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Formula: **Time Dilation**

$$\Delta t' ~=~ \frac{1}{ \sqrt{1 ~-~ \frac{\class{blue}{v}^2}{c^2}} } \, \Delta t$$
$$\Delta t' ~=~ \frac{1}{ \sqrt{1 ~-~ \frac{\class{blue}{v}^2}{c^2}} } \, \Delta t$$
$$\Delta t ~=~ \sqrt{1 ~-~ \frac{\class{blue}{v}^2}{c^2}} \, \Delta t'$$
$$\class{blue}{v} ~=~ c \, \sqrt{1 ~-~ \left(\frac{\Delta t}{\Delta t'}\right)^2}$$
$$c ~=~ v \, \frac{1}{ \sqrt{1 ~-~ \left(\frac{\Delta t}{\Delta t'}\right)^2} }$$

## Time

`$$ \Delta t' $$`Unit

`$$ \mathrm{s} $$`

This is the time interval between two events that passes on the moving clock from the point of view of an observer at rest. Since the gamma factor \(\gamma\) is greater than 1:

`$$ \gamma ~=~ \frac{1}{ \sqrt{1 ~-~ \frac{v^2}{c^2}} } > 1 $$`\(\Delta t'\) is greater than \(\Delta t\). Consequently, a stationary observer sees that more time \(\Delta t'\) has elapsed on the moving clock than on his stationary clock, \(\Delta t\).If the moving clock moves with velocity \( v = 2 \cdot 10^8 \, \frac{\text m}{\text s} \) relative to the stationary observer, the gamma factor is:
`
\[ \gamma ~=~ \frac{1}{ \sqrt{1 ~-~ \frac{(2 \cdot 10^8 \, \frac{\text m}{\text s})^2}{(2 \cdot 10^8 \, \frac{\text m}{\text s})^2}} } ~=~ 1.7 \]
`
Now, if time \(\Delta t = 1 \, \text{s}\) has passed on the clock of an observer at rest, more time has passed on the clock in motion:
`
\[ \Delta t' ~=~ \gamma \, \Delta t ~=~ 1.7 \,\text{s} \]
`

## Proper time

`$$ \Delta t $$`Unit

`$$ \mathrm{s} $$`

The proper time is the time interval between two events measured by a stationary observer on his stationary (unmoving) clock. Since \(\Delta t\) is smaller than \(\Delta t'\), less time passes on the clock at rest than on the clock moving relative to it.

## Relative velocity

`$$ \class{blue}{v} $$`Unit

`$$ \frac{\mathrm m}{\mathrm s} $$`

This is the velocity of the moving clock, from the point of view of an observer at rest. The relative velocity \(v\) is always smaller than the speed of light \(c\).

The greater the velocity \(v\) of the moving clock, the greater the amount of time \(\Delta t'\) that passes on the moving clock, as seen by an observer at rest.

## Speed of light

`$$ c $$`Unit

`$$ \frac{\mathrm m}{\mathrm s} $$`

Speed of light is a physical constant and indicates how fast light travels in empty space (vacuum). It has the following exact value in vacuum:

`$$ c ~=~ 299 \, 792 \, 458 \, \frac{ \mathrm{m} }{ \mathrm{s} } $$`