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
∆t0 – 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 c2 to get the values for ∆t and ∆t0 on their own:
After some rearranging we get:
Which is the time dilation equation where .