诱导，则 \(T\) 的李导数

\[ \mathscr{L}_K T = 0 \]

若 \(T\) 为度规场 \(T = g_{\mu \nu}\)，则

\[ \mathscr{L}_K g_{\mu \nu} = 0 \implies \nabla_\mu K_\nu + \nabla_\nu K_\mu = 0 \]

此即 Killing's equation，满足上述方程的 \(K\) 被称为 Killing vector，\(\phi\) 为 \(K\) 生成的 isometry.

\(x^\mu(\lambda)\)：geodesic，\(u^\mu = \frac{d x^\mu}{d \lambda}\)：四维速度，则

\[ \begin{aligned} u^\nu \nabla_\nu (K_\mu u^\mu) &= u^\nu (\nabla_\nu K_\mu) u^\mu + u^\nu K_\mu (\nabla_\nu u^\mu) \\ &= u^{(\nu} u^{\mu)} \nabla_{(\nu} K_{\mu)} + u^\nu K_\mu (\partial_\nu u^\mu + \Gamma^\mu_{\nu \sigma} u^\sigma) = 0 \implies K_\mu u^\mu = \text{const.} \end{aligned} \]

在 \(\mathbb{R}^n\) 中

\[ \begin{aligned} K_\mu &= C_\mu \implies K = K^\mu \partial_\mu = C^\mu \partial_\mu = \mathrm{span} \{ \partial_\mu \} \to P \\ K_\mu &= \omega_{\mu \nu} x^\nu, \, \omega_{\mu \nu} + \omega_{\nu \mu} = 0 \implies K \to L_{\mu \nu} \end{aligned} \]

Theorem

\(n\) 维的 \(M\)，最多有 \(\frac{n(n+1)}{2}\) 个线性无关的 Killing vector，即 \(\mathrm{dim} \, K \leq \frac{n(n+1)}{2}.\)

Spherically symmetric metric

\(\mathbb{R}^3\) 中的球对称

\[ d s^2 = d r^2 + r^2 d \Omega^2_2 \]