# 愛因斯坦場方程式

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

## 愛因斯坦場方程式的性質

### 能量與動量守恆

$\nabla_\nu T^{\mu \nu}= T^{\mu \nu}{}_{;\nu}=0$

$\nabla_\nu G^{\mu \nu}=G^{\mu \nu}{}_{;\nu}=0$

## 添加宇宙常數項

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

$\nabla_\nu (\Lambda g_{\mu \nu})= \Lambda \nabla_\nu (g_{\mu \nu}) = 0$

1. 此一理論所描述的靜態宇宙是不穩定的。
2. 十年後，由愛德溫·哈伯對於遠處星系所作觀測的結果證實我們的宇宙正在膨脹，而非靜態。

$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R = {8 \pi G \over c^4} \left( T_{\mu\nu} - \frac{c^4\Lambda g_{\mu\nu}}{8 \pi G} \right)$

$T_{\mu \nu}^{\mathrm{(vac)}} \equiv -\frac{c^4\Lambda g_{\mu\nu}}{8 \pi G}$

$\rho_{\mathrm{vac}} \equiv \frac{c^2\Lambda}{8 \pi G}$

## 真空場方程式

### 宇宙常數為零

$R_{\mu \nu} = {1 \over 2} g_{\mu \nu} R\$

$R \equiv R^\mu_\mu = g^{\mu \nu} R_{\mu \nu} = g^{\mu \nu} {1 \over 2} g_{\mu \nu} R$

$R = \delta^\mu_\mu {1 \over 2}R$

$R = 4 \cdot {1 \over 2} R = 2R$

$R_{\mu \nu} = 0\$

### 宇宙常數不為零

$R_{\mu \nu} = {1 \over 2} g_{\mu \nu} R - \Lambda g_{\mu \nu},\$

$R_{\mu \nu} = \Lambda g_{\mu \nu}\$

## 參考文獻

1. ^ Gamow, George. My World Line : An Informal Autobiography. Viking Adult date = April 28, 1970. [3-14-2007]. ISBN-10: 0670503762.
2. ^ Wahl, Nicolle. Was Einstein's 'biggest blunder' a stellar success?. November 22, 2005 [2007-03-14].
3. ^ Turner, Michael S. A Spacetime Odyssey. Int.J.Mod.Phys. A17S1. May 2001: 180–196 [2007-03-14].