# 勞侖茲因子

（重定向自洛伦兹因子

## 定義

${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {dt}{d\tau }}}$

${\displaystyle \alpha ={\frac {1}{\gamma }}={\sqrt {1-v^{2}/c^{2}}}\ ,}$

## 例子

${\displaystyle E=\gamma mc^{2}={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$
${\displaystyle \mathbf {p} =\gamma m\mathbf {v} ={\frac {m\mathbf {v} }{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

${\displaystyle \mathbf {p} ^{(4)}=({E \over c},\mathbf {p} )=(\gamma mc,\gamma m\mathbf {v} )=m(\gamma c,\gamma \mathbf {v} )=m\mathbf {v} ^{(4)}}$

## 數值

${\displaystyle \beta =v/c\,\!}$  ${\displaystyle \gamma \,\!}$  ${\displaystyle \alpha \equiv 1/\gamma \,\!}$
0.000 1.000 1.000
0.050 1.001 0.999
0.100 1.005 0.995
0.150 1.011 0.989
0.200 1.021 0.980
0.250 1.033 0.968
0.300 1.048 0.954
0.400 1.091 0.917
0.500 1.155 0.866
0.600 1.250 0.800
0.700 1.400 0.714
0.750 1.512 0.661
0.800 1.667 0.600
0.866 2.000 0.500
0.900 2.294 0.436
0.990 7.089 0.141
0.999 22.366 0.045

## 參考資料

1. ^ Yaakov Friedman, Physical Applications of Homogeneous Balls, Progress in Mathematical Physics 40 Birkhäuser, Boston, 2004, pages 1-21.