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李导数，等效原理，能动张量

李导数

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\(\phi: M \to N\)，切空间中的向量 \(v \in T_p M\) 可以通过 \(\phi\) 映射到 \(T_{\phi(p)} N\) 中，记为 \(\phi_* v \in T_{\phi(p)} N\)，称为 \(v\) 沿着 \(\phi\) 的 push forward（推前操作）。

对 \(f \in C^\infty(N)\)，定义 \(f: N \to \mathbb{R}\)，则 \((\phi_* v)(f) = v(f \circ \phi)\). 而 \(f \circ \phi: M \to \mathbb{R}, f \circ \phi \in C^\infty(M)\).

\[ (\phi_* v)^\alpha = \frac{\partial y^\alpha}{\partial x^\mu} v^\mu \]

对于 \(\alpha \in T^*_q N\)，定义 \(\phi^* \alpha \in T^*_p M\)，称为 \(\alpha\) 沿着 \(\phi\) 的 pull back（拉回操作）。对 \(v \in T_p M\)，有 \((\phi^* \alpha)(v) = \alpha(\phi_* v)\).

\[ (\phi^* \alpha)_\mu = \frac{\partial y^\alpha}{\partial x^\mu} \alpha_\alpha \]

如果 \(\phi\) 是微分同胚 \(\phi \in \mathrm{Diff}(M, N), \, T \in T_q^{(r, s)}(N)\)，则 \(\phi^* T \in T_p^{(r, s)}(M)\) 满足

\[ (\phi^* T)_p (\alpha_1, \ldots, \alpha_r;\; v_1, \ldots, v_s) = T_q \Big( (\phi^{-1})^* \alpha^1, \ldots, (\phi^{-1})^* \alpha^r;\; \phi_* v_1, \ldots, \phi_* v_s \Big) \]

进而

\[ \left.(\phi^* T)_p^{\mu_1, \ldots, \mu_r}\right._{\nu_1, \ldots, \nu_s} = \frac{\partial x^{\mu_1}}{\partial y^{\alpha_1}} \cdots \frac{\partial x^{\mu_r}}{\partial y^{\alpha_r}} \frac{\partial y^{\beta_1}}{\partial x^{\nu_1}} \cdots \frac{\partial y^{\beta_s}}{\partial x^{\nu_s}} \left.T_q^{\alpha_1, \ldots, \alpha_r}\right._{\beta_1, \ldots, \beta_s} \]

含参量的微分同胚：\(t \in \mathbb{R}, \phi_t: M \to M\). 特例与性质：\(\phi_0 = \mathrm{id}_M, \phi_{t_1} \circ \phi_{t_2} = \phi_{t_1 + t_2}\). 因此令 \(\phi_t := \exp(tV)\), \(V\) 是一个向量场，\(\exp(tV)\) 是 \(V\) 的流（flow）。定义 \(T_p \in T_p^{(r, s)}(M)\) 沿着 \(V\) 的 Lie derivative（李导数）为

\[ \begin{equation} \mathscr{L}_V T_p = \lim_{t \to 0} \frac{\Big(\phi_{-t}^* T_{\phi_t(p)}\Big) - T_p}{t} \end{equation} \]

举点具体例子：\(\mathscr{L}_V X\)（\(X\) 是一个向量场）和 \(\mathscr{L}_V \omega\)（\(\omega\) 是一个 1-形式）。

\(\phi_t: M \to M\)，\(\{x^\mu\}\) 是 \(p\) 点的坐标，\(\{y^\mu \}\) 是 \(q = \phi_t(p)\) 点的坐标。

弱等效原理（WEP）

物体的惯性质量 \(m_I\) 和引力质量 \(m_G\) 是相等的。也就是说，物体的加速度与其组成无关。

\[ m_I \vec{a} = \vec{F} = m_G \vec{g} \implies \vec{a} = \frac{m_G}{m_I} \vec{g} \]

\(m_{I} = m_{G} = m\).

