# 原時

## 數學的形式

### 在狹義相對論

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

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

$\tau = \int_P \sqrt {dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2}$,

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

## 在狹義相對論中的例子

### 例一：雙生子佯謬

$\Delta \tau_{A} = \sqrt{\Delta t_{A}^2} = 10\text{yr}$

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

$\Delta \tau = \sqrt{\Delta t^2-\frac{v_x^2}{c^2} \Delta t^2-\frac{v_y^2}{c^2} \Delta t^2- \frac{v_z^2}{c^2} \Delta t^2} = \Delta t \sqrt{1-\frac{v^2}{c^2}}$

## 廣義相對論的例子

### 例四：史瓦西解–地球上的時間

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