# 指标的上升和下降

${\displaystyle g^{ij}A_{j}=A^{i},}$

${\displaystyle g_{ij}A^{j}=A_{i},}$

${\displaystyle g^{ij}g_{ji}=g_{ij}g^{ji}=g_{i}^{i}=Trg=N.}$

## 广义相对论中的例子

${\displaystyle g_{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}$

${\displaystyle F^{\mu \nu }={\begin{bmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$

${\displaystyle F_{\mu \nu }=g_{\mu \kappa }g_{\nu \lambda }F^{\kappa \lambda }\,}$

${\displaystyle F_{\mu \nu }=g_{\mu \mu }g_{\nu \nu }F^{\mu \nu }\,}$

${\displaystyle F_{ij}=g_{ii}g_{jj}F^{ij}=F^{ij}\,}$

${\displaystyle F_{ii}=(g_{ii})^{2}F^{ii}=F^{ii}\,}$
${\displaystyle F_{0i}=g_{00}g_{ii}F^{0i}=-F^{0i}\,}$

${\displaystyle F_{i0}=-F^{i0}\,}$

${\displaystyle F_{\mu \nu }={\begin{bmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{bmatrix}}}$

## 参考文献

1. ^ Griffiths, David J. Introduction to Elementary Particles. Wiley, John & Sons, Inc. 1987. ISBN 0-471-60386-4.