# 原時

## 數學形式

### 狹義相對論

${\displaystyle \tau =\int {\sqrt {1-{\frac {v(t)^{2}}{c^{2}}}}}dt=\int {\sqrt {1-{\frac {1}{c^{2}}}\left(\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}\right)}}dt}$,

${\displaystyle \tau =\int {\sqrt {\left({\frac {dt}{d\lambda }}\right)^{2}-{\frac {1}{c^{2}}}\left(\left({\frac {dx}{d\lambda }}\right)^{2}+\left({\frac {dy}{d\lambda }}\right)^{2}+\left({\frac {dz}{d\lambda }}\right)^{2}\right)}}d\lambda }$.

${\displaystyle \tau =\int _{P}{\sqrt {dt^{2}-dx^{2}/c^{2}-dy^{2}/c^{2}-dz^{2}/c^{2}}}}$,

${\displaystyle \Delta \tau ={\sqrt {\Delta t^{2}-\Delta x^{2}/c^{2}-\Delta y^{2}/c^{2}-\Delta z^{2}/c^{2}}}}$,

## 狹義相對論中的例子

### 例一：雙生子佯謬

${\displaystyle \Delta \tau _{A}={\sqrt {\Delta t_{A}^{2}}}=10{\text{yr}}}$

${\displaystyle \Delta \tau _{B}={\sqrt {\Delta t_{B}^{2}-\Delta x_{B}^{2}}}=3{\text{yr}}}$

${\displaystyle \Delta \tau ={\sqrt {\Delta t^{2}-{\frac {v_{x}^{2}}{c^{2}}}\Delta t^{2}-{\frac {v_{y}^{2}}{c^{2}}}\Delta t^{2}-{\frac {v_{z}^{2}}{c^{2}}}\Delta t^{2}}}=\Delta t{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

## 廣義相對論中的例子

### 例四：史瓦西解–地球上的時間

${\displaystyle d\tau ={\sqrt {\left(1-2m/r\right)dt^{2}-{\frac {1}{c^{2}}}\left(1-2m/r\right)^{-1}dr^{2}-{\frac {r^{2}}{c^{2}}}d\theta ^{2}-{\frac {r^{2}}{c^{2}}}\sin ^{2}\theta \;d\phi ^{2}}}}$,