Equation

The equation for time dilation is as follows:

Sometimes the equation is simplified to:

The time dilation effect becomes important at speeds above 0.1 times the speed of light. Below that speed, the dilated time is nearly the same as the proper time.

Understanding

It is important to realise that an observer won’t see time slowing down in their own frame of reference, they will see time slowing down in a moving frame of reference. Imagine an observer A on earth and observer B moving at 0.8 times the speed of light (ie. 0.8c) on a spaceship. The frame of reference for observer A can be considered to be earth and the frame of reference for observer B can be considered as the spaceship. In the frame of reference of observer A, they would see a clock on earth ticking at its ‘normal pace’, but they would notice a clock on the spaceship ticking slowly. In the frame of reference of observer B, they would notice their clock ticking ‘normally’ whilst the clock on earth would be ticking slow. A general rule of thumb to remember is a moving clock runs slower. This may sound confusing and ‘unreal’ at first but that’s whole point of relativity. There is no absolute frame of reference and hence no absolute clock ticking at the absolute ‘correct’ time. The time in your own inertial frame of reference ticks away at one second per second.

Time dilation effect can be used to explain why muons are detected on Earth’s surface even though they have a short lifetime. When muons are produced in the upper atmosphere, they travel at high speeds. Hence in the frame of reference of an observer on Earth, the muons’ lifetime is dilated enough they end up at the Earth’s surface. In the frame of reference of the muons, there is a different relativistic effect taking place; length contraction, which will be explained separately in the next section.

Twin paradox | Elucidate Video

In the above video, Dev explains a very common example (the twin paradox) which is used demonstrate the time dilation phenomenon.