推广：爱因斯坦等效原理（EEP）

自由落体坐标系 \(\{\xi^\alpha\} \overset{\phi}{\longrightarrow} \{x^\mu \}\) 无重力场

自由落体：时空中的一条直线

\[ \frac{\mathrm{d}^2 \xi^\alpha}{\mathrm{d} \tau^2} = 0, \, \mathrm{d} \tau^2 = - \eta_{\alpha \beta} \,\mathrm{d} \xi^\alpha \mathrm{d} \xi^\beta \]

也就是

\[ \begin{aligned} 0 &= \frac{\mathrm{d}}{\mathrm{d} \tau} \left( \frac{\mathrm{d} \xi^\alpha}{\mathrm{d} \tau} \right) = \frac{\mathrm{d}}{\mathrm{d} \tau} \left( \frac{\partial \xi^\alpha}{\partial x^\mu} \frac{\mathrm{d} x^\mu}{\mathrm{d} \tau} \right) \\ &= \frac{\partial \xi^\alpha}{\partial x^\mu} \frac{\mathrm{d}^2 x^\mu}{\mathrm{d} \tau^2} + \frac{\partial^2 \xi^\alpha}{\partial x^\mu \partial x^\nu} \frac{\mathrm{d} x^\mu}{\mathrm{d} \tau} \frac{\mathrm{d} x^\nu}{\mathrm{d} \tau} \end{aligned} \]

这正是测地线方程

\[ \frac{\mathrm{d}^2 x^\mu}{\mathrm{d} \tau^2} + \underset{\Gamma^\lambda_{\mu \nu}}{\underbrace{\frac{\partial x^\lambda}{\partial \xi^\alpha} \frac{\partial^2 \xi^\alpha}{\partial x^\mu \partial x^\nu}}} \frac{\mathrm{d} x^\mu}{\mathrm{d} \tau} \frac{\mathrm{d} x^\nu}{\mathrm{d} \tau} = 0 \]

❓ 李导数什么时候会用到？考虑对称性的时候。

能动张量

\[ S = \int \mathrm{d}^4 x \, \mathcal{L}(\Phi, \partial_\mu \Phi), \quad x^\nu \to x^\nu + a^\nu \]

\(a\) 是一个固定的常数，但在广义相对论中最好看成一个矢量场 \(a^\nu (x)\)，则 \(V = a^\nu (x) \partial_\nu\)

Noether theorem：守恒流

\[ \left.T^{(N) \mu}\right._\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \Phi)} \partial_\nu \Phi - \left.\delta^\mu\right._\nu \mathcal{L}, \quad \mathcal{L} = -\frac{1}{4} F_{\alpha \beta} F^{\alpha \beta} \]
  • \((1,1)\) tensor. \(\partial_\mu \left.T^\mu \right._\nu = 0\)

  • 不对称：\(\left.T^{(N)}\right._{\mu \nu} = \eta_{\mu \rho} \left.T^{\rho}\right._\nu \overset{?}{=} \left.T^{(N)}\right._{\nu \mu}\)

    \[ \left.T^{(N)}\right._{\mu \nu} = - F_{\mu \rho} \partial_\nu A^\rho + \frac{1}{4} \eta_{\mu \nu} F_{\alpha \beta} F^{\alpha \beta} \]

    从上式看出关于 \(\mu, \nu\) 不是对称的。Belinfante-Rosenfeld tensor：\(\left.T^{(B)}\right._{\mu \nu} = \left.T^{(N)}\right._{\mu \nu} + \partial_\rho X^\rho_{\;\;\mu \nu}\)，其中 \(X^\rho_{\;\;\mu \nu} = - X^\rho_{\;\;\nu \mu}\).

\[ S = \int \mathrm{d}^4 x \, \sqrt{-g} \, \mathcal{L}, \quad g = \det(g_{\mu \nu}) \]