# 爱因斯坦-希尔伯特作用量

$S[g]= \int {1 \over 2\kappa} R \sqrt{-g} \, \mathrm{d}^4x$

$S[g]= \int {1 \over 2\kappa} R \, \mathrm{dV} \,$

## 导出爱因斯坦引力场方程

$S = \int \left[ {1 \over 2\kappa} \, R + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x$

\begin{align} 0 & = \delta S \\ & = \int \left[ {1 \over 2\kappa} \frac{\delta (\sqrt{-g}R)}{\delta g^{\mu\nu}} + \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}} \right] \delta g^{\mu\nu}\mathrm{d}^4x \\ & = \int \left[ {1 \over 2\kappa} \left( \frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } \right) + \frac{1}{\sqrt{-g}} \frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}} \right] \delta g^{\mu\nu} \sqrt{-g}\, \mathrm{d}^4x. \end{align}

$\frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}} \frac{\delta \sqrt{-g}}{\delta g^{\mu\nu}} = - 2 \kappa \frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}},$

$T_{\mu\nu}:= -2 \frac{1}{\sqrt{-g}}\frac{\delta (\sqrt{-g} \mathcal{L}_\mathrm{M})}{\delta g^{\mu\nu}} = -2 \frac{\delta \mathcal{L}_\mathrm{M}}{\delta g^{\mu\nu}} + g_{\mu\nu} \mathcal{L}_\mathrm{M}.$

### 黎曼张量、里奇张量和里奇标量的变分

${R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},$

$\delta{R^\rho}_{\sigma\mu\nu} = \partial_\mu\delta\Gamma^\rho_{\nu\sigma} - \partial_\nu\delta\Gamma^\rho_{\mu\sigma} + \delta\Gamma^\rho_{\mu\lambda} \Gamma^\lambda_{\nu\sigma} + \Gamma^\rho_{\mu\lambda} \delta\Gamma^\lambda_{\nu\sigma} - \delta\Gamma^\rho_{\nu\lambda} \Gamma^\lambda_{\mu\sigma} - \Gamma^\rho_{\nu\lambda} \delta\Gamma^\lambda_{\mu\sigma}.$

$\nabla_\lambda (\delta \Gamma^\rho_{\nu\mu} ) = \partial_\lambda (\delta \Gamma^\rho_{\nu\mu} ) + \Gamma^\rho_{\sigma\lambda} \delta\Gamma^\sigma_{\nu\mu} - \Gamma^\sigma_{\nu\lambda} \delta \Gamma^\rho_{\sigma\mu} - \Gamma^\sigma_{\mu\lambda} \delta \Gamma^\rho_{\nu\sigma}$

$\delta R^\rho{}_{\sigma\mu\nu} = \nabla_\mu (\delta \Gamma^\rho_{\nu\sigma}) - \nabla_\nu (\delta \Gamma^\rho_{\mu\sigma}).$

$\delta R_{\mu\nu} \equiv \delta R^\rho{}_{\mu\rho\nu} = \nabla_\rho (\delta \Gamma^\rho_{\nu\mu}) - \nabla_\nu (\delta \Gamma^\rho_{\rho\mu}).$

$R = g^{\mu\nu} R_{\mu\nu}.\!$

\begin{align} \delta R &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\\ &= R_{\mu\nu} \delta g^{\mu\nu} + \nabla_\sigma \left( g^{\mu\nu} \delta\Gamma^\sigma_{\nu\mu} - g^{\mu\sigma}\delta\Gamma^\rho_{\rho\mu} \right) \end{align}

$\frac{\delta R}{\delta g^{\mu\nu}} = R_{\mu\nu}$

### 度规行列式的变分

$\,\! \delta g = g \, g^{\mu\nu} \delta g_{\mu\nu}$

\begin{align} \delta \sqrt{-g} &= -\frac{1}{2\sqrt{-g}}\delta g &= \frac{1}{2} \sqrt{-g} (g^{\mu\nu} \delta g_{\mu\nu}) &= -\frac{1}{2} \sqrt{-g} (g_{\mu\nu} \delta g^{\mu\nu}),\end{align}

$\frac{1}{\sqrt{-g}}\frac{\delta \sqrt{-g}}{\delta g^{\mu\nu} } = -\frac{1}{2} g_{\mu\nu}$

### 运动方程

$R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R = \frac{8 \pi G}{c^4} T_{\mu\nu},$

## 宇宙学常数

$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} \,$

$S = \int \left[ {1 \over 2\kappa} \left( R - 2 \Lambda \right) + \mathcal{L}_\mathrm{M} \right] \sqrt{-g} \, \mathrm{d}^4 x$