This is the maths behind time dilation.

Feel free to check out our **time dilation** notes first. When an object travels at relativistic speeds (i.e. close to the speed of light), for each observer in a specific inertial frame, the time at which the event will occur will be different – though remember one is not more correct than the other!

**Figure 1**: Light being reflected on the train

In this derivation the following should be remembered:

*∆*** t** – time measured on the platform

*∆*** t_{0}** – time measured on the train

*v** – *velocity of train

*l **– *distance light has travelled according to passenger watching on the platform

*d**– *distance light has travelled according to passenger in the train carriage

**c – **speed of light

These can all be expressed as:

Using simple Pythagoras ** l **on the diagram can be worked out, in order to find the distance the light has travelled according to the passenger watching this experiment on the platform (lets call this (1)) :

Then we can work the distance travelled by the pulse of light according to the passenger in the actual train (lets call this (2)) :

Next, If we substitute (2)* * into (1)*, * it we get:

Rearrange by multiplying both sides of the equation by *c, *dividing by 2 and squaring to get:

Then the 2’s cancel out on both sides of the equation:

Divide through by *c ^{2 }*to get the values for

*∆*

*t and*

*∆*

*t*on their own:

_{0}After some rearranging we get:

Which is the time dilation equation where .

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