How to derive the equations for Time Dilation

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!

 

time-dilation-diagram.png

Figure 1: Light being reflected on the train

 

In this derivation the following should be remembered:

t – time measured on the platform

t0 – time measured on the train

vvelocity 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:

Screen Shot 2019-02-13 at 17.15.21

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)) :

Screen Shot 2019-02-13 at 17.15.27

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

Screen Shot 2019-02-13 at 17.15.34

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

Screen Shot 2019-02-13 at 17.15.46

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

Screen Shot 2019-02-13 at 17.15.53

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

Screen Shot 2019-02-13 at 17.15.58

Divide through by c2 to get the values for t and t0 on their own:

Screen Shot 2019-02-13 at 17.16.04

After some rearranging we get:

Screen Shot 2019-02-13 at 17.16.10

Which is the time dilation equation where Screen Shot 2019-02-13 at 17.16.15.

 

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