Simple derivation of Lorentz contraction (time dilation formula) in special relativity
The general time dilation formula in special relativity can be proven with high school algebra. Here’s how:
Consider an object moving at a speed v relative to an observer. Define a plane passing through the observer parallel to the object’s motion, and assume there is a mirror along that plane. Then we can define two time intervals t and s as follows:
t: the time in the object’s coordinates for light to a make a one way trip to the mirror
s: the same time, but in the observer’s coordinates.
In the time s, in the observer coordinates, the object moves a distance vs. In the object’s own coordinates, the distance to the mirror is ct.
by right angle geometry
which is the Lorentz / Einstein formula. In other words, the object’s time is slowed by a factor of over time in the observer coordinate system.
To get a sense of the magnitude of , suppose you are going at half the speed of light.
Which means time slows down by 15%. If you are going at 99% of the speed of light, then
So, going at 0.99 times the speed of light will slow time down by 7 times versus the observers coordinates.