# 應力-能量張量

（重定向自能量-動量張量

## 定義

${\displaystyle T^{ab}=T^{ba}\,}$

${\displaystyle \partial _{\alpha }S^{\mu \nu \alpha }=T^{\mu \nu }-T^{\nu \mu }}$

## 例子

${\displaystyle T^{00}}$

${\displaystyle T^{0i}}$

${\displaystyle T^{i0},}$

i 動量之密度。

${\displaystyle T^{ij}}$

${\displaystyle T^{ii}}$

${\displaystyle T^{ij},\quad i\neq j}$

## 作為諾特流(Noether current)

${\displaystyle \nabla _{b}T^{ab}=T^{ab}{}_{;b}=0}$ .

${\displaystyle \int d^{3}xT^{a0}}$

${\displaystyle \nabla _{b}T^{0b}=\nabla \cdot \mathbf {p} -{\frac {\partial E}{\partial t}}=0}$

## 於廣義相對論中

${\displaystyle T^{ab}=T^{ba}}$

### 愛因斯坦場方程式

${\displaystyle R_{\alpha \beta }-{1 \over 2}R\,g_{\alpha \beta }={8\pi G \over c^{4}}T_{\alpha \beta },}$

## 特殊情况下的应力-能量张量

### 孤立粒子

${\displaystyle T^{\alpha \beta }[t,x,y,z]={\frac {m\,v^{\alpha }[t]v^{\beta }[t]}{\sqrt {1-(v/c)^{2}}}}\delta (x-x[t])\delta (y-y[t])\delta (z-z[t])}$

${\displaystyle {\begin{pmatrix}v^{0}[t]\\v^{1}[t]\\v^{2}[t]\\v^{3}[t]\end{pmatrix}}={\begin{pmatrix}1\\{dx[t] \over dt}\\{dy[t] \over dt}\\{dz[t] \over dt}\end{pmatrix}}.}$

### 处于平衡状态下的流体的应力-能量张量

${\displaystyle T^{\alpha \beta }\,=(\rho +{p \over c^{2}})u^{\alpha }u^{\beta }+pg^{\alpha \beta }}$

${\displaystyle u^{\alpha }u^{\beta }g_{\alpha \beta }=-c^{2}\,.}$

${\displaystyle u^{\alpha }=(1,0,0,0)\,,}$

${\displaystyle g^{\alpha \beta }\,=\left({\begin{matrix}-c^{-2}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\,,}$

${\displaystyle T^{\alpha \beta }=\left({\begin{matrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right).}$

### 电磁应力-能量张量

${\displaystyle T^{\mu \nu }(x)={\frac {1}{\mu _{0}}}\left(F^{\mu \alpha }g_{\alpha \beta }F^{\nu \beta }-{\frac {1}{4}}g^{\mu \nu }F_{\delta \gamma }F^{\delta \gamma }\right)}$

### 标量场

${\displaystyle T^{\mu \nu }={\frac {\hbar ^{2}}{m}}(g^{\mu \alpha }g^{\nu \beta }+g^{\mu \beta }g^{\nu \alpha }-g^{\mu \nu }g^{\alpha \beta })\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi -g^{\mu \nu }mc^{2}{\bar {\phi }}\phi .}$

## 各式各樣的應力-能量張量

### 希爾伯特應力-能量張量

${\displaystyle T^{\mu \nu }(x)={\frac {2}{\sqrt {-g}}}{\frac {\delta {\mathcal {S}}_{\mathrm {matter} }}{\delta g_{\mu \nu }(x)}}}$