# 四維力

## 狹義相對論的定義

${\displaystyle \mathbf {P} =m\mathbf {U} }$

${\displaystyle \mathbf {F} \equiv {d\mathbf {P} \over d\tau }}$

${\displaystyle \mathbf {F} =m\mathbf {A} =\left(\gamma {\mathbf {f} \cdot \mathbf {u} \over c},\gamma {\mathbf {f} }\right)}$.

${\displaystyle {\mathbf {f} }={d \over dt}\left(\gamma m{\mathbf {u} }\right)={d\mathbf {p} \over dt}}$

${\displaystyle {\mathbf {f} \cdot \mathbf {u} }={d \over dt}\left(\gamma mc^{2}\right)={dE \over dt}}$

## 廣義相對論的調整

${\displaystyle F^{\lambda }:={\frac {DP^{\lambda }}{d\tau }}={\frac {dP^{\lambda }}{d\tau }}+\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }P^{\nu }}$

${\displaystyle f^{0}=\gamma {\boldsymbol {\beta }}\cdot \mathbf {F} ={\boldsymbol {\beta }}\cdot \mathbf {f} }$

${\displaystyle f^{\mu }=m{DU^{\mu } \over d\tau }}$

${\displaystyle f_{f}^{\alpha }}$自由落體參考系${\displaystyle \xi ^{\alpha }}$之中力的正確表示式，我們可以使用等效原理來描寫任意座標系${\displaystyle x^{\mu }}$之中的四維力：

${\displaystyle f^{\mu }={\partial x^{\mu } \over \partial \xi ^{\alpha }}f_{f}^{\alpha }.}$

## 案例

${\displaystyle F_{\mu }=qF_{\mu \nu }U^{\nu }}$

• ${\displaystyle F_{\mu \nu }}$電磁張量
• ${\displaystyle q}$為電荷。
• ${\displaystyle U^{\nu }}$四維速度

## 參考文獻

1. ^ Steven, Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, Inc. 1972. ISBN 0-471-92567-5.

### 延伸閱讀

• Rindler, Wolfgang. Introduction to Special Relativity 2nd. Oxford: Oxford University Press. 1991. ISBN 0-19-853953-3.